الفهرس 
Chapter 1: General IntroductionOnly 14 pages are availabe for public view
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Abstract
The main objective of this thesis, which consists of six chapters, is to
introduce an analytical and numerical treatment based on polynomial and nonpolynomial
spline functions for the solution of some types of fractional partial
differential equations. Based on the Von Neumann method, the stability of the
proposed methods is studied. The accuracy of the proposed methods is
demonstrated by several test problems. We show that the present methods give
better accuracy compared with other methods such as collocation, and spline
methods.
In chapter 1, we introduce some basic lemmas, definitions and concepts which
are needed and used throughout to develop this thesis. We also discuss some
fractional partial differential equations, which will be dealt with in the next
chapters.
In chapter 2, we propose a cubic non-polynomial spline based method to
obtain numerical solutions for linear time fractional diffusion equation. This
chapter is organized as follows: A method depending on the use of the nonpolynomial
spline is derived. The truncation error of the method is theoretically
analyzed. In addition, the stability analysis is theoretically discussed. Using the
Von Neumann method, the proposed method is shown to be conditionally
stable. Finally, two numerical examples are included to illustrate the practical
implementation of the proposed method.
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In chapter 3, a cubic non-polynomial spline based method is proposed to
obtain numerical solutions of the time fractional Burgers’ equation. A method
depending on the use of the non-polynomial spline is derived in section two.
The treatment of non-linear terms is discussed. In section three, the stability
analysis is theoretically discussed, using the Von Neumann method, where the
method is shown to be conditionally stable. Finally, two numerical examples
are presented in section four to illustrate the practical implementation as well as
the accuracy of the proposed method.
In chapter 4, a novel approach based on quartic non-polynomial spline for
solving the time fractional dispersive partial differential equation is proposed.
This chapter is organized as follows: A method depending on the use of the
non-polynomial spline is analyzed and derived. The truncation error of the
method is theoretically analyzed. In addition, the stability analysis is discussed.
Using the Von Neumann method, the proposed method is shown to be
unconditionally stable. Finally, one numerical example is included to illustrate
the practical implementation of the proposed method. The obtained
approximate numerical solutions are showed to maintain good accuracy
compared with the exact solutions.
In chapter 5, a quadratic polynomial spline based method is proposed to
obtain numerical solutions of the nonlinear space- fractional Fisher’s equation.
Derivation of the numerical method is presented and discussed in section two.
In section three the stability analysis is theoretically discussed, using the Von
Neumann method, where the method is shown to be conditionally stable.
Finally, a numerical example is included to illustrate the practical
implementation of the proposed method. The obtained approximate numerical
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solutions are in good agreement with the approximate solutions obtained using
two other methods.
In chapter 6, we propose a quadratic polynomial spline based method to
obtain numerical solutions of the linear time and space fractional Diffusion
equation. This chapter is organized as follows: A method depending on the use
of the polynomial spline is derived. The stability analysis is theoretically
discussed, using the Von Neumann method, where the method is shown to be
conditionally stable. Finally, numerical examples are presented to illustrate the
practical implementation of the proposed method. The obtained numerical
results show that the proposed method is applicable and can be used to get a set
of approximate solutions for such type of problem.