A computational method for a class of systems of nonlinear variable-order fractional quadratic integral equations

Abstract

This paper develops an accurate computational method based on the shifted Chebyshev cardinal functions (CCFs) for a new class of systems of nonlinear variable-order fractional quadratic integral equations (QIEs). In this way, a new operational matrix (OM) of variable-order fractional integration is obtained for these cardinal functions. In the proposed method, the unknown functions of a system of nonlinear variable-order fractional QIEs are approximated by the shifted CCFs with undetermined coefficients. Then, these approximations are substituted into the system. Next, the OM of variable-order fractional integration and the cardinal property of the shifted CCFs are utilized to reduce the system into an equivalent system of nonlinear algebraic equations. Finally, by solving this algebraic system an approximate solution for the problem is obtained. The main idea behind this approach is to reduce such problems to solving systems of nonlinear algebraic equations, which greatly simplifies the problem. Convergence of the presented method is investigated theoretically and numerically. Furthermore, the proposed approach is numerically evaluated by solving some test problems.