Numerical Spectral Solutions for Some Differential Equations via Special Kinds of Polynomials

Abstract

The main objectives of this thesis can be summarized in the following points: • A survey study on orthogonal polynomials in general and on Jacobi and Gegenbauer polynomials in particular. • Collecting some important formulas concerned with the Gegenbauer polynomials. • A comprehensive study on spectral methods, and their celebrated methods, namely, tau, collocation and Galerkin methods. • Establishing operational matrices of derivatives of some basis functions. • Implementing spectral algorithms for handling second-order boundary value problems. • Implementing a numerical spectral algorithm for treating the hyperbolic- telegraph type equation. • Comparing our proposed algorithms with some other ones in the literature devoted to solving the same problems aiming to demonstrate their accuracy and applicability.