Spectral Solutions of Differential Equations via Some New Classes of Orthogonal Polynomials and Special Functions

Abstract

Abstract In this thesis, we propose new efficient spectral techniques for handling certain types of partial differential equations and partial fractional differential equations such as; the one-dimensional linear hyperbolic partial differential equation of the first-order, the fractional diffusion wave equation, the heat conduction equation, the fractional initial-value problem, the time-fractional partial differential problem, the telegraph equation, and the nonlinear fractional Rayleigh-Stokes equation. In these techniques, we employ new basis functions of the shifted Chebyshev polynomials of the fifth and sixth kinds. The key idea of the presented techniques depends on transforming these equations with their underlying conditions into systems of algebraic equations in the unknown expansion coefficients. Our studies are supported by a careful convergence analysis of the suggested shifted fifth and sixth kinds Chebyshev expansions. Finally, some numerical examples are presented to confirm the accuracy and efficiency of the proposed techniques. We want to mention that this thesis consists of seven chapters and all the results of chapter two up to chapter seven are completely new. Some results obtained in this thesis are already published in four international prestigious journals with high impact factors and some others are submitted for publication and still under referring.