= sup(u; a ((7)). (3.22)
We can also show the following property for B(u>) = xi{b(uj)),
B{oj) = b(uj) uj (b(uj) b(sz)) zds
(.b(uj) b(s)) ds. (3.23)
47


Precisely we have,
B(lu) = -u ln(l -u), (3.24)
so B{u) > 0 for all 0 < u < 1, and G Ll{Q) since
[ (u/)| = f B{u)dz = (3.25)
Jn Jo 1
See Figure 3.2 for a plot of B(u).
Figure 3.2: Integral of temporal transform, B{u).
(H3) In order for a to be continuous in uj and p we require D(ui) and K{u>)
to be continuous. We see that p appears linearly and hence
(a(b(oj),pi) a(b(uj),p2)) (pi p2) = (D{uj) + K{lj)) (pi p2f > C{px p2)2
(3.26)
for some positive constant, C, since D, el, and K are bounded from above and
below. We need to restrict e1 G C2(fi) for all t and K G C'1 (0,1) (differentiably
continuous in el), so that continuity for holds.
48


(H4) This assumption places a growth condition on \a(b(uj),p)\ + \f(b(uj))\.
The bound on the diffusion term, D(ui) + K(lu), implies a(b(u>),p) = (D(u) +
A {uj))p < c(l + |p|) for some constant, c, so we need to satisfy the assumption,
\f(b(u))\ < cB{u)ll\ (3.27)
for the same constant, c, to satisfy the growth condition. Given that B(u) > 0
and consequently B{uj)1/2 > 0, and /(b(uj)) is continuous we can certainly find a
c that satisfies this assumption. In summary we are guaranteeing the following
growth condition given in [4] is satisfied,
+ l/(&M)l < c( 1 + B(lo)1/2 + |p|). (3.28)
(H5) The Dirichlet condition, el = £(max on [0,T] x T], where elmax is a
constant, is clearly in L2(0,T\W1,2(fl)) and L([0, T] x Q).
(H6) Since b(u) is surjective, b maps into the range of 6. Therefore there
is a measurable function el0 with b = b(el0).
Furthermore dt£lmax = 0 and is obviously contained in L1 (0, T; L(Q)).
We have used the conditions of the hypothesis to establish that the data
of IB VP (3.13) satisfies assumptions (H1)-(H6) of Theorem 3.2. Therefore a
solution to IBVP (3.13) exists.
3.3 Uniqueness of a Solution
In this section we show that the solution to (3.13) is unique under partic-
ular conditions on the diffusion coefficient and permeability function, D(t, r, z)
and K(t,r,z), respectively. A proof of uniqueness is provided by Alt and Luck-
haus [4] but we show the details of the proof here for the particular choice of
49


a(t, r, z,b(u),p) = A(t,r,z)p = (D(t,r,z) + K(t,r, z))p. This sum of diffusion
coefficient and permeability offers nuances to the uniqueness proof that require
an assurance that the theorem is valid for the IBVP (3.13).
Theorem 3.5 (Uniqueness Theorem) Suppose that the data satisfy (Hl)-
(H6) and the boundary and initial data assumptions with k = 2 and
a(t, r, z, b(u),p) = A(t, r, z)p = (D(t, r, z) + K(t, r, z))p, (3.29)
where A(t, r, z) is measurable in t and (r, z) such that for some a > 0
D + K o and D + K + adf(D + I\) (3.30)
are positive. Moreover assume that
|f(b(w2)) /(Mu;,))!2 < C(6M - - Wl), (3.31)
for some constant C > 0. Then there is at most one weak solution to
IBVP (3.13).
Proof: In the proof of (3.4) it was established that the data satisfy (Hl)-
(H6) as well as the boundary and initial data assumptions. Now suppose that
e[ and el2 are two weak solutions. Then let
/^=6(4)-6(£')eL2(0,T;n (3.32)
by our definition of a weak solution (3.16). The Riesz Representation Theorem
implies that there is a function v £ L2(0, T; V) such that
fT f (D + K)VvVQ= /Vc>
Jo Jn Jo
(3.33)
50


for all C L2(0, T; V). Then, letting df h be the backward difference operator,
we obtain,
2
2
h
1
h
1
h
rs+ft i rh
(d~h(},v) + j (3,v)
/s+h n ps+h
v) ~ Jx Jh (P(t~h),v)
i / rs rs+h rs+h \
+h \J0 +J Jh ('3,ty
/s+h 2 f8+h
1 f9 1 /s+h
J {(3(t + h), v(t + /?)) ~J~J (0(t),v(t + h))
If9 1 f9+h
+nLV'v) + kL (M
51


\ f f (D{t + h) + K(t + h))Vv(t + h) Vv{t + h)
h Jo Jn
~t f f (D(t) + K(t))Vv(t + h) Vv(t + h)
11 Jo Jn
Iff (D{t) + K(t))Vv(t + h) Vt,(t + h)
h Jo Jn
f f (D(t) + K(t))Vv(t)-Vv(t + h)
Jo Jn
+{[ l (D(t) + K{t))Vv{t) Vv(t)
1 rs+h
+- J (D(t) + K(t))Vv(t)Vv(t)
Iff (D(t) D(t -h) + K(t) K(t h)) Vv{t) Vv(t)
Jh Jn
+\[+ Jm) + K(t))Vv{t)-Vv(t)
+T f f (D(t) + K{t)){Vv(t + h)~ Vi;()) {Vv(t + h) Vv(t))
" Jo Jn
rs+h r
/ / d;h{D + K)Vv{t)-Vv{t)
Jh Jn
+1 f ^ ] W) + K(t))Vv(t) Vv(t)
Iff (D(t) + I<(t)){Vv(t + h) Vv(t)) (Vv(t + h)~ Vv(t))
h Jo Jn
h
2
h
+
I
Letting h> 0 we have the following:
rs+h
rs+h. r rs r
/ d^h{D + K)Vv(t)-Vv{t)= dt(D + K)Vv-Vv
Jh Jn Jo Jn
i J" J(D(t) + K(s))Vv{t) Vv(t) = JjDis) + K(s))Vv(s) Vv(s))
r f {D{t) + K{s))(Vv{t + h)-Vv{t))-(Vv(t + h)-Vv(t)) = 0,
hJo Jn
52


therefore, we obtain for s almost everywhere,
f (dt0, v) = \ ( f f dt(D + K)Vv Vv + f (D(s) + K(s))Vv(s) Vv{s)
Jo z \Jo Jn Jn
(3.34)
Now we can finish the uniqueness proof by considering the two solutions e\ and
el2 mentioned above and via Gronwalls inequality showing that they are equal
almost everywhere. By (3.33) and definition we have,
f f (D + K)VvV(el2-e[) = f (P,el2-e\) = [* [ (b(el2) -b(e[))(el2 -e[).
Jo Jn Jo Jo Jn
(3.35)
Consider our previous result (3.34),
U(D + K)Vv Vu + T f (6(4) 6(4))(4 4)
z Jn Jo Jn
= fS(dt8,v) [ dt(D + K)Vv Vu
Jo z Jo Jn
[ f (H4) -K4))(4 ~£li)
Jo Jn
+
now let v be a test function in the weak differential equation (3.16) so that the
right-hand side becomes,
- T / (D + A-)V(4 4) Vu + r f (/(6(4) f(b(£i))))v
Jo Jn Jo Jn
+ f f (b(4) b(e[))(4 4) I f / dt(D + K)Vv Vv
Jo Jn z Jo Jn
53


applying (3.35) and imposing 2ab < \a\2 + \b\2 for all a,beU yields the inequal-
ity,
f f (M 4)) f(b(z[)))v f dt(D + K)Vv Vu
Jo Jn Jo Jn
+\ j Ja iP Ja iUh + ''iVr
Imposing the hypothesis of the uniqueness theorem and Poincares inequality
[34] with positive constants, 8 and C(5), respectively produces,
\ [ 1/(44)) /(&(4))|2 + \f j M2 \[ J dt(D + K)Vv Vr
<8 j f (6(4) 6(4))( el2 4)
Jo Jn
+ \c{8) ( f Vr-Vr-J f f dt(D + K)Vv Vv. (3.36)
2 Jo Jn 2 JQ Jn
The constant for Poincares inequality, C(8), is chosen such that 0 < <5 < 1; recall
that Poincares inequality holds for some constant greater than zero. Hence, for
a given 8 < 1, we choose C(8) and fix it such that the inequality holds. Now we
consider,
\ f (D(s) + K(s))Vv(s)-Vv(s) + fS f (6(4)-6(e'))(4-4)
zJn Jo Jn
< 6 f [ (b(4) &(4))(4 4) + \ f f (C(8) dtD) Vv Vv.
Jo Jn A Jo Jn
But notice that (b(£l2)b(4))(44) 0 since b(u) is a non-decreasing function.
Hence, Jq fQ(b(4) 6(4))(4 4) 0, and (3.39) implies
1 [ (D(s) + K(s))Vv(s)-Vv(s) < l fS f (C(8)-dt(D + K))Vv-Vv. (3.37)
2 Jn 2 Jo Jq
54


We wish to apply Gronwalls inequality [34] to obtain our uniqueness result but
this step requires that the right-hand side of the inequality (3.38) contain D + K
in the integrand. Given that dt(D + K) is already present we need only make
a couple of manipulations using the inequalities present in the hypothesis of
the uniqueness theorem to obtain our result. Let a > 0 and first consider the
condition D + K + adt(D + K) > 0. It follows that
adt(D + K) < D + K
C{6) dt(D + K) <
C{S)a + D + K
a
by adding the positive constant C(6) to each side of the inequality,
impose the condition D + K a > Oto obtain
Second,
C(S) dt(D + K) <
C{5) + 1
a
(D + K).
Hence (3.38) becomes
[ (D(s) + K(s))Vv(s) Vv(s) < C'(d-)-- -1 f [ (D(t) + K(t))Vv(t) Vv{t),
Jn a Jo Jn
(3.38)
where c'^2+1 is a positive constant and JQ(D(s) + K(s))\7v(s) Vi'(s) is non-
negative. Thus, by Gronwalls inequality,
+ A'(s))Vr(s) V(s) = 0.
Therefore,
f f (&(4) Hei))(4 4) < & f f (M4) b(4))(4 4)> (3-39)
Jo Jn Jo Jn
where <5 < 1, implying that e[ = e\.
55


3.4 Analytic Solutions and Reductions of the IBVP
The proof of existence and uniqueness is an important step in the solution
process because it confirms whether or not solving the problem makes sense.
Additionally, this analysis is important because it uncovers some of the behav-
ior of the solution as well as the conditions for which we may find a solution.
As an example, under the current conditions of existence and uniqueness es-
tablished in Section 3.2 and Section 3.3, we can expect a smooth solution since
el G C (0, T; C2(Q)). In some circumstances we can actually find an analytic
solution to a nonlinear IBVP, however, the solution method often requires a
transformation such as the Cole-Hopf transformation [54], In other circum-
stances we can reduce the nonlinear PDE to a nonlinear ODE and either solve
the ODE analytically or approximate the solution using numerical techniques
or asymptotic methods. In this section we consider some cases under which we
can solve the IBVP (3.6) or reduce the PDE to an ODE and solve the result-
ing two-point BVP. The solutions to these problems will reveal the behavior to
be expected when we solve the more difficult IBVP numerically in Chapters 5
and 6.
3.4.1 Similarity Reduction of the IBVP
There are various transform methods for reducing PDEs to ODEs. For
example, a linear PDE defined (spatially) over the entire real line may be reduced
to an ODE in time using the Fourier transform. Consider a simple IBVP,
ut uxx on (x, t) G (og, oo) x [0, oo)
u(x, 0) = e~vx2, 77 > 0.
(3.40)
(3.41)
56


Under the two conditions, lim ux(x,t) = 0 V< and lim u(x,t) = 0 Vf, the
X*OQ X*OQ
Fourier transform of the PDE is the ODE (in time),
ut = (2nk)2u,
(3.42)
where
/OO
e~l2*kxu{x, t)dx,
OO
is the Fourier transform of u and k is the frequency. The solution to the ODE
(3.42) is u(x,t) Thus, the solution to the IBVP (3.40) is the
inverse Fourier transform of u,
u(x, t) = e~vx2
e-4^ei2^kxdk =
_2 1 + 4t?<
e 4t
2\fwt
(3.43)
The use of integral transform methods is prevalent for linear equations and since
the linearity of the transforms maintains the linearity for the resulting ODEs,
it also makes these methods convenient to use. On the other hand, transform
methods are less prevalent for nonlinear equations. Instead, one might group the
independent variables such that the problem reduces to a simpler PDE, which
may be linear, or an ODE. One method for determining such a group is a Lie
group transformation. Consider the Lie group (dilation) transformation,
£ *1 e = a 1u (3-44)
v = a_7u; (3.45)
r = a~Q£ (3.46)
t = a~^T (3.47)
where a > 0 is a scale factor and the parameters 7, a, and (3 are all positive
constants.
57


Note that
^ + Vs Vq(-) (Vs is the solid-phase velocity) so
Dt dr
,0

vs a(-) dt (3.48)
d(-y ae. ) (3.49)
where Vs is the transformed velocity. This last relationship implies that the
material time derivative is scaled in the same manner that the partial time
derivative is scaled.
Applying the transformation (3.44) to equation (3.1) yields,
Dsu
Dt
- rt
pa~T f e-£<-*
'0

which can further be simplified by scaling the dummy variable of integration,
Dsn . 2a-a d ( n( _7 sdu
= (l-o )o -(D(aV)^
We want the transformation from (3.1) to (3.50) to remain invariant. That is,
find values for a,/?,7 such that the parameters, {oQ, a0,~}, can be eliminated
from (3.50).
It is clear that we must choose ,3 = 0 because the parameter a0 resides in
the argument of the exponential. Hence, there is no interaction with the other
parameters implying that we cannot impose a relationship between (5 and the
remaining two exponents, a and 7. But 3 = 0 implies that a = 7 = 0 from
58


(3.50), yielding
Dt
(! )^ (d(u)
T
du
(3.50)
Equation (3.50) is identical to (3.1). Therefore, the equation (3.1) does not
transform to an ODE using (3.44).
3.4.2 Viscous Case
The viscoelastic case, where De ~ 0(1), will be solved numerically in Chap-
ters 5 and 6. However, there is a case where we can reduce the VPIDE to an
ODE and solve the equation using a numerical ODE solver. Additionally, there
is a special case for the function D(el) where we can find analytic solutions.
Solving these problems provides an opportunity to view the behavior of the so-
lutions and check the accuracy of our numerical scheme developed in Section
The integral term is a constitutive relationship modeling viscoelastic effects
in the model. Viscoelastic materials retain some or all of the energy introduced
by the stress and consequently maintain their shape for a period of time that de-
pends on a parameter called the relaxation time, r, of the material. Viscoelastic
behavior is best explained by the Deborah number, defined by equation (2.58)
but restated here
If De = 0, then the kernel of the integral term in (3.1) is zero. In this case,
equation (3.50) becomes,
5.1.
(3.51)
59


Invariance of (3.51) requires | = \ and 7 = 0 whence,

(3.52)
The case where De diffusion time dominates the relaxation time. A fluid or material that behaves
in this manner, very slow to diffuse, is viscous; viscous fluids or materials are
characterized by the property that they are resistant to flow.
The similarity variable is given by, rj = £(D0t)_"^ ^pQT, where D0 is the
diffusion coefficient associated with D(u). Without loss of generality, we can let
D0 = 1 since we can non-dimensionalize (3.1). Hence we obtain the standard
Boltzmann similarity variable,
T] =

(3.53)
such that,
dr =

2r 3D

du
dr
U^Tjr
JL
2t
2 tU'v
Direct substitution into (3.52) yields,
T] du
2 dv = {l-u)Tv{D{u)di
(3.54)
The boundary conditions and initial condition yield the requisite conditions for
solvability of the system (3.54). The Dirichlet boundary condition is imposed at
r = 1 where the transformation (3.44) yields £ = aa, but we have no information
for aa other than a > 0 which implies £ > 0 at this boundary. So letting r oo
60


u
1.0
Figure 3.3: Similarity solution to the IBVP (3.55).
implies t) > 0 and the Dirichlet boundary condition gives us u(r) 0) umax.
The initial condition provides the other boundary condition for u since r > 0
implies rj > oc and we obtain u{q = oo) = umjn. In summary, we have a
nonlinear, second-order ODE, two-point boundary-value problem,
d dr) Ml , n du + =0 1 2(1- u) dr) (3.55)
u(og) Wmjn (3.56)
?i(0) Wmax- (3.57)
We can use a simple shooting method to solve this BVP and obtain the sim-
ilarity solutions in Figure 3.3. This solution appears to showr the fluid imbibing
from the interior instead of the exterior. However, we used the initial condition
to derive the boundary condition u{r) = oc) = umjn and the Dirichlet condition
to derive the other boundary condition at u(rj = 0) = 'umax. These conditions are
the reverse of the physical boundary conditions, where the Dirichlet condition
61


is located at the exterior boundary, r = 1. So the solution in the transformed
coordinates is behaving as it should and demonstrates the expected behavior.
We anticipate the fluid to penetrate the drug-delivery device and increase the
liquid volume fraction until it reaches the steady state, umax.
3.4.3 Flory-Huggins Model
Another example of the viscous case come from the Flory-Huggins model
for swelling polymers [35] which has been a standard model used in polymer
science since 1953. In [73] the swelling model (2.53) is linked with the classical
Flory-Huggins model for swelling polymers. The analysis yields an equation
similar to (2.53) but lacks the integral term,
it = (1 tt)V (D(u)Vu), (3.58)
where D(u) Dq{ 1 u) for some constant diffusivity D0. In [73] this model
(3.58) is derived using the chemical potential for the solvent liquid, //*, assuming
only one species for each phase; only two phases are considered, liquid and solid.
It was shown in [73] that
pl po + RT\n{a), (3.59)
where a = ue1~u is the activation and //(, is the chemical potential at the initial
pressure and temperature. Weinstein writes Darcys law in terms of the Gibbs
potential and obtains the following form,
uvl,s = uplK(u)'VGi. (3.60)
Since the assumption is a single species, the Gibbs potential, Gl, is equal to the
chemical potential, so substituting (3.59) into (3.60) yields (3.58).
62


Linking the swelling model (2.53) not only validates the theory developed
in Weinstein [73], it also provides an opportunity for studying the liquid volume
fractions behavior. Recall that the drug-delivery device is immersed in a fluid
which penetrates the polymer network. We expect the liquid volume fraction to
increase until it reaches its maximum which is set at the exterior boundary. We
will solve the Flory-Huggins model analytically for a given set of initial-boundary
conditions and confirm this behavior.
Consider a one-dimensional case where the azimuthal direction has infinite
extent and we assume angular symmetry. If we nondimensionalize the PDE
using the diffusion time, t = where ro is the radius of the delivery device,
then the appropriate initial and boundary conditions are
^ (1 > f W max
=
9r r=0
u(r,0) = u0(r), (3.61)
for some function uo(r) to be determined. Transform the PDE (3.58) by letting
v = (1 it)2, then the IBVP becomes,
v = vAv
V(l,t) = (1 Umax)2
£ =
dr r=0
v(r, 0) = (1 u0(r))2. (3.62)
This IBVP is solvable by separation of variables. In order to see the behavior of
the solutions for these models, we will impose a set of initial-boundary conditions
63


and solve the problem. If we suppose that the initial condition v(r, 0) = 1 r2
and elmax = 1, then this IB VP has the solution
^ = ¥TT (3-63)
which, when we transform u back to the volume fraction, becomes
u(r^ = 1_/i^T (364)
We will use this solution to compare the numerical method used in Chapter
5. Plots of the solution are given in Figure 3.4. Notice that the liquid volume
fraction increases as time progresses and that u is tending towards a steady-state
solution, u(r, t. > oc) = 1, the Dirichlet boundary condition. This behavior thus
confirms the expectations we posited earlier.
eW)
Figure 3.4: Solution to the Flory-Huggins model (3.58) for a variety of times.
64


3.5 Discussion
In this section we demonstrated existence and uniqueness of a solution to
the IBVP (3.6) under the condition that the volume fraction reside in the space
C ([0. T]\C2(fl)) where Q is a right-cylinder and the solution exhibits angular
symmetry. Recall during the derivation of the integral term in Section 2.3 that
we computed the Laplace transform on p > 0 derivatives of el. However, exis-
tence of the Laplace transform requires that the function be piecewise continuous
in time. Hence, the requirement that el be at least continuous in time should
easily be satisfied. Moreover, since el is a volume-average quantity, it is at least
continuous [31]. The requirement for el to be twice differentiably continuous,
on the other hand, is strong as we expect the volume fraction to have corners
or behave like a ramp function. This condition arises from the requirement that
the integral term be continuous. Furthermore, we expect that uniqueness proof
can be generalized as well. That D(r, t) + K(r, t) be strictly functions of space
and time, and not el, was constrained by the use of the Riesz-Representation
theorem at the start of the uniqueness proof. Further research into the existence
and uniqueness of solutions to this equation may relax these conditions.
We also searched for similarity solutions to the IBVP (3.6) but found that
the integral term breaks the symmetry. However, consideration of the viscous
case where De = 0 permits similarity solutions. The reduced nonlinear BVP was
solved using a shooting method producing solutions that confirmed our physical
intuition. A perturbation series may reveal a boundary layer developing near
65


the exterior boundary, as the numerical solution shows, and further research
into this topic is also required. Another model of polymer swelling re-derived in
[73], the Flory-Huggins model, was reviewed and solved for a given set of initial
and boundary conditions. The solutions to this model revealed the expected
behavior of the liquid volume fraction.
66


4. Eigen-decomposition Pseudospectral Method
Pseudospectral (PS) methods are known for their spectral accuracy when
applied to smooth functions over regular geometries. The drug-delivery problem
is defined on a cylinder (regular geometry) in Lagrangian coordinates (fixed
grid) and the volume fraction, s1, is a volume-averaged quantity and as such is
continuous in both space and time [31]. Hence, the drug-delivery problem is a
candidate for pseudospectral methods.
These Volterra Partial Integrodifferential Equations (VPIDE) (2.59) are dif-
ficult to solve numerically because the integral term accumulates round-off error
as time progresses and a physically realistic exterior Dirichlet condition imposes
an initial profile with a very steep front that propagates through the spatial
domain. The former inspired the use of PS methods for solving the problem
because PS methods are well-known for their high-order accuracy and spectral
convergence for smooth functions [37]. The latter inspired the use of the Eigen-
decomposition Pseudospectral (EPS) method over standard PS methods. For
example, the geometry for the drug delivery model is a cylinder and we use an
initial condition that is initially very steep. We could have initialized the prob-
lem with a constant condition el = e[nin but this would have made the numerical
experimentation cumbersome and would not have contributed any interesting
physical attributes to the problem; the moving boundary would simply evolve
as governed by the VPIDE. By introducing the steep initial profile, we can more
easily control the initial state of the problem by incorporating a parameter that
67


adjusts the steepness. As a result, the initial condition requires a highly re-
solved solution across the spatial domain, especially as the problem evolves in
time and the liquid penetrates the drug delivery device. The moisture content
increases implying that £l is propagating towards the interior of the cylinder.
The requirement for a large number of spatial points punctuates the advantage
of the EPS method over standard PS methods.
In this chapter wre provide a review of pseudospectral methods with com-
parisons to finite differences. Additionally, we derive and demonstrate a novel
approach to a pseudospectral discretrization constructed on a polar geometry
[58]. We discuss advantages and disadvantages of this method when compared
to traditional pseudospectral methods.
Pseudospectral methods are a subclass of spectral methods, a class of spa-
tial discretizations for differential equations. The key components for their for-
mulation include linear combinations of suitable trial basis functions and test
functions. Trial functions provide an approximate representation of the solution
and the test functions ensure that the differential equation and possibly the
boundary conditions are satisfied as closely as possible by the truncated series
expansion. We then minimize, with respect to a suitable norm, the residual
produced by using the truncated expansion instead of the exact solution. In
summary:
1. Given a differential equation with boundary conditions, approximate a
solution u(x) by a finite sum v(x) = X!ib=o ak(t)k{^)\ hi the case of a time-
dependent problem, u(x,t) is approximated by v(x,t) and a^(t).
2. This series is substituted into the equation Lu = f(x) where L is the
68


x-h x x+h h = 1/N uh = T" 00 0 0
Finite Difference Method Finite Element Method
u ^<|>N

a b
1/NJ 1/N 1/N2
Spectral Method
Figure 4.1: Trial function comparison for spectral, finite-difference, and finite-
element methods. Spectral methods are characterized by one high-order poly-
nomial for the whole domain, finite-difference methods are characterized by
multiple overlapping low-order polynomials, and finite element methods are
characterized by non-overlapping polynomials with compact support one per
subdomain.
operator of the differential or integral equation, with the result being the
so-called residual function: R(x\ a0, ai,..., a^) Lv f.
3. Since R(x\ a.k) = 0 for the exact solution, the challenge is to choose the
series coefficients a*, so that the residual function is minimized.
The choice of the trial functions is one of the features that distinguishes the
early versions of spectral methods from finite-element and finite-difference meth-
ods, see Figure 4.1. To increase accuracy for finite-element and finite-difference
methods requires p-refinement (a higher order polynomial) or h-refinement (a
finer mesh). Either way, the order of the method is fixed and the error is ~ 0(hp)
where the mesh size is h = jj. Spectral methods, on the other hand, have an
error that is decreasing faster than any finite power of N [66].
69


Spectral methods are global approximation methods: one approximates
the spatial derivative by using a global interpolant through discrete data
points, then differentiating the interpolant at each point.
Finite difference/element methods on the other hand are local methods;
they employ a few neighboring grid points to make their approximations.
Finite difference/element approximations do lead to sparse matrix repre-
sentations, however, the order of their accuracy is fixed and so to increase
accuracy one must rely on a consistently finer mesh or higher order poly-
nomial interpolants.
Spectral methods use fewer points across the domain to reach spectral
accuracy and they can be implemented using the FFT and hence have a
sparse implementation.
The choice of test functions distinguishes between the three earliest types of
spectral schemes: Galerkin, collocation, and tau versions. The Galerkin method
requires a combination of the original basis functions into a newr set in which all
the functions satisfy the boundary conditions. After the boundary conditions
are satisfied, the Galerkin method requires that the residual be orthogonal to as
many of these new basis functions as possible. The tau method is similar to the
Galerkin method in the way the differential equation is enforced. However, none
of the test functions need to satisfy the boundary conditions. A supplementary
set of equations is used to apply the boundary conditions. Selecting the a*, so
that the boundary conditions are satisfied and requiring that the residual be zero
at as many spatial points as possible is called the pseudospectral (PS) method
70


[52]; the PS method is also called the spectral collocation method or the name
may be qualified by the type of trial function being used such as Chebyshev PS
(Collocation) Method or Fourier PS (Collocation) Method. The test functions
are translated Dirac delta-functions centered at the so-called collocation points
requiring that the differential equation be satisfied exactly at the collocation
points.
4.1 Pseudospectral Methods
According to Fornberg [37], when choosing the to be met:
1. v(x) = Eto ak4>k(x) must converge rapidly as N increases for reasonably
smooth functions;
2. given a*,, it should be easy to determine bk such that ^(Eib= oakk{x)) =
Ef=o M>fc(z);
3. it should be fast to convert between coefficients a*,, fc = 0,1,..., AT and the
sum value v(xk) for some set of points Xk, k 0,1,..., N.
On a periodic domain a Fourier series is often chosen for k, while on non-
periodic domains orthogonal polynomials are often used. Once the basis set
has been chosen, there are only a few optimal sets of interpolation points for
each basis. An interpolating approximation to a function f(x) is an expression
Pn- i(x), usually an ordinary or trigonometric polynomial, with N degrees of
freedom determined by the requirement that the interpolant agree with f(x) at
each of a set N interpolation points:
PN-i(xi) = f(xi) where i = 1,2,..., N.
71


One may fit any N + 1 points by a polynomial of Nth degree via the Lagrange
Interpolating Formula:
N
PN(x) = J2f{xk)Ck(x),
fc=0
where Ck(x) are the cardinal functions:
N
ck{x)= n
j=0,jjtk
X Xj
Xk Xj '
Cardinal functions have the properties that Ck(xj) = Sk] and that the inter-
polating points are not required to be evenly spaced. Consider applying the
Lagrange Interpolating Polynomial to
f(x) =
1
(5x)2 + 1
over the interval [1,1] on an equally spaced grid. One would expect that
the error in P/v would go to zero as N > oc. However, as the order of the
polynomial, N, increases large oscillations appear near the end points of the
interval, see Figure 4.2. This example is the wrell-known Runge Example.
For completeness and convenience we state the Cauchy Interpolation Error
Theorem (see [5] for example).
Theorem 4.1 (Cauchy Interpolation Error Theorem) Let f CN+l[a,b]
and let Ppj(x) be its Lagrangian interpolant of degree N. Then
f(x) PN{x) = 1/(JV+1)(^) f[(x xk) (4.1)
' k=0
for some £ [a, b].
72


N 10. Lagrange Interpolating Formula
Figure 4.2: Runges example for interpolating f(x) = (5x)2+1 with evenly
spaced nodes. As the number of evenly-spaced nodes (degree of the interpo-
lating polynomial) increases, the interpolating polynomial oscillates near the
boundaries.
If we wish to reduce the error for the Lagrange interpolation, there is nothing
that can be done about the /(JV+1)(^) factor. However, we can choose the grid
points to reduce [~[]L0(x ~ xk) and ameliorate the error. We now state the
Chebyshev Minimal Amplitude Theorem (see [15] for example) to reveal a
choice of grid points that minimizes the error.
Theorem 4.2 (Chebyshev Minimal Amplitude Theorem) Of all polyno-
mials of degree N with leading coefficient equal to 1, the unique polynomial which
has the smallest maximum value on [1,1] is Tn(x)/2n~\ the Nth Chebyshev
polynomial divided by 2iV_1.
73


Any polynomial of degree N can be factored into the product of linear
factors of the form of (x Xk) where is one of the roots of the polynomial.
In particular
1 N
Tn+1{x) = J|(x -xfc).
k=0
In order to minimize the error in the Cauchy remainder theorem, the polyno-
mial part should be proportional to TN+i(x). That is, the optimal interpolation
points are the roots of the Chebyshev polynomial of degree (N + 1). See Fig-
ure 4.3 for a qualitative example of the improvement.
Figure 4.3: Interpolating with Chebyshev nodes ameliorates Runges phe-
nomenon. As the number of Chebyshev nodes increases, the interpolation error
decreases.
An 0(hp) scheme for approximating f'(x) can be derived by differentiating a
p-point Lagrange interpolating polynomial containing the point of interest. For
example, centered differencing is an 0(h2) approximation resulting from dif-
ferentiating a 3-point Lagrange interpolating polynomial centered at the point
74


of interest. To increase the accuracy we would perform /^refinement, how'ever,
accuracy would eventually reach the limits imposed by Runge phenomena as
discussed above. If one were to differentiate a higher order polynomial and use
Chebyshev nodes, then the result would be a much more accurate differentiation
approximation, see Figure 4.4 wdiere we demonstrate the efficiency of PS meth-
ods over a couple of finite-difference methods. In this sense, PS methods may
be thought of as high-order finite-difference methods implemented at the zeros
of an appropriate orthogonal polynomial. Note that Chebyshev nodes are not
the only choice, there are evenly spaced nodes from a Fourier series for periodic
domain, Legendre nodes, Gauss-Hermite nodes, and many others.
Figure 4.4: L2 error comparison of the Chebyshev PS method versus second
and fourth order finite difference methods applied to the second derivative of
cos(7rx) over [1,1].
75


4.2 Eigen-decomposition Pseudospectral Method
The PS methods reviewed in the previous section comprise a traditional
approach to constructing differentiation matrices. We will refer to a matrix
construction derived from these interpolation-based methods as a Standard
Construction. One unfortunate consequence of the Standard Construction is
that the resulting matrices become increasingly ill-conditioned as the size of
the matrix (number of grid-points) grows [37, 71]. The increased matrix norm
dictates the size of the time-step for time-dependent problems so the Standard
Construction has limited utility for these types of problems. There are tech-
niques for reducing the matrix norm such as spectral filtering or one can use
domain decompositions for increasing the mesh resolution over regions of inter-
est. However, these techniques either diminish the accuracy of the PS method
or increase the computational complexity. In this section we will derive and
demonstrate a new class of PS methods based on the spectral decomposition
of the differential operator, the so-called Eigen-decomposition Pseudo-Spectral
Method or EPS method [58], that accommodates mesh refinement while main-
taining an optimally conditioned differentiation matrix and does not require any
additional numerical complexity over the Standard Construction.
4.2.1 Derivation
To explain the concept behind the EPS method, we first write the operator,
£, applied to / over x [a, 6] as an integral operator [58],
(4.2)
76


where the kernel, K(x,y), is defined,
OO
(4.3)
m1
where {um, vm}m=i is the complete set of orthonormal functions of £ specified by
the boundary conditions placed on / and Am is determined by the respective sin-
gular values and the normalization factor associated with the set {um,vm}^=1.
gent series. However, it can be shown that (4.3) is uniformly convergent for
a complete, orthonormal set of eigenfunctions [39]. Because of the symmetry
assumed with our model problem, see Section 3.1, we will refer to the set of
functions, as eigenfunctions. Differentiation matrix construction
occurs when the kernel, K(x,y), is truncated and a quadrature rule is selected
for the integral. We summarize these two steps in the following subsections
below.
4.2.1.1 Truncating the Sum
Proceeding as in [58], denote the rank M approximation of the differential
operator £ applied to the function /,
where {vm}^=l is the set of M eigenfunctions. Letting f(y) = vn(y) for 1 <
n < M, we obtain
For this definition to make sense wTe need to assume that K(x, y) is a conver-
(4.4)
77


where {An}^, are the M non-zero eigenvalues of VmC. Hence, the operator
Vm is a rank M projection of / onto the span of {vm(y)}^l=l.
4.2.1.2 Approximating the Integral with a Quadrature
Again, proceeding as in [58], let {#n}jtLi and [wk}£'=i be a given set of
quadrature nodes and weights and let Nc < N be an integer. We could set
Nc > N, however, we explain in Section 4.2.2 the disadvantages of exceeding
the number of quadrature points. Approximate (4.4) by,
JVC N
* £ ^ (9 k) f ()V-k (4.5)
m=l k=1
By evaluating (4.5) at the same quadrature points we obtain,
Nc N
£jk(f)(8j) = X! ^rnUm{Q]) ^2 WkVm(0k)f{9k), (4.6)
m=1 r-=i
where 1 < j < N. Thus, we can represent the operator C as the N x N, rank
Nc matrix,
nc
C-jk ^ Amum(0j)tt>/cum(0/c). (4-7)
m=l
This low rank operator may be sufficient for bandlimited functions (or approx-
imately bandlimited functions) [58], however, it may not be sufficient for non-
bandlimited functions. That is, the remaining eigenfunctions {vm(x), um(y)}m=Nc+n
may be needed to resolve the higher frequency content of the function being dif-
ferentiated. We demonstrate this concept below, refer to Figure 4.8.
Example 1. The second-derivative operator, in Cartesian coordinates, with
zero Dirichlet boundary condition has the eigen-decomposition
{-(?)-<>+<>
78


We construct the second derivative operator via its kernel
Kj£r(xV) = Xm sin(y (x + 1)) sin(y (y + 1)). (4.9)
m1
where
According to (4.7) we can now define the second-derivative matrices as follows.
Definition 4.3 (Dirchlet Boundary Conditions) Let {#/, iCf}^ denote a
set of N quadrature nodes and weights on the interval [1,1], Define the deriva-
tive matrix C = of rank Nc < N with zero Dirichlet boundary conditions as
Nc
(C)ki = sin(y (0fc + l))wi Sin(y (0J + 1)), k, l = 1,..., N.
m=1
Definition 4.4 (Neumann Boundary Conditions) Let {di,wi}j^1 denote a
set of N quadrature nodes and weights on the interval [1,1]. Define the deriva-
tive matrix C = of rank Nc < N with zero Neumann boundary conditions
as
Nc
{C)ki = Amcos(y(0fc + l))ujcos(y(0i + 1)), k,l = 1,...,N.
m= 1
79


4.2.2 Error Analysis
The rank of is Nc, indeed, if we apply (4.7) to an eigenfunction, vn
(n = 1,..., Nc), evaluated on the grid {93} (j = 1,..., N), then we ascertain,
Nc N
j) ^ ^ AmUm($j) ^ ^ ^k^m (^fc)^n (^/c)
m= 1 k=1
Nc
= ^ A,i/m(0j)(5init (Smn is the Kronecker delta)
m 1
= Kvn{9j), (4.10)
where (4.10) shows the relationship for the Nc non-zero eigenvalue-eigenfunction
pairs. The norm of (4.7) is dictated by the magnitude of An where n = 1,..., Nc.
Hence Nc regulates the condition number of (4.7). However, Nc also impacts
the accuracy of the EPS method as demonstrated through the following error
analysis.
We begin with some notation. Let the quadrature error be defined as,
Nc
equad ^ Amum(£)emn, (4.11)
m=1
where
N
£mn &mn ^ ^ U^-lm(6*/t-)cn(6*^-), (4.12)
lt=l
and £ [a, 6]. Notice that (4.12) indicates how well the quadrature rule main-
tains numerical orthogonality. The magnitude of this error (4.12) can be deter-
mined by choosing N such that, \emn\ < t. However, as we shall see, this error
cannot be regulated by choosing N alone (see Full-rank Completion below).
80


Let the truncation (tail) error" be defined as,
OO
etail
m=Nc+1
(4.13)
where
(4.14)
This error is regulated by the decay rate of fm as m becomes large. Finally, let
the approximation of the operator, via the EPS method, applied to a function,
/, be noted by,
Nc N
£/(0 = U,kVm(8k)f(8k)-
n=l n=l
(4.15)
Theorem 4.5 (EPS Error) If f C2([a,b\) with f(a) = f(b) = 0, and
{vk(x)}k=l form, a complete orthonormal set on [a, 6] such that (4-3) converges,
then given N quadrature nodes and weights {#n}fcLi and {wk}kLi respectively,
along with the integer Nc < N,
{H ^)./(£) tquad t-taili
where £ [a, b] and equad and etai{ are given by (4-11) and (4-13) respectively.
Proof: The function f(x) can be expanded as an orthonormal series of
eigenfunctions that is absolutely and uniformly convergent on [a, b] [68],
fix) = '^2fnV{x), (4.16)
71= 1
where fn is given by (4.14).
81


Consider the difference,
rb oo
pb o Nc N
(£ C)f(0 = Y XrnUm(OVm(y)f(y)dy ~ Y AroUm(0 Y WkVm(0k) f (9k)
m1 tii=\ k= 1
/b c
Y Xmum{Qvm{y)f{y)dy (truncation part)
n=Nc+1
+
Nc f fb N \
X] I / Vm(y)f(y)dy Y wkvm(dk)f(0k) I
m=l V/ fc=l /
(quadrature part)
(4.17)
We now isolate the truncation part and the quadrature part of the sum and
finish our analysis on each of these summands separately.
Consider the truncation part first. Recall that / £ C2([a,b\) mak-
ing it bounded and measurable. The vk(y) are weakly convergent since,
/ 6 C2([a,b\) C L2([a,b\) and {vk(y)}kLi complete implies
which, in turn, implies
f2(x)dx < oo,
fk = f f{x)vk(x)dx -> 0.
Hence, the eigenfunctions, vk{y), are bounded by a corollary to the Principle
of Uniform Boundedness [6], and therefore measurable as well. Moreover, if we
define,
M
S.u(y) = Y XmUm{Ovm(y)f(y), (4.18)
n=Nc+1
then S\i(y) is a bounded, measurable function which, by hypothesis converges,
lim Sm(ij) S(y), for some S(y). Thus, the conditions for the Lebesgue
Moo
82


Dominated Convergence Theorem [57] are satisfied and
pb pb pb
/ ,iim X]XmUm(Ovm(y)f(y)dy= lim Sm(y)dy = lim / SM(y)dy,
Ja Ja A/^
that is
OO pb 00
Y ^nUm{0 / vm(y)f(y)dy = Y XmUm{Ofm(y) = etail
n=JVc+l ,Aa n=iVc+l
via (4.13).
Now we analyze the quadrature part and show that it is equivalent to
(4.11). First notice that by inserting (4.16) into the difference,
rb n
/ vm(y)f(y) Y WkVm{h)f{0k)
Ja k=1
/b oo N oo
Vmiy) ^ /n^n(2/) ^ ^ WkVm(0k) ^ fnVnifik)
n=l fc=l n=1
oo / .6 JV \
^ In I / ^m(y)^n(y) ^ 'J^k^m{,8k)Vn{8k) I
n=l fc=l /
oo / N \
^ ^ /n I ^mn ^ /^k'^m^k)^n{^k) j
n=1 \ fc=l /
oo
^ fn^mw
n 1
where the switch between the integral and sum in the second step is justified
because, as pointed out in the beginning of the proof, the series is absolutely
and uniformly convergent on [a, b]. But notice that we have just shown that the
quadrature part,
Nc / rb N \ Nc
Y XmUm(0 ( / vm{y)f{y)dy Y WkVm(8k)f(8k) J = Y
m1 \ a k=1 / m=1
= 1
equad-
83


Citation
Analysis and numerical solution of nonlinear volterra partial integrodifferential equations modeling swelling porous materials

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Title:
Analysis and numerical solution of nonlinear volterra partial integrodifferential equations modeling swelling porous materials
Creator:
Wojciechowski, Keith J
Publication Date:
Language:
English
Physical Description:
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Subjects / Keywords:
Volterra equations -- Numerical solutions ( lcsh )
Integro-differential equations ( lcsh )
Swelling soils -- Mathematical models ( lcsh )
Porous materials -- Mathematical models ( lcsh )
Integro-differential equations ( fast )
Porous materials -- Mathematical models ( fast )
Swelling soils -- Mathematical models ( fast )
Volterra equations -- Numerical solutions ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D. )--University of Colorado Denver, 2011. Applied mathematics
Bibliography:
Includes bibliographical references (leaves 159-164).
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by Keith J. Wojciechowski.

