SIAMSEAS2015

The application of Homotopy Analysis Method
for the solution of time-fractional diffusion
equation with a moving boundary
Ogugua N. Onyejekwe
Department of Mathematics
Indian River State College
39th Annual SIAM Southeastern Atlantic Section
Conference
March 20-22 2015
Ogugua N. Onyejekwe Homotopy Analysis Method

Abstract
It is difﬁcult to obtain exact solutions to most moving boundary
problems. In this presentation we employ the use of Homotopy
Analysis Method(HAM) to solve a time-fractional diffusion
equation with a moving boundary condition.
HAM is a semi-analytic technique used to solve ordinary,
partial, algebraic, fractional and delay differential equations.
This method employs the concept of homotopy from topology to
generate a convergent series solution for nonlinear systems.
The homotopy Maclaurin series is utilized to deal with
nonlinearities in the system.

Abstract
HAM was ﬁrst developed by Dr. Shijun Liao in 1992 for his PhD
dissertation in Jiatong University in Shangia. He further
modiﬁed this method in 1997 by introducing a convergent -
control parameter h which guarantees convergence for both
linear and nonlinear differential equations.

Abstract
There are advantages to using HAM [4]
it is independent of any small/large physical parameters.
when parameters are chosen well, the results obtained
show high accuracy because of the convergence- control
parameter h.
there is computational efﬁciency and a strong rate of
convergence.
ﬂexibility in the choice of base function and initial/best
guess of solution.

Parameters
s (t) - diffusion front
C0 - initial concentration of drug distributed in matrix.
Cs - solubility of drug in the matrix
C (x, t) - concentration of drug in the matrix
℘ - diffusivity of drug in the matrix (assumed to be constant)
Dα
t - Caputo Derivative
R - scale of the polymer matrix

Problem Definition
Figure 1: Profile of concentration. The first picture is the initial drug
loading. The second picture is the profile of concentration of the drug
in the matrix at time t.[10]

Assumptions
We will only consider the early stages of loss before the
diffusion front moves closer to R and assume that C0 > Cs.
Perfect sink is assumed.

Introduction
Given the domain
WT = {(ξ, t) : 0 < ξ < s (t) , 0 < α ≤ 1, t > 0} (1)
The following problem is considered
Dα
t C (ξ, t) = ℘
∂2C (ξ, t)
∂ξ2
, (2)
with the initial condition
C (ξ, 0) = 0 (3)
and the following boundary conditions
C (s (1) , 1) = k1, C (s (t) , t) = Cs, t > 0, (4)
where k1 is any constant.

(C0 − Cs) Dα
t s (t) = ℘
∂C (ξ, t)
∂ξ
(ξ = s (t) , t > 0) , (5)
s (1) = k2 (6)
k2 is a constant that depends of the value of α
where Dα
t is deﬁned as the Caputo derivative
Dα
t f (t) =
t
0
(t − τ)n−α−1
Γ (n − α)
fn
(τ) dτ, (α > 0) , (7)
for n − 1 < α < n, n ∈ N and Γ ( ) represents the Gamma
function.

Reducing Governing Equations to Dimensionless
Variables
The reduced dimensionless variables are deﬁned as
x =
ξ
R
, τ =
℘
R2
1
α
t, u =
C
Cs
, S (τ) =
s (t)
R
(8)

Reducing Governing Equations to Dimensionless
Variables
The governing equation (2) subjected to conditions (3) − (5)
can be reduced to the dimensionless forms
Dα
t u (x, τ) =
∂2u (x, τ)
∂x2
(0 < x < S (τ) , τ > 0) (9)
u (S (1) , 1) = 1 (10)
where S (1) varies for different values of α and η
u (x, τ) = 1, (x = S (τ)) , τ > 0, (11)
∂u (x, τ)
∂x
= ηDα
t S (τ) , (x = S (τ)) , τ > 0, (12)
where η = C0−Cs
Cs

Solution by HAM
To solve equation (9) by homotopy analysis method, the the
initial guess for u (x, τ) is chosen as
u0 (x, τ) = (a0)−1
xτγ1
(13)
where a0 =
Γ(1−α
2 )
ηΓ(1+α
2 )
1
2
, γ1 = −α
2
The initial guess for the diffusion front is chosen as
S0 = a0τ
α
2 (14)

Solution by HAM
The auxiliary linear operator is
L [φ (x, τ; q)] =
∂2φ (x, τ)
∂x2
(15)
with the property
L [k] = 0 (16)
where k is the integral constant, φ (x, τ; q) is an unknown
function.
The nonlinear operator is given as
N [φ (x, τ; q)] =
∂2φ (x, τ; q)
∂x2
−
∂αφ (x, τ; q)
∂τα
(17)

Solution by HAM
By means of HAM,deﬁned by Liao, we construct a zeroth-order
deformation
(1 − q) L [φ (x, τ; q) − u0 (x, τ)] = qhN [φ (x, τ; q)] , (18)
where q ∈ [0, 1] is the embedding parameter, h = 0 is the
convergence-control parameter,u0 (x, τ; q) is the initial/best
guess of u0 (x, τ)

Solution by HAM
Expanding φ (x, τ; q) in Taylor series with respect to q we
obtain,
φ (x, τ; q) = u0 (x, τ) +
+∞
m=1
um(x, τ)qm
(19)
Clearly we see that when q = 0 and q = 1 equation (19)
becomes
φ (x, τ; 0) = u0 (x, τ) , φ (x, τ; 1) = u (x, τ) (20)
If the auxiliary linear operator L, the initial guess u0 (x, τ) and
the convergence-control parameter h are properly chosen so
that the series described in (20) converges at q = 1, then
u (x, τ) will be one of the solutions of the problem we have
considered.

