Approximate Spectral Solutions for Certain Types of Differential Equations Using Special Polynomials

Abstract

Abstract The thesis, in its entirety, consists of three chapters. In the first chapter, an introduction to orthogonal functions focusing on Jacobi polynomials is given. In addition, it provides a brief overview of spectral methods, namely, tau, col- location, and Galerkin. This required the presentation of definitions, concepts, relationships, and theories that were employed to serve the needs of the cal- culations done in the thesis. The second chapter presents two new types of orthogonal polynomials called shifted third and fourth kinds of Chebyshev poly- nomials and their use in constructing operational matrices of derivatives. They have been used to find spectral solutions to some boundary value problems of second-order. The effectiveness of the method was achieved in finding nu- merical results for some linear and non-linear equations and the Lane-Emden type equation. Finally, the third chapter employs the shifted Jacobi-Galerkin method to solve the telegraph equations. We needed to deduce relations for shifted Jacobi polynomials, build new basis functions, and evaluate matrices elements. Also, it discusses the study of convergence and error analysis of the method.