A new approach for solving nonlinear Volterra integro-differential equations with Mittag--Leffler kernel

In this work, we consider a general class of nonlinear Volterra integro-differential equations with Atangana–Baleanu derivative. We use the operational matrices based on the shifted Legendre polynomials to obtain numerical solution of the considered equations. By approximating the unknown function and its derivative in terms of the shifted Legendre polynomials and substituting these approximations into the original equation and using the collocation points, the original equation is reduced to a system of nonlinear algebraic equations. An error estimate of the numerical solution is proved. Finally, some examples are included to show the accuracy and validity of the proposed method.