The Müntz-Legendre Tau Method for Fractional Differential Equations

The principle result of this paper is the following operational Tau method based upon Müntz-Legendre polynomials. This methodology provides a computational technique for numerical solution of fractional differential equations by using a sequence of matrix operations. The main property of Müntz polynomials is that fractional derivatives of these polynomials can be expressed in terms of the same polynomials directly that is a fundamental property in the Tau solution of the functional equations. The fractional derivatives are described in the Caputo type. Numerical solvability of obtained algebraic system has also been discussed. The illustrative examples are provided to demonstrate the applicability and simplicity of the proposed numerical scheme. Our obtained results are compared with the some existing numerical methods on the subject and superiority of our proposed scheme is confirmed. In addition, numerical results are approved decisive preference of the Tau approximation of the fractional differential equations using Müntz-Legendre polynomials in compared with the classical orthogonal polynomials.