A new approach for solving integro-differential equations of variable order

In this work, we consider a class of nonlinear integro-differential equations of variable-order. Existence, uniqueness and stability results are discussed. For solving the considered equations, operational matrices based on the shifted Legendre polynomials are used. First, we approximate the unknown function and its derivatives in terms of the shifted Legendre polynomials. Then, by substituting these approximations into the equation and using the properties of the shifted Legendre polynomials together with the collocation points, the main problem is reduced to a system of nonlinear algebraic equations. An error bound is proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are included to show the efficiency and accuracy of the proposed scheme. (C) 2020 Elsevier B.V. All rights reserved.