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Full Text
ANALYSIS AND NUMERICAL SOLUTION OF NONLINEAR VOLTERRA
PARTIAL INTEGRODIFFERENTIAL EQUATIONS MODELING
SWELLING POROUS MATERIALS
by
Keith J. Wojciechowski
B.A., Lawrence University, 1992
M.S., DePaul University, 1998
M.S., University of Colorado, 2003
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2011


This thesis for the Doctor of Philosophy
degree by
Keith J. Wojciechowski
has been approved
by


Wojciechowski, Keith J. (Ph.D., Applied Mathematics)
Analysis and Numerical Solution of Nonlinear Volterra Partial Integrodifferential
Equations Modeling Swelling Porous Materials
Thesis directed by Associate Professor Lynn Bennethum
ABSTRACT
A nonlinear Volterra partial integrodifferential equation (VPIDE), derived
using hybrid mixture theory and used to model swelling porous materials, is
analyzed and solved numerically. The model application is an immersed porous
material imbibing fluid through a cylinders exterior boundary. A poignant
example comes from the pharmaceutical industry where controlled release, drug-
delivery systems are comprised of materials that permit nearly constant drug
concentration profiles. In the considered application the release is controlled
by the viscoelastic properties of a porous polymer network that swells when
immersed in stomach fluid, consequently increasing the pore sizes and allowing
the drug to escape. The VPIDE can be viewed as a combination of a non-linear
diffusion equation and a constitutive equation modeling the viscoelastic effects.
The viscoelastic model is expressed as an integral equation, thus adding an
integral term to the non-linear partial differential equation. While this integral
term poses both theoretical and numerical challenges, it provides fertile ground
for interpretation and analysis.
Analysis of the VPIDE includes an existence and uniqueness proof which
in


we establish under a given set of assumptions for the initial-boundary value
problem. Additionally, a special case of the VPIDE is reduced to an ordinary
different,ial equation via a derived similarity variable and solved. In order to solve
the full VPIDE we derive a novel approach to constructing pseudospectral dif-
ferentiation matrices in a polar geometry for computing the spatial derivatives.
By construction, the norms of these matrices grow at the optimal rate of 0(N2),
for Ar-by-AT matrices, versus 0(N4) for conventional pseudospectral methods.
This smaller norm offers an advantage over standard pseudospectral methods
when solving time-dependent problems that require higher-resolution grids and,
potentially, larger differentiation matrices. A method-of-lines approach is em-
ployed for the time-stepping using an implicit, fifth-order Runge-Kutta solver.
After we show how to set up the equation and numerically solve it using this
method, we show and interpret results for a variety of diffusion coefficients,
permeability models, and parameters in order to study the models behavior.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed
IV


DEDICATION
This thesis is dedicated to my wife Sharyl and my son Cole for the inspiration
they give me everyday to be a better husband, a better father, and a better
man.


ACKNOWLEDGMENT
There are several people I need to acknowledge for the help they provided in
the completion of this thesis. First, this thesis would not have been possible
without the patience and insights of Lynn Schreyer-Bennethum. Lynn taught
me the value of using physical insight to guide mathematical analysis. Kristian
Sandberg introduced me to Pseudospectral methods and showed me howT to
think creatively as a numerical analyst. His trouble-shooting acumen saved me
on many occasions and I cannot thank him enough for being so generous with
his time despite running his own company and raising a family.
I would also like extend my thanks to Bill Briggs for introducing me to
Lynn and nudging me along as I came perilously close to leaving the graduate
program. To Julien Langou, who helped me think more deeply about linear
algebra. To Leo Franca for helping to bring me to UC Denver. To Alexander
Engau, for agreeing to be on my committee despite being in the unenviable
position of being asked at the last minute. To Matthew Nabity who has been
my compadre since the start of this journey and has become a life-long friend.
To John Stinespring wrho, simply put, makes my life better. To Eric Sullivan
and Kannanut Chamsri, my academic brother and sister, for patiently listening
to my research talks and asking poignant questions. To Jinhai Chen for helping
make my arguments more mathematically sound. To Jeff Barchers for believing
in me enough to hire me at SAIC. To Richard Quanstrum who taught me, by
his example, what it means to be a real man, I carry in my heart your pride in
my accomplishments as a teacher, husband, and father you will always be my
coach. To Angela Beale and Lindsay Hiatt who kept me registered and up-to-
date on my requirments, I would not have made it through a semester without
them. To all the people whom I met over my years of education who helped me
along the way by their kindness, advice, instruction, and encouragement.
Finally, I must thank my family for their loyalty and unending support. My
mother, Eleonore, who nurtured my creativity and encouraged me to reach for


the stars, my brother Robbie who taught me the value of hard-work, my sisters
Barbara and Cheryl who always stood by me growing up and who taught me to
be a gentleman, my brother Lonnie wTho taught me that there is more to life than
work (like music!), and my father Leonard who taught me to be self-reliant.


CONTENTS
Figures ................................................................. xi
Tables................................................................. xvii
Chapter
1. Introduction......................................................... 1
1.1 Previous Work ....................................................... 5
1.2 Thesis Outline ...................................................... 7
2. Fundamental Continuity Equation..................................... 10
2.1 Brief Overview of Hybrid Mixture Theory............................. 10
2.1.1 Macroscale Field Equations........................................ 16
2.2 Derivation of the Continuity Equation .............................. 18
2.3 Darcys Law......................................................... 20
2.4 Non-dimensionalization of the Model................................. 25
2.5 Continuity Equation................................................. 27
2.6 Model Interpretation................................................ 28
2.6.1 Deborah Number.................................................... 32
2.6.2 Integral Coefficient.............................................. 35
2.7 Discussion.......................................................... 36
3. Analysis of the Volterra Partial Integrodifferential Equation ...... 39
3.1 Formulation of the Boundary Value Problem........................... 39
3.2 Existence of a Solution............................................. 46
viii


3.3 Uniqueness of a Solution....................................... 49
3.4 Analytic Solutions and Reductions of the IBVP.................. 56
3.4.1 Similarity Reduction of the IBVP.................................. 56
3.4.2 Viscous Case...................................................... 59
3.4.3 Flory-Huggins Model............................................... 62
3.5 Discussion........................................................ 65
4. Eigen-decomposition Pseudospectral Method............................ 67
4.1 Pseudospectral Methods............................................ 71
4.2 Eigen-decomposition Pseudospectral Method......................... 76
4.2.1 Derivation........................................................ 76
4.2.1.1 Truncating the Sum............................................. 77
4.2.1.2 Approximating the Integral with a Quadrature................ 78
4.2.2 Error Analysis ................................................. 80
4.2.3 Rank Completion................................................... 84
4.3 EPS Construction in Polar Coordinates............................. 89
4.3.1 Construction...................................................... 90
4.3.2 Numerical Examples................................................ 93
4.4 Discussion........................................................ 95
5. Numerical Solution of the Partial Integrodifferential Equation .... 102
5.1 Numerical Method for Solving the Swelling Equation............. 102
5.1.1 Spatial Discretization .......................................... 104
5.1.2 Time-stepping Methods............................................ 107
5.1.2.1 Semi-analytic Integration Rule Formulation.................... 109
5.1.2.2 Method-of-Lines Formulation................................... 110
IX


5.1.2.3 Pouzet Volterra Runge-Kutta Formulation....................... Ill
5.1.2.4 Examples....................................................... Ill
5.2 Numerical Solution of the IBVP.................................... 114
5.2.1 Two-dimensional Example........................................ 115
5.2.2 One-dimensional Examples ...................................... 115
5.2.2.1 Flory-Huggins Model............................................ 116
5.2.2.2 Diffusion Coefficient Comparison............................. 118
5.3 Discussion........................................................ 119
6. Model Sensitivity Analysis......................................... 126
6.1 Diffusion and Permeability Models ............................... 127
6.2 Parameter Sensitivity............................................. 130
6.2.1 Moisture Content Curves ....................................... 131
6.2.2 Viscoelastic Stress Curves..................................... 134
6.3 Discussion........................................................ 135
7. Conclusion and Future Work......................................... 144
7.1 Model Analysis, Validation, and Extension........................ 145
7.2 Generalizing the Applicability of the EPS Method................. 146
7.3 Generalizing the Existence and Uniqueness Proof for the IBVP . 147
7.4 Extending the Applicability of the Drug Delivery Model............ 147
Appendix
A. Derivation of Darcys Law.......................................... 149
B. Derivation of a Pouzet Volterra Runge-Kutta Method................. 156
References............................................................. 159
x


FIGURES
Figure
1.1 Porous polymer matrix. Photo taken from plc.cwru.edu............... 4
2.1 Averaging, Local Coordinates........................................ 13
2.2 Stress (a) versus strain (e not to be confused with the volume
fraction sl) curve for linear elastic (left) and linear viscoelastic (right)
materials........................................................... 30
2.3 Stress versus time where the loading takes place over the time interval
t< t\ and the unloading takes place over the interval t >t\...... 30
2.4 Temperature dependence of rate between transition states on tem-
perature................................................................. 34
3.1 The model geometry is a cylinder but we assume angular and az-
imuthal symmetry thus reducing the domain to a rectangle ft with
boundary T = Ti U r2................................................ 40
3.2 Integral of temporal transform, B(u)................................ 48
3.3 Similarity solution to the IBVP (3.55).............................. 61
3.4 Solution to the Florv-Huggins model (3.58) for a variety of times. . 64
xi


4.1 Trial function comparison for spectral, finite-difference, and finite-
element methods. Spectral methods are characterized by one high-
order polynomial for the whole domain, finite-difference methods are
characterized by multiple overlapping low-order polynomials, and fi-
nite element methods are characterized by non-overlapping polyno-
mials with compact support one per subdomain........................ 69
4.2 Runges example for interpolating f(x) (5x)2+1 with evenly spaced
nodes. As the number of evenly-spaced nodes (degree of the interpo-
lating polynomial) increases, the interpolating polynomial oscillates
near the boundaries...................................................... 73
4.3 Interpolating with Chebyshev nodes ameliorates Runges phenomenon.
As the number of Chebyshev nodes increases, the interpolation error
decreases................................................................ 74
4.4 L2 error comparison of the Chebyshev PS method versus second and
fourth order finite difference methods applied to the second derivative
of cos(7nr) over [1,1].................................................. 75
4.5 Sparsity pattern for orthogonality test of eigenfunctions in a Carte-
sian geometry with Neumann boundary conditions. The rank of the
differentiation matrix is Nc = N = 256................................... 85
4.6 Sparsity pattern for orthogonality test of eigenfunctions in a Carte-
sian geometry with Neumann boundary conditions. The rank of the
differentiation matrix is Nc 139 (roughly 0.54A')............ 86
xii


4.7 Sparsity pattern for orthogonality test of eigenfunctions in a Carte-
sian geometry with Neumann boundary conditions for the completed
operator. The rank of the differentiation matrix is now N 256
(7VC = 139, roughly 0.547V).............................................. 88
4.8 Comparison of the L norm relative error resulting from comput-
ing the second derivative of the functions {sin + 1))
using the EPS method with Gauss-Legendre quadrature nodes and
weights versus the Standard Construction using Cheybshev-Lobatto
nodes. Rank-completion versus a reduced rank construction the EPS
method is also compared........................................ 97
4.9 Given TV, use the EPS method to construct the polar Laplacian in-
creasing Nc from 1,2,..., TV and computing the error ||a:2 a^Hoo for
each set of Nc eigenvalues, {d2, a2})^. The colorbar is log-scaled
where the dark regions indicate smaller error and the light regions
indicate larger error.......................................... 98
4.10 Eigenvalues of Ar constructed via Chebyshev collocation versus the
exact eigenvalues, |an|2. The eigenvalues of Ar are the zeros of the
zero-order Bessel function of the first kind, J0(a) = 0 for all n =
1,2,3,................................................................ 99
4.11 Chebyshev expansion coefficients versus Fourier-Bessel expansion co-
efficients for the Gaussian pulse centered at r = 1/2.....................
xm
99


4.12 The Poisson equation example comparing the completed EPS con-
struction to the standard construction using Chebyshev collocation:
(a) The residual error, ||/ Aru||oo. (b) The error ||u A^/lloo,
and (c) L condition number.................................... 100
4.13 The maximum stable time step At for solving the radial part of the
heat equation on a disk using the explict Runge-Kutta 4 solver. Here
N denotes the size of the problem..................................... 101
5.1 The EPS differentiation matrix for the cylindrical Laplacian (5.8)
using Nr = Nz = 64 nodes.............................................. 108
5.2 Initial liquid volume fraction el(r, z, 0) with e(nin = 0.1 and elmax = 0.9.116
5.3 Liquid volume fraction, el(r,z,t), plots over the cylindrical cross-
section Q with a Kozeny-Carman permeability, K(el) = linear
diffusion coefficient, D(el) el, and model parameters // = 0.01 and
t 1. The grid size is Nr x Nz = 64 x 64 with Ncr = Ncz
42(0.65Ar) and a conventional (explicit) RK4 time-stepper was used
with constant time-step At = 10-6. Solutions are shown at (a) t = 0,
(b) t = 0.2, and (c) t = 0.4..................................... 123
5.4 Liquid volume fraction £l found by solving (5.36) using an MOL
approach with the EPS discretization and MATLABs odel5s for
time-stepping.................................................. 124
5.5 Relative L error comparing Chebyshev collocation to the EPS con-
struction over t [0,10] for the Flory-Huggins model (5.36)......... 124
xiv


5.6 Liquid volume fraction, £l(r,t), plots over the radial grid with a
Kozeny-Carman permeability, K(el) = ^7 and model parameters
H = 0.1 and r = 1. The grid size is Nr = 750 with Ncr = 450(0.60./V,.)
and a variable step-size, 5th order implicit time-stepper was used.
Solutions are shown for t [0,0.4] with (a) D(el) = 1, (b) D{el) = el,
and (c) D{el) = (e1)2
125
6.1 Lows functional form of the swelling pressure combined with the
6.2 Liquid volume fraction, sl(r,t), plots over the radial grid with a
6.3 Model comparison in terms of viscoelastic stress and moisture con-
tent writh D(el) = el and fixed parameters De = 1 and ks = 1. (a)
viscoelastic stress resulting from using a constant permeability model
(A = initial stress, o = final stress), (b) viscoelastic stress resulting
from using a Kozeny-Carman permeability model (A = initial stress,
o = final stress), and (c) moisture content comparing constant per-
meability against Kozeny-Carman permeability................. 138
6.4 Kozeny-Carman permeability as a function of liquid volume fraction. 139
6.5 Normalized moisture content curves M/AIA positive ks increases
flow whereas a negative ks inhibits flow..................... 139
129
Kozeny-Carman permeability, K1{e1) = and model parame-
ters ks = 0.1 and r = 1. The grid size is Nr = 750 with Ncr =
450(0.60Ar) and a variable step-size, 5th order implicit time-stepper
was used. Solutions are shown for t [0,0.4] with (a) D(e1) = 1, (b)
137
xv


6.6 Normalized moisture content curves (a) Fix ks = 0.1 and
compare the moisture content for De = {0,0.01,100}, (b) Fix ks
0.1 and compare the moisture content for De = {0,0.01,100} . . 140
6.7 Normalized moisture content curves M/M^: (a) Fix De = 0.01
and compare the moisture content for ks = {0.01,0.1}, (b) Fix
De = 100 and compare the moisture content for ks = {0.01, 0.1},
(c) Fix ks 0.1 and compare the cases De = {0.01,1,100}, (d)
Fix Kg = 0.01 and compare the cases De = {0.01,1,100}........... 141
6.8 Viscoelastic stress ............................................... 142
6.9 Viscoelastic stress ............................................... 143
xvi


TABLES
Table
2.1 Units for the dimensional parameters of the Darcy model (2.50). . 27
2.2 Characterization of the Deborah number, De......................... 35
2.3 Characterization of ks................................................ 36
5.1 Values of the log ||en(t)||00 error at times U = 0.25f/, <2 = 0.50 /3 = 0.75f/, and t.\ = tf comparing RI<4 and ODE45 versus SAI for
Example 1...................................................... 120
5.2 Values of the log ||en(f)||00 error at times t\ 0.25tf, t2 = 0.50tf,
£3 = 0.75tf, and <4 = tf comparing RK4 and ODE45 versus SAI for
Example 2...................................................... 121
5.3 Values of the log ||e(t)||00 error at times t\ = 0.25tf, t2 = 0.50t/,
t'i = 0.75t/, and tf comparing RK4 versus PVRK4 versus SAI
for Example 2....................................................... 122
6.1 Characterization of ks............................................... 130
6.2 Characterization of the Deborah number, De........................ 131
B.l Butcher diagram...................................................... 158
xvii


1. Introduction
Generally speaking drug delivery systems are devices used for the pro-
cess of administering pharmaceutical compounds to achieve therapeutic effects.
These systems can be divided into two major types: traditional delivery systems
and controlled released systems. Traditional delivery systems are characterized
by their immediate release of the drug which leaves absorption to be controlled
by the bodys ability to assimilate the drug concentration into different body
tissues such as the blood. Drug concentrations from these systems typically
undergo an abrupt increase followed by an abrupt decrease. Controlled release
drug delivery systems, on the other hand, are formulated to modify the drug
release profile, absorption, distribution, and elimination for the benefit of im-
proving therapeutic efficacy, safety, patient convenience, and compliance. These
systems are characterized by their maintenance of drug concentration to target
tissues at a desired level for prolonged time periods.
Depending on the release behavior, controlled release systems can be sub-
divided into three categories: passive pre-programmed, active pre-programmed,
and active self-programmed. The models we wall study focus on passive pre-
programmed release where the rate is predetermined and dependent upon the
drugs release kinetics as it interacts with the body. We leave this restriction
vague for the moment but will further restrict our focus as we derive the mathe-
matical model and impose simplifying assumptions. In lieu of definitions for the
remaining two categories, which we offer for the sake of curiosity and complete-
1


ness, we provide examples. An example of an active pre-programmed device
is an insulin pump for type 1 diabetes patients where the insulin is delivered
as a continuous flow to the body through a short tube with a needle at the
endpoint that is inserted under the skin, usually in the abdomen. The third cat-
egory, active self-programmed, is exemplified by devices that combine (provide
data interfacing) insulin pumps with glucose monitors or, as a simple example,
morphine drips.
It should be evident by the definitions and descriptions given above that
drug delivery has become more complex moving from uncontrolled release to
sustained (controlled) release and programmable (controlled) release. It has
also become more specific as we have witnessed a move from systemic delivery
to organ (as in the case of cancer radiation therapy) and cellular targeting
(as in the case of tumor-targeting delivery systems). As is the case in many
fields, a feedback loop originates where increasing complexity in the technology
drives the requirement for mathematical modeling and simulation and vice versa.
Mathematical modeling of drug delivery is in its inchoate stages, intimated by
the reviews given in [41, 60, 76], but has become an area of active and growing
research.
We will focus our attention on controlled drug delivery systems, also knowm
as controlled release systems (CRS). Furthermore, we will limit the scope of our
study to passive pre-programmed CRS and will subsequently drop this modifier.
Therefore the CRS systems that fall under our scrutiny are those that are fabri-
cated by embedding a drug in a hydrophilic (water-loving) polymer matrix such
as hydroxypropylmethylcellulose (HPMC) [76]. Matrix in this context refers
2


to the three-dimensional network containing the drug and other substances re-
quired for controlled release. The matrices can be prepared in several ways.
Two examples include (1) mixing the powdered drug with a solvent, excipient
(inactive ingredients), and pre-polymer before placing it in a polymerisation re-
actor or (2) preparing the matrix in advance and then putting it in contact with
a highly concentrated drug solution able to swell the matrix whereby the solvent
is removed afterwards (solvent swelling technique) [41]. A detailed exposition
on polymeric matrices is beyond the scope of this thesis but there are several
references available [41, 76].
However, we will briefly describe the properties of polymer matrices useful
for controlled release systems. First, polymers are viscoelastic. By the name, it
should be apparent that viscoelastic materials possess both viscous and elastic
properties. Viscosity is the characteristic of a liquid that makes it resistent to
flow. For example, honey is more viscous than water. Elasticity is the charac-
teristic of a material that returns it to its initial state (instantaneously) after
the external forces that deformed it cease. For example, a dry sponge is more
elastic than pumice. Hence, a viscoelastic material has the characteristic that
it initiates returning to its original state after the external forces that deform
it cease, but the elasticity is inhibited by the viscous behavior which resists the
change in motion.
Second, the polymer matrix is a porous material as seen by the polymeric
matrix in Figure 1.1. Polymeric matrix systems can be classified according to
their porosity (macroporous, microporous, and non-porous). We will focus our
attention to the macro and micro porous systems where pore sizes range in
3


size from 0.1 1 fjm and 50 200/1, respectively. Drug release kinetics will
be affected by polymer swelling, polymer erosion, drug dissolution/diffusion,
drug distribution (inside the matrix), drug/polymer ratio, and system geometry
(cylinder, sphere, etc.). The matrix systems under consideration will be dry and
compressed without any fluid inside. We consider the gas filled pores to be part
of the liquid phase (fluid and gas). Therefore in our models the initial conditions
will contain a small percentage of liquid phase. Once the drug delivery device
Figure 1.1: Porous polymer matrix. Photo taken from plc.cwru.edu.
is immersed in biological fluid, the surrounding fluid pressure coupled with the
hydrophilic properties of the polymer drives the fluid to penetrate the polymer
matrix as described by Darcys law for fluid flow (which we will describe in more
detail in Section 1.1 and Chapter 2); the novel form of Darcys law derive in
[73] contains a constitutive equation modeling the viscoelastic properties of the
polymer. The polymer swells as dictated by its viscoelastic properties and the
pores in the matrix enlarge until they are of a size necessary for the drug to
escape. The drug then diffuses into the surrounding fluid diminishing the drug
concentration levels present in the delivery device.
4


1.1 Previous Work
The equations considered in this thesis are nonlinear Volterra partial inte-
grodifferential equations (VPIDE) derived by Weinstein and Bennethum [73, 74]
and Singh et. al. [63]. These models are derived using Hybrid Mixture The-
ory (HMT) which involves upscaling field equations which govern the motion
of materials and include the conservation of mass, conservation of linear and
angular momentum balance, and conservation of energy from the microscale to
a larger scale via volume averaging. Restrictions are then obtained on the form
of the constitutive equations by using the second law of thermodynamics, also
formulated as a field equation, at the large scale. A variable that results from
upscaling via HMT is the volume fraction, a ratio of the volume of a particular
phase over the sum of the volumes of all phases. For example, in a two-phase
mixture with a liquid phase (indicated by l) and a solid phase (indicated by
s), the liquid volume fraction, denoted £l, is the ratio of the volume of the liq-
uid phase (Vl) to the sum of the volumes of both phases (Vs + V1). That is,
c = yr^yi- The liquid volume fraction, el, is the dependent variable considered
in the models we study in this thesis. Note that if there are only two phases, s
and /, we have el + es = 1.
In the case of the VPIDE considered, the mass conservation equation is
upscaled and coupled to a novel form of Darcys law [73]. Darcys law is a
constitutive equation relating the flux of a fluid field to a change in pressure [29].
As such it is often used to describe the flow of a fluid through a porous medium.
In the case of the VPIDE derived by Weinstein [73], time rates of change of el
are included as a constitutive variable. That is, Weinstein included the material
5


(m)l
time derivatives of the liquid volume fraction, e for m = 1,2,...,/; where p
is determined by how long of a time history we want to include in our model.
This temporal history accounts for the viscoelastic stress and the model for
the stress assumes that the strain effects are cumulative, hence it contains an
integral. That the integral term contains time derivatives of the volume fraction,
not solid-phase strain, is justified in [51] where it is shown that at moderate to
high fluid contents the normal components of the strain tensor are related to
the volume fraction of the solid phase, es. Moreover, since the solid and liquid
phases are all that comprise the material, we have that el = 1 es. Hence, at
the fluid content levels considered in these models, the time derivative of strain
may be replaced by time derivative of liquid volume fraction. As a result of this
novel form of Darcys law, the VPIDE models liquid penetration into the drug-
delivery device inhibited by the viscoelastic properties of the polymers. That is,
the VPIDE can be viewed as the sum of a liquid penetration (nonlinear diffusion
equation) term and a viscoelastic term (integral equation).
An equation similar to the one derived in [73] wras derived by Singh et. al.
[63]. Singh also numerically solved the VPIDE using a finite-element method
[64, 65]. However, the solver produced spurious oscillations that resulted in
non-physical values for the volume fraction in [65]. Even though this work was
completed in 2003, to our knowledge, it has never been revisited. Moreover, the
equation in the drug delivery context [73] has not been solved.
Models developed in the pharmaceutical literature that take into account
the viscoelastic properties of the polymer have been phenomenological [12, 47,
60, 61, 62, 76] and lack the physics of the models derived by Weinstein [73]. In
6


addition, these models are typically linear partial differential equations, such as
Ficks second law of diffusion in cylindrical coordinates [76], that can be solved
either analytically using separation of variables or numerically using elementary
methods [60]. There are diffusive models taking into account viscoelastic relax-
ation in polymers that have been developed. In particular Cohen and White [24]
extended the work of Thomas and Windle [67] in modeling sharp fronts due to
diffusion and viscoelastic relaxation in polymers. It should be noted that Cohen
and White did not solve the full partial integrodifferential equation numerically
but rather they reduced it, using a perturbation series, to a system of ordi-
nary differential equations which they solved with an Adams-Bashfort-Moulton
numerical time-stepping scheme [24],
1.2 Thesis Outline
In this work we solve the model derived by Weinstein [73] using a novel for-
mulation for pseudospectral differentiation matrices, the Eigen-decomposition
Pseudo-Spectral (EPS) method. The governing equation is a nonlinear Volterra
Partial Integrodifferential Equation (VPIDE) of the second kind. These equa-
tions are particularv difficult to solve not only because they are nonlinear but
also because the integral term poses some numerical challenges. Since this term
is cumulative, over time, it collects round-off error and can pose stability prob-
lems for a numerical solver. Singh [64, 65] solved the problem using a finite-
element method with a time-stepping technique derived by Patlashenko et. al.
[53]. However, Singhs solution for the liquid volume fraction was unsatisfac-
tory in that it attained nonphysical values. We will show that one can obtain
higher-accuracy than the method suggested in [53] by using a method-of-lines
7


(MOL) approach. In this context we have extended the work of Weinstein and
Singh as well as added mathematical rigor to their results.
In Chapter 2 we provide a review of HMT relevant to the derivation of
the governing continuity equation and nondimensionalize the model arriving at
two non-dimensional parameters, De (Deborah number [56]) and ks (ratio of
viscoelasticity to diffusivity). In Chapter 3 we analyze the VPIDE by finding
a sufficient solution space that provides a proof of existence and uniqueness
on a cylindrical geometry, under a specific set of initial boundary conditions
that match the physics of the drug-delivery application. We reduce a special
case of the VPIDE, using a derived similarity variable, to an ordinary differen-
tial equation and solve the resulting boundary value problem using a shooting
method. We find an analytic solution to the Flory-Huggins model, that was
re-derived in [73], which is another special case of the VPIDE. These solutions
provide us with an expectation for the behavior of the solutions to the model. In
Chapter 4 we provide an overview of pseudospectral methods and introduce the
EPS method. We use the same (spatial) regularity conditions imposed in the
existence-uniqueness proof to derive an error formula for the EPS method. The
numerical solver for the VPIDE is provided in Chapter 5. We use the solver on
the Flory-Huggins model with both the EPS method and a conventional pseu-
dospectral method (Chebyshev collocation) comparing the numerical solution
to the analytic solution from Chapter 4. This example allows us to compare the
EPS method against Chebyshev collocation as well. In Chapter 6 we conduct
several numerical experiments to study the behavior of solutions to the VPIDE
under a variety of conditions. We test different diffusion coefficients and per-
8


meabilitv models as well as test the models sensitivity to the non-dimensional
parameters derived in Chapter 2. Chapter 7 contains ideas for further research.
9


2. Fundamental Continuity Equation
In this chapter we review some of the results from [73] relevant to the deriva-
tion of the transport equations being solved. This work relies heavily on Hybrid
Mixture Theory (HMT) for which there are several excellent resources available
containing a rigorous and detailed overview [8, 9, 73]. As such, we will not reca-
pitulate HMT but will give a brief overview of the relevant aspects and reference
the listed resources as required. We also offer a physical interpretation for the
derived equations couched in terms of a drug-delivery model.
2.1 Brief Overview of Hybrid Mixture Theory
HMT involves upscaling field equations which govern the motion of materials
from the microscale to the macroscale via volume averaging and then obtaining
restrictions on the form of the constitutive equations by using the second law
of thermodynamics, also formulated as a field equation, at the macroscale. The
field equations often include the conservation of mass, conservation of linear and
angular momentum balance, and conservation of energy. Recall that constitutive
equations are specific to the material being modeled examples include Fouriers
Law and Darcys Law. There are several other upscaling techniques such as
homogenization, however, since we base our analysis on equations developed by
Weinstein [73] we focus on HMT.
The averaging procedure used in HMT is for any multi-scale, multi-constituent,
multi-phase material. We consider a two-scale (microscale and macroscale) two
phase material liquid (/) and solid (s), each composed of one constituent within
10


each phase. Since the equations of interest in the drug-delivery model describe
a single constituent, we will suppress any indices referring to constituents and
generally refer to phases using Greek letters, a, (3,_At the microscale the field
equations are all valid while the dependent variables such as density, velocity,
and energy are clearly defined. Moreover, the phases such as liquid and solid,
are distinguishable at the microscale. At the macroscale the phases are indis-
tinguishable and the averaging procedure provides analogous field equations at
the macroscale with variables that can be identified with their microscale coun-
terparts. The macroscale variables may be volume averaged (e.g. density) or
mass averaged (e.g. velocity) depending on the physical quantity they describe.
Hassanizadeh and Gray use the following four criteria when determining how to
average particular terms:
1. When averaging, the integrand times dv (volume differential) or da (area
differential) must be an additive quantity. For example, Edv, which is
energy times volume per unit mass, is not additive, whereas multiplying by
the density, p), yields, pEdv (units of energy), a quantity that is additive.
2. The macroscopic quantities should exactly account for the total corre-
sponding microscopic quantity. This is especially important when dealing
with a flux term across boundaries.
3. The primitive concept of a physical quantity must be preserved by proper
definition of the macroscopic quantity.
4. The averaged value should represent the same function which is most
widely observed and measured in a field situation or laboratory at the
11


macroscale. This insures applicability of the resulting equations.
Besides being a two phase, single component, two additional assumptions
concerning the material include that the interface does not contain any thermo-
dynamic properties and is massless, and that a representative elementary volume
(REV) exists on the macroscale at any point in space. The REV, as defined by
Bear [7], is characterized by the materials porosity which we define below. This
volume must be much smaller than the size of the entire flow domain, yet larger
than the size of a single pore; that is, it must be large enough to contain a
sufficient number of pores such that a statistical average can be computed.
Consider a volume, 6Vi, and let x be the centroid of this volume. Compute
the ratio (porosity)
fii =
(6Vo)i
6Vt
(2.1)
where {5V0)i is the volume of the void space within 8Vt. Gradually shrink the
size of 8V, around x such that <51) > 6V2 > 8V3 >_ The ratio n, may fluctuate
as the size of 8V, is reduced; especially in an inhomogeneous material. However,
these fluctuations will decay as 8Vi transcends below a certain level leaving only
small amplitude changes due to random changes in the distribution of the pore
sizes near x. There will be a value, call it 8V, for which n, will begin to undergo
large amplitude fluctuations again as 8 V, < 8V because 8Vt wall be small enough
to contain mostly pores or mostly solid material. Eventually, as 8Vi tends to
zero, rii will converge to 0 or 1 depending on whether x is contained within the
solid material or contained within a single pore. The volume, 8V, is the REV
with centroid x.
12


Figure 2.1: Averaging, Local Coordinates
The averaging procedure entails a weighted integration over the REV using
an indicator function of the a phase,
where r is the position vector and £ is the local coordinate referenced to the
centroid x of the REV. The position vector, r, is expressed as
Note that the weight function used in the averaging technique represents the
instrument used to measure the properties of the material [27]. Hence, using
7a may not be an appropriate weighting function in the sense that the aver-
aged value may not represent the actual values being measured. Moreover, the
presence of the characteristic function in the averaging procedure implies
distributional derivatives are required to make the process mathematically rig-
orous. We discuss these nuances as they arise and refer to the relevant sources
as necessary.
(2.2)
r = x + £.
(2.3)
13


The magnitude, denoted by | |, of the volume SV in the o-phase is defined
by
\SVQ\(x,t) = 7a(x + £,t)dv{£),
Jsv
and the a-phase volume fraction, £a, is defined as
\SVa\
£a(x,t) =
m
so that we have the relations
Ee = 1-
and
0 < £a < 1.
Define the following quantities,
£(*. 0 = TTVTT / Piri tha{r, t)dv{£)
l^bol Jsv
(ip)a(x,t) = f ip{r,t)-fa{r,t)dv{^)
\oVa\ Jsv
(2.4)
(2.5)
(2.6)
(2.7)
(average mass over SVn)
(volume average of ip)
$
a{x, *) = -aill/ I [ P(r> t)ip{r, their, t)dv(£) (mass average of ip)
Pa\t>Va\ Jsv
Since the upscaling procedure requires a weighted average (integration) of
partial differential equations, we need an averaging theorem for justifying the
interchange between the differentiation and integration. This theorem is stated
in [28].
Theorem 2.1 (Averaging Theorem) If wa/} is the microscopic velocity of
interface a,8 and na is the outward unit normal vector of 8Va indicating the
integrand should he evaluated in the limit as the a8-interface is approached from
14


the a-side then
L f^iv=I lm Lhdv\ £ w\ L, !w"s nVo
mLVhiv=v
mLhiv}+^w\Ljn,la-
This theorem outline the averaging process (see [8, 28] for a proof). If / is the
variable to be averaged, then we perform the following steps:
1. Begin with a conservation equation within a phase.
2. Multiply the equation by qQ.
3. Average each term over the REV, 6V (integrate over 5V and divide the
integral by |<5V|).
4. Apply theorem (2.1) to arrive at terms representing macroscale quantities.
5. Define physically meaningful macroscopic quantities.
The sums pj JdAa0 fw0 nVrt and £/?*, JW] ^An0 arise
from differentiating the function, qQ. When these sums are combined, they are
called an exchange term as this term represents the net change in a quantity,
such as a constituents mass, as it transitions from one phase to the other. A
specific example will be presented in Section 2.2.
15


2.1.1 Macroscale Field Equations
Recall that the mass balance equation for a single constituent is,
dp ^
+ V (pv) = pr,
(2.8)
where p is the constituent density, v is the constituent velocity, and r accounts
for the introduction or exit of the constituent mass due to chemical reactions.
We obtain the macroscale equation by formally multiplying this equation by 7Q,
integrating over 5V, and then dividing by |£F|. Using (2.1), we have for the
time derivative term,
where pa is the volume-averaged density in the a phase. A similar averaging
\SV\Jsvdtla V
1 [ dp
Is
mUrf'nda
(2.9)
process is performed on the V (pv) term to obtain the volume averaged mass
balance equation,
(2.10)
where va is the a-phase velocity and as before m"'9 is the microscopic velocity
of interface af3.
16


Dropping the overbar notation we rewrite the upscaled mass balance equa-
tion as,
(eapa) + V (eapava) = J p (wad va) nada (£), (2.11)
Aa 3
where the summation results from jumps across interface a{3. Notice that if the
interface remains stationary then ma/3 = 0, and if the flux va na is net positive,
then there is mass transfer across the interface from /3-phase to o-phase.
Define
e? = J P (wO0 ~ v) nda (£) ,
which represents the net rate of mass gained the single constituent with density
p in phase a from phase /?, then (2.11) becomes,
(eapa) + V (eapava) =
If we make the restriction that the interface does not contain mass,
/3/ct
then for a two phase system where el + es = 1 (/ and s denote the liquid and
solid phase respectively) this restriction mathematically becomes,
e? + ei = 0. (2.12)
Without mass exchange from drug to liquid we have that ef = els = 0. The
conversion of the material time derivative from liquid to solid phase is given by
the definition,
D(-) P(-) | r/..
Dt Dt
V(-),
(2.13)
17


where vl's = vl Vs is relative velocity between phases. Applying definition
(2.13) to (2.11) simplifies the mass balance equation in the liquid phase to,
Dlelpl
Dt
+ elplV -vl = 0.
(2.14)
2.2 Derivation of the Continuity Equation
In this section we show the derivation of a Volterra Partial Integrodifferential
Equation (VPIDE) that models the polymer swelling in the drug-delivery device.
We assume that there are only two phases, liquid (l) and solid (s); the liquid
phase will consist of gas (filling the pores of the polymer matrix) and fluid and
the solid phase will consist of drug and polymer. The derivation of the continuity
equation for the swelling regime essentially follows by employing the definition
(2.13).
Before continuing with the derivation it is important to note that incom-
pressibility for the liquid phase is defined as ^-t = 0. Noticing that,
Dlpl Dspl
Dt
Dt
dp1
v

V//
= -k- + va Vpl + Vl's Vpl
at
= !r
and assuming that the liquid phase is approximately incompressible ~ 0
simplifies (2.14) to,
D1-1
+ el V vl = 0. (2.15)
Dt y J
During the swelling regime we assume that the drug does not transfer to the
liquid phase, so the density of the liquid phase does not change (temporally) for
18


this regime, that is, = 0. Therefore the incompressibility assumption leads
to
vl Vpl = 0, (2.16)
and because vl ^ 0,
Vpl = 0. (2.17)
The mass balance equation for the solid phase is given by,
e* + es V vs = 0,
(2.18)
w
here we denote £
s Dses
Dt
. Equation (2.18) can be rewritten to obtain,
V -vs =
(2.19)
Notice that (2.19) can be written in terms of the liquid phase as,
A
V -vs =
(1 -£*)
Applying the definitions given in (2.13) and (2.18) to (2.15) produces,
Dl 0 = ^ + £l V vl A + Vl's Ve' + £lV V1
(2.20)
Dt
Dt
Dsel
~Dt
+ Vl'S Vs1 + £lV vl £lV Vs + £lV Vs
= il + c'V vs + vLs Ve' + £l V Vls
£l£l
= £l + ------K + Vls Ve' + e'V vLs (employing(2.20))
(1 e)
= £' + 77^7 + V-(eVs),
(1 £*)
hence we obtain the continuity equation in the form,
£' + (l-£')V-(£V'S) = 0,
(2.21)
where £lvl,s will be given by a form of Darcys law that we derive in Section 2.3
below.
19


2.3 Darcys Law
The details for the derivation of the form of Darcys law we use can be
found in [73, 74] and we review it in Appendix A. In that derivation Weinstein
postulates the dependence of the Helmholz free energy as
(n)
il>1 = xl>l{el,(me\p\Cl,T, C\C),
(2.22)
where m = 1,... ,p and n = denote material time derivatives of order
p and q, Cl] is the concentration of the jih species in the liquid phase and
j = 1,..., Ar, T is the temperature, C* is the modified right Cauchy-Green
tensor.
The form of Darcys law we employ is,
Rl eVs = V(e p) + p Ve £lpZ(t)
p
(m)l
(2.23)
where, Rl is the resistivity tensor, £(f) and
m=1
Ml
~ (m)/
(2.24)
d £
is constant. This term is the result of assuming that ^ e for m = 1,..., p are
independent variables. Making time derivatives of the volume fraction indepen-
dent variables allows for modeling viscoelastic effects since these terms retain
the time history of the volume fraction as it evolves. Implicit in the model
assumption is that these terms get weaker so that viscoelastic effects diminish
as m increases. Moreover, inclusion of these time derivatives implies that all
20


of these time derivatives of the volume fraction exist, hence el G Cp([0, oo)) in
time.
. I /
[11, 19] where ip1 is
e'T
The classical liquid pressure is given by pl
the Helmholz energy, vl is the specific volume, and T is the temperature. We
assume isothermal conditions in the biological fluid, so we will neglect T in our
derivation. Applying the definition for specific volume, vl yields,
Oil)1
Pl = (P?
dp1
elpli
(2.25)
. This pressure is the
The thermodynamic pressure is defined pl =
result of measuring the energy required to change the volume of the liquid phase
keeping the mass fixed. Hence we can write the thermodynamic pressure as a
function of the volume fraction and density, pl pl(el,elpl]).
, that
There is a third pressure called the swelling potential, it1 = i-^r
is related to pl and pl,
P,Cld
pl(el,pl,Cl>)=pl(el,£lpl) + nl(£l,pl,Cly,
(2.26)
see [11].
Now consider the pressure terms of (2.26) and rewrite them in terms of the
thermodynamic pressure and swelling potential,
-V(eV) + plV£l = -plV£l £lVpl + plV£l
= -plV£l 7r'Ve' £lVpl £lVnl + tfVs1
= -TtlV£l £lViTl £lVpl. (2.27)
We will now simplify this expression further by making some simplifying as-
sumptions for the swelling regime. Consider the boundary condition for the
21


delivery device, where Gibbs potential in the bulk liquid phase is balanced by
the Gibbs potential in the liquid phase (interior of device),
^l + H-=^B+ B,
Pl PB
(2.28)
where B denotes the bulk phase. The right-hand side is constant since we are
considering a drug delivery device that is immersed in a biological fluid.
Hence, we are assuming that the pressure is continuous across the bound-
aries. Hence we have the following,
dip1
del
Vel +
1 dp1
pl del
Vel = 0,
(2.29)
where multiplying by elpl and simplifying yields,
i i
£lp
del
v del
i i dp1
= 0.
So that,
/ dp1
--------
del
7T ~
(2.30)
(2.31)
We now assume that irl is primarily a function of £l. Moreover, following a
similar argument,
dip1
dp1
_ , , 1 dp1
vp + ly W
y)s
Vp' 0,
where multiplying by (p )2 and simplifying yields,
(Pl)
K 2
dp1
, / dp1 i dp1
+ P t: el dp1 -p = p jn el dPl
= 0,
applying (2.25). Hence,
dp1
dp1
e
= 0,
(2.32)
(2.33)
(2.34)
22


and we conclude that the classical pressure, pl, is primarily a function of el.
Next consider the gradient of the thermodynamic pressure,
vp'=aJ-
P del
_ ( dp
Ve + tt-t
dp1
Vpl
(2.35)
Equation (2.17) holds justifying the following assumption,
dp1
del
_ ; dp1
V£ >y w
VPl
(2.36)
Now one may argue that the thermodynamic pressure can change rapidly with a
change in density such that the coefficient of Vpl in (2.36) will make these terms
the same order of magnitude as one another. The counterargument to this (well-
justified) point is that the incompressibility assumption forces Vp( 0 such that
(2.36) is justified. Therefore, applying (2.17) to (2.35) simplifies the pressure to,
V(eV) + fVe' =
i i
7 +£ a?
+ e?l
j del
Ve1.
(2.37)
One final simplifying assumption is that, while in the swelling regime, the change
in thermodynamic pressure relative to the change in liquid content is small
compared to the change in the swelling potential relative to the change in liquid
content, el

del
Therefore (2.37) finally simplifies to
V{elpl)+plVel = -P0W + e1^
Ve',
(2.38)
where the constant / o has units of pressure and is written to indicate that
7r' is a dimensional quantity with units of pressure. We will eventually non-
dimensionalize the resulting partial integrodifferential equation, so labeling these
dimensional quantities will prove convenient for notational reasons.
23


