The operational Tau method, a well known method for solving functional equations is employed to approximate the solution of nonlinear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. The unique solvability of the linear Tau algebraic system is discussed. In addition, we provide a rigorous convergence analysis for the Legendre Tau method which indicate that the proposed method converges exponentially provided that the data in the given FIDE are smooth. To do so, Sobolev inequality with some properties of Banach algebras are considered. Some numerical results are given to clarify the efficiency of the method.

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.

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In this paper, the spectral collocation method is investigated for the numerical solution of multi-order Fractional Differential Equations (FDEs). We choose the orthogonal Jacobi polynomials and set of Jacobi Gauss–Lobatto quadrature points as basis functions and grid points respectively. This solution strategy is an application of the matrix-vector-product approach in spectral approximation of FDEs. The fractional derivatives are described in the Caputo type. Numerical solvability and an efficient convergence analysis of the method have also been discussed. Due to the fact that the solutions of fractional differential equations usually have a weak singularity at origin, we use a variable transformation method to change some classes of the original equation into a new equation with a unique smooth solution such that, the spectral collocation method can be applied conveniently. We prove that after this regularization technique, numerical solution of the new equation has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method.