Approximate solution of nonlinear fractional integro-differential equations using fractional alternative Legendre functions

In this paper, a new set of functions called fractional alternative Legendre is defined for solving nonlinear fractional integro-differential equations. The concept of the fractional derivative in this problem is in the Caputo sense. To solve the problem, first we obtain the operational matrix of the Riemann–Liouville fractional integration of these functions for the first time. Then, this matrix and collocation method are utilized to reduce the solution of the nonlinear fractional integro-differential equations to a system of algebraic equations. Also, the convergence analysis of the proposed method is investigated. Finally, some examples are included to demonstrate the validity and applicability of the approach.