Solution by HAM
Differentiating the zero-order deformation equation (18) m
times with respect to q and then dividing it by m! and ﬁnally
setting q = 0 , we obtain an mth-order deformation equation
L [um (x, τ) − χmum−1 (x, τ)] = hVm
−−−−−−−→
um−1 (x, τ) (21)
where
χm =
0 if m ≤ 1;
1 if m > 1.
and
Vm
−−−−−−−→
um−1 (x, τ) =
∂2um−1 (x, τ)
∂x2
−
∂αum−1 (x, τ)
∂τα
(22)

Solution by HAM
We have
um (x, τ) = χmum−1 (x, τ) + hL−1
Vm
−−−−−−−→
um−1 (x, τ) + k (23)
and the integration constant k is determined by the boundary
condition equation (10).
Looking at equation (23), the values for um (x, τ) for
m = 1, 2, 3, ... can be obtained and the series solution gained.

Solution by HAM
The approximate analytic solution is gained by truncating the
following series
u (x, τ) =
m
i=0
ui(x, τ). (24)
Equation (24) contains the convergence-control parameter h,
which determines the convergence region and rate of the
homotopy-series solution.
The convergence-control parameter h is obtained by setting
(u(S (1) , 1)HAM = (u(S (1) , 1)exact
The diffusion front S (τ) is obtained by setting
(u(S (τ) , τ))HAM = 1

Comparison between Approximate and Exact
Solutions when x=0.75
The exact solution for u (x, τ) and S (τ) are given as follows
[10].
u (x, τ) = H
∞
n=0
x
τ
α
2
2n+1
(2n + 1)!Γ 1 − 2n+1
2 α
(25)
S (τ) = p.τ
α
2 (26)

Figure 2: Drug Distribution in tissue when η = 0.5 and α = 1,
u (x, τ)HAM is in red and u (x, τ)EXACT is in green

Figure 3: Diffusion Front in tissue when η = 0.5 and α = 1, S (τ)HAM
is in red and S (τ)EXACT is in green

Figure 4: Drug Distribution in tissue when η = 0.5 and α = 0.75,

Figure 5: Diffusion Front in tissue when η = 0.5 and α = 0.75,
S (τ)HAM is in red and S (τ)EXACT is in green

Figure 6: Drug Distribution in tissue when η = 0.5 and α = 0.5,

Figure 7: Diffusion Front in tissue when η = 0.5 and α = 0.5,
S (τ)HAM is in red and S (τ)EXACT is in green

Figure 8: Drug Distribution in tissue when η = 1 and α = 0.5,

Figure 9: Diffusion Front in tissue when η = 1 and α = 0.5, S (τ)HAM

Conclusion
When calculating the values for u (x, τ) for a fixed value of
η and varying values of α, the higher the value of α, the
smaller the relative error.
For fixed values of η and varying values of α, the
approximate and exact values of S (τ) are in direct
agreement with each other.
Similarly for fixed values for α and varying values of η, the
approximate and exact values of S (τ) are in direct
agreement with each other.
Whereas for fixed values for α and varying values of η, the
values of u (x, τ) are in more agreement than they were for
u (x, τ) for a fixed value of η and varying values of α. The
relative error is smaller.

Conclusion
We have shown that HAM can be used to accurately predict
drug distribution in tissue u (x, τ) and the diffusion front S (τ)
for different values of α and η.

References I
A.K. Alomari.
Modiﬁcations of Homotopy Analysis Method for Differential
Equations: Modiﬁcations of Homotopy Analysis Method,
Ordinary, Fractional,Delay, and Algebraic Equations.
Lambert Academic Publishing,Germany, 2012.
S. Liao.
Homotopy Analysis Method in Nonlinear Equations.
Springer,New York, 2012.
S. Liao.
Beyond Perturbation: Introduction to the Homotopy
Analysis Method.
Chapman and Hall/CRC,New York, 2004.

References II
S.Liao
Advances in The Homotopy Analysis Method
World Scientiﬁc Publishing Co.Pte. Ltd, 2014.
Rajeev, M.S. Kushawa
Homotopy perturbation method for a limit case Stefan
Problem governed by fractional diffusion equation.
Applied Mathematical Modeling,37(2013),3589-3599.
S.Das, Rajeev
Solution of Fractional Diffusion Equation with a moving
boundary condition by Variational Iteration and Adomain
Decomposition Method.
Z. Naturforsch,65a(2010), 793-799.

References III
S. Liao.
Notes on the homotopy analysis method - Some deﬁnitions
and theorems.
Common Nonlinear Sci.Numer.Simulat, 14(2009),983-997.
V.R.Voller
An exact solution of a limit case Stefan problem governed
by a fractional diffusion equation.
International Journal of Heat and Mass
Transfer,53(2010),5622-5625.
V.R.Voller, F.Falcini, R.Garra
Fractional Stefan Problems exhibiting lumped and
distributed latent-heat memory effect.
Physical Review,87(2013),042401.

References IV
X.Li, M.Xu, X.Jiang.
Homotopy perturbation method to time-fractional diffusion
equation with a moving boundary condition.
Applied Mathematics and Computation,208(2009),434-439.
X.Li, S.Wang and M.Zhao
Two methods to solve a fractional single phase moving
boundary problem.
Cent.Eur.J.Phys.,11(2013),1387-1391.

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