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Homotopy
Analysis Method
for Solving the
Backward Problem
for the
Time-Fractional
Diffusion
Equation
Mohammad F. Al-Jamal
(1) Department
of Mathematics
and Natural
Sciences,
College of
Sciences and
Human Studies,
Prince Mohammad
Bin Fahd
University, Al
Khobar 31952,
Saudi Arabia
Email address:
maljamal@pmu.edu.sa
(2) Department of
Mathematics, Faculty
of Science, Yarmouk
University, Irbid
21163, Jordan
Email address:
mfaljamal@yu.edu.jo
Doi :
https://doi.org/10.47013/16.4.9
Cited by :
Jordan J. Math &
Stat.,
16 (4) (2023),
763 - 788
PDF
Received on:
Oct. 16,
2022;
Accepted
on: Dec. 13,
2022
Abstract. This paper
deals with the
backward problem
of a
nonhomogeneous
time-fractional
diffusion
equation, that
is, the problem
of determining
the past
distribution of
the substance
from present
measurements. By
the separation
of variables
method, exact
solutions of the
forward and
backward
problems are
obtained in
terms of
eigenfunctions
and
Mittag-Leffler
functions.
Contrary to the
forward problem,
i.e.,
determining the
present solution
from given
initial data,
the backward
problem, i.e.,
the problem of
recovering the
initial
condition from
noisy
measurements of
the final data,
is proved to be
ill-posed and
highly unstable
with respect to
perturbations in
the final data,
and thus, some
regularization
technique is
required. The
novelty of the
current work
stems from
utilizing the
homotopy
analysis method
as a tool to
obtain a
regularization
scheme to tackle
the instability
of the backward
problem.
Stability and
convergence
results of the
proposed method
are proved, and
optimal
convergence
rates of the
regularized
solution are
given under both
a priori and a
posteriori
parameter choice
rules. The
resulted
algorithm is
very efficient
and
computationally
inexpensive.
Numerical
examples are
presented to
illustrate the
validity and
accuracy of the
proposed
homotopy method.
Keywords: Inverse
problems,
fractional
diffusion,
backward
problem,
homotopy
analysis.
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