An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations

Abstract

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation method to reduce the equations into a system of equations. Error bounds of the presented method are obtained. Five test problems are solved. The obtained numerical results show the accuracy, efficiency, and applicability of the proposed approach.

Keywords

• the mixed Riemann–Liouville integral
• fixed-point theorems
• shifted Jacobi polynomials
• operational matrices
• collocation method
• error bound