الفهرس 
Introduction and PreliminariesOnly 14 pages are availabe for public view
 |  | from | 188 |  |  |
| | | from | 188 | | |
Abstract
It’s well known that fractional calculus is an extension of the classical calculus, since,
in the fractional case we can de ne the derivative and integral with any non interger
order by using one of the famous de nitions such as, Riemann-Liouville, Grunwald-
Letnikov and Caputo de nitions [66]. Fractional-order derivatives and integrals provide
useful tools for description many natural phenomenons [66].
The main objective of this thesis is to develop new ecient numerical methods
and supporting analysis, based on the Shifted Chebyshev Polynomials, Shifted Legendre
Polynomials, Variational Iteration Method (VIM), Pade approximation VIM and
Crank-Nicolson nite di erence method for solving fractional Logistic di erential equation
with two di erent delays, fractional di usion equation, fractional Riccati di erential
equation, fractional Logistic equation, linear and non-linear system of fractional
di erential equations, the fractional order SIRC model associated with the evolution of
in
uenza A disease in human population and solving the fractional di erential equation
generated by optimization problem are implemented. Also, we introduced Legendre
collocation method for solving a wide class of fractional optimal control problems.
One important contribution of this thesis is the demonstration of how to choose different
approximation techniques for di erent fractional derivatives. Another objective
of this thesis is to derive the analytical solutions for some fractional systems di erential
in one and two dimensions. By doing so, we can ascertain the accuracy of our proposed
numerical methods.
iv
The stability and convergence of our proposed numerical methods are also investigated.
Numerical experiments are carried out in support of our theoretical analysis.
We also emphasise that the numerical methods, we develop are applicable for many
other types of fractional di erential equations.
This thesis is presented by publications. Our original contribution to the literature
is listed in six published papers. This thesis consists of six chapters:
Chapter one.
This chapter consists of 8 sections. In this chapter, we give the de nitions and
results which we use throughout this thesis.
Chapter two.
In this chapter, the approximate formula of the Caputo fractional derivative using
Chebyshev polynomials series is introduced. Special attentions are given to study the
convergence analysis and estimate the upper bound of the error of the proposed formula.
This formula is implemented to obtain approximate solutions of some models which
represented by fractional di erential equations such as, logistic di erential equation
with two di erent delays, Riccati di erential equation, linear and non-linear system of
di erential equations, the order SIRC model associated with the evolution of in
uenza
A disease in human population.
Chapter three.
In this chapter, the approximate formula of the Caputo fractional derivative using
Legendre polynomials series is introduced. Special attention is given to study the
convergence analysis and to estimate the upper bound of the error of the proposed
formula. This formula is implemented to obtain the approximate solutions of some
models which represented by fractional di erential equations such as, fraction di usion
equation, fractional wave equation, logistic di erential equation with two di erent dev
lays, linear and non-linear system of di erential equations.
Chapter four.
In this chapter, numerical studies for the fractional order di erential equations using
di erent classes of nite di erence methods (FDM) such as the Crank-Nicolson
FDM (C-N-FDM) using the Grunwald-Letnikov^s de nition are presented. A discrete
approximation to the fractional derivative D is applied. C-N-FDM is used to study
some real life models problems, such as, the fractional Riccati di erential equation,
fractional logistic equation, time fractional di usion and time fractional wave equations.
The obtained results from the proposed method are compared with the result
using Pade-variational iteration method (Pade-VIM) and VIM respectively. Stability
analysis and the truncation error are presented for time fractional di usion and time
fractional wave equations. In the end linear and nonlinear system of the fractional
di erential equation are solved using C-N-FDM.
Chapter ve.
In this chapter, Chebyshev, Legendre collocation method and nite di erence method
are implemented to solve the fractional di erential equation generated by optimization
problem. Also, we introduced Legendre collocation method for solving a wide class of
fractional optimal control problems. In the end of this chapter, we apply the classical
control theory to a fractional di usion equation in a bounded domain. Some theorems
and lemmas to study the existence and the uniqueness of the solution of the fractional
di usion equation in a Hilbert space are given in [62].
Chapter six.
Conclusions and Future Research.