We now focus on writing an expression for Darcys law that eliminates the
time derivatives of zl in the £(t) term and expresses V(eV) + ^X7:1 in terms
of liquid diffusion resulting from the bulk fluid penetrating into the delivery
device. Begin by taking the Laplace transform of £(f) [45],
___ y r
m = Y,M(r)v /
m=1
e e dt
m 1
P
m=l
where we assume that all initial gradients are zero,
= 0 for all m.
V(me)l
(2.39)
(2.40)
(2.41)
(2.42)
t=o
Now apply the convolution theorem [1] to £(t) to obtain
m=1
,_1 Jo
m=l
Letting
(2.43)
(2.44)
Bv(t) = B0Bv(t S) = B0J2 - f)
m1
wTe obtain the following form of (2.43),
£(t)= f Bv(t t')Veldt'. (2.45)
Jo
The constant Bo is inserted for the purposes of dimensional analysis to be per-
formed below.
Therefore, Darcys law can be written as
B!.elvls = -Ll + e1^
] Ve1 ip1 f Bv(t t')Vildt', (2.46)
yl J JO
24


and letting
(2.47)
Ve/ k\el)elpl [ Bv(t t')Veldt'. (2.48)
In [63] the permeability function is assumed to be constant and the entire co-
efficient of the integral term is absorbed into the kernel. But we can see from
the integral. However, el is a function of space and time, hence this combination
of the integral term.
2.4 Non-dimensionalization of the Model
As an aid to interpreting the physics of these models and solving them, we
consider the dimensional parameters associated with some of these terms. Non-
dimensionalizing the model yields non-dimensional parameters that appear as
coefficients of some terms in the equation. Hence, these parameters scale some
of the terms in the equation and permit us to determine their relevance.
According to [7] the coefficient of the permeability function is where
Mg is the specific surface area and vl is the viscosity of the liquid phase. So we
can write the permeability as,
(2.48) that this maneuver translates to Kl plel combining with Bv(t t') under
of terms found in [63] is invalid and we leave the function, K plel, as a coefficient
(2.49)
25


where Kl(el) is a non-dimensional function to be determined. Notice that (2.48)
can be written with these dimensional parameters as,
Po
elv11
My
BoP1
K'W) d{En')
dsl
rt
Ve1
(2.50)
+ -
f-K\e1)e1 [ Bv(t t')VeldA .
0 Jo )
A units check of the parameters is given in Table 2.1 where we notice that the
coefficients and have units of diffusivity, L2/T, and non-dimensional
respectively. We should note that the terms B0pl and P0 both have units of
pressure. The coefficient of the integral term, B0pl, is a modulus of elasticity
(which has units of pressure) that behaves as an amplitude for the viscoelastic
effects of the model. On the other hand, flow* driven by the swelling pressure
occurs instantaneously and does not retain any time history so the pressure term
(2.38) accounts for the elastic effects of the model and the coefficient, P0, signifies
the magnitude of those effects. Note that this term does encompass fluid-solid
interactions and will account for some viscoelastic effects. Achanta et. al. [3]
showed that anomalous flow results from this term, and we will demonstrate this
result when we show the solutions to the model under a variety of conditions.
Define the dimensionless parameter ks = and the parameter with units
of diffusivity (L2/T) k0 = then write Darcys law (2.50) as
£lVl,S = -K0
AV) a{eV)
del
rt
Ve1
(2.51)
+ksK1(e1)e1 [ Bv(t-t')Vildt'
Jo
Note that ks is a ratio of the modulus of elasticity to the swelling pressure
coefficient. For the purposes of physical interpretation, we will make ns a signed
26


Parameter Units
Po M LT2
AI2a 1 L2
ul M LT
Pl M L3
Bo ( ) m\t2>
Table 2.1: Units for the dimensional parameters of the Darcy model (2.50).
number, ks E (00,00). A negative ks is akin to the restoring force and indicates
the polymers resistance to deformation. Whereas a positive na indicates a
materials compliance to deformation. We provide a brief analysis regarding
this parameter in Section 2.6 and show the impact of ks when we conduct a
parameter sensitivity study in Chapter 6.
2.5 Continuity Equation
The continuity equation we will consider models polymer swelling as fluid
imbibes into the polymer matrix, i.e. the swelling regime. Here the drug
delivery device is immersed in a biological fluid. The initial assumptions for this
regime are that there is no transfer of drug to the liquid phase and that the
solid phase, s, is composed of both polymer and drug; that is, this continuity
equation is a two phase model such that es + sl = 1 and such that there is
no diffusion (species transport into the liquid phase) taking place in this model
regime.
27


Let the function modeling the liquid penetration into the solid phase be
denoted
D(£') = KV) a{eV)
del
(2.52)
Incorporating (2.50) and (2.52) into the continuity equation with no mass-
transfer between phases (2.21) yields the following Volterra partial integrod-
ifferential equation (VPIDE):
il = K0(l e1)'V yJ(£l)V£l + KsKl(£l)£l Bv(t t')Vildt'j , (2.53)
where we need to ascertain functional forms for 7t1(e1) and Kl(el). The use
of the term diffusion requires some clarification. Engineers often refer to
the term diffusion when they are considering a species propagating into a fluid
whereas mathematicians refer to diffusion as the process by which a substance
satisfies the time-evolution of a diffusion equation. We adopt the mathematical
point of view and refer to the liquid penetrating the device as diffusing into
the device, and hence refer to D(e1) as a diffusion coefficient rather than a
liquid-penetration coefficient or liquid diffusion coefficient.
2.6 Model Interpretation
In this section we offer a physical interpretation for the model (2.53). We
also provide a nominal expression for the kernel Bv found in (2.53); the functional
form for Bv provided in this section will be used throughout the thesis. Non-
dimensionalizing the model will yield two parameters, the Deborah number (De)
and integral term coefficient (ks). These non-dimensional parameters wall be
used to analyze and study the model as we perform numerical experiments in
Chapter 6. We will define them and provide interpretations below.
28


By the name, one can infer that viscoelastic materials exhibit both viscous
and elastic properties. Elastic materials are characterized by the property that
if a load is introduced to the material and it deforms, then upon unloading,
the material immediately returns to its initial state. This property is shown
in Figure 2.2 where the stress versus strain curve for the elastic material (left)
is identical during the loading and unloading phases. Linear elastic materials
follow Hookes law which states that stress is linearly proportional to strain.
A simple model describing Hookes law is the spring equation where the force
exerted by the spring is proportional to the length that the spring is stretched
from its initial (resting) position.
Viscoelastic materials on the other hand retain some or all of the energy.
Figure 2.2 show's the stress versus strain curves for a viscoelastic material (right)
following different paths during the loading and unloading phases. During the
unloading phase the stress curve sags beneath the loading curve signifying the
decay in stress over time; this phenomenon is called hysteresis. Figure 2.3 shows
an example of stress versus time where constant stress is introduced during the
loading portion, t < t\, then finished at t\ thus commencing unloading, t > t\.
The stress curve for an elastic material would drop to zero instantaneously;
however, for a viscoelastic material it decays. The simplest models, such as the
Maxwell or Kelvin model, use a decaying exponential as a means to capture this
behavior, see [38] for example.
Physically, the integral term in (2.53) is derived from a constitutive rela-
tion that accounts for the viscoelastic effects of the solid material being mod-
eled. Typically the viscoelastic stress is written using the stress-strain relation-
29


Stress a
Stress Figure 2.2: Stress (a) versus strain (e not to be confused with the volume
fraction el) curve for linear elastic (left) and linear viscoelastic (right) materials.
Stress(T
Figure 2.3: Stress versus time where the loading takes place over the time
interval t < t\ and the unloading takes place over the interval t>t\.
30


ship [23],
(2.54)
where Es is the solid-phase strain tensor.
This model for the viscoelastic stress assumes that the strain effects are
cumulative, hence integral. This model follows the so-called Boltzmann Super-
position Principle and can be derived using the Riesz Representation Theorem
[23]. However, the integral term (3.8) contains time derivatives of the volume
fraction, not solid-phase strain, Es. There is a relationship between these twro
variables. In [51] it is shown that at moderate to high fluid contents the normal
components of the strain tensor are related to the volume fraction of the solid
phase, es. Moreover, since the solid and liquid phases are all that comprise the
material, we have that el 1 es. Hence, at these fluid contents the depen-
time derivatives of volume fraction, el, were chosen as independent variables
for the constitutive equation (2.22) and appear in the series (2.24) which was
subsequently converted to the integral term using the Laplace transformation.
The kernel, Bv, is the function that accounts for the decay of the viscoelastic
stress and is aptly named the compliance function; it indicates how well the
material complies with the stress introduced. Following Christensen [23] we
write the compliance function as a decaying exponential function Bv(t t') =
exp[^] where r is the relaxation time, the time it takes the model to attain
its 4 point. Now (2.53) becomes
(m) (m)
dence of ip in (2.22) on the strain Es may be replaced by es Recall that the
il = k0(1 e')V D(ei)Ve' + KsKl{el)el
31


Notice that by letting
t =
kq t
r0
(2.56)
where r0 is some characteristic length depending on the model geometry such
as the radius of a cylinder for a drug-delivery model or the radius of a sphere
for a soybean model for example, we can non-dimensionalize (2.55),
il = (1 e1)'V (D{el)V£l + Ks£lK\el) [ exp[- (f t')]VildA (2.57)
V Jo K0T )
where the dot (il) operator now signifies the material time derivative with re-
spect to t (non-dimensional time). Now introduce the non-dimensional Deborah
number [56],
De = -^, (2.58)
tD
where to = ^ is the diffusion time to obtain,
£' = (1-£')V-
D(£l)V£l + ks£1K1(£1) [ exp[ t\vsxdt'
Jo De
(2.59)
2.6.1 Deborah Number
The Deborah number, De [56], is conventionally known as the ratio com-
paring the relaxation time to the observation time, but in this case the diffusion
time suffices as the observation time because once this time is reached the
material is saturated and the model is no longer valid. We provide a famil-
iar example to facilitate an understanding of the Deborah number. Consider
a dry strand of spaghetti which is said to be in the glassy-state; the strand
can be bent slightly under a small amount of stress while increasing the stress
eventually breaks it. In this state the time scale for observing creep or flow
32


(relaxation time, r) is long compared to the observation time, that is, De 2> 1.
In this state the spaghetti strand instantaneously recovers to its initial state
after it experiences a period of stress; that is, it is an elastic material. When the
strand is placed in a pot of boiling water, the temperature increase changes its
molecular architecture and the strand enters what is called a transition state.
Roughly speaking the free volume increases; the space available for molecular
segments to comply increases. The critical temperature at which this transi-
tion state occurs is called the glass transition temperature, Tg [26]. These rates
of conformational change can often be described with reasonable accuracy by
Arrhenius-type expressions of the form
rate oc exp
E
W'
where E is the activation energy of the process, R is the gas constant, and T is
the temperature. See Figure 2.4 where it is evident that as T increases the rate
of conformational change increases and the material becomes more compliant.
The strand in the boiling water, at a temperature above Tg, enters the rubbery
state (softens) and if we were to extract the strand from the water and place
it under stress, it would easily deform. Depending on how far the strand has
progressed in the rubbery state determines how close to its initial state it returns
after a period of stress. This slow recovery to the initial state is indicative of the
relaxation time, r. Since the strand is pliable (viscous) but recovers (elastic) it
is said to exhibit viscoelastic properties; the relaxation and observation time are
of equal order so De ~ 1. After the spaghetti strand has been heated for a long
enough period of time and its temperature is high enough above Tg it enters the
rubbery state and does not recover to its initial state after stress. Hence, the
33


Figure 2.4: Temperature dependence of rate between transition states on
temperature.
time scale for observing creep or flow is very short compared to the observation
time, De -C 1, and the strand has entered the viscous state. We summarize
the properties in Table 2.2 [56, 69, 72]. If De ~ 0( 1), then the VPIDE (2.59)
maintains its present form which we will refer to as the Viscoelastic Case. If the
Deborah number is very small, De is approximately zero and the integral term may be neglected, yielding
il = (1 el) V (D(£l)Vel), (2.60)
which we will call the Viscous Case. Finally, if the Deborah number is very
large, De 1 (elastic material), then the argument of the exponential is roughly
34


De 1 material is viscous
De > 1 material is elastic
De 0{ 1) material is viscoelastic
Table 2.2: Characterization of the Deborah number, De.
zero implying that the integral term takes the form,
which can be integrated to yield
ve1 v4-
However, recall the assumption (2.42), Ve10 = 0. Thus, the Elastic Case is
given by,
il = (1 el)V (D(e1)Ve1 + kse1K\e1)Ve1) . (2.61)
2.6.2 Integral Coefficient
The coefficient, ks (00,00), is a ratio of the modulus of elasticity to
the swelling pressure coefficient. It appears in front of the integral term in
(2.59); however, using integration by parts we can move this parameter and
show how it impacts the models diffusion. Consider the integral term and
perform integration by parts to obtain,
£l = (l-£l)V-
{D{£') + ks£1K1(£1)) Ve'
(t-t')
De
1
~D~e
V£ldt'
(2.62)
35


Ks < 0 Ks > 0
\ks\ <§; l increase flow inhibit flow
\ks\ > 1 inhibit flow increase flow
Table 2.3: Characterization of ks.
and notice the diffusion term
D(£l) + KsSlKl(£l),
where D(el) > 0, Kl(el) > 0, and el > 0. Hence, a change in sign and magnitude
of ks can affect the models diffusion properties. We summarize the impact in
Table 2.3. Certainly the integral term will also have an effect on el, especially
as De changes. However, diffusion (generally) has the most dramatic impact on
2.7 Discussion
In this chapter we reviewed the derivation of a coupled system of nonlinear
equations from [73]. The VPIDE models the swelling effects of the drug-delivery
device. The wrork presented here is not new', but wre discovered an error in earlier
work, present in [63] and further propagated in [64, 65] where the equation is
written in the form,
il = (1 e')V ^D(£1)Ve1 + jf Bv(t t')Veldt^j . (2.63)
The integral term should contain a volume fraction as a coefficient. In [63]
the author claims that el gets absorbed into the integral; however, there is no
36


mathematical justification for this maneuver. The model appears correctly in
(2.59), where we see £lKl(el). The permeability Kl(el) appears in the equation
as stated in [73] and our calculations, which follow closely the development in
[73], confirm its place as a coefficient of the integral term.
Since the equations have been formulated in Lagrangian coordinates we do
not notice the swelling by observing the dependent variable, el. However, if wre
consider the relative change in volume of the two-phase material, we can see
the swelling effects. Let Vo denote the initial volume of the material and let V
denote volume at some later time. Then let Vq, Vqs, V1, Vs denote the respective
volumes of the material in the liquid and solid phases. We have,
£
l
V1
V0S + V1
and
_ VS
0 Kf + q-
Now compute the relative volume,
(2.64)
(2.65)
V -V0 V1 + Vs Vk V0S V1 V0S
Vo ~ V[+V~o 4 + Kf
since Vs = because the volume of the solid phase does not change (no solid
mass is introduced into the system). Substituting (2.64) and (2.65) into the
relative volume and simplifying yields,
V Vh V" V?
J J
t t0
(2.66)
Vo Vg + Kf 4(1 -e')'
For example, if 4 = 0.1 and el 0.2, then the relative change in volume of the
material is 125%.
37


Let fl be the domain modeling the drug-delivery device (e.g. cylinder or
sphere), then the moisture content is defined as,
M{t) = I £l{x,t)dn, (2.67)
Jq
and provides an additional physical interpretation for el. Since M(t) is an ag-
gregate and 0 < el < 1 holds, the moisture content is a nondecreasing function.
Moisture content provides another useful metric for studying model behavior;
we analyze M{t) after we compute solutions to (2.59).
38


3. Analysis of the Volterra Partial Integrodifferential Equation
The Volterra Partial Integrodifferential Equation (2.59) from Chapter 2 was
derived to model fluid transport through swelling porous materials where the
solid phase is viscoelastic [63, 74, 75]. Examples of such systems include foodstuff
such as soybeans and pasta and polymers used in drug-delivery systems. By
inspection we see that this transport equation contains the sum of a nonlinear
diffusion term and an integral term, also known as an hereditary term. In other
words the VPIDE can be viewed as a nonlinear diffusion equation with a memory
term. The hereditary term poses both theoretical and numerical challenges yet
provides fertile ground for interpretation and analysis.
Before beginning our analysis we select an initial condition and bound-
ary conditions for the drug-delivery model. These conditions coupled with the
VPIDE (2.59) comprise an initial-boundary value problem (IBVP). Once the
IBVP is stated we establish existence and uniqueness and perform Lie group
analysis to derive approximate solutions. In general it is difficult to prove ex-
istence and uniqueness for nonlinear PDEs, but nonlinear VPIDEs pose even
more of a challenge due to the integral term. This work requires a proof that
relies on previous results showing the existence and uniqueness of quasilinear
parabolic equations [4], In Chapter 5 we solve the problem numerically.
3.1 Formulation of the Boundary Value Problem
The governing equation is
(3.1)
39


where, for convenience, we write K(el) instead of kss1 K(el). Note that el in-
dicates the material time derivative, but since we are considering the model in
A |
Lagrangian coordinates, this time derivative behaves exactly as
We choose a model geometry that is consistent with the drug delivery ap-
plication so we consider a right cylinder. However, we assume angular and
azimuthal symmetry so the domain, Q Q U F, is a rectangle where Q is the in-
terior and T is the outer boundary, see Figure 3.1. Since el is a volume-averaged
Figure 3.1: The model geometry is a cylinder but we assume angular and
azimuthal symmetry thus reducing the domain to a rectangle Q with boundary
r = Tj u r2.
quantity it follows that el is continuous [31]. Additionally since we assume the
material can never be all liquid or all solid wre assume that
0<£L<£'<4a* (3.2)
40


where elmin and £lmax are constants. For example, there are gas pores inside the
polymer network and the gas phase is considered to be part of the liquid phase.
Additionally, for the time regime over which we are solving this problem, there
will be polymer and drug (solid phase) present; the polymer would not have
corroded away over the time interval considered.
In the models it is assumed that the material is immersed in a single con-
stituent liquid and that the material imbibes the fluid from the exterior only
making the exterior boundary condition Dirichlet along Fj. Hence the boundary
and initial data for (3.1) are
4 = 4.ax alonS [0. T\ x ri (3-3)
de*
= 0 along [0, T] x T2 (3.4)
and
e{r,z,0) =e0(r,z) (3.5)
respectively, where we assume that £l0(r,z) C2(fl), 4|r2 = 4,im 4|r, = eL>
and T G (0, oc] is some arbitrary time.
In summary, the IBVP for the swelling portion of the drug-delivery model
can be written,
£l = (l-£l)V
D(e')Ve' + K{£1)
f
(t-t') \ ,
e ^ V£ldt
in [0, T] x
4 = e'max on [0,r] x Tj (3.6)
= 0on [0, T] x r2
£l(r, z, 0) = 4(a z) G [e'min, e'max] V (r, z) H,
41


where 0 < T < oo.
We will establish existence and uniqueness for the IBVP (3.6) by employing
two theorems proven in [4] under the following conditions:
1. el(r, z, t) £ C ([0, T]; C2(Q)) where Cl is a closed cylinder with unit radius
(assumptions of angular symmetry and azimuthal symmetry in the axial
direction will reduce the problem to two spatial dimensions, (r, z) £ [0,1] x
[0,1])
2. 0 < el < 1
3. D(£l) > 0 is continuous in £l
4. K(£l) > 0 is differentiably continuous in el
5. De £ [0, oo] (see Remark 3.1).
First we write the IBVP and reformulate it to conform to the IBVP given
in [4] and second we show under what conditions the coefficients, initial value,
and boundary conditions satisfy the assumptions of the hypotheses stated in the
existence and uniqueness theorems. The IBVP as written in [4] is
dfb{u) V a(b(u), \7u) = f(b(u)) in [0, T] x Q
b(£l) = b on {0} x f!
u ud on [0, T] x Ti (3.7)
a(b(u), Vu) v = 0 on [0, T] x r2,
where u is the dependent variable in [4], The prescribed initial and Dirichlet
boundary conditions in (3.7) are b and up respectively.
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Rewrite the IBVP (3.6) in a form suitable for [4] using integration by parts
on the hereditary term to yield,
V = V (K(e1)Ve1) (3.8)
-V.(aV)/V^V£v).
Now reformulate the IBVP (3.6) by defining the following terms,
b(el)t-V-a(b(£l),V£l) = f(b(e1)), (3.9)
where
a{b(u),p) = (D(uj) + K{u))p, (3.10)
b(uj) = ln(l ui), (3.11)
and
f{b(e1)) = -ksVe1 (^K(el) j* Vs1 dt'^) . (3.12)
The IBVP (3.6) now takes a form similar to the one given in [4],
b(el)t V a(b(£l), Vs1) = f(b(e1)) in [0, T] x Q
b(el) = b(£lQ) = el0(r, z) on {0} x Q (3.13)
£l = 4ax on [0, T] x Ti
a(6(ei), Ve1) v = 0 on [0, T] x T2.
The no flux condition along T2 given in (3.3) guarantees that a(b(El), Ve() v =
0 at T2 holds and represents the angular and azimuthal symmetry. The initial
and Dirichlet boundary conditions in (3.13) are b(£l0) = £l0(r, z) and elD = £lnmx
respectively where £lmax is the constant upper bound for £l(r,z,t).
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Remark 3.1 The cases of the IB VP (3.6) where the Deborah numbers attain
the values De = 0 and De oo may permit a less restrictive function space
for existence and uniqueness than the one we posit in this chapter. However,
these cases do not invalidate the proof given. If De = 0, then the integral term
vanishes and we can neglect terms involving K(el) such as the forcing term
f(b(e1)), yielding a nonlinear PDE
i = (1 -el)V D{el)Vel.
If De = oo, then the integral term reduces to K(e1)^£1 and f(b(e1)) may be
neglected, yielding
il = (1 e')V (D{el) + I<{£1)) Ve'.
In either case the VPIDE reduces to a nonlinear diffusion equation.
Let
V = {v W^iD) : u|r = 0} , (3.14)
where IL1* denotes a Sobolev space, then a weak solution, el elD + Lk(0, T; V),
of the IB VP (3.7) satisfies the following two properties [4]:
1. b(el) L(0, T; L!(Q)) and dtb(el) Lk* (0,T,V*) with the initial values
6, that is,
f me1), 0 + f [ (b(el) ~ me = 0 (3.15)
Jo Jo Jn
for every test function £ Lk(0, T; V) p| IV1,1 (0, T; L(Q)) with ((T) = 0.
2. a(b(£l), Vs*), f(b(e1)) Lk" ([0, T] xfl) and el satisfies the differential equa-
tion, that is,
[T(dtb(el),0 + fT f a(b(£l),Vel) VC = T f f(b(el))( (3.16)
Jo Jo Jq Jo Jn
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for every £ Lk(0, T; V).
The six assumptions the IBVP needs to satisfy in [4] are:
(HI) C R x E be open, bounded, and connected with Lipschitz boundary,
T C dVl which is measurable with trace (T) > 0, and 0 < T < oo.
(H2) b is a monotone vector field and a continuous gradient, that is, there
is a convex C1 function T : Mm > M with b(u>) = VT. The function 4> leads to
the definition of a function, B(ui) := b(uj) u 4>(o;) + 4>(0).
(H3) a(b(uj),p) is continuous in u> and p and satisfies the following ellipticity
condition,
{a(b(uj),pi) a{b(uj),p2)) (pi p2) > c{px p2)k. (3.17)
for some constant, c, and 1 < k < oo. The function f(b(u)) is continuous in u.
(H4) The following growth condition is satisfied,
\a(b(u),p)\ + |/(&M)| < c(l + + H*-1). (3.18)
In general, the coefficients a, b, f may also depend on t and x.
(H5) elD e Lk (0, T; VT1,fc(fi)) p) £([0, T] x fi).
(H6) T(b) G and b maps into the range of b such that there is a
measurable function sl0 with b = b(el0).
The existence and uniqueness theorems are stated here for convenience:
Theorem 3.2 (Alt and Luckhaus [4], Theorem 1.7) Suppose that the data
satisfy (Hi)-(H6) and assume that dtelD Lx(0, T; L(Q)). Then there is a weak
solution to IBVP (3.13).
45


Theorem 3.3 (Alt and Luckhaus [4], Theorem 2.4) Suppose that the data
satisfy (H1)-(H6) with k = 2 and
a{t,r,z,b(uj),p) = A{t,r, z)p + e{b{u)), (3.19)
where A(t, r, z) is a symmetric matrix and measurable in t and (r, z) such that
for some a > 0
A al and A 4- adtA (3.20)
are positive definite. Moreover assume that
\e(b(u>2)) e{b{u}i))\2+\f(b(ui2))f (b(uji))\2 < C(6(u;2)-6(wj))^-^), (3.21)
for some constant C. Then there is at most one weak solution to IBVP (3.13).
3.2 Existence of a Solution
We begin this section with an existence theorem for IBVP (3.13), Theo-
rem 3.4. The strategy for proving this theorem is to show that the conditions
stated in its hypothesis imply that the data in IBVP (3.13) satisfy assumptions
(H1)-(H6) for Theorem 3.2. Once these assumptions are satisfied then so is
the hypothesis of Theorem 3.2 and we conclude that a solution to IBVP (3.13)
exists.
Theorem 3.4 (Existence Theorem) Suppose that the following conditions
hold for IBVP (3.13).
1. Yl open, bounded, and connected with Lipschitz boundary, Y C (XI is mea-
surable with H(T) > 0;
46


2. 0 < T < oo;
3. 0 < e' < 1;
4 D(cj) is continuous and K(lo) is differentiably continuous;
5. f(b(e1)) defined in (3.12) is continuous;
6. el(t,r,z)eC([0,T}-,C2m,
then there exists a weak solution to IBVP (3.13).
Proof: We proceed by justifying each of the assumptions (HI) through
(H6).
(HI) The cylinder fi C RxMis certainly open, bounded, and connected with
Lipschitz boundary, T C dQ is measurable with H(T) > 0, and 0 < T < oo.
(H2) b is continuous and is strictly monotone increasing for 0 < uj < 1,
the domain of definition. Moreover, 6(0) = 0 and there is a function $(w) =
(1 uj) ln(l uj) + uj such that d> C2{0,1), '(u;) = 6( since 0 for all u < 1.
Since such a exists, we can define
^h(u;) = sup (uj b(so)) ds
Full Text

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A ALYSIS A D UMERICAL SOLUTION OF NONLINEAR VOLTERRA PARTIAL INTEGRODIFFERENTIAL EQUATIONS MODELING SWELL! G POROUS MATERIALS by Keith J. Wojciechowski B A., Lawrence University 1992 M.S. DePaul University 1998 M.S. University of Colorado 2003 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 2011

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This thesi:s for the Doctor of Philosophy d egree by Keith J. Woj c iechow ski h as been approve d by !fril .25'.J.O/i D ate 7

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Wojciechowski Keith J. (Ph.D., Applied Mathemati cs) Analysis and Numer i ca l Solution of Nonlinear Volterra Partial Int egrod iff erentia l Equations Modeling Swelling Porous Materials Thesis directed by Associate Professor Lynn Bennethum ABSTRACT A nonlinear Volterra partial int egrodifferent i a l equation (VP IDE) derived using hybrid mixture theory and used to model swe llin g porous materials is analyzed and so lv ed numerically. The model application is an immersed porous material imbibin g fluid through a cy lind er s exterior boundary. A poignant example comes from the pharmaceutical industry where controlled release, drugdelivery systems are comprised of materials that permit nearly constant drug concentration profiles. In the cons id ered application the release is controlled by the viscoelastic properties of a porous polymer network that swells when imm ersed in stomach fluid conseq u ently in creasing the pore sizes and a llowin g the drug to escape The VPIDE can be viewed as a combination of a non-linear diffusion equation and a constitut iv e equation modeling the viscoelastic effects. The viscoelastic model is expressed as an integral equation, thus adding an int egra l term to the non-linear partial differential equation. While this int egra l term poses both theoretical and numerical challenges, it provides fertile ground for in terpretation and anal ysis. Analysis of the VPIDE includes an existence and uniqueness proof which iii

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we establish under a given set of assumptions for the in i tial-boundary value prob l em. Additionally, a special case of the VPIDE is reduced to an ordinary differentia l equation via a derived simi l arity variable and solved. In o r der to solve the full VPIDE we derive a novel approach to constructing pseudospectral dif ferentiation matrices in a polar geometry for computing the spatial derivatives. By construction, the norms of these matrices grow at the optimal rate of O(N2), for N by N matrices, versus O(N4 ) for conventiona l pseudospectral methods. This smaller norm offers an advantage over standard pseud ospect ral methods when solving time-dependent problems that require higherr esolution grids and, potentially, larg e r differentiation matrices. A meth od of lines app r oach is em ployed for the time-stepping using an implicit fifth-order Runge-Kutta solver. After we show how to set up the equation and numerically solve it using this method we show and int erpret results f o r a variety of diffusion coefficients, permeability models, and parameters in order to study the model's behavior. This abstract accurately r e presents the content of the candidate's thesis. I recomm en d its publication. Signed lV

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DEDICATION This thesis is dedicated to my wife Sharyl and my son Cole for the in spirat i on they give me everyday to be a better husband a better father a nd a better man.

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ACKNOWLEDGMENT There a r e seve r a l p eo pl e I need to ac knowl e dge f o r the h e lp t h ey p rov id e d in t h e com p leti o n o f thi s t h es is. Fir st, t hi s t h es i s would not h ave bee n p oss ibl e w i t h o u t t h e pati e n ce a n d insi g h ts o f L ynn Sc hr eye rB e nnethum. L y nn tau g h t m e t h e va lu e o f u s in g physical in s i g ht to g uid e m athe matical a n a l ys i s Kristi a n Sandbe r g in t r oduce d m e to P se ud ospect r a l meth o d s a nd s howed m e how to t hink creat ivel y as a num e ri ca l a n a l yst. Hi s trou b l e-s h oot in g ac um e n save d m e o n m a n y occas i o n s a nd I ca nnot t h a nk him e n o u g h for b e i ng so gen e r o u s with his t im e d es pite runnin g hi s o wn co mp a n y a nd r a i s in g a f a mil y I woul d a l so lik e exte nd m y t h a nk s to Bill Bri ggs f o r i ntroduc in g m e to L y nn a nd nud g in g m e a l o n g as I ca m e p e ril o u s l y clo se to l eav in g t h e g r aduate p rog r a m T o Juli e n L ango u wh o h e lp e d m e t hink m o r e deepl y a b o u t lin ear a lgebr a T o L eo Fr a n ca f or h e l p in g to brin g m e to U C D e n ver. T o Ale x a nd e r Enga u f o r agree in g to b e o n m y co mmittee d espite b e in g in t h e un e n v i a bl e po s ition o f b e in g as k e d at t h e l as t minute. T o M a t t h e w Na bity w h o h as been m y comp a d re s ince t h e star t o f t hi s j o urn ey a nd h as b eco m e a lif e -l o n g frie nd. T o J ohn Stin es prin g wh o s impl y put, m a kes m y lif e b etter. T o Eri c Sulli va n a nd K a nn a nu t C h a m s ri m y aca d e mi c b rot h e r a nd s i s t e r f o r pati e n t l y liste nin g to m y researc h talk s a nd as kin g p o i g n a n t qu est i o ns. T o Jinha i C h e n f o r h e lpin g m a k e m y a r gume n ts m o r e math e matically so und T o J eff B arc h ers f o r b eliev in g in m e e n o u g h to hir e m e at SAI C T o Rich a rd Qua n strum w h o tau g h t m e b y hi s exa mpl e w h a t it mean s to b e a r ea l m a n I carry in m y heart your p r id e in m y acco mpli shments as a teac h e r hu s b a nd a nd f athe r -you w ill a l ways b e m y coac h. T o Angel a Beal e a nd Lind say Hiatt w h o k e p t m e reg i stere d a nd upto d a t e o n m y re quirm e nts, I would n o t h ave m a d e i t throug h a se m es t e r without t h em T o all t h e p eo pl e w h o m I m et over m y years o f educati o n w h o h e lp e d m e a l ong t h e way b y t h e ir kindness a d v ice, instr uct i o n a nd e ncourage m ent. Fin ally I mu s t tha nk m y f a mil y f o r the ir l oyalty a nd un e ndin g s upp ort. M y m ot h e r Eleon ore, wh o nur ture d m y c reati vity a nd e n co ur age d m e to "reac h f o r

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the stars," my brother Robbi e who taught me t h e value of h a rd-w ork m y s i sters Barbara a nd Chery l who a l ways sto od b y me g rowin g up a nd who taught m e to b e a gentl e m a n m y brother L on ni e who tau g h t m e t hat there i s more to lif e than work (like music!) a nd m y fath e r Leon ar d w ho taught me to be se lf-r elia n t.

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Figures Tables Chapter 1. Introd u ction 1.1 Prev iou s Work 1.2 Thesis Outlin e CONTENTS 2. Fundamental Continuity Equation 2.1 Brief Overview of H ybrid Mixture Theory 2.1.1 Macroscale Field Equations ..... 2.2 D er ivation of the Continuity Equation 2 3 Dar cy's Law . . . . . . . 2.4 Non-dimensionalization of the Model 2.5 Continuity Equation 2.6 Model Int erpretation 2.6.1 D eborah Number 2.6.2 Integra l Coefficient 2.7 Discussion ..... 3. Analysis of the Volterra Partial Integrodifferentia l Equation 3.1 Formulation of the Boundary Value Problem 3.2 Existence of a Solution . . . . . . Vlll XI XVII 1 5 7 10 10 16 18 20 25 27 28 32 35 36 39 39 46

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303 niqueness of a Solution 0 0 0 0 0 0 0 0 0 0 0 0 0 3.4 Analytic Solutions and Reductions of the IBVP 30401 Similarity Reduction of the IBVP 0 3.402 Viscous Case 0 0 0 0 0 3.403 Flory-Huggins Mode l 0 305 Discussion 0 0 0 0 0 0 0 0 40 Eigen-decomposition Pseudospectral Method 401 Pseudospectral Methods 0 0 0 0 0 0 0 0 0 0 402 Eigen-decomposition P seudos p ect r a l Method 0 40201 D er ivation 0 0 0 0 0 0 0 4 0201.1 Truncating the Sum 40201.2 Approximating the Integral with a Quadrature 40202 Error Analysis 0 0 40203 Rank Completion 0 403 EPS Constructi on in Polar Coordinates 0 40301 Construction 0 0 0 0 40302 Numerica l Examples 4.4 Di scussion 0 0 0 0 0 0 0 50 Numerical Solution o f t h e Parti a l Integrod iff erent i a l Equation 501 Numer ical Meth od for Solving t h e Swelling Equation 501.1 Spatial Di scret ization 0 501.2 Time-stepping Methods 501.2 0 1 Semi-analytic Integration Rule Formulation 501.202 Method-of-Lines Formulation 0 0 0 0 0 0 0 0 I X 49 56 56 59 62 65 67 71 76 76 77 78 80 84 89 90 93 95 102 102 104 107 109 110

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5.1.2.3 5.1.2.4 5.2 P o u zet Volt erra Runge-Kutta Formulation Examples .......... um e rical Solution of the IBVP 5.2.1 Two-dim e nsion a l Ex a mple 5.2.2 On e -dim e nsion a l Example s 5.2.2 1 Fl o r yHu gg in s Model ... 5.2.2.2 Diffusion Coefficient Comparison 5.3 Disc ussion ....... 6. Model S e nsitivity An a l ys i s 6.1 Diffu s ion and P er m ea bilit y Model s 6.2 Param ete r S e nsitivit y . . 6.2.1 Moisture Content Curves 111 111 114 115 115 116 118 119 126 127 130 131 6.2.2 Viscoe lasti c Stress Curves 134 6 3 Di sc ussion . . . . . 135 7. Conclusion a nd Future Work 144 7.1 Model Anal ys is, V a lid atio n a nd Exte n s i o n 145 7.2 G e neralizing the Appli cab ilit y of the EPS Method 146 7.3 G e n e r a lizin g the Existence and Uniqueness Proof for the IBVP 147 7.4 Exte nding the Appli ca bilit y of the Drug D e livery Mod e l 147 App e ndix A. D er ivation of Dar cy's L aw . . . . . . . B. D e rivation of a Pouzet Volterra Runge-Kutta Method Ref ere n ces . . . . . . . . . . . . . X 149 156 159

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F IGUR ES Figur e 1.1 Porou s polymer matri x. Photo taken from plc.cwru. edu. 4 2.1 Av erag in g Local Coordinates . . . . . . . 13 2. 2 Stress ( O") versus str a in ( E -not to be co nfu se d with the volume fract ion c:1 ) c urv e for lin ear e lasti c (left ) a nd lin ea r viscoelastic ( right ) materi a ls. . . . . . . . . . . . . . . . . 30 2.3 Stress versus time where the loa din g takes place over the time int e rv a l t < t1 and the unlo ad in g tak es pl ace over the int e rv a l t 2: t1 . . 30 2.4 T e mp erature d e p e nd e n ce of rate b etwee n transition states on temp e rature. . . . . . . . . . . . . . . . . 34 3 1 The mod e l geometry is a cy lind e r but we ass um e angular and az imuth a l symmetry thus r educ in g the dom a in to a r ecta ngl e D with boundary r = rl u r2 . . . . 40 3 2 Int eg r a l of t e mpor a l transform B(w). 48 3.3 Similarity so lution to the IBVP (3.55) 61 3.4 Solution to the Flor y -Hu gg in s mod e l (3.58) for a variety o f times. 64 Xl

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4.1 Trial function compar i son for spectral finite-difference and finite e l ement methods. Spectral methods are characteri zed by one high order polynomial for the whole domain finite-difference methods are characterized by multiple ove rl app in g low-ord er polynomials, and finite e l ement methods are characterized by non-overlapping polynomials with compact support one per subdomain. 4.2 Runge 's examp l e for int erpo latin g f(x) = (sx)12+1 with even l y spaced nodes. As the number of even l y-spaced nodes (degree of the int erpo lating polynomial) increases, the interpo l ating polynomial osc ill ates 69 near the boundaries. . . . . . . . . . . . . . 73 4.3 Interpo l ating with Chebyshev nodes ame lior ates Runge's phenomenon. As the number of Chebyshev nodes increases the interpolation erro r decreases. . . . . . . . . . . . . . . . . 7 4 4.4 L2 error compar i son of the Chebyshev PS method versus second a nd fourth order finit e difference methods applied to the second derivative of cos(1rx) over [-1, 1]. . . . . . . . . . . . . 75 4.5 Sparsity pattern for orthogonality test of eigenfunct ion s in a Cartesian geometry with Neumann boundary cond i tions. The rank of the differentiation matrix i s Nc = N = 256. . . . . . . . . 85 4.6 Sparsity pattern for orthogona lit y test of e i genfunction s in a Cartesian geomet r y with Neumann boundary condit ions. The rank of the differentiation matrix is Nc = 139 (rough l y 0 .54N). . . . . 86 xu

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4.7 Sparsity pattern for orthogona lit y test of eigenfunctions in a Carte sian geometry with Neumann boundary conditions for the comp l eted operator. The rank of the differentiation matrix is now N = 256 (Nc = 139, roughly 0.54N). ................ 4.8 Comparison of the L00 norm relative error resulting from comput in g the second derivative of the fun ctions {sin ( rr;_1r (x + 1))} :=-N using the EPS method with Gauss-Legendre quadrature nodes and weights versus the Standard Construction using Cheybshev-Lobatto nodes. Rank-completion versus a reduced rank construction the EPS 88 method is a l so compared. . . . . . . . . . . . 97 4.9 Given N, use the EPS method to construct the polar Laplacian increasing Nc from 1 2, ... Nand computing the error I Ia;,-a;,lloo for each set of Nc e i genva lu es, The co lorbar is logsca l ed where the dark regions indi cate smaller error a nd the light regions indi cate l arger error. . . . . . . . . . . . . . 98 4.10 Eigenvalues of l::!.r constructe d via Chebyshev co llo cation v e rsus the exact eigenva lu es, lanl2 The eigenva lu es of l::!.r are the zeros of the zero-order Bessel function of the first kind J0 ( an) = 0 for a ll n = 1 2 3 ,.. .. . . . . . . . . . . . . . . . . 99 4.11 Chebyshev expansion coefficients versus FourierBessel expans i on co efficients for the Gaussian pulse centered at r = 1 /2. . . . . 99 Xlll

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4.12 The Poisson equation examp l e c omparing the completed EPS con struction to the standard construction using Chebyshev collocation: (a) The residual error llf(b) The error lluA;:-1 flloo, and (c) L00 condition number. . . . . . . . . . . 100 4.13 The maximum stable time step for solv in g the radial part of the heat equation on a disk using the explict Runge-Kutta 4 solver. Here N denotes the size of the problem. . . . . . . . . . 101 5.1 The EPS differentiation matrix for the cylindrica l Laplacian (5.8) using r = N z = 64 nodes. 108 5.2 Initial liquid volume fraction c:1(r, z, 0) with = 0.1 and = 0.9.116 5.3 Liquid volum e fraction c:1(r, z t), plots over the cylindrical cross section [2 with a Kozeny-Carman permeability K(c:1 ) = linear diffusion coeffic i ent, D(c:1 ) = c:1 and model parameters J..L = 0 .01 and T = 1. The grid size is Nr X N z = 64 X 64 with N c r = Nez = 42(0.65Nr) and a convent ional (explicit) RK4 time-stepper was used with constant time-step = 10-6 Solutions are shown at (a) t = 0 (b) t = 0.2 and (c) t = 0.4. ....... ... 123 5.4 Liquid volum e fraction c:1 found by solving (5. 36) using an MOL approach with the EPS discretization and MATLAB s ode15s for time-stepping. . . . . . . . . . . . . . . . 124 5.5 Relative L00 error comparing Chebyshev coll ocation to the EPS con struction overt E [0, 10] for the Flory-Huggins model (5.36). . . 124 XlV

