الفهرس 
BASIC DEFINITIONS IN FRACTIONAL CALCULUSOnly 14 pages are availabe for public view
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Abstract
Summary The main aim of this thesis is to develop and implement spectral
wavelets numerical algorithms for solving systems of ordinary/fractional differential, integrodifferential linear/nonlinear equations in addition to some applications
in physics and biology, whenever possible we discuss in details the convergence of
the presented techniques, also we compare our results with other existing techniques in literature.
The thesis is organized as follows:
Chapter one:
We give a detailed introduction to numerical methods for solving differential equations, orthogonal polynomials and wavelets and some mathematical tools are given.
Chapter two:
A new numerical scheme is presented to solve systems of integrodifferential equations are discussed. The derivation of this the scheme is essentially based on constructing the shifted second kind Chebyshev wavelets collocations methods. One
of the main advantages of the presented scheme is its availability for application
on both linear and nonlinear systems of integrodifferential equations. Another
advantage of the developed scheme is that high accurate approximate solutions
are achieved using a few number of the second kind Chebyshev wavelets. The
obtained numerical results are comparing favourably with the analytical ones.
Chapter three:
A numerical scheme is presented to solve a system of fractional differential equation is discussed. The derivation of this scheme is essentially based on constructing
the shifted third kind Chebyshev wavelets collocations methods. One of the main
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advantages of the presented scheme is its availability for application on both linear and nonlinear systems of fractional differential equations. Another advantage
of the developed scheme is that high accurate approximate solutions are achieved
using a few number of the third kind Chebyshev wavelets. The obtained numerical
results are comparing favourably with the analytical ones.
Chapter four:
This chapter is devoted to implementing a Chebyshev wavelets method for obtaining the numerical solution for one of the well-known fractional SIRC, PreyPredator models. In the SIRC case study, we concerned to discuss and study the
effect of the rate of progression from infective to recovered per one year to the
population density functions for various fractional Brownian motions and also for
standard motion N = 1. We compared the obtained solutions with those obtained
using 4th Runge-Kutta method. from this comparison, we can conclude that the
obtained numerical solution using the suggested wavelets method is in complete
agreement with the numerical solution using RK4 method.
Chapter five:
In this chapter, four algorithms for obtaining numerical spectral wavelets solutions for Telegraph Equation and KdV, Burger type equation were analyzed and
discussed. Chebyshev polynomials of four kinds are used. One of the advantages
of the developed algorithms is high accurate approximate solutions are achieved
using a few number of the Chebyshev wavelets. The obtained numerical results
are comparing favourably with the analytical ones.