Numerical Solution of Fractional Order Integro-Differential Equations via Mntz Orthogonal Functions

In this paper, we derive a spectral collocation method for solving fractional-order integro-differential equations by using a kind of Muntz orthogonal functions that are defined on 0,1 and have simple and real roots in this interval. To this end, we first construct the operator of Riemann-Liouville fractional integral corresponding to this kind of Muntz functions. Then, using the Gauss-Legendre quadrature rule and by employing the roots of Muntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. We also construct an upper bound for the truncation error of Muntz orthogonal functions, and we analyze the error of the proposed collocation method. Numerical examples are included to demonstrate the validity and accuracy of the method.