The Numerical And Approximate Analytical Solution Of Parabolic Partial Differential Equations With Nonlocal Boundary Conditions
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Date
2011-12
Authors
Ghoreishi, Seyed Mohammad
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
Many scientific and engineering problems can be modeled by parabolic partial dif-
ferential equations with nonlocal boundary conditions. Examples of such problems
can be found in chemical diffusion, thermoelasticity, heat conduction processes,
nuclear reactor dynamics, inverse problems, control theory and so forth. In the
last two decades, the development of numerical and approximate analytical tech-
niques to solve these equations has been an important area of research due to the
need to better understand the underlying physical phenomena. There is a need
to develop new and more accurate techniques and this is the area of focus of this
thesis. In this thesis, we propose new finite difference methods and study approxi-
mate analytical methods for solving linear and nonhomogeneous parabolic partial
differential equations with nonlocal boundary conditions. We have introduced a
new explicit finite difference method and a new (3,3) Crandall- formula method
and have discussed the obtained results. In addition, we have also studied sev-
eral approximate analytical methods- Adomian Decomposition Method, Variation
Iterative Method, Homotopy Perturbation Method, Homotopy Analysis Method,
Optimal Homotopy Asymptotic Method and have applied the standard approach
and modifications to solve linear and nonhomogeneous parabolic partial differential
equations with nonlocal boundary conditions. It is known that the approximate
analytical methods solve diferential equations by using the initial condition only.
Thus, we also proposed a new modification of Adomian Decomposition Method to
solve linear and nonhomogeneous parabolic partial differential equations with non-
local boundary conditions by using nonlocal boundary conditions. We also show
that the finite difference methods developed and approximate analytical methods
considered are capable of accurately solving linear and nonhomogeneous parabolic
partial differential equations with nonlocal boundary conditions.
Description
Keywords
The numerical and approximate analytical solution , of parabolic partial differential equations