The Müntz-Legendre Tau method for fractional differential equations

The main result obtained in this study is the following operational Tau method based on Müntz-Legendre polynomials. This method provides a computational technique for obtaining the numerical solutions of fractional differential equations using a sequence of matrix operations. The main property of Müntz polynomials is that fractional derivatives of these polynomials can be expressed directly in terms of the same polynomials, which is a fundamental property of the Tau solutions of functional equations. The fractional derivatives are described as the Caputo type. We also discuss the numerical solvability of the algebraic system obtained. Illustrative examples are provided to demonstrate the applicability and simplicity of the proposed numerical scheme. The results obtained are compared with those obtained by existing numerical methods, thereby confirming the superiority of our proposed scheme. In addition, our numerical results indicate a clear preference for the Tau approximation of the fractional differential equations using Müntz-Legendre polynomials compared with the classical orthogonal polynomials.