الفهرس 
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Abstract
Thesis title: “The Numerical Treatment of Fractional order Partial
Differential Equations”
Fractional order partial differential equations (FPDE) are considered as one
of the recent subjects although the concepts of fractional calculus are old as the
beginning of classical calculus with Newton’s and Leibentz.
The thesis in general considers two models of the main categories: the elliptic
type class (Poisson’s equation) and the parabolic class (diffusion equation). In
this thesis we consider the fractional order counterparts of these models with
addition of an integral coefficient in the parabolic case (integro-differential
equation). Also, we considered the finite difference method to approximate the
partial differential equations by an algebraic systems. Because of the special
properties of the resultant algebraic systems many authors like Young and
Evans have considered the algebraic structures in the integer order case; we
have considered these structures in the fractional order Poisson’s equation. Also,
we considered the method of lines combined with a spectral method based on
shifted Chebyshev polynomials.
This thesis consists of four chapters as follows:
Chapter1:Fractional Calculus: Fundamentals and Motivations
In this chapter, a brief historical review about the starting and the
developments of fractional calculus and its applications in many fields is
considered. Also a review of important definitions and theorems to provide
subsequent chapters are established. General formulas which have been
modified and used in the numerical solution of fractional order differential
equations are given.
Chapter2: Boundary Value Problems in the Plane from Integer to
Fractional Orders:
In this chapter the algebraic systems which are the final step in the finite
difference treatment of fractional elliptic differential equations in two
independent variables are deeply studied. Poisson’s equation and its fractional
counterpart represents the model problem for this chapter. The structures of the
coefficient matrices of the algebraic systems corresponding to two different
methods of labeling grid points (Lexicographical ordering and the Chequerboard
ordering) are given. Differences between the systems of algebraic
equations resulting from the finite difference treatment of the classical integer
case and the corresponding fractional order are considered. from the structures
of the coefficient matrices of the fractional orders we illustrated the memory
and hereditary properties of the fractional order derivatives.
The spectral properties of the iteration matrices of the successive overrelaxation
(SOR) and its variant KSOR are considered. Precondition technique is
discussed with application to the resultant algebraic systems. Two types of
precondition are applied to the resultant algebraic systems appear in the
Lexicographical ordering. Also, we introduced a precondition for the system
appears in the Chequer-board ordering. Comparison of the spectral radius of the
iteration matrices and the selection of the relaxation parameters for the SOR and
the KSOR in the two systems after and before applying the precondition is
made. To assist in clarifying what has been reached, programs are designed with
(MATLAB 7.10), the results are introduced in terms of tables and figures
illustrating the relations between the fractional orders and the relaxation
parameters.
Chapter3: Fractional Diffusion Equation
The numerical solution of the simple parabolic PDE or integro-differential
equation is considered with the essential properties of the numerical method
(Finite difference method). Fractional derivatives with respect to time only,
space only, space and time, and space and time with existence of integral terms
are considered. Explicit and implicit finite difference schemes with their
conditions of stability are established. We used the direct way method and the
shifted Grünwald-Letnikov (G-L) approach in approximating the fractional
derivatives. Also, we used the trapezoidal method in approximating the integral
terms. The matrix structures for the implicit and explicit schemes are
established. Also, the conditions of stability are discussed in comparison with
the classical integer cases. The results of running Matlab programs (MATLAB
7.10) for three examples are illustrated in terms of tables and graphs.
Chapter4: The Method of Lines Combined With Chebyshev Spectral
Method
In this chapter one of the weighted residual methods (WRM), the collocation
method, is used with the shifted Chebyshev polynomials of the first kind in
approximating the sequential fractional space boundary value problems results
from using the finite difference method in approximating the fractional time
derivatives. Also, an algorithm is introduced. Application of the algorithmic
steps for solving fractional partial differential and integro-differential equations
is considered. Graphs and tables to illustrate the behavior of the solution are
introduced.