Spectral Solutions for Differential Equations via Special Polynomials

Abstract

The main objectives of this thesis can be summarized in the following points:

A survey study on special polynomials such as; generalized Fibonacci and Lucas polynomials.

A theoretical study on fractional di erential equations subject to initial and boundary conditions.

A comprehensive study on spectral methods, and in particular tau, col-location and Galerkin methods.

Finding explicit formulae for the operational matrices of derivatives for certain basis functions that satisfy the homogeneous initial and boundary conditions, on the problems under investigation.

Implementing spectral algorithms for handling some certain types of dif-ferential equations involving some speci c problems such as; fractional Bagley-Torvik equation and one-dimensional telegraph type equation..

Comparing our algorithms with some other algorithms to show the ac-curacy and the applicability of the proposed methods.