Fractional Differential Equations and Chebyshev Polynomials and Their Numerical Solution

Chebyshev Polynomials and their Numerical Solution

Abstract

In this paper we present an advanced set of polynomials that can be generalized to Chebyshev polynomials. Some basic properties of Chebyshev polynomials and their variables, as well as formulas related to generalized polynomials, will be presented. Orthogonal polynomials are used to solve linear polynomial fractional differential equations (FDEs) which arise from many applications. The proposed algorithm was deduced using a novel method of the power formula for Chebyshev polynomials and the Galerkin formula. The method transforms FDE differential equations with initial or boundary conditions into a system of linear equations that can be solved efficiently and accurately by solving the appropriate numerical solution. The paper includes some examples and comparisons with other methods to prove the effectiveness and usefulness of the proposed algorithm