Numerical solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations using shifted Jacobi operational matrices via collocation method

Abstract

In this paper, an efficient numerical method is presented to approximate solutions of two-dimensional nonlinear fractional Volterra and Fredholm integral equations. We derive new operational matrices of fractional-order integration and product based on two-variable shifted Jacobi polynomials. These operational matrices via shifted Jacobi collocation method are utilized to reduce the understudy equations to systems of linear or nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the error bound and convergence analysis of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing methods in the literature.