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5.6 Liquid v o lum e fracti o n c1(r, t), pl ots ove r t h e r a di a l gr id with a K oze n y -Carma n p ermeabili ty, K(c1 ) = and m o d e l p a r a m e t e r s p, = 0 1 and T = 1. The g rid s ize i s Nr = 7 5 0 with Ncr = 4 5 0 ( 0 .60Nr) a nd a vari a bl e s t e ps i ze, 5th o rd e r impli c i t t im e s t eppe r was u se d. S o lu t i o n s a r e s hown f o rtE [ 0 0.4] w i t h (a) D(c1 ) = 1 ( b ) D(c1 ) = c1 and (c) D(c1 ) = (c1)2. . . . . . . . . . . . . . 125 6.1 Low 's fun c tion a l f orm o f the s w e llin g press ur e c ombin e d with the . l l exp[ l -j. I K oze n y -Carm a n p ermeab1ht y fun c tiOn D(c) = E 1_1 129 6.2 Liquid v o lum e fracti o n c1(r, t), pl o t s ove r the r a di a l gr id with a K oze n y-Carm a n p ermeability, K1(c1 ) = a nd m o d e l p a r a m e t e r s K.8 = 0 1 a nd T = 1. The grid s ize i Nr = 7 5 0 wit h Ncr = 4 5 0 ( 0.60 Nr) and a v a ri a bl e s t e ps i ze, 5th orde r impli c it t im e s t eppe r w as u se d Sol uti o n s a r e s h o wn fortE [ 0 0.4] wit h (a) D(c1 ) = 1 ( b ) ex p [ 1-t I D(c1 ) = c1 (c) D(c1 ) = (c1 ) 2 and (d ) D(c1 ) = K1( c1 ) 7'. . . 137 6 3 Mod e l compa ri s on in t erms o f vi sc o e l as ti c str ess a nd m o i sture c on -ten t wit h D(c1 ) = c1 and fixe d p a r a m e t e r s D e = 1 and K.8 = 1. (a) vis coe l as ti c stress r es ultin g fro m usin g a constan t p ermeabilit y mod e l = initi a l stress, o = fin a l stress), ( b ) viscoe l astic str ess r es u lting fro m u s in g a K oze n y -Carm a n p ermeability ini t i a l str ess, o = fin a l stress ) a nd (c) m o i sture conte n t compa rin g co n s t ant p e r meabi l it y again s t Kozen y -C ar m a n p ermeabilit y . . . . . . 13 8 6.4 Kozen y -Carma n p ermeabilit y as a function o f liquid volum e fracti on 13 9 6.5 Normalized m o i sture conte n t c urv es !11 / M00. A po iti ve K.8 in c r eases flow wh e r eas a n ega tiv e K.8 inhibit s flow. . . . . . . . 139 XV

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6.6 Normalized moisture content curves M/M00: (a) Fix K8 = 0.1 and compare the moisture content for De= {0 0.01 100}, (b) Fix Ks = -0.1 and compare the moisture content for De= {0, 0.01 100} 140 6.7 Normalized moisture content curves 111/!1100: (a) Fix D e = 0.01 and compare the moisture content for K8 = { -0.01, -0.1}, (b) Fix De= 100 and compare the moisture content forKs= { -0.01, -0.1}, (c) Fix Ks = -0.1 and compare the cases D e = {0.01, 1 100} (d) Fix K8 = -0.01 and compare the cases De= {0.01 1 100} 141 6.8 Viscoelastic stress 142 6.9 Viscoelastic stress 143 XVI

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TABL ES T a bl e 2 1 Units for the dim e nsion a l p a r a m ete r s of t h e Dar cy mod e l (2.50). 27 2.2 Characterization of the D ebora h numb er D e 35 2.3 Characterization of K,8 . 36 5.1 V a lu es o f the lo g lle n (t)lloo erro r at t im es t1 = 0.25tf, t2 = 0 .50tf, t3 = 0 .75t 1 a nd t 4 = tf compar in g RK 4 an d ODE45 vers u s SAl for Example 1. . . . . . . . . . . . . . . . 120 5.2 Values o f the log llen(t)lloo error at t im es t1 = 0 25tf, t2 = 0.50tf, t3 = 0 75tf, and t 4 = t1 compa rin g RK 4 a nd ODE4 5 versus SAl for Exampl e 2. . . . . . . . . . . . . . . . 121 5.3 V a lu es of the log llen(t)lloo e rror at times h = 0 25tf, t2 = 0.50tf, t3 = 0.75tf, a nd t 4 = t1 co mparin g RK 4 vers u s PVRK4 versus SAl f or Example 2. . . 6.1 C h aracter i zat ion of K,8 6.2 C h a r acte ri zat ion of t h e D ebora h number D e B 1 Butch e r diagram ................ XVll 122 130 131 158

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1. Introduction Generally speaking "drug delivery systems are devices used for the pro cess o f admini ster in g pharmaceutica l compounds to ac hi eve therapeutic effects. These system s ca n b e divid e d into two m a jor types: traditi onal d e liv e r y systems and co ntroll e d released systems. Traditi o n a l d e liv ery syste m s are c h aracterized by their immediate release of the drug which l eaves absorpt i on to be co ntroll ed by the body s abilit y to ass imilate the drug concentration into different body tissues s u c h as t h e blood Dru g concentrati ons from these systems typically und e rgo a n abrupt in c r ease followed b y a n abrupt d ec r ease. Controlled r e l ease drug delivery syste m s on the oth e r h a nd are formul ate d to modif y the drug r e l ease profile a bsorption distribution a nd e limin a tion for t h e benefit of im provin g therapeutic efficacy saf ety, patient co nv e ni ence a nd co mpli a nce. These systems are c h aracterized by t h e ir maintenan ce of drug co n centrat ion to target tissues at a desir e d l eve l for pro l onged time p e riod s D epe ndin g on the release behavior controlled release syste m s ca n b e s ub divid ed in to three ca t egor ies: passive pre -pro gramme d act iv e pre -pro g r amme d a nd act ive se lf-pro g r ammed. The mod e l s we w ill study foc u s o n passive pre pro grammed r e l ease where the rate i s pr edeter min e d a nd dependent upon the drug's re l ease kin e ti cs as it int e racts with the b o dy. W e l eave this r est riction vague for the moment but will furth e r r estr ict our foc u s as we d e riv e the math e mati ca l mod e l and impo se s implifyin g ass umptions. In lie u o f d efinitions f or the r e m a inin g two catego ri es which we off er f or the sake o f c urio s it y a nd compl ete -1

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ness we provide examp l es. An examp l e of an active pre-programmed device is an in su lin pump for type 1 diabetes patients where the in s ulin is delivered as a continuous flow to the body through a s hort tube with a needle at the endpoint that is inserted under the s kin usually in the abdomen The third cat egory, active se lf-pro grammed is exemplified by devices that combine (provide data int erfacing) insulin pumps with g lu cose monitors or as a simp l e examp le, morphine drips. It s hould be ev id ent by the definitions and descriptions given above that drug delivery h as become more comp l ex moving from uncontrolled release to s ustain ed (contro lled) release and programmable (co ntroll ed) release. It has a l so become more specific as we have witnessed a move from system i c delivery to organ (as in the case of cancer radiation therapy) and cellular targeting (as in the case of tumor-targeting delivery systems). As is the case in many fie ld s a feedback loop originates where in creas in g comp l exity in the technology drives the requirement for mathematical modeling and s imulation and vice versa. Mathematical modeling of drug delivery i s in its inchoate stages, intim ated by the reviews given in [41, 60, 76], but h as become an area of act iv e and grow in g research. We will focus our attention on controlled drug delivery systems, a lso known as controlled release systems ( CRS). Furthermore we will limit the sco p e of our study to passive pre-programmed CRS and will subseq u ent l y drop this modifier. Therefore the CRS systems that fall under our scrutiny are those that are f abricated by embedding a drug in a hydrophilic (waterloving) polymer matrix suc h as hydroxypropylmethylcellulose (HPMC) [76]. Matrix" in this context refers 2

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to the three-dimensional network conta inin g the drug a nd other s ub stances re quired for co ntroll e d r e l ease. The matrices can be prepare d in severa l ways. Two examp l es includ e (1) mixing the powd ered drug with a solv ent, exc ipi e nt (inactive in g r e di e nt s) and pre-polymer before pl ac in g it in a pol ymer isation r e actor or (2) preparin g the matrix in advance a nd then putting it in contact with a hi g hl y concentrate d drug so lution ab l e to swe ll the matrix whereby the so lv e nt i s r emoved afte rw a rd s (so lv e n t swe llin g technique) [41 ] A detailed expos iti on on pol ymer i c matrices i s beyond the sco p e of this thesis but there are severa l r efer e n ces ava ilable [ 41, 76]. How eve r we will bri efly describe the properties of polymer matrices u sefu l for co ntroll ed r e l ease system First, pol ymers are viscoelastic. B y the n a m e it should be apparent that viscoelastic material s possess both viscous a nd e lasti c properti es. Viscosity i s t h e c har acte risti c of a liquid that mak es i t resistent to flow. For exa mple hon ey i s more viscous than water. Elasti c it y i s the c h a r ac t e ri st i c of a mate rial that returns it to its initi a l state (instantaneously) a fter the externa l for ces that deformed it cease For examp le, a dry sponge i s more e lasti c than pumice. H ence a viscoelastic material h as t h e character i stic that it initi ates returnin g to it s original state a ft e r the externa l for ces that d eform it cease but the elasti c it y i s inhibit e d b y the viscous b e h av ior which resists the c hange in motion. Second the pol y m er matrix i s a porou s materi a l as seen by the pol y m e ri c matrix in Fi g ur e 1.1. P o l ymer i c matrix systems can be classified acco rdin g to their porosity (mac rop o r ous microporous a nd non-porous). W e will foc u s our attention to the mac ro and mi cro porous system s wh e r e por e sizes range in 3

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size fro m 0 1 1p ,m a nd 5 0 200 A, r es p ect ively. Dru g r e l ease kin e tics will b e affecte d b y pol y m e r swe lling, pol y m e r e rosi o n drug di sso luti o n / diffu s i o n drug distributi o n (in s id e t h e m a trix) drug/p o l y m e r ratio a nd sys t e m geom etry ( cy lind e r sphe re, e tc.) The matri x s ys t e m s und e r co n s id e r a ti o n will b e dr y a nd compresse d wit h o u t a n y fluid in s ide. W e co n s id e r t h e gas fille d p o r es t o b e part o f the liquid phase (fluid a nd gas). The r ef o r e in o ur mod e l s t h e ini t i a l c ondi t i o n will c ontain a s m a ll p e rcentage of liquid phase. Once the drug d e liv e r y d evice F igure 1.1: Porou s p o l y m e r matrix. Photo t a k e n from plc. c wru.edu is imm e r se d in biolo g i ca l fluid the s urroundin g fluid press ur e co upl e d with the h y drophili c prop ertie s o f the pol y m e r driv es the fluid to p e netr a t e the p o l y m e r m a trix as d esc rib e d b y D a r cy's l a w f o r fluid flow ( whi c h w e will d esc rib e in m o r e d e t ail in S ect i o n 1.1 a nd C h apte r 2); the n ove l f orm o f D a r cy s l a w d e rive in [73] c ont a in s a c on stitutive e qu a ti o n mod e lin g the vi sc o e l as ti c prop e rti es o f the pol y m e r. The pol y m e r s w ells as dictate d b y it s vi sc o e l as ti c pr op e rti es a nd the por es in the m a trix e nl arge until they a r e o f a s ize n ecess ar y f o r the dru g to esca pe. The drug the n diffu ses into the surro undin g fluid dimini s hin g the dru g c oncentra ti o n l e v e l s pr ese n t in the d e liv e r y d ev i ce 4

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1.1 Previou s Work The equations considered in this thesis are nonlinear Volterra partial inte grodifferentia l equations (VPIDE) derived by Weinstein and Bennethum [73, 74] and Singh et. al. [63]. These models are derived using Hybrid Mixture The ory (HMT) which involves upscaling field equations which govern the motion of materials and includ e the conservation of mass conservation of linear and angular momentum balance and conservation of energy from the microscale to a l arger scale v i a volume averag ing. Restrictions are then obtained on the form of the constitutive equations by using the second l aw of thermodynamics, also formulated as a field equation, at the l arge scale. A variable that resu lts from upscaling via HMT is the volume fraction a ratio of the volume of a particular phase over the sum of the volumes of all phases. For example, in a two-phase mixture with a liquid phase (indicated by l) and a solid phase (indicated by s), the liquid volume fraction denoted c1 i s the ratio of the volume of the liq uid phase (V1 ) to the sum of the volumes of both phases (V8 + V1). That is c1 = The liquid volume fraction c1 is the dependent variable cons id ered in the models we study in this thesis. Note that if there are only two phases s and l we have c1 + c8 = 1. In the case of the VPIDE considered the mass co nservation equation is upscaled and coup l ed to a novel form of Darcy's law [73]. Darcy's l aw is a constitutive eq uation relating the flux of a fluid field to a change in pressure [29]. As such it is often used to describe the flow of a fluid through a porous medium. In the case of the VPIDE derived by W e instein [73], time rates of c hange of c1 are includ ed as a constitutive variab le. That is, Weinst e in includ ed the material 5

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time d er ivativ es of the liquid volume fract i on for m = 1 2 ... p where p is determin ed by how lon g of a time hi story we want to includ e in our model. This temporal hi sto r y accounts f or the viscoelastic str ess and the mod e l for the stress ass um es that the strain effects a r e c umulativ e h e nce it contains a n inte g r al. That the int egra l term conta in s time d er ivativ es of the volume fraction not solid-phase strain i s justifie d in [51 ] where it is shown t hat at moderate to high fluid contents the normal co mpon ents of t h e strain tensor are related to the volume fraction of the so lid phase E:8 Moreover, s ince the solid and liquid phases a r e all that compri se the mate ri a l we h ave that r::1 = 1 E:8 H e nce, at the fluid content l eve l s co n s id ere d in these models the time derivative of strain may be replaced by time derivative of liquid volum e fract i on As a result of this nov e l form of D arcy's l aw, the VPIDE models liquid p e netrati on into the dru g d e liv e r y d ev ice inhibit e d by the viscoelastic properties of the polymers. That is, the VPIDE can b e viewed as the sum of a liquid p enetration (nonlinear diffusion equation) term and a viscoelastic term ( int eg r a l eq uation). An eq uati o n s imilar to the one d er iv ed in [73] was d er iv ed by Singh et. al. [63] Singh a l so num er i cally so lv ed the VPIDE u s in g a finite-element method [64, 65]. How eve r the so lv er produced spur iou s oscillations that resulted in non-physi ca l values for the volume fract ion in [65]. Even though this work was completed in 2003, to our knowl e d ge, it h as n eve r been r ev i s it ed Moreover the equati on in t h e drug delivery context [73] has not been so l ved. Models developed in the pharmaceutical lit e ratur e that take into account the v i scoe l astic properties of the polymer h ave been phenomenolog i ca l [12, 47, 60, 61, 62, 76] a nd lac k the phys ics of the models d e rive d by Weinstein [73] In 6

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addition, these models are typica ll y lin ear partial differential equations, such as Fick 's second l aw of diffusion in cy lindrical coordinates [76], that can be solv ed either analyti cally u s in g separation of variables or numerically using e l ementary methods [60 ] There are diffusive models taking into account viscoelastic relaxation in polymers that have been developed. In particular Cohen and White [24] extended the work of Thomas a nd Windle [67] in modeling sharp fronts due to diffusion and viscoelastic relaxation in polymers. It shou ld be noted that Cohen a nd White did not so lv e the full partial int egrod iff erent i a l equation numerically but rather they reduced it, using a perturbation series, to a ystem of ordi nary differential eq u ations which they solv ed with an Adams-Bashfort-Moulton numerical time-stepp in g scheme [24] 1.2 T h esis Out line In this work we so lv e the model derived by Weinstein [73] using a novel for mulation for pseudospectral differentiation matrices the Eigen-decomposition Pseudo-Spectral (E PS) method. The governing equation is a nonlinear Volterra Partial Int egrodifferential Equation (VP IDE ) of the second kind. These equa tions are particulary difficult to so lve not only because they are nonlinear but a lso because the integral term poses some numerical challenges. Since thi s term is cumulative over time it collects round-off error and can pose stabilit y prob l e m s for a numerical so lv er. Singh [64, 65] so lv ed the problem using a finite e l ement method with a time-stepping technique derived by Patl ashenko et. al. [53]. H owever Singh 's solution for the liquid volume fraction was unsatisfac tory in that it attain ed nonphysical values. We will show that one can obtain higher-accuracy than the method suggested in [53] by using a method-of-lines 7

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( MOL ) a ppr oac h In t hi s contex t we h ave exte nd e d the work o f W e in ste in a nd Sin g h as w ell as adde d math e m a ti ca l rigor to the ir resul ts. In C h a pter 2 w e prov id e a r eview o f HMT r e l e v a n t to t h e d e rivati o n o f the g overnin g co ntinuit y e quati o n a nd n o ndim e n s i o n alize the m o d e l ar ri v in g at two n o n-dim e n s i o n a l p a r a m ete rs, D e ( D e b o r a h numb e r [56]) a nd r;,8 ( rati o o f viscoe lasti c it y t o diffu s ivit y). In C h apte r 3 we a n a l yze the VPIDE b y findin g a suffic i e n t so lu t i o n s p ace t hat prov id es a p roo f o f ex i ste n ce a n d uniqu e n ess o n a cy lindri ca l geo m etry und e r a s pecific se t o f initi a l bound a r y c ondi t i o n s tha t m atc h the phys ics o f t h e drug -d e liv e r y a ppli ca ti o n W e r educe a s peci a l case o f t h e VPIDE, u s in g a d e rived s i milar i ty va ri a ble, to a n o rdin ary diff e r ent i a l e quati o n a nd so lve the r es ul t in g bo und a r y va lu e prob l e m u s in g a s h oot in g m etho d W e find a n a n a lyti c s oluti o n to t h e Flor y-Hu gg in s m o d e l tha t w as r e -d e rived in [73], whi c h i s anothe r s peci a l case o f t h e VPIDE. These solut i o n s pro vid e u s w i t h a n e xpectati o n f o r t h e b e h av i or o f t h e so lu t i o n s to t h e m o d el. In Ch apte r 4 we prov id e a n overvi e w o f p se udosp ectra l meth o d s a nd in t roduce t h e EPS meth o d W e u se t h e sam e (spati a l ) r eg ul a rity condi t i o n s imp ose d in t h e existe nce-uniqu e n ess proo f to d e rive a n e rr or f o rmul a f or t h e EPS meth od. T h e num e ri ca l so lv e r f o r the VPIDE i s pro v id e d in C h apter 5. W e u se the s olv e r o n t h e Fl ory -Hu gg in s m o d e l w i t h both t h e E P S meth o d a nd a co nven t i o n a l p seudo s p ect r a l meth o d ( Ch e b ys h ev collocat i o n ) comp a rin g t h e nu mer ical solu t i o n to the a n a lyti c soluti o n f ro m C h apte r 4 Thi s exa mpl e allo w s u s to compa r e the EPS meth o d aga in st C h e b ys h e v co ll ocat i o n as well. In C h a pter 6 we conduct seve r a l n ume ri ca l ex p e rim e n ts to study t h e be h av i o r o f solu t i ons to t h e VPIDE und e r a va ri ety o f condi t i o ns. W e test diff e r e n t diffu s i o n c o effic i e nt s a nd p e r-

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meability models as well as test the model 's sensitivity to the non-dimensional parameters derived in Chapter 2. Chapter 7 contains ideas for further research 9

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2. Fundamental Continuity Equation In this c hapter we r ev i ew some of the results from [73] relevant to the d er iv a tion of the transport eq uation s being so l ved. This work relies h eavily on H yb rid Mixture Theory ( HMT ) f o r which there are severa l excellent resources availab l e conta inin g a rigorous a nd detaile d overview [ 8, 9, 73]. A s s u c h we will not r eca pitulate HMT but will g iv e a brief overv iew of the r e l evant aspects a nd r ef e r e nce the list e d r eso ur ces as r e quir e d. W e a l so offer a phys i ca l interpretation for the d er iv e d eq uation s co u c h e d in terms of a dru g -d e liv ery model. 2.1 Brief Overview of Hybrid Mixture Theory HMT inv o l ves up sca lin g fie ld equatio n s w hi c h govern t h e motion o f m ate ri a l s from the mi crosca l e to the m acrosca l e via volume ave r ag in g a nd then obtaining r es triction s on the form of the co nstitutive e quations b y u s in g the seco nd l aw of the rmod y nami cs a l so formulated as a fie ld eq uation at the macroscale. The fie ld eq u atio n s often includ e t h e conse rvati on o f mass co n servation o f lin ear and a n g ul a r momentum b a l ance, a nd co n se rvati o n o f e n e r gy R ecall t hat con st itutiv e eq uati o n s are s pecific to the material b e in g m o d e l e d examp les includ e Fouri e r s L aw a nd D arcy s Law. There a r e seve r a l oth e r upscalin g techniques s u c h as homogenizati o n how ever s ince we base o ur a n a l ys i s on eq u ations d eve lop e d b y W e in ste in [73] we f oc u s on H MT. The ave r ag in g pro cedure u se d in HMT i s f o r a n y multisca l e multi-constituent multi-ph ase material. W e co n s id e r a two-scale (microsca l e an d m acroscale) two phase materi a l liquid ( l ) a nd solid (s) eac h compo se d of o n e constitue nt within 10

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eac h phase. Since t h e equations of interest in the drug-delivery mode l describe a s ingl e constituent, we will suppress any indi ces referring to constituents and generally refer to phases using Greek letters, a {3, .... At the microsca l e the field eq uations are a ll valid while the dependent variab l es such as density, ve l ocity, and energy are clearly defined. Moreover the phases s u c h as liquid and solid are distin g ui s habl e at the microsca le. At the macroscale the phases a r e indis tinguishable and the ave r aging procedure provid es anal ogous fie l d equations at the macrosca l e with variab l es that can be identified w ith their microsca l e co unterparts. The macroscale variables m ay be vo lum e averaged (e.g. density) or m ass ave r aged (e.g. velocity) depending on the physical quantity they d escribe. H assanizadeh and Gray use the followin g four crite ri a when determinin g how to ave r age particular terms: 1. When averag ing the integrand times dv (volume differential) or da (area diff erent i a l ) must be a n a dditive quantity. For example, Edv, w hi c h i s e n ergy times volume per unit mass, i s not additiv e, whereas multipl y in g by the density, p), y i e ld s pEdv (units of energy), a quantity that i s additive. 2. The macroscop i c quantities shou ld exact l y acco unt f or the total corre spondin g microscopic quantity. This i s espec i ally important when dealin g with a flux term across boundaries. 3 The primitive concept of a phys ical quantity must be preserved by proper d e fini t i on of the macroscopic quantity. 4. The averaged va lu e s hould r epresent the same functi o n which i s most wide l y observed and measured in a fie ld sit uati on or laboratory at the 11

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m ac r osca le. Thi s in sures a pplicabilit y o f t h e res ul t i ng eq uati o ns. Besid es be in g a two phase, s in g l e co mpon e nt, t w o a ddi t i o n a l assumpt i o n s co ncernin g t h e m a t e ri a l includ e t hat the interf ace d oes n ot contain a n y t h ermo-d y n a mi c p r op e r t i es a nd i s m ass l ess, a nd t hat a r e pr ese ntative e l e m enta r y volum e ( REV ) ex i sts o n t h e m ac r osca l e at a n y p o in t in s p ace The R E V as d e fin e d b y B ea r [7], i s c h a r ac t e rized b y t h e materi a l 's p o r os ity w hi c h we d efine b e low. This volum e must be mu c h small er t h a n the s ize o f t h e e n t ir e flo w d o m a in yet l a rger t h a n t h e s ize o f a s in g l e p o re; t h a t is, i t must b e l a rge e n o u g h to conta in a suffic i e n t nu m b er o f p o r es s u c h tha t a s tatisti ca l ave r age ca n be compute d Co n s id e r a volume, 6Vi, a nd let x b e t h e centroid o f t his volume. Compu te t h e rati o (poros i ty) ( 2 .1) w h e r e (6Vo)i i s t h e volum e o f t h e void s p ace w i t hin 61/i. Gr ad u ally s hrink t h e s i ze o f 6Vi a r o und x s u c h t hat 6V1 > 6 V2 > 6V3 > .... The rati o ni m ay flu ctuate as the s i ze o f 6Vi i s r educed ; es peci a ll y in a n inh o m oge n eo u s mater i al. H o w eve r these fluctuati o n s will d ecay as 6Vi t r a n scen d s be low a ce rtain l eve l l eav in g o nl y s m all a mpli t ud e c h a n ges d u e to rand o m c h anges in t h e distribu t i on o f the por e s i zes n ear x. The r e will b e a va lue, call i t 6V f or w hi c h ni w ill beg in t o und e rgo l a rge a mpli t ud e flu ctuati o n s aga in as 6Vi < 6V beca u se 6Vi w ill b e s m all e n o u g h t o co ntain m ost l y p o r es o r m ost l y solid materi al. Even t u ally as 6Vi t e nd s to zero, ni w ill co nverge to 0 o r 1 d e p e ndin g o n w heth e r x i s contain e d within t h e so lid materi a l o r conta in e d w i t hin a s in g l e p o re. The volume, 6V, i s t h e REV wit h cen t r o id x. 12

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Figure 2.1: Averaging, Local Coordinates The averaging procedure entail s a weighted int egrat ion over the REV using a n indi cator function of t h e a phase, { 1 for r EVa l a(r, t) = 0 for r E V.e, (3 f= a (2.2) where r i s the position vector and e is the local coor dinate referenced to the centroid x of the REV. The position vector r i s ex pre sse d as (2.3) Note that the w e i ght function u se d in the averag in g technique represents the instrument u sed to m easure the properties of the material [27]. Hence, usin g Ia may not be a n appropriate weighting function in the se n se that the aver-age d value may not represent the actua l va lues being m easured Moreover, the pre se nce of the characte risti c fun ction Ia in the a v e r ag in g procedure implie s distribution a l derivatives are r eq uir e d to m a k e the pro cess mathematically ri g orous. W e di sc u ss these nu ances as they a ri se a nd r e f e r to the relevant sources as n ecessary 13

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The magnitude, denoted by I I, of the volume SV in the a-phase is defined by ISVal(x t) = r la(X + e t)dv(e), lov (2.4) and the a-phase volum e fraction ca, i s defined as a( ) ISVal c x t = ISVI' (2.5) so that we have t h e relations (2.6) and (2.7) D efine the f o llowin g quantities (average mass over 6Va) (vo lum e average of 'ljJ) (mass average of 'lj;) Since the upscaling procedure requires a weighted average (integrat i on) of partial differential eq uati ons we need an averaging t h eorem for justifying t h e interchange betwee n the differentiation and integration. This t h eorem i s stated in [28]. Theorem 2.1 (Averaging Theorem) If waf3 is the microscopic vel ocity of interface a/3 and n a is the outward unit normal vector of 6Va indicatin g the integrand should b e evaluated in th e limit as the a/3 -inter fac e is approached from 1 4

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the a side then This theorem outline the averag in g process (see [8, 2 ] for a proof). Iff i s the variable to be ave r aged, then we p er form the followin g steps: 1. B eg in with a conservat i on equation within a phase. 2. Multiply the equati on by Ia 3. Average eac h term over the REV, b'V (integrate over b'V and divide the integral by lb'VI). 4. Apply theorem (2.1) to arriv e at terms representing macroscale quantities. 5. D efine physically meaningful macroscopic quantities. The sums L:.a#a lb'VI f&Aaf3 fwa.B n ada and L: .afa I&VI f&Aaf3 fnada arise from differentiating the function Ia When these sums are combined, they are called an "exchange term'' as this term represents the net change in a quantity, s u c h as a constituent's mass, as it transitions from o n e phase to the other. A specific exampl e will be presented in Section 2.2. 1 5

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2.1.1 Macroscale Field Equations R ecall that the mass bal ance eq uation for a s in g l e constituent is, + v (pv) = pr, (2.8) where p i s the constituent density, v i s the co nstitu ent vel ocity and r accounts for the introducti on or ex it of the constituent m ass due to c h e mi ca l r eact i ons. W e obtain the macrosca l e eq uation by form ally multipl y in g t hi s eq uation by / a integratin g over 6V and t h e n dividin g b y I6VI. Using (2.1), we hav e for the time d er ivative term, wher e [P i s the volume average d d e n s ity in the a phase A s i mi lar averag in g process i s performed on the V (pv ) term to obtain the volu me averaged mass b a l a n ce eq uation (2.10) where v a i s the a-phase velocity a nd as b e for e wcrf3 i s the microscopic velocity of int er f ace a/3. 16

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Dr o ppin g t h e o v erba r notatio n we rewr ite t h e up sca l ed m ass b a l a nce e qu a tion as, ( 2 .11) wh e r e t h e summa tion r es ul ts fro m jumps ac r oss int e rf ace a/3. o tice that if the int erface re m a in s s t a ti o n ary t h e n wetf3 = 0 a nd if the flu x vet net i s n e t posi t ive, the n t h e r e i s m ass t r a n s f e r ac r oss the int e rf ace fro m /3-ph ase t o a -ph ase. D efine = J p (wetf3v). netd a ( e), Aa/3 whi c h r eprese nts the net rate o f m ass ga in ed t h e s in g l e co nsti t u e n t with d e n s it y pin phase a from phase /3, the n ( 2 .11) b eco m es, If w e m a k e t h e r e stri c ti o n that the int e rf ace d oes n o t conta in m ass, the n for a two phase syste m wh e r e c1 + c5 = 1 ( l a nd s d e note the liquid a nd so lid phase r es p ect iv e ly) this restri c ti o n math e mati ca ll y becom es, = 0 ( 2 .12) Wi t h o u t m ass e x c h a nge from dru g t o liquid w e h ave t h a t q = = 0. The c onver s i o n o f the m a t e ri a l time d e riv a tive fro m liquid to so lid phase i s g iv e n b y the d e fini t i o n ( 2 .13) 17

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where v1 8 = v1 -v s i s relative velocity between phas es. Applying d e finition (2.13) to (2.11) s implifi es the mass b a l a n ce eq u atio n in the liquid phase to (2.14) 2.2 Derivation of the Continuity Equation In this sect ion w e s how the d e riv a tion of a Volt erra Partia l Int eg rodiff e r e nti a l Equation (VPIDE) that models the polymer swe llin g in the drug-delivery device. W e ass um e that there are only two phas es, liquid ( l ) a nd so lid ( s); the liquid phase will cons ist of gas ( fillin g the por es o f the p o l y m er matrix) a nd fluid a nd the so lid phase will co n s ist o f dru g a nd pol ymer. The d e rivati on of the continuit y equat i on for the swe llin g r eg ime esse ntiall y follow s b y e mplo y in g the d efinition (2.13). B efor e cont inuin g with the d er iv a tion it i s important to note that in compress ibilit y for the liquid phase i s d efin e d as t = 0. Noticing that, an d ass uming that the liquid phase i s approximatel y in compress ibl e 0 s implifies (2.14) to, (2.15) Durin g the swe llin g r eg im e we assume that t h e drug does not tran s f e r to the liquid phase, so the d e n s it y o f the liquid phase d oes not c h a nge (te mpor a ll y) for 1 8

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this regime that is, = 0. Therefore the in compress ibilit y ass umption l eads to and b ecause v1 =I 0 The mass balance e qu atio n for the so lid phase i s g iv e n by wh ere we d enote E:8 = Equation (2.18) ca n b e r e written to obtain Notice that (2.19) can b e written in terms of the liquid phase as, I V s e V = ( 1 el)" Appl y in g the d e finition s g iv en in (2.13) an d (2.18) to (2.15) produces, Dle l D sel 0 = --+ e1V v 1 = --+ v 1 '8 V e1 + e1V v 1 D t D t D sel = --+ v1 '8 V e1 + e1V v1 -e1V V8 + e1V V8 Dt = E;l + e i V v s + vl,s. Vel+ eiV vl,s (2.16) (2.17 ) (2.18) (2.19) (2.20) elf: l = E;1 + + v1 8 V e1 + e1V v1 '8 (employ in g(2.20)) ( 1 e1 ) e lf:l = E;l + + V (el VI,s) (1 e1 ) hence we obtain the continuit y eq uation in the form (2.21) where e1v1 8 will be g iv en by a form of D arcy s law that we d er iv e in Section 2 3 below 19

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2.3 Darcy's Law The details for the derivation of the form of Darcy s law we use can be found in [73, 74] and we review it in Appendix A. In that derivation Weinstein postulates the dependence of the H e lmh o l z free energy as (n) ,.,, t = ,.,,t( t (m)t t ct, T C s Cs) oy oy E, c p , , (2.22) where m = 1 ... ,p and n = 1 ... q denote material t im e derivatives of order p and q C1i is the concentration of the lh species in the liquid phase and J = 1 ... N, T is the temperature, C8 i s the modifi e d right Cauchy -Gr een tensor. The form of Darcy s law we emp l oy is, (2.23) w h ere R1 is the resi tivity tensor = t M/m) and m=l (2.24) is constant. This term is the resu l t of assuming that for m = 1 ... p are in dependent variables. 1aking time derivatives of the volume fraction independent variab l es a llows for modeling viscoelastic effects since these terms retain the time history of the volume fraction as it evolves. Impli cit in the model assumption is that these terms get weaker so t hat viscoe l astic effects diminish as m in creases. Moreover inclusion of these time d e rivatives implies t hat a ll 20

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of these time derivatives of the volum e fraction ex ist, hence c:1 E CP([O, oo)) in time. The class ical liquid pressure is g iv e n by p1 = I [11, 19] where 'lj;1 is E l T the H e lmh o l z energy, v1 i the specific volum e and T i s the temperature. We assume isothermal cond i t i ons in the biological fluid so we will n eg lect T in our derivation. Applying the definition for specific volume, v1 = -Ir yie lds, p l = ( 1)2 8'1/Jll P P a l p E l (2.25) The thermodynamic pressure i s defined ti = 8 8*1 I 1 This pressure i s the Elp J result of measuring the energy required to change the volum e of the liquid phase keeping the mass fixed. Henc e we can write the thermodynam i c pressure as a function of the volum e fraction and density ti = f.} (c:1 c:1 p1i) There is a third pressure ca ll ed the swelling potential 1r1 = 8:/j/ I that pl,Cld is related to p1 and ti, (2.26) see [11]. Now co nsider t h e pressure terms o f (2.26) and rewrite them in terms o f the t h ermodynamic pressure and swe llin g potential (2.27) We will now simp lif y thi s ex pression further by making some simp lif y in g as s ump tions for the swe llin g regime. Consider the boundary condition for the 21

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delivery device where Gibbs potential in the bulk liquid phase is balanced by the Gibbs potential in the liquid phase (interior of d e vic e), (2.28) wher e B denotes the bulk phas e The right-hand side is constant since we are considering a drug delive ry device that is immers e d in a biological fluid. H ence, we are assuming that the pr essure is continuous across the boundanes. Hence we have the following (2.29) where multiplying by c1 p1 and simp lif ying yie lds, (2.30) So that, l [)pll 7f rv -;:;-[ u E pl (2.31) W e now assume that 1r1 is primarily a function of c1 Moreover, following a similar argument, [)'lj} I l ( 1 [)pl I pl ) l 7fT V p + 1 at --( 1)2 V p = 0 p e:l p p e:l p (2 .32 ) where multiplying by (p1 ) 2 and s implif y in g yields (2.33) applying (2.25). H ence, (2.34) 22

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and we conclude that the classical pr ess ure p1 is primarily a function of c1 Next consider the gradient of the thermodynamic pressure -l fYti I l fJti I l V p = 7Ji V c + fjl V p c pl p c;l (2.35) Equation (2.17) holds justifying the following assumption, fJti I l fJti I l 7Ji Vcfjl V p c c;l p c;l (2.36) Now one may argue that the thermodynamic pressure can change rapidly with a change in density such that the coefficient of V p1 in (2.36) will make these terms the same order of magnitud e as one another. The counterargument to this (well justified) point is that the incompressibility assumption for ces V p1 ;:::::: 0 such that (2. 36) is justified. Therefore apply in g (2.17) to (2.35) simplifies the pressure to (2.37) One final simp lif ying assumption is that, while in the swe llin g regime the change in thermodynamic pressur e relative to the change in liquid co nt e nt is small compared to the change in the swe llin g potential relativ e to the change in liquid content, c1 IP1 c1 IP1 There fore (2.37) finally simplifies to (2.38) where the constant P0 has units of pressure and is written to indicate that 1r1 is a dimensional quantity with units of pressure. We will eventually non dim e nsionalize the resultin g partial int eg rodiffer e ntial equation, so l abe lin g these dim e nsional qu a ntities will prove co nveni ent for notational r easo ns. 23

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W e now foc u s on writing a n express ion for D a r cy's l aw that e limin ates the time d e rivativ es of c:1 in the term a nd exp r esses V(c:1p1 ) + tiVc:1 in terms of liquid diffusion r es ultin g from the bulk fluid p e netrating into the d e liv e r y d ev ice. B eg in by taking the L a p l ace tra n s form of [45], @ = t M[m) V loo m=l -oo p = L M[m )s(ml l v;L m=l m=l wh ere we assume that all initi a l gra di ents a r e zero (m)!l V E t=O = 0 for all m. (2.39) (2.4 0 ) ( 2.41 ) (2.4 2 ) p Now a ppl y the convolution theorem [ 1 ] to@= L to obtain m=l (2.43) Lettin g p Bv(t) = BoBv(t-t') = Bo L M[m)6(m-l)(tt') (2.44) m=l w e obtain the followin g f orm of ( 2.43) (2.45) The co nstant B0 i s in serted f or the purposes o f dimensional a n a l ys i s to be p er-form e d below The r efore, D a r cy's l aw can be writt e n as R1 c:1v1 '8 = ( n1 + c:1 an: I ) Vc:1 -c:1p1 t B v(tt') V i1dt' 8c pi l o (2.46) 24

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a nd lettin g (2.4 7 ) w h e r e K1 = K1(c:1 ) i s t h e p ermeability, we o btain ( 2 .48) In [63] the p ermea bili ty functi o n i s assumed to b e co nstan t a nd t h e e n t i re co-effic i e n t o f the int eg r a l term is a b sorbe d into t h e k ernel. But we can see fro m (2.48) t hat t hi s m a n e uver t r a n s l a t es t o K1p1c:1 co m b inin g w i t h B v(tt ) un de r the int eg r al. H o w e v e r c:1 i s a fun c tion of s p ace a nd time, h e nce this combin a ti o n o f ter ms f o und in [63 ] i s i n va lid a nd w e l eave t h e fun ct i o n K1 p1c:1 as a coeffic i e n t o f t h e in teg r a l term. 2.4 Non-dimensionalization of the Model A s a n a id to int erpretin g the phys i cs o f t h ese m o d e l s a nd so lvin g t h e m we co n s id e r t h e dim e n s ion a l p a r a m ete r s assoc iated with s om e o f these terms. ondim e n s i o n aliz in g t h e m o d e l y i e ld s non-dim e n s i o n a l p a r amete r s t hat appea r as coeffic i e nts o f so m e ter ms in t h e e quati o n H e nce, t h ese p arameters sca l e some o f t h e ter ms in t h e e quati o n a nd p e rmi t u s to d eter min e t h e ir r e l e v a nce. A cco rdin g t o [ 7 ] the coeffic i ent of t h e p ermea bilit y functi o n i s M; 1 wh e r e sll M? i s t h e s p ecific surface a rea a nd v1 i s t h e v i scos ity o f t h e liquid phase S o w e ca n wr ite t h e p e rm ea bil ity as ( 2 .49) 25

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where K1(c:1 ) is a non-dimensional function to be determined Notice that (2.48) (2.50) A units check of the parameters is g iv en in Table 2.1 where we notice that the coeffic i ents 1 a nd have units of diffusivity, /T, and non-dimensional s v r-o respectively. We should note that the terms B0p1 and P0 both have units of pres sure The coeffic i ent of the integral term, B0p1 i s a modulus of e l asticity (which has units o f pressure) that behaves as an amplitude for the viscoelastic effects of the model. On the other hand, flow driven by the swe llin g pressure occurs instantaneously and does not retain any time history so the pressure term (2.38) accounts for the e l astic effects of t h e model and the coeffic i ent P0 signifies the magnitude of those effects. Note that this term does encompass flui d sol id int eractions and will account for some viscoelastic effects. Achanta et. al. [3] s how ed that anoma lou s flow results from this term, and we will demonstrate this result when we show t h e so lu tions to the model under a variety of cond itions. D efine the dimensionless parameter "'s = and the parameter with units of diffu siv it y (L2 /T) "'o = 1 then write D arcy's law (2.50) as v 'v'' -"" ( ( K '( ') V' (2.5 1 ) +"'sK1(c:1)c:1 1 t B v(tt') V i1dt') Note that K,8 i s a ratio of the modulus of e l astic ity to the swe llin g pressure coeffic i ent. For the purposes of physical interpretation, we will make "'s a signed 26

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Parameter Units Po M LT2 M2 1 s vl M LT pl M Bo e ( ) M T2 Table 2.1: Units for the dimensional parameters of the Darcy model (2.50) number K-8 E ( -oo, oo). A negative K-8 is akin to the restoring force and indi cates the polymer 's resistance to deformation Whereas a positive K-8 indicates a material 's compliance to deformation. We provide a brief analysis regarding this parameter in Section 2.6 and s how the impact of K-8 when we conduct a paramete r sensit ivit y study in Chapter 6. 2.5 Continuity Equation The continuity equation we will consider models polym e r swell in g as fluid imbibes into the polymer matrix, i.e. the "swelling regime." H e re t h e drug d e liv ery device is imm ersed in a biological fluid The ini t i a l assumptions for t hi s regime are that t h ere is no transfer of drug to t h e liquid phase and that the so lid phase, s, is composed of both po l ymer and drug ; that is, this cont inuity eq uation is a two phase model such that c8 + c1 = 1 and such that there is no diffusion (species transport into the liquid phase) taking p l ace in this model regime. 27

