الفهرس | Only 14 pages are availabe for public view |
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 v 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. |