NEW SPECTRAL APPROACHES FOR SOLVING VARIOUS TYPES OF FRACTIONAL DIFFERENTIAL EQUATIONS USING JACOBI POLYNOMIALS

Abstract

The main aim of this thesis is to investigate the features of the Galerkin spectral methods to construct new approaches for solving various types of linear fractional differential equations based on shifted Jacobi polynomials. In addition, we introduce and develop new Galerkin algorithms based on shifted Legendre polynomials technique for solving different types of ordinary and partial delay fractional differential equations. Moreover, we aim to construct novel operational approaches for multi-dimensional time-fractional partial differential equations using Galerkin spectral method. Furthermore, the convergence and error analysis of the suggested approaches are discussed. In Chapter 1, we present a general introduction to the fractional calculus and spectral methods with their advantages over the standard numerical methods. We also clarify the differences between the three most commonly used spectral methods, namely, the Galerkin, collocation and tau methods. Furthermore, the orthogonal polynomials and their properties are introduced. As well as some general properties and important relations concerned with the Jacobi polynomials and shifted Jacobi polynomials are considered. The primary point of Chapter 2 is to execute a new modified Galerkin algorithm based on the shifted Legendre polynomials for solving fractional differential equations and a system of fractional differential equations governed by homogeneous and nonhomogeneous initial and boundary conditions. In addition, we apply the new algorithm for solving fractional partial differential equations with Robin boundary conditions and the time-fractional telegraph equations. The key thought for obtaining such an algorithm depends on choosing trial functions satisfying the underlying initial and boundary conditions of such problems. Some illustrative examples are discussed to ascertain the validity and efficiency of the proposed algorithm. Also, some comparisons with some other existing spectral methods in the literature are made to highlight the superiority of the new algorithm.