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Let the function modeling the liquid p e netration into t h e so lid phase be d enoted D ( cl) = K l(cl) I (2.52) O E pl In co rporating (2.50) a nd (2.52) into the co ntinuit y e quation with no masstransfer b etwee n phases (2.2 1 ) y i e l d s the followin g Volterra p a rti a l int egro d iff e r e nti a l eq uation (VPIDE): i1 "
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B y the n a me, one ca n inf e r that viscoelastic materi a l s ex hibit both viscous a nd e l ast i c properties. Elastic materials are character i zed by the prop erty that if a l oad is introduced to the material and it deforms then upon unlo ad in g the material imm e diatel y returns to its initi a l state. This property is shown in Fig ur e 2.2 where the stress versus strain curve for the e lasti c material ( l eft) is identical during the load in g a nd unlo ading phases. Linear e lasti c materials follow Hook e's l aw which states that stress is lin ea rl y proportional to strain. A simp l e model describing Hook e's law is the spr in g eq uation where the force exerted by the spr in g i s proportional to the l engt h that the spr in g i s stretched from its initial ( r est ing) position. Viscoe lasti c m ate ri a l s on the other hand retain so m e or all of the e n ergy. Figure 2.2 shows the stress versus strain curves for a viscoelastic material (r i ght) followin g different paths during the load in g and unloading phases. Durin g the unloadin g phase the stress c urv e sags beneath the load in g c urv e s i g nifyin g the d ecay in stress over time; this phenomenon i s calle d hysteresis. Figure 2.3 s how s a n examp l e of stress versus time where constant str ess is in troduce d during the loadin g portion, t < t1 then finished at t1 thus commenc in g unlo a ding, t > t1 The stress curve for an e lasti c material wou ld drop to zero instanta n eous l y ; how eve r for a viscoe lasti c material it d ecays. The s impl est models s u c h as the Maxwell or K e lvin mod e l u se a decaying exponentia l as a means to capture this behavior see [38] for exam ple. Physically, the int egra l term in (2.53) is derived from a constit utiv e rela tion that accounts for the v i scoe lasti c effects of the so lid material being mod e l e d Typically the viscoe lasti c str ess i s written u s in g the stress -strain relation-29

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Stress
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ship [23], 100 d E8(t') a(t) = 0 G(t-t') dt' dt', (2.54) wh e r e E8 is the solid-phase strain tensor. This model for the viscoelastic stress assumes that the strain effects are cumu l ative hence integral. This model follows the so-called Boltzmann Sup e rposition Principle and can b e derived using the Riesz Repres e ntation Theor e m [23]. However the integral term (3.8) contains time derivatives of the volum e fraction not solid-phase strain, E8 There is a re l ationship between these two variabl es. In [51 ] it is shown that at moderate to high fluid contents the normal components of the strain t enso r are related to the volume fraction of the solid phase, E8 Moreover since the so lid and liquid phases are all that comprise the mate rial we have that c1 = 1 E8 H ence at these fluid contents the depen-(m) (m) dence of 'lj;e in (2.22) on the strain E8 may b e r eplace d by E8 Recall that the time derivatives of volume fraction c1 were chosen as ind ependent variables for the constitut iv e equation (2.22) and appear in t h e series (2.24) which was subsequently converted to the integral term using the Laplac e transformation. The k e rnel Bv, is the function that accounts for the d ecay of the viscoe l astic stress and is aptly name d the comp l iance function; it indicates how well the material complies with the stress introdu ced. Following Christensen [23] w e write the compliance function as a decaying exponentia l function B v (l -t') = exp[where T is the relaxation time the time it tak es the model to attain its point Now (2.53) b eco m es E1 "
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otice that by l etting K-ot t=-T o (2.56) w h e r e T o i s some characte risti c l ength d e p e ndin g on the mod e l geo metr y suc h as the radius of a cy lind e r for a drug-deliv e ry mod e l or the r a diu s of a sphe r e for a soy b ea n mod e l for exa mple, w e can non-dim e nsion alize (2.55), w h e r e the dot (i1 ) operator now s i g nifies the materi a l time derivative with r e s p ect tot ( n o n-dim e n s i o n a l time). Now introdu ce the non-dim e n s ion a l D e bor a h numb er [56], T De= tn' w h e r e tn = !:!!. i s t h e diffu s ion tim e to obtain "'o 2 .6.1 Deborah Number ( 2 .58) The D ebo r a h numb er, D e [56], i s co nv e ntion ally known as the ratio com p a rin g the r e l axation time to the observation time but in this case the diffu s ion time suffices as the observation time" b eca u se once this t im e i s r eac h e d the materi a l i s satur ate d a nd t h e mod e l is no l o nger va lid We provide a famil i a r exa mpl e to f aci litate a n unde r standing of the D e bor a h number. Consider a dry str a nd o f s p ag hetti which is said to b e in the "gl assystate"; t h e str a nd can b e b ent slig htly under a small a mount of stress while in creas in g the str ess e v entually brea k s it. In thi s stat e the time sca l e for ob se rvin g c reep or flow 32

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(re laxation time, T) i s l ong compared to t h e observat ion time that is, De 1. In this state t h e spag hetti strand instantaneo u s l y r ecovers to its initi a l state after it experi ences a period o f stress; that i s, i t i s a n e lasti c material. When the str and i s placed in a pot of boiling water, the temperature in crease c h a n ges its molecular architecture and the str and enters what i s call ed a transition state. Roug hl y speakin g the free volume" in c r eases; the space ava ilabl e for molecular segments to comp l y in creases. The c ritical temperature at which this transi-tion state occurs i s call ed the g l ass transition temperature, T9 [26] These rates of conformat ion a l c h ange can often b e d escr ib ed with reasonable accuracy by Arrhenius-type express i ons of the form -E rate ex: exp RT where E i s the activati on energy of the process R i s the gas constant, and Tis the temperature. See Figure 2.4 where it i s ev ident that as T in creases the rate of conformatio n a l c h a n ge in creases and the material becomes more compliant. The strand in the boilin g water, at a temperature above T9 enters the rubbery state (softens) and if we were to extract t h e strand from the water and place it under stress, it would eas il y deform. D epend in g on how far the strand has progressed in the rubbery state determines how close to its initi a l state it returns a ft er a period of str ess. This s low recovery to the initi a l state i s indicativ e of the relaxation time, T. Since t h e strand i s pliabl e (v i sco u s) but recovers (elastic) it i s said to exh ibit viscoelastic properties ; the relaxation and observation time are of equ a l order so D e 1. After the s paghetti strand h as been heated for a lon g enough period of time and its temperature i s hi g h eno u g h above T9 it enters the rubbery state and does not recover to its ini t i a l state a ft er stress H ence, the 33

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Glassy rate T Figure 2.4: T e mp e ratur e d e p e nd e nce of rate between t r a n s ition states on temperature. time sca l e for observing c reep or flow is very s hort compare d to the observation time, D e 1 a nd the strand h as e nt e red the viscous state. W e summarize the pro perti es in T a bl e 2.2 [56, 69, 72]. If De"" 0 ( 1 ) then the VPIDE ( 2 .59) m a int a in s its pr esent form w hi c h we will r ef e r to as the "Viscoe lasti c Case." If the D e bor a h number i s very small, D e 1 (v i sco u s materi a l ) then t h e ex pon e nti a l i s a ppr ox imatel y zero a nd the int egra l term m ay be n eg l ected y i e ldin g ( 2.60 ) whi c h w e will call the "Visco u s Case." Fin ally, if the D e b o r a h numb e r i s very l a rge, D e 1 (el as ti c m ate ri a l ), then the argument of the ex pon e nti a l i s roughly 34

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D e 1 materi a l i s v i sco u s D e 1 materi a l i s e lasti c D e 0 ( 1 ) materi a l i s viscoe lasti c Table 2.2: Characterizat i on of t h e D e b o r a h numb er, D e zero impl y in g that. the int eg r a l t erm takes the form which ca n be int egrate d to y i e ld How eve r r eca ll the ass ump t i o n ( 2.42 ), Vc:b = 0. Thus, t h e Elastic Case 1 s g iv e n by, (2.6 1 ) 2.6.2 Integral Coefficient The coeffic i e nt "'s E ( oo, oo), i s a rati o o f the modulus o f e l ast icit y to the swe llin g pressure coeffic i ent. I t appears in front o f t h e in teg r a l term in (2.59) ; h oweve r u s in g int eg rati o n b y p a r ts we ca n m ove t hi s p a r a m eter a nd s how how i t imp ac t s t h e mode l 's diffu s i o n Consider t h e int eg ral t e rm a nd p erfo rm int eg ration b y p a rts to obtain 35

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"-s < 0 "-s > 0 l"-sl 1 increas e flow inhibit flow l"-s l 1 inhibit flow increase flow Table 2.3: Characterization of "-s and notice the diffusion term where D(c1 ) > 0 K1(c1 ) > 0 and c1 0. Hen ce, a c hange in sign and magnitude of "-s ca n affect the model s diffusion properties W e summarize the impact in Table 2.3 Certain l y the integral term will also have an effect on c1 especially as D e changes How ever, diffusion (generally) has the most dramatic imp act on El. 2.7 Discussion In this chapter we reviewed the d e rivation of a coup l ed system of nonlinear equati ons from [73]. The VPIDE models the swe llin g effects ofthe drug-delivery device. The work presented here is not n ew, but we discovered an error in earlier work, present in [63] and further propagated in [64, 65] where the equation is written in the form (2.63) The integral term should contain a volum e fraction as a coeffic i ent. In [63] the author claims that c1 gets absorbed into the integral; however there i s no 36

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math ematica l justificati o n f or this m a n e u ver. The model appears corr ect l y in (2.59) where we see c:1K1(c:1). The p ermea bili ty K1(c:1 ) appears in t h e e quation as stated in [73] a nd our calc ulations whi c h follow clo se l y the d eve l opment in [73], confirm it s place as a coeffic i ent of the int eg r a l term. Sin ce the e qu a tion s h ave b ee n formulated in L ag r a n g i a n coo rdin a t es we do no t notice the swe llin g b y o b ser vin g the d e p e nd ent variable, c:1 How eve r if we co n s id e r the r e lativ e c h a nge in volume of the two-phase material, we ca n see the s w e llin g effects Let V0 denote the initi a l volum e of the material a nd let V d e n ote volume at some later t ime. The n let VJ, VQ8 V1 v s d enote the r es p ect ive volumes o f the materi a l in t h e liquid a nd so lid phases. W e h ave, and Now compute the r e l a tiv e volume, V-Vo Vo ( 2.64 ) (2.65) s ince v s = V0 because the volum e of the so lid phase do es not c h a nge (no so lid m ass i s in t r od u ced into t h e syste m). Substi tuting (2. 64) a nd (2. 65) into t h e r e lati ve volume a nd s implifyin g y i e lds, V-Vo Vo Vl v;s 0 v;t + v;s 0 0 ( 2.66 ) For exa mple, if E b = 0 1 a nd c:1 = 0 2 the n the r e lativ e c h a nge in volum e of t h e mate ri a l i s 125%. 37

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Let n be the domain modeling the drug-delivery device (e.g. cy lind er or sphere), then the moisture content i s defined as, (2.67) and provides an addi t ional physical int erpretation for c1 Since M(t) i s an ag gregate and 0 ::; c1 ::; 1 holds the moisture content is a nondecreasing function. Moisture content provides another useful metric for studyin g model behavior; we anal yze M(t) after we compute so luti ons to (2.59). 38

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3. Ana lysi s of the Volte rra P a rti a l Integrodiffe r entia l Equation The V o lterr a P a rti a l In teg r o diff e r e n t i a l E quati o n ( 2 .59) fro m C h apte r 2 was d e rived to m o d e l fluid t r a n s port t hr o u g h swe llin g por o u s materi a l s w h e r e t h e so l id phase i s v i scoe lasti c [63, 7 4 75]. Exampl es o f s u c h s y ste m s in cl ud e f oo dstuff s u c h as soy b ea n s a nd pasta a nd p o l y m e r s u se d in drug -d eliver y sys t e ms. B y in s p ect i o n w e see t hat t hi s t r a n s p ort e quati o n co ntain s t h e sum o f a n o nlin ear diffu s i o n te rm a nd a n in teg r a l term a l so known as a n h e r e ditar y te rm. In othe r words t h e VPIDE ca n b e v i e w e d as a n o nlin ear diffu s i o n eq uati on w i t h a m e m o r y t erm. The h e r e ditar y term po ses both theo r e t i ca l a nd num e ri ca l c h alle n ges yet pro vid es f e r t il e g r ound f or interpretati o n a nd a n a l ys is. B ef o r e b eg innin g o u r a n a l ys i s w e se l ect a n ini t i a l co ndi t i o n a nd b o unda r y co ndi t ion s f o r the dru g -d e l iver y m o d el. These co ndition s co up l e d with t h e VPIDE (2.59) compri se a n i ni t i a l b oundary va lu e p ro b l e m ( IBVP). On ce t h e IBVP i s stated we establi s h ex i ste n ce a nd uni q u eness a nd p er f orm Lie g r o up a n a l ys i s to d e rive approx imate so luti o ns. In gen e r a l it i s diffi c ul t t o prove ex i s t e n ce a nd uniqu e n ess f o r n o nlin ea r PDEs, but n o nlin ea r VPIDEs p ose eve n m ore o f a c h alle nge due to t h e in teg r a l term. Thi s work r e qui res a pr oo f that r elies o n prev i o u s res ults s ho w in g t h e e xistence a n d unique n ess o f qu as ilin ea r p a r a b olic e qu a ti o n s [4]. I n C h apte r 5 we so lv e t h e pr o b l e m num e ri cally. 3.1 Formulation of the Boundary V a lu e Proble m The gove rnin g e quati o n i s 39 (3.1)

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wh e re, for co nv e nience, we write K(c:1 ) instead o f K,8c1K(c:1). Note that c;l indi cates the material time derivative, but s ince we a r e co n s id er in g the model in L ag r a n g i a n coordinates, this time d e riv ative b e h aves exact l y as a;t1 W e c hoo se a mod e l geomet r y that i s co n sistent with the drug deliv e r y a pplication so we co nsid er a right cy lind e r. H oweve r we assume a n g ular and az imuthal sym metry so t h e domain n = n u r i s a rectan g l e where n i s the in t e rior and r i s the o uter bounda ry, see Fi gure 3.1. Sinc e c:1 i s a volume averaged I L r I / r r Figure 3.1: The mod e l geometry i s a cy lind e r but we ass um e a n g ul a r and azimutha l symmetr y thus reducing the dom a in to a rect a n g l e D w ith bounda r y r = r1 u r2. quantity it follow s that c:1 i s co ntinuou s [31]. Addition a ll y s ince we ass um e t h e m a t e ri a l can never be a ll liquid o r a ll so lid we assume that (3.2) 40

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wh e r e a nd a r e constants. For exa mple, t h e r e a r e gas p o r es in s id e t h e pol y m e r n e twork a nd the gas phase i s co n s id e r e d to b e p art o f t h e liquid phase Addi t i o n ally, f o r t h e t im e reg im e ove r w hi c h we are so l v in g t hi s pro bl e m t h e r e will b e p o l y m e r a nd drug (so lid pha s e) present; t h e p o l y m e r w o uld not h a v e corro d e d away ove r the t i me int e r va l co n s id e r e d In t h e m o d e l s i t i s ass um e d t h a t t h e m a t e ri a l i s imm e r se d in a s in g l e co n stituent liquid a nd t h a t t h e m a t er i a l imbibes t h e fluid from the exte rior o nl y m a kin g t h e exter i o r b o un dary co ndi t i o n Diri c hlet a l o n g r 1 H e nce t h e b o und a r y a nd initi a l data f o r (3.1) a r e a nd c-1 = a l o n g [ 0 T ] X r l EJcl av = 0 a lon g [ 0 T ] X f2 ( 3 3 ) (3.4) (3.5) r es p ect ively, w h e r e we ass u me t hat cb(r, z) E C2(f2), c&lr2 = c&lr1 = a nd T E ( 0 oo] i s so m e a rbi t r ary tim e In s umm a ry, the IBVP f o r the s w e llin g p o r t i o n o f the drug deliver y m o d e l ca n b e w ritten = (1c-1) V [ D(c-1) Vc-1 + K(c-1 ) fat in [ O T ] X n c1 = o n [ 0 T ] X rl act av = 0 o n [ O ,T] X r 2 c-1(r,z, O ) =c-&(r,z) E V (r,z) E f2, 4 1 (3.6)

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where 0 < T oo. We will establish existence and uniqueness for the IBVP (3.6) by employing two theorems proven in [4] under the following conditions: 1. e:1(r, z, t) E C0 ([0, T]; C2(0)) where n is a closed cylinder with unit radius (assumptions of angular symmetry and azimutha l symmetry in the axia l direction will reduce the problem to two spatial dimensions (r, z ) E [0, 1] x [ 0 1]) 2. 0 < c1 < 1 3. D(e:1 ) :2 0 is continuous in e:1 4. K(e:1 ) :2 0 is differentiably continuous in e:1 5. DeE [ 0 oo] (see Remark 3.1). First we write the IBVP and reformulate it to conform to the IBVP g iv en in [4] and second we show under what conditions the coefficients, initial va lue, and boundary cond iti ons satisfy the assumpt i ons of the hypotheses stated in the ex i stence and uniqueness theorems. The IBVP as written in [4] is Otb(u)-v a(b(u), Vu) = f(b(u)) in [0, T ] X n b(c1 ) = b0 on {0} X 0 u = UD on [ 0 T] X r 1 a(b(u), Vu). 1/ = 0 on [0, T] X r2 (3.7) wher e u is the dependent variable in [4]. The prescribed initi a l and Dir ichlet boundary conditions in (3. 7) are b0 and uD respectively. 42

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Rewrite the IBVP (3.6) in a form suitable for [4] using integration by parts on the hereditary term to yield (3.8) Iow reformulate the IBVP (3.6) by defining the following terms (3.9) where a(b(w),p) = (D(w) + K(w)) p (3.10) b(w) = -ln(1-w), (3.11) and f(b( e1)) = -1),8 V e1 ( K(e1 ) 1 t e-
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R emar k 3 1 The cases of the IBVP (3.6} w h ere the D e borah numbers attain th e values D e = 0 and D e = oo may permit a l ess restrictive function space for existence and uni qu eness than the one we posit in this chapter. How ever, th ese cases do not invalidate th e proof gi ven. If D e = 0 th en the int e gral term v ani s h es and we c an neglect terms invo l vin g K(c1 ) suc h as the for cin g term f(b(c1)), yielding a nonlinear PDE I f D e = oo, th e n th e integral t erm reduces to K(c1) V c1 and f(b(c1)) may b e neglected, yielding In ei ther case th e VP IDE reduces to a nonlinear diffusion e quat i on. Let V = { v E W1 k(O): vir= 0}, (3.14) wh e r e W1k d e notes a Sobol ev space, then a weak so lution c1 E + Lk (0, T ; V), of the IBVP (3.7) satisfies the foll owing two prop e rties [4]: 1. b(c1 ) E L00(0,T;L1(0)) and Btb(c1 ) E Lk*(O T ; V *) with the initia l v a lues b0 that is 1 T (8tb(c1), () + 1 T 1 ( b(c1)-b0)8t( = 0 (3.15) for eve r y test fun ct ion ( E Lk(O, T ; V) n W1 1(0, T ; L00(0)) with ((T) = 0. 2 a(b(c1 ) V c1 ) f(b(c1)) ELk* ([0, T ] x 0 ) and c1 satisfies the diff ere nti a l equa tion that is, 44

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for every ( E Lk(O, T ; V). The six assumptions the IBVP needs to sati sfy in [4] are: (H1) n C lR x lR be open, bounded, and connected with Lip schitz boundary, r can wh i ch i s measurable with trace H n-1(f) > 0 and 0 < T:::; oo. (H2) b i s a monotone vector fie l d and a continuo u s gradi ent, that is, there is a convex C1 fun ction : !Rm-; lR with b(w) = V. The function l eads to the definition of a fun ction, B(w) := b(w) w(w) + (O). ( H3) a(b(w),p) i s contin uou s in wand p and sati sfies the following e llipticity condition, for some constant, c, and 1 < k < oo. The function f(b(w)) i s cont inuous in w. (H4) The following growt h condition is satisfied, ia(b(w), p)l + lf(b(w))l :S c( 1 + B(w)(k-l)/k + IPikl ). (3.18) In genera l the coeffic i ents a, b f may a l so depend on t and x. ( H 5) ELk (O,T; W1k.(n)) n L ([O,T] x n). ( H6 ) w(b0 ) E L1(n) and b0 maps into the range of b such that there i s a measurable function Eb with b0 = b(c&). The exi tence and uniqueness theorems are stated here for convenience: Theorem 3.2 (Alt and Luckhaus [4], Theorem 1.7) Suppose that the data satisfy (H1}-(H6 } and assume that E L1(0 T ; L00( n)). Th en there is a weak so l ution to IBVP (3.13}. 45

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Theorem 3.3 (Alt and Luckhaus [4], Theorem 2.4) Suppos e that th e data satisfy (H1} (H6} with k = 2 and a(t r z b(w),p) = A(t, r z)p + e(b(w)), (3.19) where A(t, r z) is a symmetric matrix and measurable in t and (r, z ) such that for some a> 0 (3.20) ar e posit ive d e finit e Mor eover assume that fo r some c on stant C Th en th ere is at most on e weak solution to IBVP (3.1 3}. 3.2 Existence of a Solution W e b eg in t hi s sect i on wit h an ex i stence theo rem for IBVP (3.13) Theo r em 3.4. The strategy for proving this t heorem i s to s how that the co nditi ons stated in its hypothesi s impl y that the d a t a in IBVP (3.1 3) satisfy ass umpti ons (H1) ( H6) for Theo r e m 3.2 Once these ass umption s are sati s fied t h e n so is the h ypothes i s o f The or e m 3 2 and we conclude t hat a so lu t i on to IBVP (3.13) ex i sts Theorem 3 4 (Existence Theorem) Suppo se that th e follo win g c onditions hold for IBVP (3.1 3}. 1. 0 open bound e d and connecte d wit h L ipsc h i t z boun dary r c 80 is mea surable with H0 ( f ) > 0 ; 46

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2. 0 < T oo; 3. 0 < c1 < 1 ; 4. D(w) is continuous and K(w) is diff eren tiably continuous; 5. f(b(c1)) defined in (3.12} is continuous; then there exis ts a weak solution to IBVP (3.1 3}. Proof: We proceed by justifying each of the assumptions (H1) through (H6). (H1) The cy lind er n C !Rx!R is certain l y open bounded and connected with Lips ch itz boundary, r c 80 is measurable with H0(f) > 0, and 0 < T oo. (H2) b is continuous and is strictly monotone increasing for 0 < w < 1 the domain of definition. Moreover b(O) = 0 and there is a function cl>(w) = (1-w) ln(1-w) + w such that ci> E C2[0, 1), ct>'(w) = b(w), and cl>(w) is convex s in ce cl>"(w) = 12w > 0 for all w < 1. Sin ce such a ci> exists we ca n defin e wb(w) =sup t (w-b(sa)). ds e7EIR l o = s up(w a-cl>(a)). (3.22) e7EIR We can a l so show the following property for B(w) = w (b(w)) B(w) = b(w) wcl>(w) = 11 (b(w)-b(s z) ) z ds = 1 w (b(w)-b(s)) ds. (3.23) 47

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Prec i se l y we have, B(w) = -w ln ( 1 -w), (3.24) so B(w) 0 for a ll 0::::; w < 1, and w(w) E L1(S1) s in ce { /ll!(w)/ = t B(w)d z = J n J o 2 (3. 2 5) S ee Figure 3 2 f or a plot o f B(w). 35,----.---..----,----,---,.-.,...-----, 30 25 ,.-,. 20 CQ IS 10 0ol_-"O.,--I -0. 6 __,.0 7--:;:0.8=-0 9 Cll F i g ur e 3.2: Integral of tempor a l tran s f orm, B(w). (H3) In order for a to b e continuous in wand p w e r equire D(w) and K(w) to be continuous. W e see that p appears linearl y and h ence (a(b(w),pl)a(b(w),p2)) (PI-P2) = (D(w) + K(w)) (PI-P2)2 C(piP2)2 (3.26) for so m e positive co nstant, C, since D E 1 and K are bounded from a bov e and b e l ow. W e need to restrict c1 E C2(0) for a ll t and K E CI(O 1 ) ( diff e r entiabl y continuo u s in c1 ) so that continuity for J(b(w)) h o ld s 48

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( H4) This assu mption pl aces a g rowth co ndition on Ja(b(w),p)J + Jj(b(w))J. The bound on the diffu s i o n term D(w) + K(w), impli es a(b(w),p) = ( D(w ) + K(w))p:::; c ( 1 + JpJ) f o r some constant, c, so we need to ati sfy t h e ass umption Jf(b(w))J :S c B(w)112 (3.27) for the same co nstant c, to sati s f y the growt h conditi o n Given that B(w) > 0 and co n se qu e ntl y B(w)112 > 0 a nd f(b(w)) i s co ntinuou s we ca n certainl y find a c that sati sfies this ass umpti o n In summary we are g uaran tee in g t h e f o llowin g g rowth condi t ion g iv e n in [4] i s sati sfie d Ja(b(w),p)J + J j(b(w))J:::; c( 1 + B(w)112 + JpJ). (3.28) ( H5) The Diri c hl e t co ndition c1 = o n [ 0 T] x f1 where i s a co nstant, i s clearly in (0 T; W1 2( 0)) and L ( [ 0 T ] x 0). (H6) Since b(w) is s urj ect ive, b0 m aps into the r a nge o f b Therefore there i s a m eas ur a bl e fun c tion E b wit h b0 = b ( cb) Furthermo r e = 0 a nd i s obviously contain e d in 1(0, T; L00( 0 )). W e h ave u se d t h e con dition s of the h ypothes i s to establi s h that the data of IBVP (3.13) sati sfies ass umption s (H1) ( H6) o f Theore m 3.2. The r e for e a so lu t ion to IBVP (3.13) ex i sts. 3.3 Uniqueness of a Solution In this sect ion we s how that the so lu tion to (3. 13) i s uniqu e und er particul a r co ndi t i o n s on the diffu s ion coefficient a nd p e rm eab ilit y functi o n D(t r, z) a nd K(t, r z ) r es p ect iv e ly. A proo f o f uniqu ene s i s provid e d by Alt a nd Lu ckh a u s [4] bu t we s how the detai l s o f the proof h ere for the parti c ul a r c hoice o f 49

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a(t,r,z, b(w) p) = A(t, r, z )p = (D(t,r,z ) + K(t, r,z))p. This sum of diffusion coefficient and permeab ili ty offers nuances to the uniqueness proof that require an assurance that the theorem is valid for the IBVP (3.13). The or e m 3.5 (Unique ness Theore m ) Suppose that the data satisfy (H1}( H6} and the boundary and initial data assumptions with k = 2 and a(t, r z, b(w) ,p) = A(t, r z )p = (D(t, r z ) + K(t, r, z ))p, (3. 29) where A(t, r, z) is measurable in t and (r, z ) such that for some a> 0 D + K a and D + K + a8t ( D + K) (3.30) are positiv e Moreover assum e that for some constant C > 0. Then th e r e is at most one weak solution to IBVP (3.13}. Proof: In the proof of (3.4) it was established that the data satisfy (H1) ( H6) as well as the boundary and initial data assumptions. Iow suppose that c:i and c:& are two weak solutions. Then let (3.32) by our definition of a weak solution (3.16). The Ri esz Representation Theorem impl i e that there is a function v E (0 T ; V) such that (3. 33) 50

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for all ( E (0 T ; V). Then, letting o;-h be the backward diff e r e nce operator we obtain r+h 1 t 2 J h (8"thf3, v) + h J o ((3, v) 2 1s+h 2 1s+h =-((3, v) --((J(t-h), v) h h h h 1 ( r ls+h rs+h ) +h J o ((3, v) + s ((3, v)-J h ((3, v) 1 r+h 2 r+h = h J h ((3, v) h J h ((J(t-h), v) 1 r lls+h +h J o ((3, v) + h s ((3, v) 1 r 2 r = h J o ((J(t + h) v( t +h))h J o ((J(t), v( t + h)) 1 r lls+h +h J o ((3, v) + h s ((3, v) 51

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= A 1 s 1 (D(t +h)+ K(t + h)) Vv(t +h) V v( t +h) -A 1 s 1 (D(t) + K(t))Vv(t +h) Vv(t +h) + A 1 s 1 (D(t) + K(t))Vv(t +h) Vv(t + h) 1 s 1 (D(t) + K(t))V v(t) Vv(t + h) + A 1 s 1 (D(t) + K(t))V v(t) V v( t) lis+h +h s (D(t) + K(t))V v( t) V v(t) 1 r+h r = h j h J n (D(t)-D(t-h) + K(t)-K(t-h)) V v(t). V v( t) lis+h r + h s J n (D(t) + K(t))V v(t) V v(t) + A 1 s 1 (D(t) + K(t))(Vv(t +h)V v( t)) ( Vv(t +h)V v(t)) r+h r = j h J n 8t:h(D + K)V v(t) V v(t) lis+h r + h s J n (D(t) + K(t))V v(t) V v(t) + A 1 s 1 (D(t) + K(t))(Vv(t +h)V v( t)) ( Vv(t +h)V v(t)) Lettin g h ---t 0 we hav e the following: 1s+h 1 a;:h(D + K)V v( t) V v( t) = 1 s 1 8t(D + K)V v V v lis+hl l (D(t) + K (s)) V v(t) V v(t) = (D(s) + K(s))V v(s) V v(s)) h s ll ll f ( (D(t) + K(s))(V v( t +h)V v(t)) ( V v (t +h)V v(t)) = 0 h J o J n 52

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therefore we obtain for s almost everywhere 18 (8tf3, v) = (18 1 Bt(D + K)V v V v + 1 (D(s) + K(s)) V v(s) V v(s) ) (3.34) ow we can finish the uniqueness proof by considering the two solutions c:i and mention e d above and via Gronwall 's inequality showing that they are equal almost everywhere. By (3.33) and definition we have 181 (D + K)V v c:i) = 18 ((3, c:i) = 1 s 1 c:i). (3.35) Consider our previous result (3.34) f (D + K)V v V v + t f c:i) 2 l n l o l n = t (8tf3 v)t f Bt(D + K) V v V v J o 2 l o J n + 1 8 1 c:i) now let v be a test function in the weak differential equation (3.16) so that the right-hand side becomes, 181 (D + c:i) V v + 181 f(b( c:i))))v + 181 c:i)-1 s 1 Bt(D + K)V v V v 53

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a ppl y in g (3.35) a nd imp os in g 2ab ial2 + lbl2 f or a ll a b E R y i e ld s t h e in e qu a l -ity, 15 in f ( b ( ci)))v-15 in Ot(D + K ) V v V v 15 f ( b ( c i})l2 18 inivl2 -18 in 8t( D + K ) V v V v. Imposin g t h e h y p othesi s o f t h e uniqu e n ess t heor em a nd P o incar e's in e qu ality [ 34] wit h p os i t ive c onstants, 8 a nd C(8) r espect i ve l y produces, The co nstant f or Poin ca r e s in e qu a lity, C(8) i s c h ose n s u c h t hat 0 < 8 < 1 ; r ecall that P o in care's in e qu ality h o ld s f o r some constan t g r ea t e r t h a n zer o H e n ce, f o r a g iven 8 < 1 we c h oose C(8) an d fix i t s u c h t hat t h e in e qu ality h o lds. Now we co n s id er { ( D (s) + K (s)) V v (s) V v (s) + t { ci) 2 J n J o J n 8 15 in c i ) + 15 in (C(8) -OtD ) V v V v But notice t hat ( b ( c&) -b(ci)) ( c &c D ;::: 0 s ince b(w) i s a n o n-d ec r eas in g functi on. H e nce, J ; f n( b ( c&)b (ci))( c & c D ;::: 0 and (3.39) i m pli es { ( D (s) + K (s)) Vv(s) Vv(s) t { (C(S)-8t( D + K )) V v V v. (3. 3 7 ) 2 J n 2 J o J n 54

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We wish to appl y Gronwall 's inequality [34] to obtain our uniqueness result but this step requires that the right-hand side of the inequality (3.38) contain D + K in the integrand. Given that Bt(D + K) is already present we need only make a coup l e of manipulations using the inequalities present in the h ypothes i s of the uniqueness theorem to obtain our result. Let a > 0 and first consider the condition D + K + a8t(D + K) > 0. It follows that -a8t(D+ K) < D + K C(8)-Bt(D + K) < C(8)a + D + K a by adding the positive constant C(8) to each side of the inequality. Second imp ose the condition D + K a > 0 to obtain C(8)-Bt(D + K) < C( 8 ) + 1 (D + K). a H ence (3.38) becomes { (D(s) + K(s)) Vv(s). V v(s) C( 8 ) + 1 t { (D(t) + K(t)) V v(t) V v(t) J n a J o J n (3.38) where is a positiv e constant and fn(D(s) + K(s)) V v(s) V v(s) i s non negative. Thus, by Gronwall 's inequ ality, 1 (D(s) + K(s)) V v(s) V v(s) = 0 Therefor e, where 8 < 1 implying that c:i = 55

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3.4 Analytic Solutions and R eductions of the IBVP The proof of existence and uniqueness is an important step in the so lu tion process because it confirms whether or not solving the problem makes sense. Additionally this anal ys i s is important because it uncovers some of the behavior of the solu tion as well as the conditions for which we may find a solution. As an examp le, und er the current conditions of existence and uniqueness established in Section 3.2 and Section 3 3, we can expect a smooth so l ution since c:1 E C0 (0, T; C2(0)). In some circumstances we can actually find an analytic solution to a nonlinear IBVP however, the so lution method often requires a transformation such as the Co le-Hopf transformation [54]. In other circum stances we can reduce the nonlinear PDE to a nonlinear ODE and either so lv e the ODE analyti cally or approximate the solution using numerical techniques or asymptotic methods. In this sect ion we consider some cases under which we can so lv e t h e IBVP (3.6) or reduce the PDE to an ODE and so lv e the resulting two-point BVP. The so lu tions to these problems will reveal the behavior to be expected when we solve the more difficult IBVP numerically in Chapters 5 and 6. 3.4.1 Similarity Reduction of the IBVP There are various transform methods for reducing PDEs to ODEs. For examp le, a lin ear PDE defined (spat iall y) over the entire rea l l ine may be reduced to an ODE in time using the Fourier transform. Consider a s impl e IBVP Ut = Uxx on (x, t) E ( -oo, oo) X [ 0 oo) ( 0) --ryx2 0 u x, e r; > 56 (3.40) (3.4 1 )

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Under the two conditions lim u x (x t) = 0 Vt and lim u( x, t) 0 Vt, the X-+00 X--+00 Fourier trans form of the PDE is the ODE (in time) (3.42) where u = 1 : e-i21rk x u( x, t)d x, is the Fouri e r transform of u and k is the frequency The solution to the ODE (3.42) is u(x, t) = e (41r2k2t +7)x2l Thus, the solution to the IBVP (3.40) is the inverse Fourier transform of u, 00 -x21 + 4ryt 2 1 2 kz k e 4t u( x, t) = e-1Jx e -41r t e'21r xdk = oo 2vSi (3.43) The use of integral transform methods is prevalent for linear equations and since the linearit y of the tran forms maintains the linearit y for the resulting ODEs it also makes these methods c onvenient to use. On the other hand transform methods are less prevalent for nonlinear equations. Instead one might group the independent variab l es such that the problem reduces to a simpler PDE, whi c h may b e linear or an ODE. One method for dete rmining such a group is a Lie group transformation. Consider the Lie group (dilation) transformation (3.44) (3.45) (3.46) (3.47) where a > 0 is a scale fa c tor and the parameters {, a and {3 are all positiv e constants. 57

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Note that = 8J/ + V8 V0(-) ( v8 i s t h e so lid-ph ase vel oc i ty) s o (3.48) (3.49) wh e r e v i s t h e t r a n s f o rm e d vel ocity Thi s last r e lati o n s hip impli es tha t t h e m a t e ri a l tim e d e rivative i s sca l e d in t h e sam e m anne r tha t the p a rti a l t im e d e ri vat ive i s sca l e d Appl y in g t h e t r a n s formati o n (3 .44) to e qu a ti on (3.1 ) y i e ld s whi c h ca n fur t h e r b e s implifi e d b y sca lin g t h e dummy va ri a bl e o f in teg rati o n W e wa n t t h e t r a n s formati o n fro m (3.1) to (3. 5 0 ) to re m a in in va ri a n t That i s find va lu es f o r a, /3,1 s u c h t hat t h e p a r amete r s { a00, ai3, a-r} ca n be e limin ate d fro m (3.5 0 ). I t i s clear t h a t we must c h oose f3 = 0 beca u se the param ete r ai3 r es id es in the argument of the ex p o n e n t i al. H e nce, the r e i s n o int e raction wit h the ot h e r p a r a m ete r s impl y in g that we ca nnot imp ose a r e lati o n s hip b etwee n f3 a nd t h e r e m a inin g two ex p o n e n ts a a nd f. Bu t f3 = 0 impli es t hat a = 1 = 0 from 58

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(3.50) y i e ldin g D8u a ( au DT + r eri.(r-i) dt)) J o Dt (3. 50) Equation (3. 50) is identical to (3.1). The r efore, the equation (3.1) do es not transform to a n ODE u s in g (3.44). 3.4.2 Viscou s Case The viscoelastic case, w h ere De rv 0 ( 1 ), w ill be solv ed num e ricall y in Chapters 5 a nd 6. How eve r there i s a case where we can r ed uce the VPIDE to an ODE a nd so lv e the e quati o n u s in g a num er i ca l ODE so lv er. Additionally there i s a spec ial case for the fun ctio n D(c1 ) w h ere we ca n find analyti c so lutions. Solvin g t h ese problems provides a n oppor t unit y to view the behavior of the so lution s and c h ec k the accuracy of our numerical sc h eme d eve lop e d in S ect ion 5.1. The int eg ral term i s a co nstitutive r e lati o n s hip mod e lin g viscoelastic effects in the m o d el. Vi scoe lasti c materials retain some o r a ll of the e n ergy introdu ced by the str ess a nd c ons eq u ent l y m a int a in t h eir s h ape for a p er iod of time that d e p ends on a p a r a m ete r called t h e r e laxati on time, T o f the material. Viscoelastic b e h av ior i s best ex pl a in ed by the D ebora h number, defined b y eq uation (2.58) but restated h e r e T De=-. tD If D e = 0 then the k erne l of t h e int eg r a l term in (3. 1 ) i s zero. In this case, e quation (3. 5 0 ) b eco m es, (3.5 1 ) 59

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In va ri a nce o f (3.5 1 ) r eq uir es = a nd 1 = 0 w h e nce, -=(1-u)-D(u). D8u a ( au) D T (3.5 2 ) The case w h ere D e 1 i s eq ui va l ent to t h e case w h e r e T tD, t h a t i s t h e diffu s i o n t im e do min ates t h e r e laxati o n t im e A fluid o r materi a l t h a t b e h aves in t hi s m anne r ver y s low to diffu se, i s v i sco us; v i sco u s fluid s o r m ate ri a l s are c h a r acte rized b y t h e prop erty t hat they a r e r es istan t t o flow. The s imil a rity var iabl e i s g i ve n by, 'f} = = 7kr, w h e r e D0 i s t h e diffu s i o n coeffic i e nt assoc i ate d wit h D ( u). Wi t h o u t l oss o f gen e r ality, w e ca n let D0 = 1 s in ce we ca n n o n-dim e n s i o n alize (3.1 ) H e nce w e o btain t h e stand a rd B o ltzm a nn s imil arity v ar i a ble, 'f} = .fi' (3. 53) s u c h t hat, Direct s ubsti t u t i o n int o (3.52) y i e lds, ---= ( 1 -u)-D(u). 'f}du d ( du) 2 d'f} d'f} d'f} (3.54) The b o und a r y condi t i o n s a nd ini t i a l condi t i o n y i e ld t h e r e qui s ite co nditi o n s for s olv a bility o f t h e sys t e m (3.54). The Diri c hlet bo und ary co ndi t i o n i s imp ose d a t r = 1 w h ere t h e tra n s f ormati o n (3.44) a0 but w e h ave n o inf o rmati o n f o r a0 oth e r t h a n a0 > 0 w hi c h > 0 at t hi s b ounda r y S o lettin g T-+ oo 6 0

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u 1.0 Figure 3.3: Similarity so lu tion to the IBVP (3. 55). implies ry -t 0 and the Dirichl e t boundary condition gives us u(ry = 0) = Umax The initial condit ion provides the other boundary condition for u since T -t 0 implies ry -t oo a nd we obtain u(ry = oo) = Umin In summary we have a nonlinear second-order ODE, two -point boundary-value problem d (D du) ry du -(u)-+ o dry dry 2(1-u)dryu(oo)=umin u(O) = Umax (3.55) (3.56) (3.57) We can use a simp l e shooting method to so l ve this BVP and obtain the similarity solutions in Figure 3.3. This solu tion appears to show the fluid imbibin g from the in terior in stead of the exterior. However we used the ini t i a l condition to derive the boundary condition u( ry = oo) = Urn in and the Dirichlet condition to derive the oth er boundary cond i tion at u(ry = 0) = Umax These co ndition s are the reverse o f the phys i ca l boundary conditions where the Diri ch let condition 61

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is located at the exterior boundary, r = 1. So the solution in the transformed coordinates is behaving as i t shou ld and demonstrates the expected behavior. We anticipate the fluid to penetrate the drug-delivery device and increase the liquid volume fraction until it reaches the steady state, Umax 3.4.3 Flory-Huggins Model Another examp l e of the viscous case come from the F l ory-Huggins model for swe llin g polymers [35] which has been a standard model used in polymer sc i ence s ince 1953. In [73] the swe llin g model (2. 53) is link ed with the classical Flory-Huggins model for swe llin g polymers. The analysis yields an equation simi lar to (2.53) but l acks the integral term, u = (1u)V (D(u) Vu), (3.58) where D(u) = D0(1 -u) for some constant diffusivity D0 In [73] this model (3.58) is derived using the chemica l potential for the so lv ent liquid p,1 ass uming only one species for each phase ; on l y two phases are considered, liquid and solid. It was shown in [73] that p,1 + RTln(a) (3.59) where a = ue1-u is the activation and J-tb is the chemica l potential at the initial pressure and temperature. Weinstein writes Darcy 's l aw in terms of the Gibbs potential and obtains the following form (3.60) Since the assumption is a single species, the Gibbs potential G1 is equa l to the c h emica l potential so sub tituting (3.59) into (3.60) yie ld s (3.58). 62

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Linkin g the swe llin g model (2.53) not only validates the theory developed in W e inst e in [73], it a l so provid es a n opportunity f or stud y in g the liquid volume fracti o n 's behavior. R ecall that the drug-delivery d ev ice is imm e r se d in a fluid which p e n etrates the pol ymer n etwork. We expect the liquid volume fraction to incr ease until it r eac h es its maximum which i s set at the exte rior bound a ry W e will so lv e the Flor yHu gg in s mod e l a nalytically for a g iv e n set of initi a l-bound a r y co ndition s a nd co nfirm this behavior. Consider a one-dimensional case where the az imuthal dir ect ion h as infinit e extent and we assume a n g ular symmetry. If we nondimensionalize the PDE 2 u s in g the diffu s ion time t = ifot, where ro i s the radius o f the delivery device, then the a ppropriate initi a l a nd boundary co ndition s are u(1, t) = Umax aul = 0 8r r=O u(r, 0) = uo(r) (3.61) for some function u0(r) to be d eter min ed Transform the PDE (3.58) by lettin g v = (1-u)2 then the IBVP b ecomes v = vb.v v ( 1 i) = (1-Umax)2 avl 0 8r r=Ov (r 0 ) = (1-u0(r))2 (3.62 ) This IBVP i s so lv a bl e by se paration of variables. In orde r to see the behavior of the so luti o n s for these mod e l s we will imp ose a set of initi a l-boundar y con dition s 63

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a n d so lve t h e pro bl e m If we suppose t hat t h e ini t i a l co ndi t i o n v(r, 0 ) = 1 -r2 a nd = 1 t h e n t hi s IBVP h as t h e so luti o n 1-r2 v(r, t) = ---, 4t + 1 w hi c h w h e n we tra n s f orm u back to t h e vo lum e fracti o n b ecomes u(r, t) = 1 --. 4t + 1 (3.63) (3. 64) W e w ill u se t hi s s oluti o n to compa r e t h e nume rical m etho d u se d in C hapter 5 Plots o f t h e so lu t ion a r e g iv e n in Fig ur e 3.4. Not ice t h a t t h e liquid v o lum e fracti on i nc r eases as t im e p r og r esses a nd t hat u i s te ndin g towards a s tead y -state so l u t i o n u(r, i--t oo) = 1 t h e Diri c hlet b o und a r y co ndi t i o n This b e h av i o r t hu s c onfirm s the e xp ec t a ti o n s we posit e d earli e r. e1(r,/) 1.0 0 8 r 0 2 r 0 0 0 2 0.4 0 6 0 8 1.0 Fig ur e 3 .4: S o lution t o t h e Fl o r y Huggin s m o d e l (3.58) for a va ri e t y o f times. 64

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3.5 Discussion In t hi s sectio n we demo nstrated ex i s t e nce and uniqu e n ess o f a solu t i o n to the IBVP (3.6) und e r t h e co ndi t i o n t hat t h e volum e frac ti o n res id e in t h e s p ace C0 ( [ 0 T]; C2(0) ) wh e r e 0 i s a ri g htcy lind er a nd t h e solu t i o n ex hi b i ts a n g ul ar symmetr y R ecall d uri ng t h e d e ri va ti o n o f t h e in teg r a l term in Sect i o n 2.3 t hat we co mputed t h e L a pl ace tran s f orm o n p 2: 0 d e rivati ves o f H oweve r ex i s tence o f t h e Lapl ace tran sfo r m r eq uir es t hat t h e functi o n b e pi ecew i se co n t inu o u s in t ime. H e n ce t h e re qui rement t h a t b e at l east co n t inu o u s in t im e s h o uld eas il y b e sati sfie d. M o r eove r s in ce i s a volum e ave r age qu a nti ty i t i s a t l east c ontinu o u s [31]. The r e quir e m e n t f o r to be twice diff e r e n t i a bl y continu o u s on the othe r h a nd i s stro n g as w e expect t h e volum e fract i o n t o h a v e corne r s o r b e h ave lik e a r amp functi on. This condi t i o n a ri ses fro m t h e r e quir e m e nt t h a t the in tegra l ter m b e co n t inu o us. Fur t h ermo r e w e ex p ect t hat uniqu e n ess proo f can b e gen e r alized as well. That D(r, t) + K(r, t) b e strictl y fun ctio n s o f s p ace a nd t im e and n ot was co nstr a in e d b y t h e u se of t h e Ri esz-Represe ntati o n t heor em at t h e start o f t h e uniqu e n ess proo f Fur t h e r r esea r c h in to t h e ex i ste nce a nd unique n ess o f so lu t i o n s to t hi s e quati o n m ay r e l ax t h ese co ndi t i o ns. W e a l so searc h e d f or s imil a rity so lu t i o n s to the IBVP (3 .6) but f o und t hat the int eg r a l term brea k s t h e symmetry. H oweve r co n s id e r a ti o n o f the v i sco u s case wh e r e D e = 0 p e rmi ts s imil a rit y s olu t i o ns. The r educe d n o nlin ea r BVP w as s olv e d u s in g a s h ooting meth o d produc in g so luti o n s that co nfirm e d our phys i ca l intuiti o n A p erturbatio n se ri es m ay r e veal a b ounda r y l aye r d eve l opin g nea r 65

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the exte ri o r b ounda ry, as t h e num e ri ca l so lu t i o n s h o w s a nd fur t h e r r esea r c h into this topi c i s a l s o r e quir e d An othe r m o d e l o f p o l y m e r swe llin g r e -d e riv e d in [73], t h e Flo r yHu gg in s m o d e l w as r eviewed a nd s olved f o r a g iven set o f ini t i a l and b ounda r y conditi o ns. The soluti o n s to t hi s mod e l r evea l e d t h e expect e d b e h avio r o f t h e liquid v o lum e fract ion 66

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4. Eigen-decomposition Pseudospectral Method P se udosp ect r a l ( PS) meth o d s a r e known f o r t h e ir s pectral accuracy w h e n a ppli e d to smoot h fun c ti o n s over r eg ul a r geo m et ri es. The drug -d eliver y probl e m i s d efine d o n a cy lind e r ( r eg ul a r geo metr y) in L ag r a n g i an coo rdin a t es (fixed g rid ) a n d t h e volum e fracti o n c:1 i s a volum e ave r age d quantity an d as s u c h i s c on t inu o u s in both s p ace a nd t im e [31]. H e nce, t h e dru g -d elivery pro bl e m i s a ca ndidate f o r p se udo s p ect r a l m etho d s These V o lterra Parti a l In tegrod iff e r e n t i a l Equ a ti o n s (VPIDE) ( 2 .59) a r e dif fic ul t to so lve n um e ri cally b eca u se t h e i ntegra l term acc umul ates ro undoff erro r as t im e prog r esses a nd a phys i cally r ealisti c e xteri o r Diri c hlet cond i t i o n imp oses a n ini t i a l pr ofile with a ve r y steep fro n t t hat prop agates thro u g h t h e s p a ti a l dom a in The form e r in s pir e d the u se o f PS m etho d s for so l v in g the probl e m b eca u se PS m etho d s are well known f or t h e ir hig horde r accuracy a nd s pectra l c onvergence f o r s m oot h functi o n s [ 37]. The latter in s pir e d t h e u se o f t h e Eigen d eco mposi t i o n P se udosp ectra l (EPS) meth o d ove r stand ard P S met h o ds. F o r e x a mpl e t h e geo metr y f o r the dru g d eliver y m o d e l i s a cy lind e r a nd w e u se a n ini t i a l co ndi t i o n tha t i s ini t i a ll y ver y s t ee p. W e co uld h a v e ini t i alized t h e prob l e m wit h a co n s t a nt c ondi t i o n c:1 = but t hi s w o uld h ave m ade t h e num e ri ca l e xp e rim e ntati o n c umb e rsom e an d would not h ave co ntributed a n y int e r est in g phys i ca l attri b u tes t o t h e prob l e m ; t h e mo vin g b o und a r y would s impl y evo lve as gove rn e d by t h e VPIDE. B y in t r oduc in g t h e steep ini t i a l profile we ca n m o r e ea sil y con t r o l the initi a l stat e of the pr o bl e m b y in corpo ratin g a p a r a m e t e r tha t 6 7

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a djusts t h e steepness. A s a r esult, t h e ini t i a l co nditi o n re qui res a hi g hl y r e so lv e d so lu t i o n ac ro ss t h e s pati a l d o m a in es peci a ll y as t h e pro bl e m e v o l ves in time a nd t h e liq uid p e netr a t es the dru g d e liv e r y d evice. The m o istur e conte n t in c r eases impl y in g that c1 i s pro p aga tin g towards t h e interi o r o f t h e cy lind er. The r e quir e m ent for a l a rge numb e r of s p a ti a l point s punctua t es the a d vantage of the EPS m ethod ove r standard PS m ethods. In t hi s c h a pter w e p rovid e a r ev i e w o f p se udosp ect r a l meth o d s with comp a rison s to fini te diff e r e n ces. Additi o n a lly, w e d e rive a nd d e m o nstrate a n ove l a ppr oac h to a p se ud os pectral di sc r e trizati o n c onstruc t e d o n a p o lar geo m etry [58]. W e d i sc uss a d va n tages a nd disad va n tages o f t hi s meth o d w h en comp are d to t r a di t i o n a l p se udosp ect r a l meth o d s P se udosp ect r a l meth o d s are a s ubcl ass o f s p ect r a l meth o ds, a cl ass o f s p a ti a l di sc retizati o n s for diff e r e n t i a l e qu a ti o ns. The k ey compo n e nt s f o r the ir f o r mul a ti o n includ e lin ea r co mbin a tion s o f suita bl e tri a l b as i s functi o n s a nd t est fun ctions. Tri a l fun c tion s provid e a n a pproxim a t e r eprese nt a tion of the s olution a nd t h e test functi o n s ens ur e t hat the diff e r e n t i a l e quati o n a nd p oss ibl y t h e bound a r y co ndi t i o n s are sati sfie d as cl ose l y as p oss ibl e b y t h e trun ca t e d se ri es expa n s i o n W e t h e n minimi ze, with r es p ec t to a suita bl e n o rm t h e residu a l produ ced by u s in g t h e t run cate d ex p a n s i o n instead o f t h e exact so luti on. In s umm a ry: 1. Giv e n a diff e r e n t i a l e qu a ti o n with bound a r y c ondi t i o ns, a ppr ox im a t e a so lu t i o n u(x) b y a fini te sum v(x) = in t h e case o f a t im e d e p en d e n t pro bl em u(x, t) i s appro ximated b y v(x, t) a nd ak(t). 2 This se ri es i s s ub titute d into the e qu a tion L u = f (x) w h e r e L i s t h e 6 8

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u = 'P n x-h x x + h h = 1/N F i n ite D iffere nce Meth od F i n ite Ele ment Meth od b 1 / N Spect r a l Method Figure 4 .1 : Tri a l functi o n c omp a ri s on f o r s pectra l finite-diff e r e nce, and finit e e l e m ent methods. Spectra l m etho d s a r e c h a r ac t e riz e d b y o n e hi g h-ord e r pol y nomi a l f o r the wh o l e d o m a in finit e -diff e r e n ce method s are c h aracte rized b y mul t ipl e overl a ppin g l ow-o rd e r p o l y nomi a ls, a nd finit e e l e m e n t m etho d s are c h a r ac t e riz e d b y non-ov e rl a ppin g polynomi a l s with c omp act support on e p e r subdo m a in op e rator o f the diff e r e n t i a l o r int eg r a l e quation with t h e result b e in g the so-calle d r es idu a l functi o n : R(x; a o a1 ... aN)= L v -f 3. Since R(x; ak) = 0 f o r the exac t s oluti o n the ch alle nge i s to c hoo se the se ri es coeffic i ents ak so t hat t h e r es idu a l functi o n i s minimized The c h o i ce o f t h e t ri a l functi o n s i s on e o f t h e f eatures t hat di stinguis hes the ea rl y v e r s i o n s o f s pectra l m etho d s from finit e e l e m ent a nd finit e -diff e r e nce m e tho ds, see Fi g ur e 4.1. T o in c r ease acc ur acy f o r finite-e l e m ent a nd fini te -diff e r e nce m etho d s r e quir es p-r e fin e m e n t (a hi g h e r o rd e r p o l y nomi a l ) o r h-r efine m ent (a fine r m es h). Eithe r w ay t h e o rd e r o f the m etho d i s fixe d a nd t h e e rror i s rv O ( hP) wh e r e t h e m es h s ize i s h = tt. Spectra l meth o d s o n the ot h er h a nd h ave a n e rr o r tha t i s decr eas in g f aste r t h a n a n y finite power o f N [66] 69

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Sp ect r a l m ethods a r e g l o b a l appro ximati o n m etho ds: o n e a ppr o xim a t es the s p a ti a l d e riv a tiv e b y u s in g a g l o b a l int e rp o l ant thro u g h di sc r e t e d a t a p o int s t h e n diff e r e n t iatin g t h e in terpo l a n t at eac h p o in t. Fini te diff e r e nce/el e m e n t meth o d s on t h e oth e r h a nd a r e l oca l m ethods ; they e mploy a f ew n e i g hb o rin g gr id p o in ts t o m a k e t h e ir appro xim a tion s Fini te diff e r e nce/el e m ent a pproxim a ti o n s d o l ea d t o s p arse matrix r epre se ntati o n s ho weve r t h e orde r o f t h e ir acc ur acy i s fixe d a nd so to in c r ease acc ur acy on e mu s t r e l y on a co n s i ste ntl y fin e r m es h o r hi g h e r orde r pol y n o mi al int e rpol a n ts Sp ect r a l m ethods u se f e w e r point s ac r oss the dom a in to r eac h s pectra l acc ur acy a nd they ca n b e impl e m ente d u s in g the FFT a nd h e nce h a v e a s p arse impl e m e ntati o n The c h o ice o f test function s di sting ui s h es between the three earliest t y p es of s p ect r a l sc h e mes: G a l e rkin collocati o n a nd tau ver s i o ns. The G a l er kin meth o d r e quir es a co mbin a ti o n o f t h e ori g in a l b as i s functi o n s into a n ew set in whi c h all the functi o n s sati s f y t h e b o und a r y co nditi o n s A fte r the b o und a r y co nditi o n s a r e sati sfie d t h e G a l e rkin m etho d r e quir es t hat t h e res idu a l b e o r t h ogo n a l to as m a n y o f t h ese n e w b as i s funct i o n s as p oss ible. The tau m etho d i s s imil a r to t h e G a l e rkin meth o d in t h e way t h e diff e r e n t i a l e quati o n i s e nf o rced H o w e v e r n o n e of the t est function s need to sati s f y t h e b o und a r y co ndition s A s uppl e m e nt a r y set o f e quati o n s i s u se d to a ppl y t h e b o und ary condi t i o ns. S e l ect in g t h e ak so that t h e b ounda r y co ndi t i o n s a r e sati sfie d a nd r e quirin g that t h e r es idu a l b e z e r o a t as m a n y s pati a l p o in ts as p oss ibl e i s calle d t h e p se udospectra l ( PS ) m etho d 70

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[52]; the PS method i s a l so called the spectra l collocation method or the nam e m ay b e qualified b y the type of trial function b e in g u se d s u c h as Chebyshev PS (Collocation) Method or Fourier PS (Co llocation ) Method The test function s are translated Dir ac d elta -fun c tion s cente r e d at the so-calle d co llocation point s requirin g that the diff e r e nti a l e qu a tion be satisfied exact l y at the collocation points. 4.1 Pseudospectral Methods A cco rdin g to Fornb e r g [37], when c hoo s in g the k three r eq uir e m e nts need to b e m et: 1. v(x) = akk(x) must co nv e rge rapidly as N in c r eases for r easo n ab l y s mooth functions; 2. g iv e n ak, it s hould b e easy to d ete rmin e bk s u c h that akk(x)) = b kk(x); 3. it should b e f as t to conv ert betwee n coeffic i e nts ak, k = 0 1 ... Nand the s um value v (xk) f or some set of point s Xk, k = 0 1 ... On a p er i od i c domain a F ou ri e r se ri es i s ofte n c ho se n f o r k while on non periodi c dom a in s orthogonal pol y nomi a l s a r e often u sed. On ce the basis set has been c ho se n there a r e only a f ew optimal sets of int e rpolati o n points for each b as is. An int e rpolatin g a pproximation to a function J(x) i s an express ion P N _1 ( x), u s u ally a n ordinary or trigonometric pol y nomi a l with N d eg ree s of fre e dom determined by the r eq uir eme nt that the int erpo lant agree with f(x) at eac h of a set N int e rpolation point s : PN-J (xi) = f(xi) wh e r e i = 1 2 ... N. 71

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On e may fit any N + 1 poin ts by a p o l y n om i a l of Nth degree v i a the L ag r a nge Inter polatin g F o rmula: N P N(x) = L f (xk)Ck(x) k = O where Ck(x) are the ca rdin a l functions: N Ck(x)= IT x-xj Xkx j = O ,jf.k J Cardinal functi o n s h ave the prop e rti es that Ck(Xj) = c5kj a nd that the int e r po latin g points a r e not re quir e d to b e eve nl y space d Consider appl y in g the L agra nge Int erpo latin g P o l ynom i a l to 1 f(x) = (5x)2 + 1 over t h e int e rv a l [-1, 1 ] o n a n eq u ally spaced gr id One would ex p ect that the e rror in P N wou l d go to zero as N ---+ oo. How e v e r as the order of the po l y nomi a l , in c r eases l arge oscillations appear near the e nd p o int s o f the int e rv a l see Figure 4.2. This exampl e i s the well-known "Runge Example. For co mpl eteness and conve ni e n ce we state the Cauchy Int erpo lati o n Error Theor e m (see [5] for exam pl e). The or e m 4. 1 (Cauchy Interpolation Error The or e m ) L et f E CN+l[a, b ] and l et P N(x) b e i t s Lagrangian interpolant of degree N Then ( 4.1) for E [a,b]. 72

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I 0 Lagrange Interpolating Fonn ula 3 5 3 2 5 1.: 2 X 1 5 : :::, \ I I 05 \ 0 __ -/,,.==-= -0 5 1 I -0 5 0 X ---p10( x) L..___:::....._ ___Ji 0 5 ' ' / Fig ur e 4 .2 : Runge' s exa mpl e for int e rpolatin g f(x) (5x )\+1 with even l y s p ace d nod es. A s the numb er of even l y-spaced nodes (deg ree of the in ter po latin g pol ynomia l ) in creases the int erpo latin g polynomial oscillates n ear the boundari es. If we wish to reduce the e rror for the L ag r ange int e rpo lati o n there i s nothin g that can be done about the f actor. H oweve r we ca n c hoo se the g rid point s to reduce -xk) a nd ame liorate the error. W e now state the "Ch e b ys h ev Minima l Amplitud e Theore m (see [ 15] for exam ple) to r evea l a c hoice of gr id points that minimizes the error. The or e m 4. 2 (Che byshev Minima l Amplitude Theor e m ) Of all polynomials of degree N with leading coefficient equal to 1 the unique polynomia l which ha s th e small est maximum value on [ 1 1 ] is TN ( x) j2N -1 the Nth Chebyshev polynomial divided by 2N -1 73

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An y p o l y n o mi a l o f d eg r ee ca n b e f actore d in to t h e product o f lin ear f acto r s o f t h e f orm o f (xxk) w h ere Xk i s o n e o f t h e r oots o f t h e p o l y n o mi al. In p a r t i c ul a r 1 N 2N TN+I(x) =: IJ(x-Xk) k=O I n ord er t o minimiz e t h e er r o r in t h e C a u c h y r e m a ind e r theor e m t h e p o l y no-mi a l part s h o uld b e p ro p o r t i o n a l to T N+I(x). That i s t h e optima l in terpo lati on p o in ts are t h e r oo t s o f t h e C h e b ys h ev p o l y n o mial o f d eg ree (N + 1). See Fi g ur e 4.3 f o r a qu alitativ e exampl e o f the impr oveme n t I ..agrange Interpol a tion or Runge Example a l ("h e bysh ev !\odes 10
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of inter est. T o in c r ease t h e accuracy w e w o uld p e rf or m pr efine m e n t h o w eve r acc ur acy w o uld e v entua ll y r eac h the limits imp ose d b y Runge phe nom e n a as di sc u sse d a b ove. If o n e wer e to diff e r e n t iate a hi g h e r orde r p o l y n om i a l a nd u se Che b ys h ev n o d es, t h e n the resul t w ould b e a mu c h m o r e acc ur ate diff e r e ntiati o n appro ximati o n see Fig ur e 4 4 w h e r e w e d e m o n strate t h e effic i e n cy o f PS method s ove r a coupl e of finite-diff e r e nce meth o ds. In this se n se PS m e thod s m ay b e tho u g h t o f as hi g ho rd er finite-diff e r e n ce meth o d s impl e m e nted a t t h e zeros o f a n appro priate orthogo n a l p o l y n o mi al. Note t hat C h e b ys h ev n o d es a r e n ot the o nl y c h o i ce, t h e r e ar e e v e nl y s p ace d n o d es fro m a F o uri e r se ri es f o r p e riodi c d o m a in L egendre nod es, G a uss-H e rmite n o d es, a nd m a n y oth e rs. JO' ,------------.----r==::;==::::::;] -tid Order FD ...... 4th Order FD -Chebyshev PS 10 10 10 I 10 1 01 102 1 01 N Figure 4.4: L2 erro r co mp a ri so n o f the C h e b ys h ev PS meth o d ver s u s seco nd a nd f ourth orde r finite diff e r e n ce m etho d s a ppli e d to t h e seco nd d e rivative o f c o s (1rx) ove r [-1, 1 ] 7 5

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4.2 Eigen-decomposition Pseudospectral M ethod The PS methods reviewed in the previous section comprise a traditional approach to constructing differentiation matrices. We will refer to a matrix construction derived from these interpo l ation based methods as a Standard Construction. One unfortunate consequence of the Standard Construction is that the resulting matrices become increasingly illconditioned as the s ize of the matrix (number of grid-points) grows [ 37 71]. The increased matrix norm d ictates the size of the time-step for time dependent problems so the Standard Construction has lim i ted uti l ity for these types of prob l ems. There are tech niques for reducing the matrix norm such as spectra l fil tering or one can use domai n decompos i t i ons for i ncreasing the mesh reso lu tion over regions o f i nter est. H owever these techniques either dim i nis h the accuracy of the PS method or increase the computationa l comp l ex i ty. I n this section we w ill derive and demonstrate a new class of PS methods based on the spectral decomposition of the differentia l operator the so-called Eigen decomposition Pseudo-Spectral Method or EPS method [58] that accommodates mesh refinement whi l e maintaini ng an optimally conditioned differentiation matrix and does not require any additiona l numerica l complexity over the Standard Construction. 4.2.1 Derivation To exp l ain the concept behind the EPS method we first write the operator , applied to f over x E [a, b] as an integra l operator [58], f = 1 b K(x, y)f(y)dy, (4.2) 76

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where the kernel K(x,y), is defined 00 K(x, y) = L AmUm(x)vm(y), (4.3) m=l where { um, is the comp l ete set of orthonormal functions of specified by the boundary cond ition s placed on f and A m is dete rmined by the respective sin gular values and the normalization factor associated with the set { um, For this definition to make sense we need to assume that K(x, y) is a conver gent series. However it ca n b e shown that (4.3) is uniformly convergent for a comp l ete, orthonorma l set of eigen function s [39]. Because of the symmetry assumed with our model problem, see Section 3.1, we will refer to the set of functions { Um, as eigenfunctions. Differentiation matrix construction occurs when the kernel K(x, y), is truncate d a nd a quadrature rule is se l ecte d for the integral. We summarize these two steps in the following subsections below. 4 2 1.1 Truncating the Sum Proceeding as in [58], denot e the rank M approximation of the differential operator applied to the function j b M P111{(J)}(x) = 1 L AmUm(x)vm( Y)f(y)dy a m=l M b = L AmUm(x) 1 Vm(Y)f(y)dy m=l a (4.4) wher e is the set of M e i genfunctions. L etting f(y) = vn(Y) for 1 :S n ::; M, we obtain M b Ad PM{ (vn)}(x) = L AmUm(x) 1 Vm(y)vn(y)dy = L AmUm(x)8mn = AnUn(x), m=l a m=l 77

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wh e r e a r e t h e M n o nzero e igen va lu es o f P M H e nce, t h e o p e r a t o r P M i s a r a nk M pro j ect i o n off onto t h e s p a n o f 4.2.1.2 Approximating the Integral with a Quadrature A ga in procee din g as in [58], l e t { a nd { b e a g iven set o f q u adrature n o d es a nd w e i g hts a nd let N c :S N b e a n in teger. W e c ou l d set N c > N h oweve r w e ex pl a in in S ect i o n 4.2.2 t h e disad va n tages o f e x cee din g t h e nu mbe r o f qu adrature p o in ts. Approx imate (4 4) b y N c N P N c { (J )}(x) L AmUm(x) L Vm(Bk)f ( Bk)wk. ( 4.5) m = l k=l B y eva luatin g (4.5) at t h e sam e qu a drature p o ints we obtain N c N Ljk(J)(B j) = L AmUm(B j) L Wkvm(Bk)J ( Bk), ( 4.6) m = l k=l wh e r e 1 :S j :S N. Thus we ca n r e pr ese n t t h e op e rator as t h e N x N r a nk Nc m a trix N c Ljk = L AmUm(Bj)WkVm(Bk) (4. 7 ) m=l Thi s low rank op e r a t o r m ay b e suffic i e n t f o r b a ndlimited functi o n s (or approxim a t e l y b a ndlimit e d functi o n s) [58] h oweve r i t m ay not b e suffic i e nt f o r n o n b a ndlimited fun c ti o ns. That is, the r e m a inin g e igenfuncti o n s { vm(x), um(Y ) };;;=Nc+l, m ay b e need e d to r eso lve t h e hi g h e r fre qu e n cy conten t o f t h e functi o n b e in g diff e r e ntiated W e d e m o nstrate thi s co ncep t b e low, r ef e r to Fi g u re 4.8. Example 1. The seco nd-d e ri vat ive op erato r in Cartes i a n coord in ates w i t h zer o Diri c hlet bound a r y co nditi o n h as t h e e igen-d ecompos i tio n ( 4 .8) 7 8

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We construct the second derivative operator via its kerne l N Kgjr(x y) = LAm sin(; (x + 1)) sin(; (y + 1)). ( 4.9) m=l where According to ( 4 7) we can now define the second-derivat iv e matrices as follows. Definition 4.3 (Dirchlet Boundary Conditions) L e t { B1 w1 }f::1 denot e a set of N quadrature nodes and weights on the interval [ -1, 1]. D e fine the deriva tive matrix = :!:2 of rank Nc :S N with zero Dirichlet boundary conditions as Nc ( L)kl = LAm sin(; (Bk + 1))wl sin(; (Bl + 1)) k l = 1 ... N. m=l Definition 4.4 (Neumann Boundary Conditions) Let {B1,w1}f::1 denote a set of N quadrature nodes and weights on the interval [ -1, 1]. D e fine the deriva tive matrix = :!:2 of rank Nc :S N with zero Neumann boundary conditions as Nc m m ( L)kl = cos( 2(Bk + 1 ))wl cos( 2(81 + 1)) k, l = 1 ... N. m=l 79

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4.2.2 Error Analysis The rank of Ljk i s Nc, indeed if we ap pl y ( 4. 7 ) to a n e igenfunction Vn (n = 1 ... c), eva lu ated on t h e g rid {B1 } (j = 1 ... N), then we ascerta in Nc N L jk(vn)(Bj) = L AmUm(Bj) L WkVm(Bk)vn(Bk) m=l k=l N c = L AmUm(Bj)<5mn (<5mn i s the Kroneck e r delta) m=l (4.10) where ( 4.10) s h ows the r e lati o n s hip for the Nc non-zero e i ge nv a lu e e igenfunction pairs. The norm of ( 4. 7) i s dictated by the m agnitude of A n wh e r e n = 1 ... N c H e nce c regulates the con diti o n numb e r o f ( 4. 7). H owever c a l so imp acts the acc ur acy of the EPS method as d e monstrated through the followin g e rror anal ys is. W e b eg in with som e notation. Let the "quadrature e rr or" be d efine d as, N c Equa d = L AmUm(OEmn, (4.11) m=l wh ere N Emn = <5mn-L WkVm(Bk)vn(Bk), (4.12) k=l a nd E [a, b]. Notice that ( 4.12) indi cates how w e ll the qu a drature rul e m a in tains num e ri cal orthogonality. The m ag nitud e o f this error ( 4.12) ca n b e d ete rmin e d b y c h oos in g N s u c h t h at, kmnl :::; E H owever, as we s h all see, this e rror ca nnot b e r eg ul ated by c hoosin g N a lon e (see Full-rank Comp l etion b elow). 80

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Let the "truncation (tail) e rror be d efine d as, Etail = L (4.13) m=Nc+l wh e r e (4.14) This e rror i s r eg u l ated b y the d ecay rate of f m as m be co m es l a r ge. Fin a lly, let the a pproximation of the op erator, via the EPS method, ap pli e d to a function f b e note d by, Nc N LJ(O = L L WkVm(fh)J ( Bk) (4.15) n=l n=l Theore m 4.5 (EPS Error) I f f E C2([a, b]) with f(a) = f(b) = 0 and { vk(x)}k=l form a comp l ete orthonormal set on [a, b ] such that (4. 3) conver g es, then given quadratur e nodes and weights { and { respectively along with the int e ger N c :S N, ( -L ) J(O = Equad + Etai[, E [a,b ] and Equad and Etail are given by (4.11) and (4.13) respectively Proof: The function J(x) ca n b e expande d as an orthonorma l series of e igenfun ctio n s that i s abso lut e l y a nd uniforml y conv e r gent o n [a, b] [68], f(x) = L fnvn(x), ( 4.16 ) n=l where f n i g iv e n b y (4. 14). 81

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Consider the difference bOO N ( -L)J(O = 1 L Amum(Ovm(Y)f(y)dy-L Amum(O L Wkvm(Bk)f(fh) a m=l m=l k=l jb 00 = L (truncation part) a n=Nc+l + Amum(() ( l Vm (y )f(y )dy WkVm( e,) !( e,)) (quadrature part) ( 4.17) We now isolate the "truncation part" and the "quadrature part' of the sum and finish our analysis on each of these summands separately. Consider the truncation part" first. Recall that f E C2([a b]) making it bounded and m eas urable. The vk(Y) are weakly convergent since, f E C2([a,b]) C L2([a,b]) and { vk(y) }k'?:1 complete implies ft = 1 b f2(x)dx < oo which in turn, imp l ies Hence, the eigenfunctions vk(y), are bounded by a corollary to the Principle of Uniform Boundednes s [6], and therefore measurab l e as well. Moreover if we define M S111(y) = L A mum(Ovm(Y) f(y) (4.18) n=Nc+l then S l\1 (y) is a bounded measurable function which by hypothesis converges l im S111(y) = S(y), for some S(y). Thus, the conditions for the Lebesgue M--> oo 82

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Domin ated Convergence Theorem [57 ] are sati sfied and 1b M b b lim L A mum(Ovm(Y)f(y)dy = 1 lim SM(y)dy = lim 1 SM(y)dy a M---> m=l a A1--->oo Af---> a that i s L 1 b Vm(y)J(y)dy = f = E tail n=Nc+l a n=Nc+l via (4.13). Now we anal yze the "quadrature part" and s how that i t i eq uiv a l ent to (4.11). First notice that by in serting (4.16) into the diff e r ence b N 1 Vm(y)J(y); WkVm(fJk)j(fJk) b oo N oo = 1 Vm(Y) L fnvn(Y)-L WkVm(Bk) L fnvn(Bk) a n=l k=l n = l oo ( b N ) = f n 1 Vm(y)vn(Y); WkVm(Bk)vn(Bk) = t, f n (mn-t. w,vm(B,)v.(B,)) 00 = LfnEmn n=l where the sw itch b etween t h e int egra l a nd s um in t h e seco nd step i s justified b eca u se as pointed o u t in the beginning o f the pro o f the ser i es i s abso lut e l y a nd unif o rml y co nv ergent o n [a, b ] But notice t hat we h ave just s h own that the quadrature part," >-mu..(<) ( l Vm(Y)f(y)dy-t. w,vm(B,)f(B,) ) = = E quad 83

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An i m p o rtant qu es ti o n s ur ro undin g t h e EPS meth o d i s how to se l ect t h e truncati o n p a r a m e t e r N c A s we h ave see n E qu a d i s se n s i t ive to t h e va lu e o f N c b eca u se i t d e t e rmin es how well t h e i n n er-product o f t h e e igenfuncti o n s approx im a tes t h e id entity. If N c i s too l arge, t hi s approx im a ti o n i s p oo r ; comp are the s p a r s i ty pattern in Fi gure 4.5. On t h e oth e r h a nd if N c i s too s m all, t h e n Etail co ntain s m o r e terms Mor eove r t h e lower i n d exe d term s in t h e se ri es are t h e l a r ges t in m ag nitud e b eca u se the coeffic i e n ts d ecay as the ind e x in c r eases. The answe r to this qu est i o n de p e nd s o n t h e op erato r s d eco m pos i t i o n W e w ill d e m o n trate t h e re l a tion s hip b e tween N c a nd N in Secti o n 4.3 w h e n w e con struct t h e radi a l L a pl ac i a n in p o l a r coo rdin ates. 4.2.3 Rank Completion The e igenfunction s l ose num e ri ca l o r t h ogo n ality as Nc a ppr oac hes N a n d t hi s d ev iati o n w ill in c r ease t h e erro r in t h e E PS meth o d. Co n s id e r t h e e igen va lu e e igenfun c ti o n p a ir s in a Cartes i a n geometr y corres pondin g to a seco nd-d e riv a ti ve, diff e r e n t iati o n m a tri x wit h e um a nn b o und a r y co ndi t i o ns: Let t h e in ner product be, N ( f g ) = 2:::.: fn9nWn (4. 1 9) n = l w h ere we u se G a u ss -L ege n d r e wei ghts {wn};;'=1 a nd n o d es X n E [-1, 1], n = 1 2 ... N. Let w = <>mnWn b e the di ago n a l matrix wit h wn's o n the di ago n al. If we compute 84

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Figure 4 5: Sparsity pattern for orthogonality test of eigenfunctions in a Carte s i an geometry with Neumann boundary conditions. The rank of the differenti ation matrix is Nc = N = 256. where v = { :=1 we sho uld obtain an approximation to the id entity matrix I However, notice the sparsity matrix, iuwuTI > E, in Figure 4.5 resulting from this calcu l ation where the error tolerance for the off-diagonal e lements is E = 10-13 When n is l arger than some N N, the off-diagona l values become larger than E and we observe the matrix "fill-in in the bottom-right portion. In other words this deviation from the identity shows the impact in the growth of Emn that is part of the quadrature error, Equad (4.11). In order to reduce this error, we can truncate the kernel at Nc < N, thus reducing the rank (and condition number) of the differentiation matrix see Figure 4.6. otice that this sparsity pattern indeed conforms to a rank Nc matrix and provides a better approximation to t h e id entity. The truncation parameter, Nc, is generally less than then number of quadrature nodes N. In ome ca es the function being differentiated requires that the rank of the constructed operator be higher than 85

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nz = 13 9 Fig ur e 4 .6: Sp a rsit y p attern for ortho g on a lit y tes t o f e igenfun c tion s in a C arte sia n geo m etry with N e um a nn bound a r y co nditi o n s The r a nk o f the diff e r e nti a tion matri x is N c = 139 ( r o u g hl y 0.54N). Nc, say full-r a nk N. If the function b e in g diff e r e nti a t e d i s b a ndlimited the n t h e truncate d co n struc tion m ay b e suffic i ent; n ote t h a t N c limits the b a ndwidth of the e igenfuncti o n r eprese ntati on. How e v e r if t h e fun c tion b e in g diff e r e nti a t e d i s n o t b a ndlimit e d in suffic i e ntl y s mo oth, hi g hl y osc ill atory o r conta in s stee p g r a di ents the n t h e construc ti o n m ay r e quir e t h e trunca t e d e igenfunction s to r eso lv e the hi g h fre que n cy co ntent in the functi o n s r eco n structi o n In t h ese cases, w e need to o r t h ogo n alize t h e r e m a inin g N N c e igenfun c ti o n s { U m D enote these orthog on alized e igenfun ctio n s as { Um, the n the ( r a nk ) c ompl ete d di sc r e tized op e rator becomes t h e m a tri x N c N Ljk = L AmUm(Bj)WkVm(B k ) + L ( 4.20 ) m=l The Gr a m-S chmidt orthogo n alization pro cess m ay b e u se d to o r t hogon alize these e igenfuncti o ns, how eve r w e mu st t a k e ca r e tha t t h e initi a l se t i s w ell8 6

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c onditi o n e d," othe rwi se t h e Gr a m-S chmidt orthogo n a lization i s un st a ble. A stable m ethod for ortho g on a lizin g the r e m a inin g N N c vector s i s the QR alg orithm wh e r e the c olumn s o f the Q m a trix provid e the orthonorm a l se t o f vect o r s { Um, vm};;:=Nc+l W e provid e a n e x a mpl e of the MATLAB c omm a nd s n ee d e d f o r the c ompletion ste p of the seco nd-d e riv ative m a trix with N e um a nn bound a r y c ondition s in t h e co d e blo c k b e low This e x a mpl e d e mon stra t es how w e utilize MATLAB 's QR a l g orithm whi c h e mplo ys Hou se hold e r r eflection s, for c omputin g a full-r a nk diff e r e nti a tion m a trix lik e the on e g iv e n b y ( 4.20 ) % lnFuts: ;,( N quadrature size N by 1 % w N quadrature weights, size 1 by N % Nctruncation parameter, integer less-than-or to N % % Set rndex m = 1 : N ; m = m( : ) ; \ S1Ze N by 1 % N by N diagonal weight matrix W diag (w); % Define eigenfunctions with Neumann f = cos((pi/2)(x+1)m' ) ; % Nrramlrzation constant: e1genvalueorthonormality constant lambda= -pi.2m 2 / 4 ; % Rani< reduced second-derrvatrve matr1x L = (f (:, 1 : Nc) diag(lambda (1 :Nc))) (f (:, l:Nc) 'W); Full Rani< Ccnstruction % Orthcgonalize the remairing vector. % usrng H useholder reflections from QR algor1thm [Q, R] = qr(W(1/2)f); fhi = w (-1/2)Q(:,Nc+1:N); % "e t r are rth n rmal the normalization not needed % Kernel ..... 1effi<'ient is simpl}' the eigenvalJe. L = L + (fhi(-diag(((pi/2)m(Nc+l:N)) 2)))(fhi'W); 8 7

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100 150 200 nz = 256 Figure 4 7: Sparsity pattern for orthogonality test of e igenfunction s in a Carte sian geometry with Ieumann boundary cond iti o n s for t h e co mpl eted operator. The rank o f the diff e r e ntiati o n matri x is now N = 256 (Nc = 139 rou g hl y 0 .54N). For an exam pl e of the effect iv e n ess of this technique we return to the e igen functi ons with Neumann conditions, v = { }:=1 an d co mpl ete the operator, then we see in Fi gure 4. 7 that within the to l erance of 10-1 3 the inn e r-pr od uct (4.19) ac hi eves orthogonality. Now cons id e r Fi g ur e 4.8 (locate d at the e nd of the c h apter) where we h ave plotted the L00 norm r e lativ e e rror resulting from computing the seco nd d e riv a tive o f the fun ctio n s { s in (m1r ( x + 1 ) ) } N 2 m=-N This plot conta in s a co mp a ri so n the EPS method with G a u ss -L egendre qu adra-ture nodes and weights versus the Standard Construction u sing Cheybshev Lobatto nodes. It a l so conta in s a com p arison o f t h e reduced rank (Nc) construction versus a rank-completed construction (wh ere the rank of the operator i s N 88

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in s t ea d o f c -thi s techniqu e i s to b e d esc rib e d b e l o w ) Not ice t hat t h e r a nkco mpl e t e d EPS m ethod m a intain s acc ur acy f o r a wid e r s p ectrum o f fre qu e n c ies t h a n b oth t h e r educed-r a nk constructi o n a nd t h e Stand a rd (C h ebys h ev) con-structi o n The r eas on t h e co mpl e t e d op e rator ac hi eves a hi g h e r acc ur acy i s t hat r a nk co mpleti o n r educes t h e qu adrature e rr o r E qu a d (4.11) b y r educ in g Emn (4.12). 4.3 EPS Construction in Polar Coordinates In this sectio n w e u se t h e EPS m etho d f o r co n structing a d e ri vat iv e matrix for t h e r a di a l p art o f the L a pl ac i a n o p e r ator o n a di sk. Thi s co nstruc ti o n i s t h e o n e u e d t o c ompute the L a pl ac i a n d iff e r e n t i a ti o n matri ces e mpl oye d in the num e ri ca l s oluti o n o f t h e IBVP 3.6 W e f oc u s o n t h e co n struc ti o n o f t h e L a pl ac i a n whil e the g r a di ent f ollo w s a s imil a r co nstructio n The L a pl ac i a n in p o lar coo rdin a t es i s g iven b y a nd w e d efin e t h e r a di a l p a r t as [)2 1 [) /:).r = !:'! 2 + u r r u r ( 4 .21) ( 4.22 ) The op e r a t o r ( 4 .22) appea r in seve r a l diff e r e n t i a l e quation s s u c h as the B esse l e igen va lu e pr o bl e m wh e r e i t i s a s p ecia l case o f t h e S t urm-Li o u v ill e Pro bl e m w h e r e av, n i s t h e nth zer o o f t h e v o rd e r B esse l fun c ti o n o f t h e first-kind ; the p a r ax i a l wave equa ti o n fro m op t i cs ou !:). u+ 2 2 k = O r [) z 89

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or the Poisson equation on a disk .6.ru = f(r, u). 4.3.1 Construction The rank N radial Laplacian operator with zero Diri c hlet boundary condition has the eigen-decomposition (4.23) where am is the mth zero of the zero-order Bessel function of the first kind J0 ( r). In order to construct the kerne l for the radial Laplacian we first note the following orthogonality relation for zero-order Bessel functions : We construct the second derivative operator via its kernel N p) = L A m Jo(a mr) Jo(amp) m=l w h ere 2a2 Am=(4.24) ote that here we have includ e d the normalization constants in the Am values.) A cco rding to (4.7) we can now define the radial Laplacian. D e finition 4.6 L et { B1 wt} l=l denote a set of N quadrature nodes and weights on the interval [0, 1 ] Defin e the d eriv ative matrix .6.r of rank N c :=:; N with zero Diri chle t boundary conditions as Nc (.6.r)kl = L AmJo(amBk)wzJo(amBl)el k, l = 1 ... N. m=l 90

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By adding the rank comp letion step from the S ect ion 4 2 .3, we d e fin e the rank co mpl ete d radia l L a pl ac i a n matrix as follows: Definition 4. 7 L e t { 81 denote a set of N quad rature nodes and weights on th e interva l [ 0 1]. D efine the d eriv ative matrix D.r with zero D iric hl e t boundary con d itions as fork, l = 1. ... N. Th e set {um(Bk)};';:=Nc +1 comes from orthogonali zi ng the last Nc functions in the set { Jo( amBk)} with respect to the inner product (4.19} We ca n est imate a value for Nc that provid es maxima l acc ur acy b y comput-in g the e i genva lu e e rror for the rank comp l ete d construc tion of D.r H e re, we compute (4. 25) wh e r e
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inadequate. In this case it may be necessary to increase N c to as large as value as possible that is, to the dark border a l ong the threshold (light gray to white region of the figure) and construct a rank completed operator. 92

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4.3.2 Numerical Examples In orde r to di sc r et i ze t h e express ion s fro m the pr evio u s sect i o n we u se a set o f L ege ndr e qu a dratur e n o d es { a n d wei g hts { f o r t h e in te rv a l [ 0 1 ] a nd se l ect Nc = 0.6N. In o ur first expe rim e n t we co nstruc t the r a nk co mpl e t e d r a di a l L a pl ac i a n op e rator condi t i o n s acco rdin g to D e fini t i o n 4. 7 u s i ng C h e b ys h ev qu adratur es F o r c omp a rison we construct p se ud os p ect r a l d e ri vat ive m a tri ces as in [ 36 37 70]. W e first t r a n s f orm t h e in te r va l [ 0 1], t o [ 1 1 ] v i a r = 2 r -1 a nd di scar d h a lf t h e p o int s b y u s in g symmetry, a nd se l ect C h e b eys h e v p o l yno mi a l s as t h e b as is. The r es ul t i s a m atrix t h a t op e r ates ove r t h e r a di a l inter va l [ 0 1], w i t h clu ste rin g n ea r the ri ght b o undary. W e r e f e r to this co n structi o n as the s t an dard cons t ruc t i on. A s a lread y m e ntion e d o n e unfortuna t e c on se que n ce of the s t a ndard co nstruc ti o n i s t hat t h e num e ri ca l e igen va lu es o f t h e L a pl ac i a n op e r a tor g r o w as O(N4)[71], instead of t h e op t im a l O(N2), as the s ize o f the m a trix in c r eases (see Fi gure 4 .10 l ocate d a t t h e e nd o f t h e c h apte r). W e a ppl y t h e d e rivativ e matrix f o r the two con structi o n s to the functi o n u(r) = for N in the r a nge b e tween 300 a nd 1000 n o d es. For eac h s i ze, w e m eas ur e the di sc r ete L2 n orm o f t h e r es idu a l f o r P o i s o n s e quati o n t::.ru = f (4. 2 6) 9 3

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wh e r e J(r) = v = 10000. Not e t h a t w e c ompu te the r es idu a l u s in g t h e di sc r ete L2 no rm (4.27) wh e r e 'Ar d e n otes t h e approx imate r a di a l L a pl ac i a n a nd h = i s the n omin a l m es h s ize. The resul t is s hown in Fi g ur e 4.12a ( l ocate d at t h e e nd o f t h e c h a p ter). W e n o t e t hat both erro r c urv es di s pl ay t h e b e h av i o r o f s pectra l con-vergen ce, w h e r e t h e e rr o r w i t h r es p ect to t h e stand ar d co nstructi o n co nv erges slig h t l y f aste r t h a n the error f o r t h e EPS m etho d Co n i de r t h e C h ebys h ev a nd F o uri e r B esse l expa n s i o n coeffic i ents f o r t h e fun c tion u(r) = in Fi gure 4.11 f o r N = 1 2 . 500 n o d es. Not i ce t hat the C h e b ys h ev coeffic i e n ts d ecay f aste r t h a n the F o uri e r B esse l coeffic i e n ts w i t h a c r oss ove r at r o u g hl y N = 1 35. Thi s r es ult i s co n s i s t e nt wit h t h e cla im r ega rdin g F o uri e r-B esse l se ri es m a d e in [40]. H o w eve r t h e s peed o f convergen ce i s prob l e m d e p e nd e n t a nd r es ul ts m ay va ry. The maj o r a d va n tage to t h e EPS met h o d r es id es in i ts a bility to in c r ease r eso luti o n wit h out the p e n a lty o f a n illco ndi t i o n diff e r e n ti-ati o n m a tri x, the norm a n d co ndi t ion num be r o f the op e r ator u s in g the EPS metho d i s cons id e r a bl y s m alle r t h a n f o r t h e stand ard co nstructi on. A s a res ul t w h e n so l v in g t im e -d e p e nd e n t pr o bl e ms, t h e in c r ease in r eso lu t i o n (a nd h e nce co ndi t i o n numb e r ) certainl y in c r eases t h e computati o n a l cost b u t t h e p e n a l ty i s l ess seve r e t h a n in the case o f t h e stand a rd co nstructi o n w h e r e a (si g nifi ca n t l y) s m alle r t im e ste p mu s t a l so be u se d. W e de m o nstr ate these two p o in ts be l o w Man y a pplicati o n s r e quir e t hat we in vert t h e L a pl ac i a n matrix e i t h e r dir ect l y o r it e r ative l y ( in whi c h case we ca n co mpu te a n approx im a t e r es idu a l at eac h 94

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iteration). In order to study the effect of the derivative matrices on inversion we next solve Poisson's equation. Since both constructions are at full rank we can invert them directly see Figure 4.12b (located at the end of the chapter). In this case we note that the EPS method outperforms the standard construction in terms of the error llu-6.;1 Jlloo This can be understood by studying the condition number of these matrices which we plot in Figure 4.12c (located at the end of the chapter). The matrices built with the EPS method enjoy anal ytic eigenvalues whereas the matrices built via the standard construction are ill conditioned. We consider the time step needed to ensure stability for solving the (radial part) of the heat equation Ut = flu on the unit disk. In Figure 4.13 (located at the end of the chapter) we have plotted the maximum time step ensuring stability for solving this equation using the explicit Runge-Kutta 4 so l ver. As expected, the l argest stable time step is (asymptotica lly) orders of magnitude l arger for the EPS method compared to the standard construction. 4.4 Discussion In this chapter we demonstrated a novel construction to PS differentiation matrices the EPS method [58]. PS methods were briefly reviewed while the advantages and disadvantages of the EPS method compared to conventiona l PS methods was discussed. It was noted that conventiona l PS methods, based on the orthogonal expansion of the function to be differentiated (i.e. interpolation based methods), while efficient and spectrally accurate for smooth functions on regular geometries, suffer from poorly conditioned differentiation matrices ; 95

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the e igenvalues eventually grow as O(N4). The EPS method on the other hand permits an increase in the number of gr id points and hence in the size of the differentiation matrix without the limits of a large condition number. The increase in the number of grid points permits differentiation on a high resolution gr id while keeping the condit i on number of the matrix near it s optimal value; in the case o f the EPS method the eigenva lu es grow as O(N2 ) One advantage of the l ower condition numb er for the EPS method versus the standard construction is that it permits a larger time-step for time dependent problems. An error anal ys i s of the EPS method was comp l eted revealing that the accuracy of the method dep e nds on a truncation parameter c < N where N is the number of quadrature nodes The integer c contro l s the cond i tion number of the differentiation matrix but i t a l so limits the rank of t h e operator. This limi t may be adeq uate for bandlimited functions where Nc acts as the bandlimit but for non-bandlimited fun ctions there may be a l oss of accuracy as the error resulting from truncation Etail m ay no t diminish below a suitab l e tolerance. In this case we ortho gonalize the remaining N-Nc eigenvectors using Gram Schmidt or a modified Gram-Schmidt pro cess, and construct a rank comp l eted operator. We demonstrated some of the advantages of a rank comp l eted operator in terms of improved accuracy. 96

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,.. 10' r-_, _, ,.. to!. to' "21 0 .. 1 I 1 0-t t ! ,.. f to. toa -<() _,. Nc1 6 N 0 m/2 (a) Nc64 N 1 28 EPS(CIIIT1piN)l.gendre SlendardChebythev 0 iii m/2 (c) ,.. ,.. f-to', '")tO .. L. i ,.. 10 ... t o a, " _., ,,. to' -...! .. -_,. Ncl2 N S.. -"""( b ) 0 m/2 Sl8ndwd ChebyaMY I ( d ) 0 ""' 50 Figure 4 8 : Co mp a ri so n o f t h e L00 n orm re lative e rr o r resul t in g fro m co m puting the seco nd d e ri vat ive o f t h e functi o n s { s in + 1 ) ) u s in g t h e EPS meth o d wit h G a u ss -L ege ndr e qu adratur e n o d es a nd wei g h ts ver s u s t h e St a nd ard Co n struc tion u s in g C h ey b s h e v L o batt o n o des. Rank -co mpl etio n ver s u s a r educed r a nk con structi o n the EPS meth o d i s a l so co mp a r e d. 97

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-1. ':b :0.. -1-0 -5 -1--10 z -15 -25 3 0 -35 1 50 100 5 0 Nc (Kemol Truncation) Figure 4 .9: Given N u se the EPS meth o d to co nstruct t h e p o lar L a pl ac i a n in c r eas in g c fro m 1 2 ... N a nd co mpu t in g t h e erro r I I I f o r eac h se t o f N c e igen va lu es { ::::1 The co l o r b a r i s logsc a l e d w h ere t h e d a rk r eg i o n s i n di cate s m alle r er r o r a nd t h e lig h t r eg i o n s indi cate l a rger erro r. 98

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X 10' 7 -2 -20 .6r ----St a n d ard Construction ( -Exact ;: ( / J 4 0 60 N 80 100 110 Figure 4.10: Eigenv a lu es o f D.r co n struc t e d v i a C h e b ys h ev colloca tion ver s u s t h e exact e i ge n va l u es, 12 The e i ge nv a lu es o f D.r a r e t h e zero s o f t h e zero-ord e r B esse l functi o n o f the first kind J o(an) = 0 f o r all n = 1 2 3 .... Expansion CoeffiCients 10 Founer Bessel ExpansiOn 10"' 10 .. 10 ' \ Chebyshev Expansion 100 200 300 400 500 600 700 800 900 1000 N Figure 4.11: C h e b ys h ev ex p a n s i o n coeffic i ents ver s u s F o uri e r-B esse l e xp a n s i o n c o effic i e n t f o r t h e G a u ss i a n pul se cente r e d at r = 1/2. 99

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-+---rJ>S S t and.Jrd < 'on!lln taion e -2 :f ... -<1 HI 300 "l(l 500 900 1 000 (a) -+---EPS -+-Stand.ard < 'omtrucbon fi :f -6 --12 -14 300 Oil ""' 6011 7011 8011 1 000 (b) I 2"' 1-+---EPS sWtWr d t 'Dnitruo.;tion 1 0 1 00 200 .\00 400 .SOO 600 iOO 800 900 1000 (c) Figure 4.12: The Poi s son equation examp l e compar in g the completed EPS constru c tion to the standard construction using Chebyshev collocation : (a) The residual e rror II{(b ) The error llu-Li;:-1 flloo, and (c) L condition numb e r. 100

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1\ = 1 .2:\ 10 ... ::::= 1 0 .. I 00 200 300 400 500 600 700 800 900 1000 N Figure 4.13: The mCL'
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5. Numerical Solution of the Partial Integrodifferential Equation V olterra in teg r o diff ere n t i a l e qu a ti o n s ( VIE ) h ave b ee n so lved effect iv e l y over t h e years [17, 18, 22, 3 0 48, 53, 55]. An y sc h e m e u se d to so lve t h ese V o lterr a P artia l In teg r o diff e r e n t i a l E quati o n s ( VPID E) mu st in co rp orate s p a ti a l d e riv a t i ves whi c h produce r o undoff e rr o r s t hat acc umul a t e as t h e integral term i s co mputed ove r the g iv e n t im e int e r val. H e nce, i t i s n ecessa r y to se lect a s pati a l diff ere n t iati o n sc h e m e that h as a suffic i e n t or d e r o f acc ur acy beca u se e rr o r acc umulati o n ove r l o n g in teg rati o n times co uld eve n t u a ll y l ead to p oo r acc ur acy P se udo s p ect r a l method s pro mi se hig h or d ers o f accuracy f o r s m oot h fun ct i o n s o n r eg ul a r d o m a in s [21, 3 7 4 0]. B eca u se t h e volum e fracti o n i s con t inu o u s a nd w e assume at l east CJ(O ) r eg ul a rity a nd t h e soy bean a nd dru g -d eliver y pro bl e m s provid e t r ac t a bl e geo m etries s u c h as di s k s o r spheres t hi s pro bl e m i s id eally s uit e d f o r p se udo s p ect r a l m etho d s Additi o n ally t h e pro bl e m i s solved in L ag r a n g i a n (fixe d ) coo rdin ates Decidin g to u se a p se udosp ect r a l meth o d f o r the sa k e o f acc ur acy lead s to t h e co n s id e rati o n o f hi g ho rd e r te mp o r a l sc h e m es so t h e ga in s in s p atia l acc ur acy a r e not offset b y l osses in temp o r a l accuracy. S eco ndorde r sc h e m es m ay b e a d e qu a t e if o n e ca n ac hi e v e suffic i e n t preci s i on u s in g p se udosp ect r a l meth o d s h o w e v e r we will se l ect a f o ur t hor d e r sc h e m e b e ca u se i t i s o n t h e sa m e ord e r o f co mpl ex ity as t h e sec ondo rd e r sc h e m e impl y in g tha t w e can ac hi e v e hi g h e r acc ur acy f o r n ea rl y t h e sam e computati o n a l cost. 5.1 Numerical Method for Solving the Swelling Equation A geo metri c m o d e l f o r a d r u g -d eliver y d ev ice i s a ri g h t cy lind e r. W e as sume az imu t h a l a nd ang ular symmetry whi c h r educes the pr o bl e m d o m a in to a 102

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rectan g l e ( polar coordinates in the radial direction and Cartesian coo rdinates in the azimutha l direction) n = n u r where n is the interior and r is the outer boundary as indicated in Figure 3.1. The non-dimensionalized initial-boundary va lu e probl e m (IBV P ) is, = (1c1) V [n(c1) V c1 + K.8c1K(c1 ) fat in [ 0 T] X n c1 = on [ 0 T ] X rl fJcl av = 0 on [0, T] X r2 c1(r, z, 0) = c&(r, z) E \:f (r, z) E 0 where 0 T oo and 1. (r, z) E [ 0 1 ] x [0, 1]) 2. 0 < El < 1 3. D(c1 ) 2: 0 is continuous in c1 4 K(c1 ) 2: 0 is differenti ab ly continuous in c1 5. T E [ 0 oo] 6 K.8 2: 0 7. DeE [ 0 oo]. (5.1) The boundary co nditions are mixed: Dirichlet at the exterior and eumann at the center of the cy lind er. 103

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5.1.1 Spatial Discretization The integral term has a cumulative effect on the error so it is beneficial to not only select a time stepping scheme with high-order accuracy but to select a spa tial discretization with high-order accuracy as well. Several options are avai l able includin g high-order finite difference schemes high-order finite element schemes spectral element schemes, and pseudospectral schemes. In this case pseudospectral methods were chosen because it can be shown that pseudospectral methods enjoy superior accuracy to the other methods mentioned when used for func tions that are continuous on smooth domains [15, 21, 37, 40, 42, 58, 70]. When so lu tions are continuous l y differentiable pseudospectral methods converge ex ponentially [66]. The proposed geometry for the models under consideration i s a cy lind er and c:1(r, t) E C0 ([0, T ] ; C2(0)) thus making pseudospectral methods a reasonable choice. The Eigen-decomposition Pseudospectral (EPS) method derived in Section 4.2 will be used to construct t h e spatial derivative approximations. Based on the behavior of the so lu tions from Section 3 4 and the work performed in [24], where sharp fronts where observed we expect the IBVP to be stiff. Given this expectation we will employ the EPS method becau e of its advantages over traditional pseudospectral methods with respect to time-step requirements. The assumption of angu lar symmetry in the cylindrical coordinate system simplifies 104

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the spatial operators as follows V:= or o z (5.2) 02 02 1 0 02 6, = b.r + :::"! 2 = :::"! 2 + "!l + :::"! 2 u z ur r ur u z (5.3) where we are already familiar with the radial Laplacian, b. n from Section 4.3. The boundary conditions are mixed so we have to choose eigenfunctions for the EPS method that ex hibit suitabl e behavior at the exterior and int erior boundary points. The kernel for the EPS representation of the grad ient and Laplacian can be expanded in B esse l functions of the first-kind sine functions, and cosine functions. We list the kernels for the EPS representation of (5.2) and (5.1.1) below Nr K ;1) (r, p) = -2 Jb7;m) Jl(amr)Jo(amp) (5.4) (5.5) Nr 2 J(;2l(r, p) = -2 Jo(amr)Jo(amp) (5.6) N ( 1 ) ( 1 ) cos (m + 2) z cos (m + 2)( (5.7) where Nr is the number of nodes in the radial direction and N z is the number of nodes in the azimutha l direction The kernel expansion (5.6) also satisfi es the Neumann boundary condition since oJo(r) I = -JJ (0) = 0. Or r = O Hen ce, the EPS representation for the radial Laplacian naturally sati sfies the mixed boundary conditions. It s hould be clear that (5.7) satisfies the Neumann 105

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boundary condition s i n ce 8cos(z) I =sin(O) = 0. OZ z=O For the sake of cl arity and compl e tion we d e fin e the op erators separately below. Definition 5.1 (Discrete Gradi ent Ope rator) L e t { 81 wt} f':t d enote a se t of eit h e r r or Nz quadratur e no d es and wei ght s on th e interval [ 0 1]. D e fin e th e components of th e d eriv at ive matrix for (5.1 1 } wi th mixed boundary c ondit i on s as ( Vr)kl = l; -2 Jl()..m Bk)wtJo(>.mBl)el + a;.um( Bk)W(Vm(Bt)el, ( V ,)" -2 ( m + D s i n ( (m + w, cos ( (m + + t ( m + 2 um(Bk)wtvm(Bt), m=Nc,+l fork, l = 1 ... N. Th e set { Um ( Bk), Vm ( Bk)} comes from orthogonalizing th e las t Nr-Ncr fun ctions in th e set and th e set { Um, comes from orthogonali zi ng th e last Nz-Nez fun c t ions in th e se t { s in ( (m + cos ( (m + wit h respect to the inner produ ct ( 4 .19}. Definition 5.2 (Discrete Laplacian Ope rator) L e t { 81 w1} {":t d en ot e a se t of quadratur e nodes and wei ght s on th e int erva l [ 0 1 ] where Nr is th e number of nod es in th e radia l direction and N z is th e number of nodes i n th e a zi muthal 106

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direction. D efine th e components ofthe d eriva tiv e matrix for {5.1.1 ) wit h mixed boundary con d i t i ons as N c r 2 Nr = -2 Jo(>.mBk)wlJo(>.mr)j)el + t, -2 ( m ((m + w1 cos ( (m + + t ( m + 2 vm(Bk)wlvm(Bl), m=Nc,+I fork, l = 1 ... N. Th e set comes from orthogona l izing th e last NrNcr function s and th e set comes from orthogonali zi ng th e last Nz-Nez fu nctions { cos ((m + wit h respect to th e inner produ c t ( 4.19). These matri ces are co nstru cted u s in g the Kroeck e r tensor product. For example let b e the Nz x N z id entity matrix and b e the Nr x Nr id entity matrix then the two-dimens ional cy lindri ca l L a pl ac i a n ca n b e co nstru cted by computing x(z) (b. ) ( a2 ) x(r) Uij T kl + az2 kl Uij l (5. ) see Figure 5 .1. 5.1.2 Time-stepping Methods Time step pin g m etho d for lin ea r a nd n on lin ear p a rtial differ e n t i a l e quation s (PDEs) a r e m a nifold. For e xample, there a r e impli c i t a nd ex pli c it Eul e r meth ods a l t ernatin g -dir ect ion implicit ( ADI ) methods a nd the Cranki c hol son method f or p arabolic prob l e ms a nd the L eapfrog method L axW e ndroff method a nd backward-difference in time method for h ype rboli c probl e ms. Se l ect ion de-p ends up on the p artic ular nu a n ces of the problem: desir e d acc ur acy, stabilit y 107

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nz = 52019 2 Fig ur e 5 .1: The EPS differentiation matrix for the cy lindri ca l L ap l ac i a n (5.8) usin g Nr = N z = 64 nod es. (stiffness) etc In m a n y cases one ca n u se a method-of-lin es (MOL) a ppro ac h and e mplo y a Runge-Kutta (RK) method as a dvo ca t e d in [42]. RK sc h emes can b e m a d e implicit to improv e stabi lit y and can b e exte nd e d to obtain hi g h e r orders ( mor e than seco nd-ord e r ) of accuracy. RK a lgorithms developed to solv e Volterra equat ion s are calle d Volterra Runge-Kutta (VRK) sc h e mes. Just as there are explicit a nd impli c i t RK method s t h ere are explic i t a nd impli c it VRK methods. In particular o n e may u se ex pli cit VRK methods o f P o u zet (PVRK) type or B e l tyukov (E BVRK ) [ 18] type or impli c i t VRK methods of de Hoo g a nd W e i ss ( HVRK) [ 30 ] A more detaile d expos ition with stabilit y theory in clud ed can b e found in [ 1 8 55]. We provid e a brief d e rivati o n of a n ex pli c i t PRK method for conve ni ence a nd compl ete n ess in App e ndix B 108

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We advocate solving the IBVP (3.6) using a MOL approach. In particular we will exp l ore and compare by way of examp le, three t i me-stepping schemes: PVRK4, RK4 and semi-analytic integration (SAl) [53] Our reason for choosing the SAl method is that Singh used this time-stepping algorithm in [65]. The model prob l ems will take on one of two forms Int e grodiffer e ntial Equation ( I DE) dy t dt = J(y) + J o e-r (t-s)y(s)ds (5.9) Volterra Integrodiffer e ntial Equation (VIDE) dy = J(y) + t e-r (t-s)dy ds. dt }0 ds (5.10) We choose these examp l es because they most close l y resemble the VPI DE we are considering and because they offer us a direct comparison to the examples given in [53]. 5 1.2 1 S emi-anal ytic Integration Rule Formulation I n developing SA l one approximates the dependent variable Yn = y(t n ) with a two-point linear interpo l ation formula on [ tn_1 tn] then integrates exactly to obtain the quadrature weights [53], D = e xp( -r6.t) B = (r6.t + exp( -r6.t)-1)/ (r26.t) A = (1-exp( -r6.t)(1 + r6.t)) j(r2 6.t). 109 (5.11) (5.12) (5.13) (5.14)

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Now SAl e mpl oys a f o r wa rd Eul e r m etho d to solve t h e ID E a nd VIDE s as follows, Yn+l = Yn + D..t ( D + BYn-1 + Ayn) (5.15) f o r the IDE a nd (5. 16) f o r t h e VIDE w h e r e int eg rati o n b y p a rts was u se d to conver t t h e V I DE to a n IDE 5.1.2.2 Method-of-Lines Formulation In [53] t h e a u t hor s cl a im t hat a twop o in t in teg r a ti o n f o rmul a i s, "the best p oss ibl e integrati o n rule ... s i nce n o a ddi t i o n a l inf o rm a ti o n ... i s availa bl e in t h e int e r va l [tn_1 tn]" How e v e r i t i s unnecessa r y to r est rict o ur se l ves t o tw o -poin t in teg rati o n b eca u se w e ca n u se Runge -Kutta sc h e m es whi c h in ter p o late b y u s in g a w e i g hted approx imati o n to t h e s l op e o f the tan gent l i ne to t h e c urv e y(t) ove r t h e in terva l [tn-1, tn] In o rd e r to appl y RK sc h e m e we w rite t h e IDE a nd VIDE as syste m s o f e quati o ns. W e b eg in b y wr i t in g t h e in teg r a l as a n ODE in t im e v(t ) = 1 t e-r( t -s) y(s) d s, wh i c h t r a n s l ates to the OD E, dv dt = -rv + y. Now t h e IDE a nd V IDE b eco m e t h e firstor d e r lin ea r system s dy dt = J (y) + v d v = -rv+y dt 110 (5. 17 ) (5.18) (5. 1 9) (5. 20 )

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a nd resp ect ivel y = f(y) + y(t)-e-rtYo-rv d v = -rv+y d t 5.1.2.3 Pouze t Volte rr a Runge-Kutta Formul ation (5.21) (5. 2 2 ) PVRK meth o d s a r e u se d to sol v e in teg r a l e qu a ti o n s so we f o rmu late t h e V I D E as a VIE (t) = J (y) + 1 t e-r( t s)(s) d s (5.23) t h e n so lve t h e r es ultin g lin ear, a u to nom o u s ODE, dy dt = (t) (5. 24) u s in g a n RK solver. I n so m e instan ces i t i s conv e ni e nt to u se t h e sam e RK p a r a m ete r s f o r b oth the PVRK p art a nd t h e RK p ar t W e e mpl oy t hi s m et h od o l ogy in o ur exa mpl es b y u s in g t h e conven t i o n a l RK 4 p a r a m ete r s App e ndi x B c ont a in s a d e riv a ti o n of the PVRK m etho d see [17, 55] f o r m o r e r ef e r e n ces a nd variati o n s 5.1.2.4 E xamples The purpose o f t h e f ollowin g exa mpl es i s to s how t h e s up er i o rit y o f t h e MOL approac h u s in g a n RK 4 t im e steppe r over the SA l meth o d W e tak e three exa mples fro m [53] a nd co mp a r e the co n ve nti o n a l RK 4 solver MATLAB 's OD E45 solver t h e PVRK4 solver a n d t h e SAl so lver. R es ul ts co mp a rin g t h e l o g II e n ( t) I I 00 n orm a r e prese nted in the t a bl es at t h e e nd o f t h e c h apter. 111

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Example 2. Vve start with a s imple, lin ear probl e m as a baseline for compa rison then add m ore compl exity to o ur examples Con s id er the lin ea r IDE dy t dt = )..y + f.L J o e-r(t-s)y(s)ds, y(O) = 1 (5.25) taken from [ 53], where).. = 0.5 1 J.L = -5, r = 10, a nd t E [ 0 10]. This exam pl e has a n exact sol ution y(t) = 0 0477e-10 5t + 0.9523e-0 010t. In this exa mpl e we co mp are two RK4 so lv e r s versus SA l with fl. t = [ 0 .01, 0 04, 0.07 0.1, 0.2 0.3]. It should be note d for this exam pl e that the lin ear stabilit y r eg ion of co nv e ntion a l RK 4 promises stabilit y f or roughly fl.t < 0 .27 and num e ri ca l exper im ents bear this out, see Tabl e 5.1. An a dv antage of the SAl method i s that it r e m a in s stabl e f o r time ste p s mu c h lar ger t h a n the conve ntion a l RK 4 limit However if stability i s a n i ss u e then o n e can u se MATLAB 's ODE45 function which e mplo ys the D orma nd-Prince p a ir [32]. T a bl e 5.1 r evea l s that for t h e sam e time-steps stabi lity i s m a int a in e d w i t h s up e rior accuracy for the MOL approac h. Example 3. ow co n s i d e r the nonl in ea r VIDE dy t dt = -y2-J o e-r( t -s) y(s)ds y(O) = 1 (5.26) wh e r e r = 2 a nd t E [ 0 10]. This exa mpl e h as a n exact so lution y(t) = e-t. The r es ult s in T ab l e 5.2 s how that the MOL approac h i s orders o f magnitude m o r e accurate than the SA l method. Example 4. Finally co n s id er the non-lin ear VIDE with a temporal d e rivativ e in the int eg r a nd (simila r to the VPIDE b e in g solv e d in S ect ion 5.2 ) = -y2 + fat e-r(t-s) ds y(O) = 1 (5.27) 112

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where r = 2 which has exact so lu t i on y(t) = e-t. To so lve this VIDE u s in g RK4 a nd SAl requires in tegratio n by parts to e liminate t h e t im e derivative in the int egrand dy t dt = -y2 + y-e-rty(O)-r J o e-r ( t -s) y(s)ds, y(O) = 1 (5.28) The RK4 method then so l ves t h e VIDE as t h e system (5.21) with J(y) = -y2 where SAl numerically computes t h e int egra l using the quadrature weights (5.12). ote that the PVRK4 method does not require such an int egrat i on a nd t hi s family of solv ers can prove to be a compe llin g c h o i ce in t h e case of k erne l s that are intractabl e or non-differentiable. VRK methods are typically computationa ll y expe n s iv e O(N3 ) but Prete et. al. [22, 55] formulated an O( N log N) a l gorithm u s in g a numerical sc h eme f or the inv erse Lapl ace transform Example 3 a l so reveals that t h e M OL approac h with an RK4 so lv er ex hibit s compet itiv e performance for smoot h kernels see Table 5.3. On e can ex p e rim ent with diff erent RK p arameters for the PVRK4 method to improv e acc ur acy but i t would still be comparab l e to that o f the MOL approac h and would still be more computationally expensive; recall that even if one uses a fast VRK so l ver as in [22], it still r e quir es an extra RK step (5.24). These examp l es s ho w that the MOL approach with a high-order int eg r ator s u c h as RK4 is a reasonable c h o ice for t im e-stepping. It shou ld be noted that the comparisons made h ere are to demonstrate the impr oveme nt in accuracy. W e c ho se explic i t so lv ers with constant t im e steps so that we cou ld make compar i sons under simi l a r cond i t i ons. The IBVP (5.1) lik e many diffusion problems ca n be stiff for certain initi a l con dition s coeffic i ents and parameters making an 113

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impli c i t so lv e r a judic iou s c h o ice. In S ect i o n 5.2 we will s how a so luti o n r es ul t in g from t h e u se of a conv e ntional RK4 solver but we will a lso s how solution s re s ultin g from u s in g a n impli c it, 5th order, variable time-step int egrator [59 ] The purpose of the d e mon stration i s still r e l evant b eca u se it s how s that the MOL approac h with a hi g h-ord er sol ver i s s up er i or in accuracy to two-point meth o d s s u c h as t h e SA l method. The only quest ion i s whether a n impli cit so l ver i s mor e effic i ent than a VRK m ethod. But g iv e n the stiffn ess of the prob l e m we would h ave to resort to a n impli c i t VRK method as well a nd we still would need to solv e a r es idu a l e quation for 1 thus a ddin g to the computationa l c ompl exity. H e nce, the MOL a ppro ac h with the implicit 5th orde r variable time-stepper r e m a in s t h e method of c hoice for so lvin g t h e IBVP (5.1 ) t empo r ally 5.2 Nume rical Solution o f the IBVP In this sect ion we so lv e the IBVP (5.1) u s in g the MOL a ppro ac h with t h e EPS method u se d to dis c retize the spatial op erators. The int egra nd of (5. 1 ) l e nd s itse lf to int eg ration b y parts y i e lding, l t 1 t ( t-s) 1 l E eveu0 -e-oeE ds. 0 D e (5.29) Note t hat in Singh [63] a nd W e in ste in [73] it i s assumed that V c1(r, 0 ) = 0 a nd we will invok e this assumption h e re. Thi s maneuver h as the adde d b e n efit o f e liminatin g the time d e rivative in the int eg r a nd and thus y i e ldin g t h e firstorder lin ea r syste m : = (1c1) V (D(c1) V c1 ) + K:s(1-c1) V c1K(c1) V ( c1 v ) (5.30) 1 l v= D e ( E -v), (5.31) 114

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where 1t 1 (ts ) l v (t) = e-l'iec; ds. 0 D e 5.2.1 Two-dimensional Example (5.32) We begin by so lvin g the problem in two dimensions on the cy linder 's cross section the rectangle n. In this case we insert a Kozeny-Carman permeability function [ 7], (5.33) lin ear diffusion coefficient, D(c:l) = c;l and model parameters p, 0.01 and T = 1. The grid size is Nr X N z = 64 X 64 with N c r = Nez = 42(0.65Nr) and a conventional (explicit) RK4 time-stepper was used with constant time-step tlt = 10-6 The initi a l condition is a cosine bell 1 1 El(r, z, 0) = 2(1 +cos( 1r(r1))) )" (1 2 (1 +cos( 1r(z1))) )" (5.34) with CJ = 100 see Figure 5 .2. Figure 5.3 shows the so lution at three simulation times t = {0 0.2 0.4}. otice that as time progress the liquid volume fraction increases as expected. 5.2.2 One-dimens ional Examples In order to s implif y the compari son between conventiona l pseudospectral methods and the EPS method, we consider a one dimensional case. Her e we cari make the assumption that the cy lind er has infinite extent, effectively e liminating the azimuthal variable. We will consider two sets of examples. First we revisit the FloryHuggins model summarized and so l ved in Section 3.4 and compare the analytic so lution to those obtained by the EPS method as well as a conventiona l 115

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K o z e n y C arman P ermeabilit y D ( t1 )""'E1 0 0 Fig ur e 5 .2 : Ini t i a l liquid volum e frac ti o n c:1(r, z, 0 ) w i t h = 0 1 a nd = 0.9. p se udosp ect r a l solv er. S econ d we will solve t h e IBVP (5.1 ) incorpo r ating t h ree d iffe r ent d iffus i o n c o effic i e nts, D ( c:1 ) I n eac h case the t im e steppe r u se d was MATLAB s od e 1 5s a n i m p l i c i t 5tho rd e r so lver. 5.2.2. 1 Flory-Huggins Mode l Co n s id er t h e Fl o ry-Hu gg in s m o d e l in r a di a l coo rdin ates w i t h t h e ini t i a l b o u n d a r y conditi o n s g uid e d b y (5.1 ), 1 = (1c:1) V ( (1c:1) Vc:1 ) c:1( 1 t) = 1 = o OT r = O 116

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where we assume that t h e mod e l h as a lr eady been non-dim ens i ona liz ed. The analyti c so lution to (5.35), re-written h e r e f o r conven i ence i s (5.35) The PDE i s transformed by lettin g u = (1c:1 ) 2 then the IBVP becomes u = u fl. u u(1 ,t)=O 8u I 0 EJr r=Ou(r,0)= 1-r2 (5.36) This IBVP (5.36) is so lv ed u s in g two competing spati a l discretizations for fl., Chebyshev collocatio n w ith the co nstruction advocated by Fornberg and Tre feth e n [36, 70] versus the EPS construction (5.2) with ::2 neglected. Figure 5 4 i s a plot of the so lution s to (5.36) over t E [0, 1 ] u s in g the EPS constructi on and transformed back to liquid volume fraction space c:1 = 1 .jU. Notice that these c urv es com p are to those shown in Fi gure 3.4. The IBVP (5.36) was so l ved aga in u s in g Chebyshev co llo cation. In both cases Cheybshev collocat ion and the EPS method, the num er ical so lution s denoted and lkps respectively, were compa r ed to the analyti c so lution (5. 35) using the L00 norm at eac h time step tk, (l1 s i gnifies the numerical solution ) Figure 5.5 s h ows a compar i son of this r e lativ e error u s in g N = 64 nodes. The relative error as opposed to absolute error was u sed to reveal how well t h e numerical solution m a int ains pace with the anal ytic solution through a diffusive process. It i s clear in this examp le, that the EPS method maintains better accuracy over the time int e rv al. 117

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5.2.2.2 Diffu sion Coefficient Comparison Figure 5 6 s hows three sets of so lu tions c:1(r, t), in the radi a l dir ect ion In thi s case we once again insert a Kozeny-Carman permeability c:1K(c:1 ) = w ith mode l para meters J.L = 0 1 and T = 1. The initial condition i s a G a u ss i a n curve s hift ed to r = 1 to mode l the initi a l liquid t hat h as penetrated t h e d elivery d ev i ce. In Figure 5.6 we compare so lu t i ons resu l t in g from diff e rent diffusion coefficient functions, D(c:1 ) = {1,c:1 (c:1 ) 2 } and see t hat the l a r ger the exponent for c:1 t h e s l ower the liquid p e netration into the delivery d ev i ce. The run time for eac h case span s t E [0, 0.4 ] but c:1 manages to reach diff e r ent l eve l s at the c ompletion of the s imulation This behavior i s ex p ecte d because 0 < c:1 < 1 and s in ce D(c:1 ) co ntrol s the liquid p enetration into the d ev i ce, in creas in g the value o f t h e ex ponent d ecreases t h e magnitu de o f D(c:1 ) and h en ce retards the liquid volume fracti o n s progress. Addi t ionally, the nature o f the curves is slightly diff er ent in eac h case Apparently in creas in g the va lu e of the exponent a l so increases the steepness of the curves An interpretati on for this b e h av ior i s t hat t h e l arger exponent a l so magnifies the cont rast between values of c:1 thus present s h arper front s as see n in Figure 5.6(c). Notice that t h e behavi or of the so lutio n s in Figure 5.6 i s com parabl e to the c h a racteristi cs revea l e d in Figures 3.3 and 3.4 So the answers w e see f or c:1 match our phys i ca l intuition as well as satisf y t h e expectations put forth b y our anal ys is. 118

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5.3 Discussion In t hi s c h a pter w e d escr ib e d the num e rical a ppro ac h em pl oye d f o r so l v in g t h e IBVP (5.1). The EPS co nstru c ti o n f o r t h e cy lindri ca l L a pl ac i a n a nd g r adie n t w i t h mix e d Diri c hlet a nd eumann, boundary condi t i o n s w as d efine d. It w as a lso d e m o n stra t e d t hat t h e MOL a ppro ac h with a hi g h-ord e r time st eppe r s u c h as a var i a bl e time ste p impli c it 5th o rd e r ODE int eg rator ( f o r exam pl e MATLAB s o d e 1 5s [59]) i s a goo d c h o ice in term s o f b ot h acc ur acy a nd stability. The M OL a ppro ac h com bin e d with t h e E PS co nstructi o n was u se d to so lve (5.1 ) u s in g so m e se l ect diffu s i o n coeffic i e n ts, p ermeabilit y functi o ns, a nd m o d e l p a r a m ete rs. Plots and a br i e f a n a l ys i s o f t h ese so luti o n s were offe r e d 119

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M ethod tl t 2 t3 t 4 t::.t = 0.01 RK4 0.0011 0 0022 0.0032 0.0041 SAl 0.0013 0 0002 0 0009 0 0019 ODE4 5 0.0011 0 0022 0 .003 2 0 .004 1 t::.t = 0 04 RK4 0 0011 0.0022 0 0032 0.0041 SAl 0 0089 0 0076 0.0063 0.0052 ODE45 0.0011 0.0022 0 0032 0 0041 t::.t = 0 07 RK4 0 0011 0.0021 0 0032 0 0041 SAl 0 0171 0 .015 7 0 0142 0.0129 ODE4 5 0.0011 0.0021 0 0032 0 .004 1 t::.t = 0.1 RK4 0.0011 0.0022 0 0032 0 0041 SAl 0.0261 0 0244 0 022 8 0.0212 ODE4 5 0.0011 0 0022 0 0032 0 0041 t::.t = 0.2 RK4 0 0011 0 0022 0 0032 0 0041 SAl 0 0603 0.0 577 0.0552 0.0 5 30 ODE4 5 0 0011 0 0022 0.0032 0.0041 t::.t = 0.3 RK4 a NaN a NaN SAl 0.0998 0 0 966 0.0930 0 0 899 ODE45 0.0011 0 0021 0 0032 0 0041 Table 5.1: V a lu es o f t h e log llen(t)ll erro r at times t1 = 0.25tf, t2 = 0.50 t f t 3 = 0 75t1 a nd t 4 = t f comparin g RK4 a nd ODE45 ver s u s SAl f o r Exampl e 1. 120

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Method tl t2 t 3 t4 tlt = 0.01 RK4 8.1085e-012 1.0541e-011 1.7017e-012 2.0851e-013 SAl O.OOll 0.0003 6.14 79e-005 9 .19lle-006 ODE45 1.9 69e-006 7 .84 25e-006 2 7505e-006 4.3136e-007 tlt = 0.04 RK4 1 4940e-009 2. 7 461e-009 4.3919e-010 5.4810e-Oll SA l 0 0046 0.0013 0.0002 3.4459e-005 ODE45 2.5902e-006 7.8425 e -006 2 7169e-006 4.3136e-007 tlt = 0 07 RK4 6. 7778 e -009 2.6672 e -008 4.3212e-009 5.5585e-010 SAl 0.0083 0.0023 0.0004 5.9139e -005 ODE45 3.6379e-006 7. 9623e-006 2.76 5e-006 4.5477e-007 tlt = 0 1 RK4 3.3590 e -008 1.1 090e-007 1.8279e-008 2.2555 e -009 SAl O.Oll7 0.0032 0 0005 7.4569e-005 ODE45 1.9 69e 006 7 .8 425e-006 2 7505 e -00 6 4.3136e-007 tlt = 0.2 RK4 1. 5420e -006 1 .865 0 e -006 2.9128 e -007 3.9252e-008 SA l 0.0248 0 0064 0.0009 0 0001 ODE45 5.6325e -006 7.8425e-006 2.6ll5e -006 4 3136 e -007 tlt = 0.3 RK4 1.5958 e -005 1 0935 e -005 1. 7129e-006 2.3455e-007 SAl 0.0383 O OllO 0.0014 0 0001 ODE45 5.6325e-006 1. 0 198e 005 2. 7269e-006 4 6755e-007 Tabl e 5.2: Values of the lo g llen(t)ll e rror at times t1 = 0 .25tf, t2 = 0.50tf t3 = 0.75tf and t4 = tf comparing RK4 and ODE45 versus SAl for Example 2. 121

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M ethod tl t 2 t 3 t 4 tlt = 0 .01 RK4 8.10 85e -012 1.0541 e -011 1.7017 e -012 2.0851 e 013 SA l 0.0011 0.0003 6 1479 e -005 9 1911 e -006 PVRK4 2.6035 e -010 2.972 8e -010 3.0159 e -010 3.0205 e -010 tlt = 0 04 RK4 1.494 e -009 2 746 e -009 4 3919 e -010 5 481 e -011 SA l 0 0046 0 0013 0 0002 3 4459 e -00 5 PVRK4 6 4202 e -008 7.3 5 48 e -00 8 7.4622 e -00 8 7.4734 e -008 tlt = 0.07 RK4 6. 777 8e009 2 6672 e -00 8 4 3212 e -009 5.5585 e -010 SA l 0 00 8 3 0.0023 0.0004 5.9139 e -005 PVRK4 5 7918e -007 6 6634 e -007 6 7639 e -007 6. 77 43 e -007 !lt = 0.1 RK4 3. 3 59e -008 1.109 e -007 1 8279 e -00 8 2 2555 e -009 SAl 0.0116 0 0032 0 0006 7.4569 e -00 5 PVRK4 2 3462 e -006 2 6832 e -006 2 7228 e -006 2 727 e -006 tlt = 0 2 RK4 1.5 42 e -006 1.86 5e -006 2 912 8e -007 3.9252 e -00 8 SAl 0 .0248 0 0064 0.0009 0.0001 PVRK4 3.2964 e -005 3 8 313 e -005 3.8894 e -00 5 3.8949 e -00 5 tlt = 0.3 RK4 1.5958e -005 1 0 9 35e -00 5 1.7129 e -006 2.345 5e -007 SAl 0 .038 3 0.0110 0.0014 0.0001 PVRK4 0.0001 0.0002 0.0002 0 0002 Tabl e 5.3: V a lues of the l og llen(t)lloo erro r a t times t 1 = 0 .25t f t 2 = 0.50tf, t 3 = 0. 7 5 t 1 and t4 = t f compa rin g RK4 v e r s u s PVRK4 v e r s u s SAl for Exa mpl e 2. 122

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Kozcny-cannan J>cnncability, D(1 ) -1 K o zcny-Cannan Pcnncability. D(1 ) e 1 (a) ( b ) Kozcny-Cannan Penncab1lity, D(1 ) 1 (c) Figure 5.3: Liquid volum e fract i o n c1(r, z, t) pl ots over t h e cy lindri ca l c r oss sec ti o n D w i t h a Kozen y -Carm a n p ermeabili ty K ( c1 ) = lin ea r diffu s i o n c o efficient, D ( c1 ) = c1 a nd mod e l p a r a m e t e r s J.L = 0.01 a nd T = 1. The g rid s ize i s Nr X N z = 6 4 X 64 wit h N cr= Nez= 4 2 (0.65Nr) a nd a co nven t i o n a l (expli c i t) RK4 time steppe r w as u se d wit h co n s t a nt t im e ste p t:l t = 10-6 S o lu t i o n s a r e s ho w n at (a) t = 0 ( b ) t = 0 2 a nd (c) t = 0.4. 123

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09 0.8 0 7 06 -.., Figure 5.4: Liquid volume fract ion c:1 f ound by so lvin g (5.36 ) u s in g an MOL a ppro ac h with the EPS discr e tization a nd MATLAB 's ode15s for t im e ste pping. JO' ' flol") -l-luggins Model 64 ,. -------... --Chcbysh c' Collocauoo EPS 10. '-------'-----'-___c __._ __._ _._ _,_ _,_ __l_ __J 0 10 Figure 5.5: R e l ative L e rror comparin g Chebyshev collocati o n to the EPS construction over t E [ 0 10] for the Flor y-Huggin s mod e l (5.36). 124

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.: -;; 0.2 OJ 0 4 (a) .: -;; Ol ;: -;; Oj (c) Kouny-carm a n Prrmnblt hy. l)t':1)ae1 ( b ) Fig ure 5 .6: Liquid vo lum e fracti o n c:1(r, t), plots over t h e r a di a l g rid w i t h a Ko ze n y-Carm a n p ermeability, K(c:1 ) = and m o d e l parameter s J.t = 0 1 and T = 1. The g rid size i s Nr = 7 5 0 with Ncr = 4 5 0 ( 0.60 Nr) and a v a ri a bl e ste p size, 5th order impli c i t t im e s t epper was u se d. S olutio n s are s h o wn f o r t E [ 0 0.4] wi t h (a) D(c:1 ) = 1 ( b ) D(c:1 ) = c:1 and (c) D(c:1 ) = (c:1 ) 2 125

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6. Model Sensitivity Analysis In this chapter we study the impact of using different functional forms for the diffu s ion coefficient, D( c1), the permeability tensor, K1(c1), as well as testing the mod e l 's sensitivity to the coe fficient of the integral term, K,8 and the Deborah numb er, De. The governing IBVP i s given by [ t i!.=.Q] c1 = (1c1)\7 D(c1)\7c1 + K,8c1K(c1 ) J o e-D e \7 c1dt' in [ 0 T ] X n c1 = on [ 0 T ] X rl act av = 0 on [ O T ] X r2 c1(r,z,0) E V (r,z) E f2. (6.1) R ecall that the model is a combination of a non-linear diffusion equation and a con titutive equation modeling viscoelasticity (integral term). The behavior of c1 varies great l y depending on our c hoi ces for these functions and parameters. For exa mple as we will show in the sections below the magnitudes of K,8 and D e impact the rate at which the fluid penetrates the polymer matrix. Additionally the sign of K,8 dictates whether the material inhibits sorption (K,s < 0) or increases sorption (K,s > 0). The IBVP (6.1) was solved using the method described in S ect ion 5.1 where we emp l oy a method-of-lines (MOL) approach using an impli cit 5th order, vari ab l e time-stepper (MATLAB ode15s [59]) and an Eigen-decomposition Pseu dosp ect ral (EPS) co nstruction for the spatial derivati ves. The spatial grid co n126

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tained = 750 nodes the truncation parameter was set to 0.60 = 450 and we ran the sim ulation s fortE [0, 1] (except when stated otherwise). 6.1 Diffusion and Permeability Models We begin our stud y by cons id er in g some functional forms for D(c:1 ) and K1(c:1). Weinstein [73] arrives at D(c:1 ) = K1(c:1 ) and S i ngh c hoo ses a linear form, D(c:1 ) = c:1 In [49], Low recommends the swe llin g pressure (for clay) to be, (6.2) If this form of the swelling pressure is substituted into (2.52), then we obtain In summary, [ 1 el] D(c:l) = Kl(c:l)exp -T (c:l)2 Dw(c:t) = ( c:lf Kt(c:l) 1 c:l Ds(c:l) = E l D ( l) = Kl( L E E (c:l)2 where the subscr ipt s represent the last names of the founders. (6.3) (6.4) (6.5) (6.6) I otice the appearance of the permeability K1(c1), in two of the coeffic i ents. Determining a suitab l e functional form for the permeability depends on the application area ( h ydrology, pharmaceutical biomaterials etc.) and i s a topic for further researc h In the exa mpl es exam in ed in this chapter we will cons id er two permeability models: (1) constant permeability model and (2) Kozeny-Carman permeability mode l [7], (6.7) 127

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Fur t h ermo r e, we w ill s i m plif y t h e diffu s i o n coeffic i e n t, d eri ve d by W e instein as D w(c1 ) = (c1 ) 2 to m ake i t m ore t ractabl e f or nume ri ca l ex p erime nts. W e d ro p the subscripts f ro m t hi s p o in t f orwar d a n d s impl y w rite o u t t h e functi o n a l f orm o f t h e diffus i o n coeffic i e n t. Fi gure 6.1 co ntain s a plot o f (6. 6 ) w i t h K1 g i ve n b y (6. 7). In L o w 's m o d e l wh en c1 1 liq uid diffu s i o n i s l a r ge b eca u se t h e materi a l swe ll s to tak e in m o i s -ture; swe lli ng p ressure i s l a r ge pus hin g liq uid into t h e materi al. This swe llin g r eg im e i s f o ll owe d b y a r eg i o n w h e r e D(c1 ) tap ers b eca u se t h e materi a l i s sat-uratin g and t h e s we llin g pressure subs id es; noti ce the ex p o n e n t i a l fun ction a pproac h es 1 as c1 tends t o 1. Aft e r this t a p e r liquid diffu s i o n in c r eases again as c1 approac h es 1 b eca u se t h e materi a l i s n ea rl y saturated and t h e liquid can flow eas il y The first example e ntail s revi s i t in g t h e diffu s i o n coeffic i e n t com p a ri so n co mpl e t e d in Sect i o n 5.2.2. Figure 6.2 co ntain s t h e pl o t s o f c1 f o r a va riety o f diffu s i o n c o effic i en ts; we u se the K oze n y -Carma n m o d e l (6.7 ) f o r t h e p ermeability. The case u s in g t h e Low m o d e l s h ows muc h faster liquid p e netrati o n t h a n t h e oth er three cases. The va lues f o r D ( l) = lexp [1f f 1 l -[ r e m a in a b ove 1 whil e the set o f D(c1 ) = {1, c1 (c1 ) 2 } all remai n l ess t h a n 1 t hu s y i e ldin g t h e ex p ecte d r es ults in Figure 6.2. Recall the d e fini t i o n o f m o isture co nten t (2.67) w h ic h we restate h e r e f or conven i e n ce (6.8) 128

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D 20 10 ,. 00 I -02 0 6 Figure 6.1: Low's functi o n a l form of the swellin g pr ess ur e combined with the Koz e n y -C ar m a n p e rmeabilit y fun ct ion D (cl) = E l The moisture con t ent, as a function of time, indi cates the flow rate a nd is u seful in d ete rminin g if a p a ram ete r diffu s ion model or p e rmeabilit y model in crease or inhibit flow. The viscoelastic str ess i s d efine d as, (6.9) The fun c tion V(r, t) i s esse nti ally the int egra l term of (6.1 ) and as s u c h yie ld s the s pati a l (and temporal) stru cture o f the stress indu ce d as the liquid p e n et r ates t h e polym er matrix. Fi g ures 6.3(a) a nd 6.3(b) s how a comp ar ison o f the viscoe l ast i c stress (6.9) resulting from settin g D (c1 ) = c1 De= 1 a nd Ks = 1 in (6.1 ) w hil e c h a n g in g the p ermea bilit y models. Bo t h mod e l s constant p ermea bility a nd Koz e n y -C a rm a n p ermeability, y i e ld similar results in t e rms of the viscoelastic str ess. The Kozen y-Carman mod e l produ ces more stress as ca n be see n by the f act that it dip s l ower below -0.7 n ea r r = 1 a nd b e low 0.4 n ea r r = 0. The (slig htly) mor e dramati c differ ence a pp ears in Fi g ur e 6 3(c) wh e r e the Koz e n y -C arman model pr od u ces a hi g h er moisture content over the constan t perm ea bilit y mod e l esp ecially later in the s imulation The n o nlin earity of the 129

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/'\,8 < 0 1'\,s > 0 1/'\,sl 1 in c r ease flo w inhibit flow 1/'\,81 1 inhibit flow in c r ease flo w Tabl e 6.1: Characte rization o f /'\,8 Kozen y-Carm a n m o d e l m a k es i t diffi cult to interpret this r es ult, h oweve r if we co n s id e r a plot o f t h e K oze n y-Carma n m o d e l as a fun c ti o n o f c:1 see Fi gure 6.4, t h e n we see t h e r a pid in c r ease in p ermea bilit y as c:1 approac h es 1 (sat urati o n ) wh e r eas t h e oth e r p ermeability m o d e l r e m a in s constant 6.2 Parameter S ensitivity In this sec ti o n w e p e rform a study of the m o d e l 's ( 6 1 ) se n s i t ivit y t o t h e non dim e n s i o n a l param e t e r s /'\,8 a nd D e d e riv e d in S ec tion s 2.3 a nd 2 6 r es pectiv e ly. W e se t D (c:1 ) = c:1 a nd u se (6 7 ) for K (c:1 ) for all expe rim e nts. R eca ll tha t a nd m o dulu s of e l astic it y /'\, = ----------'--8 swe llin g pressure coeffic i e n t D e = r e laxati o n t im e o b se rvati o n t im e W e summa ri ze t h e impact o f t h ese n o n-dim e n s i o n a l p a r a m ete r s fro m S ect i o n 2 6 in T a bl e 6.1 a nd T a bl e 6.2 The num e ri ca l ex p e rim e nts b e low provid e e vid e n ce in support of t h ese int erpre t a tion s thro u g h plots of the m o isture c ont e n t (6.8) a nd v i scoe l as ti c stress (6.9 ) 130

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D e 1 m ate ri a l i s viscous D e 1 material i s e lasti c De'"'"' 0 ( 1 ) material is viscoelastic T a bl e 6 2 : Characterization o f the D ebora h numb e r De. 6 2.1 Moi sture Content C urves In this sect ion w e co mp a r e the normaliz e d moistur e conte nt M(t)/Moo wh ere Misgiven b y (6.8) a nd Moo= max IM(t)l. Thi s qu antity i s u se d quit e t exte n s iv e l y in the pha rm ace uti ca l literatur e see the r ev i ews in [60 76] for in-stance. B y d e finition this quantity i s time-dependent and we find it instru c tiv e to plot the c urv es as a function of d. This c hoice of ex pon ent follow s from the Hig u c hi e quation [43, 44 6 0 ] w h e r e M Moo= kyt. The ex p o n ent eq u a l to 0.5 indi cates Fi ckia n diffusion [62] an d c urv es d ev iatin g from this pow er l aw indi cate a nom a lou s diffu s i o n which we expect g iven the vis coe lasti c prop erties of the pol y m e r. The followin g plot s of M / M00 will confirm our physical intuition. Figure 6.7 conta in s plots o f M(t)/Moo, where the norm alization M00, is not on l y the maximum for eac h case of moisture co nt e nt but a l so the m ax over all cases In Fi g ur es 6.7(a) a nd 6.7(b) the D ebo r a h numb e r i s fixe d a nd we 131

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c on s id e r M(t)/Moo for t h e cases K,8 = { -0.1, 0.01}. I t i s well-kn o wn tha t the pol y m e r s in the drug -d e liv e r y d e vi ces w e a r e co n s id e rin g w ill r es i s t the fluid int a k e [62 ] ( w e di sc u ss t h e str ess stra in r e lati o n s hip in t h e f ollo win g sect ion ) R ecall tha t t h e int eg r a l t erm i s d e riv e d from a con stitutive e qu a ti o n mod e lin g the visc o e lasti c stress. H e nce, r es i s t a nce to flow i s mod e l e d by settin g "'s < 0 s o t h a t the int e gr a l t erm acts a r estoring f o rce. The m ag ni t ud e of r es i s t a nce i s se t b y "'s so tha t the l a rger the m ag nitude, I "'s I, the s l o w e r the s orption. Thi s b e h a vior i s co nfirm e d in Fi g ur es 6 7 (a) a nd 6 7 ( b ) Not ice t h e lin ea r b e h avio r for s mall time, indi cating Fi c ki a n flow, a nd the d e vi a tion fro m t h e p o w e r-l a w as time pro g r esses indi cating n o n-Fi ckia n b e h av i o r [62]. I t i s a l so in this ea rl y tim e r eg ion wh e r e the moistur e content c urves t e nd to o v e rl a p b eca u se the v i sc o e l as ti c stress ( non-Fi ckia n b e h avio r ) h as not h a d tim e t o influ e n ce t h e b e h a vior of c:1 In the n ext sec tion w e di sc u ss the influ e nce o f stress mor e t h o r o u g hly. The c o effic i ent, K,8 i s not the only p a r a m e t e r d e scribin g the polym e r 's c on form a tion a l attributes The D e bor a h numb e r De, a l s o d ete rmin es the pol y m e r s c ompli a nce (th e k erne l in the int eg r a l t e rm i s so m etimes calle d the "compli a nce functi o n [23]). W e off e r e d a n in-d epth d esc rip t i o n of D e in S ect ion 2.6 whi c h w as di s till e d into t h e T a bl e 6.2 a b o ve; r o u g hl y t h a t D e d ete rmin es a m a t e ri a l s a bilit y to flow [56]. W e r evie w the imp ac t o f D e o n (6. 1 ) fro m S ect i o n 2.6 h e r e a nd not e h o w it work s in co n ce r t with "'s to influ e n ce the b e h av i o r o f c:1 A v e r y s m all D e bor a h numb e r D e 1 (v i sc ou s materi a l ), implies the k erne l a nd h e nce the int eg r a l i s rou g hl y zer o a nd w e obtain ( 6.10) 132

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We see a similar result if K.8 1 that is, the elasticity (restoring force) is very weak relative to the fluid pressure. On the other hand, a very large D eborah number, D e 1 (elasti c material), implies the kernel is roughly unity and we can integrate the time derivative of c:1 to obtain (6.11) In this form we can easily see that the sign of K.8 influences the moisture content see Figure 6.5. Now co n s id er Figure 6.6(a). In the case of K.8 = 0.1 it is clear that the larger Deborah number D e = 100, increases flow versus the case where De= 0. In the case of K.8 = -0.1, we see the reverse behavior that the smaller D eborah number De= 0, provides a faster flow than the case where De= 100 see Figure 6.6(b). This latter result is in line with conventional expectations regarding polymers. From a modeling perspective there are advantages and disadvantages to depending on two dimensionless parameters especia ll y when comparing the model to experimenta l data. On the one hand having only one parameter makes it eas i er to fit the output to data. On the other hand having two parameters can a llo w for finer tuning of the output. It also can permit a more thorough understanding of the physics. The model is sensit iv e to both D e and K.8 as can be seen in Figure 6.7 How ever the small er values for one parameter can diminish any sensit ivit y the model has to the other. See Figures 6.7(a) and 6.7(d) for instance, where De= 0.01 or K.8 = 0 .01 are small enough to make the model in sensitive to parameter c h anges. On the other hand in Figure 6.7(b) the higher va lu e D e = 100, magnifies the difference between the values for K.8 Similarly in Figure 6.7(c) K.8 = -0.1, is large eno u g h to s how a distinction between De= {0.01, 1 100}. 133

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Hence there is a threshold for these parameters that must be exceeded before the model shows any sensitivity to change 6.2.2 Viscoelastic Stress Curves An increase in viscoelastic stress is induced by high strain rates, which in our case is replaced by the time-rate of change in volume fraction. Given the initial liquid volume fraction profile with a steep gradient near the boundary, we expect higher viscoe l astic stress near the boundary f or early times (closer to the initial value) of the simu lation; Figure 6.8 clearly shows this behavior. While stress curves such as the ones g iven in Figure 6.8 are instructive for cons id ering polymer behavior they are not convenient for viewing the g lob a l impact of D e and K-8 Instead we consider plots such as Figure 6.9 which contains plots of both IV(r t*)l and IV(r*, t)i where t* i s the time at which max IV(r t)i tE[O T] occurs and r* is the locati on at which occurs. max IV(r t)l rE[O l] Not i ce that there is little qualitative difference when comparing the curves between K-8 = -0.1 and K-8 = 0 .1. Recall from Sec tion 2.6 that a positive value for n,8 increases diffusion and hence the time-rate of change in c:1 Thus, the viscoelastic stress will be higher for a l arger value of K-8 Also noti ce in Figures 6.9(a) and 6.9(b), that for a fixed value of K-8 the viscoe l astic stress increases for a l arger value of D e In the cases of De= {0.01 1}, where the pol y mer is more 134

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viscous or viscoelastic, the stress actu ally d ecreases over time which i s con s istent with the d ecay in g expo n ent i a l model for t h e k ernel. Whe r eas when D e = 100 the str ess i s not only hi g h er but increases over time. This behavior is consi stent with the f act that the polymer in this case, i s mor e e l as ti c The l arge r D ebora h numb er indi cates that the relaxation time i s lon ger than the diffu s ion time. In this num er i ca l ex p e rim ent the D ebo r a h numb er, D e = 100, i s large e nou g h so that the polym e r do es not h ave time to relax relative to diffusion. H e nce, the st r ess in c r eases over time. Figures 6.9(c) a nd 6.9(d) s impl y co nfirm our ex pectation that the viscoelastic stress will be hi g h er near the boundary where the initi a l liquid volume fraction profi l e i s steeper than s ub sequent profiles. 6.3 Discussion A comp a ri so n of diffu s ion a nd p ermea bilit y models w as conducte d as well as a param eter se n s itivit y stud y in order to see how the mod e l (6.1) behaves under a variety of co nditions. The diffu s ion a nd p er m ea bilit y models h a d a g reat imp act on b e h av ior of c:1 es p ecia ll y the Kozen y-C a rm a n m o d e l ( 6. 7 ) which i s unbound e d as c:1 tends to 1 For exa mpl e we ca n see t hat t h e most r ap id flow i s d e riv e d by u s in g a Low model diffu s ion coeffic i ent with a Kozen y-C arma n p ermea bility. The most restrictive flow entails u s in g D(c:1 ) = (c:1 ) 2 with a co nstant permeability. The parameter se n sit ivit y a n a l ys i s y i e ld ed an import ant co nclu s ion in that the int eg r a l term h as a s i gnificant impact on the so rption properties of the d e l iv e r y d evice. One may be a bl e to sca l e D(c:1 ) ap propriatel y a nd resort to a phenome nolo g i ca l mod e l that match es t h e data accurately. However, this a ppro ac h i s limit e d in two ways: ( 1 ) it l ac k s the constitutive eq uation mod e lin g the polymer physics so it does not p ermit int erp retation s that would otherwise 135

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follo w dir ect l y from this m o d e l (su c h as argume n ts in terms o f D e ) a nd ( 2 ) i t l ac k s gen e r ality. H e n ce a n e xp e rim e ntalist m ay b e a bl e to m o dif y a coeffic i e n t to match the d a t a but w ill b e trea tin g t h e m ode l as a bl ac k-b ox makin g educate d g u esses r ega rdin g t h e p o l y m e r imp act w hil e eventua ll y r eac hin g the limit s o f what ca n b e ac hi e v e d b y a djustin g a s in g l e p a r a m ete r. F o r exa mple, will t h e diffu s i o n c o efficie n t need m o difyin g as t h e p o l y m e r p asses thro u g h t h e tra n s iti o n phases (gl assy leath e r y rubbe r y)? The EPS meth o d proved to b e a s t a bl e a nd acc urate solver f o r t hi s study It s h o uld b e n o t e d that at t hi s r eso luti o n N = 7 5 0 standard p se udosp ect r a l m etho d s would b e diffi c ul t t o impl e m e n t b eca u se o f the ir p oo rl y co ndi t i o n e d diff e r e nti a tion m atrices, e v e n within the d e f a ult t o l e r a n ce of the impli c it s olv e r MATLAB o d e 1 5s. 136

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Koztny-cumn l)t:1)-1 02 Ol 0 4 OS 07 Ol (a) (c) (b) (d) F igure 6 .2: Liquid volume fraction c1(r, t), plots over the radial grid with a Kozeny-Carman permeability K1(c1 ) = and model parameters "'s = 0.1 and T = 1. The grid size is Nr = 750 with Ncr = 450(0.60Nr) and a variable step-size, 5th order impli cit time-stepper was used. Solutions are shown for t E [ 0 0.4 ] with (a) D(c1 ) = 1, (b) D(c1 ) = c1 (c) D( c1 ) = (c1 ) 2 and (d) [ l t I D(cl) = Kl(cl) ex p E(' 137

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Constant: K5 = I De = I Kozeny-cannan: K 5 = I. D e = I 08 09 0 I 0 2 O .J 0 4 o.s 06 07 08 09 0 1 02 OJ 0 4 OS 0 6 07 08 09 (a) ( b ) K5=I.De= l 09 , 0.8 , , 0.7 , , 06 I o.s 04 O J 02 0 1 00 0 I 02 OJ 0 4 OS 06 07 08 09 tl"l (c) Figure 6.3: Mode l compariso n in t erms o f v i scoe lasti c stress and m o isture co ntent wi t h D (c1 ) = c1 and fixe d parameter s De= 1 and "'s = 1. (a) v i sc o e l astic stress r es ul t in g from u s in g a co nstant p ermeabilit y m o d e l ( b. = initi a l stress, o = fin a l stress), ( b) vi scoe lasti c str ess r esulting fro m u s in g a Koz e n y -C arma n p ermeabilit y m o d e l ( b. = initi a l str ess, o = fin a l str ess), and (c) moi sture c ontent comparing co nstant p ermeability again s t K oze n y -Carm a n p ermeabilit y 138

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8 0 7 0 --;;; N) I 5 0 r 40 30 lO 1 0 0 0 I 0 2 0.3 0 4 0.5 0.6 0 7 0.8 09 Figure 6.4: K oze n y-Carman permeability as a function of liquid volum e frac tion. De = I 08 , 07 , , Ed 06 , I , I ., , ::; , 0 4 , , OJ , , Ol 0 I 0 0 0 I 0.2 0.3 0. 4 ., 0 6 07 08 til Figure 6.5: Normalized moisture content curves M/M00. A pos i tive "'s m creases flow whereas a negative "'s inhibit s flow. 139

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1(1 = 0 1 K s =-Q.I 09 09 08 08 0 7 07 06 06 I I OS OS :!; 04 04 OJ OJ OJ OJ 0 I 0 I 0 0 0 I OJ OJ 04 OS 06 07 08 09 0 0 0 I OJ OJ 04 OS 06 07 OS 09 lin 1 112 (a) (b) Figure 6.6: Normalized moisture content curves M/M00: (a) Fix "'s = 0.1 and compare the moisture content for D e = {0, 0.01 100}, (b) Fix "'s = -0.1 and compare the moisture content for De= {0 0.01 100} 140

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De-0.01 o 08 07 Ob I OS ::;: 04 03 02 0 I ,t/2 (a) 08 01 06 04 01 02 0 I 0oL_-:':o __ _J (c) De= 100 08 01 06 I OS 04 03 02 01 til 08 01 06 .. OJ 02 01 00 ( b ) K 1 = -0.01 OJ 02 OJ 0.4 OS 06 07 08 oq 11'2 (d) Fig ur e 6.7: ormalized moisture content c urves M/Nf. (a) Fix De= 0.01 and com p are the moistur e content for "'s = { -0.01, 0 .1}, (b) Fix D e = 100 and com p a r e the moistur e content for "'s = { -0.01, 0.1} (c) Fix "'s = -0.1 and com p are the cases D e = {0.01, 1 100}, (d) Fix "'s = -0.01 a nd comp are the cases De= {0.01, 1 100} 141

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Kozcny-cannan : K5 = 0 .1. De= I 0 7 08 0 9 01 02 03 04 0.5 06 0.7 0 8 0 9 Figure 6.8: Vis coe lasti c stress 142

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.. --0.1 lh-"'IM 01 01 02 02 0 I 0 I o, ... ... %1 02 OJ 0 4 Ol 06 07 08 09 %1 0 2 OJ 0 4 Ol 06 07 08 (a) (b) ... ---o. l .. -0.1 08 08 07 07 o 06 Ol Ol "" 'Lo.-'L 0 4 :E: :E: OJ 0.1 02 02 0 I 0 I 00 0 I 02 0.1 0 4 ., 07 08 00 0 I 02 01 0 4 Ol 07 08 09 (c) (d) Fig u re 6. 9: Viscoelastic str ess 143

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7. Conclusion and Future Work A n o nlin ea r V o t erra P a r t i a l Integ rodiff e r e n t i a l Equ a ti o n (VPIDE) o f t h e seco nd kind w as a n a l yze d a nd so lv e d on a cy lindrica l geometry. The a n a l ys i s pro vid e d so m e in s i g hts into the m o d e l s u c h as r eg ul a rity r e quir e m ents f o r ex i s t e nce a nd unique n ess o f a solu t i o n as w e ll as so lu t i o n b e h av i o r over l o n g t im e p e ri o ds. In o rd e r t o solve t hi s m o d e l we so u g h t a m etho d t hat provid e d hi ghaccuracy s pati ally a nd tem po r ally a nd a rrived at a n o v e l p se ud os p ect r a l meth o d in s pir e d b y S andbe r g a nd W o j c iech owski [58]. Thi s meth o d was vette d wit h a ri g or o u s erro r a n a l ys i s a nd exte nd e d to a p o lar geo m etry so t hat i t co uld b e a ppli e d to t h e cy lindri ca l coo rdin a t e s of the dru g -d eliver y pr o bl e m In o rd er to s olv e the m o d e l w e need e d t o d e t e rmin e s uitabl e functi o n a l f orms f o r t h e diffu s ion c o efficient and p ermea bilit y mod el. Once s om e ca ndid a t e mod e l s w e r e se l ecte d num e ri ca l e xp e rim e n ts w e r e run to o b se rv e the soluti o n s b e h avio r. The work d o n e in t hi s t h es i s f o llowed a conventi o n a l p ath f o r a ppli e d m athe matics: mod e lin g a n a l ys is, a nd approx imation The work d o n e in t hi s t h es i s h as adde d m ore mathemati ca l ri gor to t h e p revio u s work d o n e b y W e in ste in [73] a nd Sin g h [ 6 3 64, 65] a nd l a id t h e g r o undwork f o r fur t h e r stud y o f t h ese m o d e ls. For e x a mpl e W e in ste in [ 7 3 ] a d vocated for num e rical so lu t i o n s to t h ese e qu a ti o n s in o rd e r to va lidate t h e a ppli ca ti o n of this t h eo r y to drug d eliver y sys t e m s In fulfillm e n t o f t hat prop osa l w e h a v e d eve l op e d a so lv e r tha t i s f ast, acc urate, sca l a ble, a nd easy t o impl e m ent. If prov id e d a diffu s i o n coeffic i e nt, D(c1 ) a nd p e rm ea bility m o d e l K1(c1), w e 1 4 4

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can find so lu tions for a variety of purposes. For examp le, the non-dimensional parameters can be determined by making comparisons to data and once the parameters are obtained we can perform simu l ation. The parameters are indeed r e l ated to expe rimentally measurable quantities such as modulus of e l asticity, polymer relaxation time [26], and diffusivity [12, 62]. So once these quantities are measured we can compute "'s and D e with some possible adjustments made to "'s to account for the externa l (b ulk ) fluid pressure. There are severa l avenues for future r esearch wh i c h we propose in this sec tion Some o f these topics entail usin g the numerical so lv er to validate the models while others are extensions to the theoretical and numerical work done in this thesis. The topics covered by this continued research span a wide va riety of fields includin g mathematical modeling functional anal ysis, numerical anal ys is, computation, materials sc i ence, and fluid mechanics. 7.1 Model Analysi s Validati o n and Extension The mathematical models derived in [73] describe drug delivery device dur in g a swe llin g regime on a cy lindri ca l geometry. How ever, the biological fluid event u ally corrodes the polymer and cy lind ers are not the on l y geometr ies availab l e for drug models. Additionally, the drug delivery model is not the on l y model availab le, there are other competing models suc h as those found in [ 16, 60] for examp le. These models do not coup l e the concentration equation to a swe llin g equation because they do not incorporate Darcy s law. In stead the polymer character i stics are combined into the drug concentration's diffusion coefficient with parameters derived through ex perim ent that is, they use a phenomenolog i ca l model. Work that st ill needs to be accomplis h ed for the drug-delivery 145

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model includes: 1. modeling and solving the problem when the polymer is corroding and finding an appropriate numerical so l ver by exp l oring meshless methods such as a l gebraic multigrid or examining smoothed particle hydrodynamic simu l ations; 2. solving the contro ll ed release drug delivery problem on a variety of geome tries such as spheres and e llip so id s ; 3. performing stability anal ysis of numerical techniques based on t h e ir sen sitivity to the magnitude of the drug delivery model s nondimensional parameters ; 4. vetting the drug delivery model by testing it against pharmaceutical data and compare the results to competing models; 5. performing perturbation anal ysis to determine soluti ons at the onset of the boundary-layer near the exterior of the device; 7.2 Generalizing the Applicability of the EPS Method The EPS method offers severa l advantages over standard pseudospectral methods without requiring preconditioning of the differentiation matrices. The method is new and still requires further numerical anal ysis as well as an extension to its range of applicability. Work that still needs to be accomplished for the EPS method includes: 146

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1 exten ding the EPS method on a variety of geometr i es such as sphe res a nd e llipsoids (this i t e m dovetails with m y research involvin g the drug d e liv e r y model) ; 2 incorporating the EPS method into dom ai n d ecompos ition tec hniqu es (thi s item would b e u se ful for so lvin g probl ems with unu s u a l geomet ri es); 3 providing e rror analysis for a more genera l function s p ace ; 4. d ete rminin g a formul a or at the very l east, a n h e uristi c for the truncation parameter's, Nc, impact on a n operator discr e tization 's accuracy; 7.3 G e n e r a li z in g the E x i s t e nce and Unique ness P roof f or t h e IBVP Solutions with jump di sco ntinuiti es would provid e for a mathematically in teresting proof and are required for a pplicati o n s in multiph ase flow. An ave nu e for r esea r c h wou l d b e to prov e ex ist e n ce and uniqu e ne ss of the IBVP 6.1 for a mor e ge neral function s p ace s u c h as Hg ( S1) a nd for di sco ntinuou s diffu s ion and permeabilit y models. 7.4 E x t ending the Applicability of the Drug D e liv e r y Mode l As m e ntion e d earlier VPIDEs can b e u sed to mod e l diffu s ion eq uation s with memory. Applications ari se in a number of fie lds s u c h as l aser heatin g where acc urate models for heat transport with m emo r y are cr u c ial or s oft-tissues lik e a rteri es where viscoelasticity i s r e l eva nt. Anoth er ave nu e for research would b e to exp lor e these a ppli cation areas a nd exam in e potential modifications for (6.1 ). The work performed in this thesis h as produ ce d a num e rical method a n a l ysis, a nd so lution s to some probl e m s posed. In other words we h ave d eve lop e d 147

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so m e too l s f o r a n swer in g q u est i o n s a n d u se d t h ose too l s to generate some a n s w e rs. H oweve r as indi cated by t h e list o f prop ose d id eas f o r fur t h er resea r c h we h ave pr od uced m o r e quest i ons t h a n a n swe rs. These qu est i o n s set u p a r esearc h age nd a a nd the num e ri ca l too l s a nd a n a l ys i s pro duced in t hi s t h es i s pro vid e a m ea n s f or a ddr ess in g these to pics. 148

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Appendix A. Derivation of Darcy's Law This append ix conta in s t h e relevant computation s for deriving the novel form of D arcy's l aw stated i n Chapter 2. The calc ulation s rely h eavily upon the work done in [73] and so we refere nce it frequent ly. We begin o ur derivation o f D arcy's law by consider in g three fiel d eq uations: the conservat ion of mass the co n servation of lin ear momentum, and entropy balance. The bal ance eq uation s are g i ven below: Conservation of Mass D a(capa) Dt + ca pa( V va) = {3-#a (A.l) where we are familiar with t h e terms of this eq uati on from Chapter 2. Further, s ince the interface is ass um ed to be mass l ess, we h ave the r estr iction : (A.2) B a l ance of Lin ear Momentum (A.3) h as no lin ear momentum: (A.4) 149

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where t a is the partial Cauchy stress tensor, g a is gravity and ca paT f3 represents the rate momentum i s transferred from phase f3 to phase a. Conservation of Energy capa----rJt-cato: V va-V (c0Qa)-capaha = 2..:::ca paQ 3 (A.5) f3#a with the restriction that the interface has no energy density: [ca paQ 3 + c0 paT; v a + ca pae{; ( e0 + + [c/3 + c/3 vf3 + c/3 ( ef3 + = 0 (A.6) Entrop y Bal ance with restriction (A.8) The definitions of all m acrosca l e variab l es in terms of microscale quantities can be found in [73]. W e assume there are only two phases a so lid a = s, a nd liquid a= l with negligible int erfacia l effects. Further, we ass um e a form of loca l equi librium wherein there is one temperature for all phases i.e. ra(x, t) = T(x, t). 150

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This effect iv e l y states that t h e rate of heat transfer between phases i s mu c h fast e r than the time scales of int eres t to the probl e m. B y the seco nd l aw of thermodynamics the net e ntrop y generate d must be non-n egat iv e for the tota l body i.e pA = L c0l xA
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in eq u ality: Examining the fie ld eq uations, we r ecog nize the unknown s as l a a T a E, p V , q e, ( A .12) ( A .13) ( A.14 ) The variables in the first row ( A.13 ), a r e t h e prim a r y unknowns; E8 i s no t includ e d s in ce E8 = 1c:1 V a ri a bl es, ( A 1 4), are con s id e r e d constitutive a nd a r e a function of constitutive ind e p e nd e nt variab l es. An a ddi t ion a l co nstitutive e quation f or eac h o f these variables i s r e quir e d in orde r to close t h e syste m o r h a v e the sam e numb e r of eq uati o n s as unknown s A care ful count in g of equation s a nd unkn owns r evea l s that we are one eq u ation s h ort -the o n e correspondin g to the unknown c:1 This i s known as the closure probl e m [ 13], a nd it a ri ses from the homogenization of the mi crosco pi c geo metr y a nd is present in all up sca lin g t ec hniqu es. W e follow [ 2 10, 14] to close the syste m a nd view the materi a l time r ate of c h a n ge of the volum e fract ion D8c:1 / Dt as a const i t utiv e v a ri ab le, that is, that the rate at which t h e volume fraction c h a n ges i s a function of the co mpo s i t i o n o f the m e dium a nd other co nsti t utiv e ind epe nd e nt variab l es The c h o i ce of consti tutive ind e p e nd ent variables i s b ase d o n expe ri e nce and knowl e dge o f the syste m being modeled ; the c hoice of ind e p ende nt variables 152

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is what defines the materia l being mod e l ed. In [73] it i s p ost ul ate d that the Helmhol z free e n e r gy is d e p e nd ent upon the followin g ind e p e nd e nt v a ri a bles (n) = l (m)l l Cli T C s Cs) ' c:, c: p ' (A.15) ( A .16) wh ere m = 1 ... p and n = 1 ... q a r e materi a l time derivatives of order p a nd q C11 i s the co ncentrati o n o f the lh s p ec i es in the liquid phase and j = 1 ... T is the temperature C8 i s the modifi e d ri ght Cauchy-Green tensor. How eve r Wein ste in t h en po sits that the materi a l i s imm e r sed in a fluid where the hi g h (n) fluid co nt e nt a llow s on e to n eg l ect the s h ea r stresses, that is i gnore C8 a nd C8 [73]. Moreover if co n s id er only the pol y m e r swe lling, then we ca n i g nor e the c oncentration terms as well a nd reduce the list (A .1 7) to = l (m)l l T) ' c:, c: p ( A .17) (A.18) Using t h e c hain rule on the materi a l time d e rivativ e of the H e lmh o ltz potential we obtain a n ew form for t h e entro p y in e qu a lity. The g r o upin g of term s i s c ho se n so that the ex ploit a t i on of the e ntropy inequ a l it y in the sense o f Co l eman and Noll [25] i s eas i er to follow. The conse rv ation of mass for eac h phase ( A 1 ) i s enforced weakly u s in g L agra nge multipli ers, >.a The variables which appea r in the e ntrop y in eq uality in [73] that a r e not in this list a re: ( A.19 ) These variables are appear lin ea rl y in the e ntrop y in eq u a lit y a nd a r e n either con st itutiv e n or independ ent thus they ca n vary ind e p e nd e ntl y in s u c h a way as 153

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to violate t h e in e qu a lity. Consequently we set their coeffic i ents eq u a l to zero a nd in d o in g so we obtain noneq uilibrium r es ults. These r es ult s hold at e quilibrium and n o neq uilibrium. If we d e not e the variables in ( A .19) as z then the di ss ipative portion of the entrop y in e qu a lit y ca n then be written as f z where f is a vee-tor conta inin g the coeffic i ents of eac h Zj. At eq uilibrium the entro p y gen er ated i s maximum and h ence the p artia l of the total e ntrop y gen e r ate d with r espect to Zj must be zero. Since eac h va ri ab l e appears as Zjjj(z 1 z 2 ... ), fj(z1 z 2 ... ) + zj ( 8 f j ( z 1 z2, ... )/ (o zj) must be zero at e quilibrium or s in ce Zj = 0 at eq uilibrium f(z 1 z 2 ... ) = 0 at eq uilibrium H ence t h e coeffic i ent of eac h of these var i ab l es must be zero Nea re quilibrium r es ult s a r e obtained b y linearizin g about e quilibrium Thus, u s in g the notation from a bov e, we h ave (A.20) where fj(z)leq i s zero from the eq uilibrium results. The material coeffic i ents, ai (which m ay be sca l a rs, vectors, or tensors) are a fun ct ion of a ll t h e indep e nd ent variables n ot listed in (A.19). D a r cy's l aw i s obtained by d ete rminin g const itutiv e eq uati ons for t1 a nd T8 and comb inin g them with the stead y-state, liquid phase, lin ear mom entum eq uation (A.21) 154

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The coeffic i e n t o f t h e r e lative vel oc ity term, v l s, in t h e e n tro p y in e qu ality f o und in [ 73], w i t h t h e lis t o f var i a bl es g i ve n in (A.l7) i s (A.2 2 ) wh e r e ti i s d e fin e d in Cha pter 2 W e o btain t h e e quilibrium r es ul t b y lin ea ri z in g a b o u t v l s to o btain ( A .23) Addi t i o n a ll y f o r t1 w e h ave t h e co nsti t u t ive eq uati o n [9], ( A 24) Substi t u t in g e quati o n s ( A 23) a nd ( A 24) in to e quati on ( A .21) we o btain N eg l ect in g g r avity a nd ass uming n eglig ibl e s h ea rin g effec t s w e a r r ive at D arcy's l a w [73], ( A 25) Note t hat e quati on ( A .25) coin c i des w i t h eq uati o n (2.48) in C h a pter 2 155

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Appendix B. Derivation of a Pouzet Volterra Runge-Kutta Method In t lli s appendi x we derive an explicit Pouzet Volterra Runge-Kutta (PVRK) method. This derivation is included for completeness and ease of reference. For a more comp lete description including other var ieties of VRK methods see [17, 18, 55] Select a discretization of the temporal interval [to, T], call it h = { t1 = t 0 t2 t3 ... tN = T}, with cardinalit y N. For ease of exposition suppose the discretization is uniform with spacing h = t n+l -tn for n = 1 ... N -1. An RK method solves an initial va lu e problem such as du dt = F(u t) u(t =to) = uo via the estimation, (B.1) (B.2) (B.3) w h ere Un = u(tn) tn E h and is a weighted approximation to the s l ope of the so lu t i on over an interval of length h. I ow consider the foll owing Volterra integral equati on (VIE) y(t) = f(t) + 1 t I<(ts)y(s)ds. (B.4) The VIE (B.4) can be rewritten by relating it to the mesh h as (B.5) 156

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where 1tn Fn(t) = f(t) + 0 K(t-s)y(s)ds ( B.6) i s the l ag term a nd n(t) = J.t K(t-s)y(s)ds tn ( B.7) will denote the in c r e m ent function The l ag term conta in s the hi story of the phenomena and s ince it conta in s t it must be computed f or eac h time-step. Consequently the cost in creases as time passes thus l ag term comp utati on makes the num e ri ca l int egrat ion o f VIE s computation ally ex p e n s ive. Approximating the VIE at tn+l yie l ds y( tn+I) = Yn+l, Yn+I = Fn(tn + h)+ n(tn +h), n = 0 ... N -1 ( B .8) where t h e in c r ement function h as the f orm m n(t) = h L b iK(t, tn + cih Yn,i), (B.9) i=l with m Yn,i = Fn(tn + Ci_h) + h L ai, sK(tn + Ci,h, tn + C s h Yn,s) i = 1 ... m. ( B.10 ) s=l H ere the Ci and b i are vector compon e nt s and the ais are written as matrix e ntri es of the Butc h er di agram Table B.1 for ODEs. 157

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Table B.l: Butcher diagram .. 7 158

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