Collocation method for solving two-dimensional nonlinear Volterra-Fredholm integral equations with convergence analysis

This paper presents a method for solving the two-dimensional nonlinear Volterra- Fredholm integral equation. The main idea of the method is to use the Lagrange interpolation function to approximate the unknown solution and the Legendre-Gauss quadrature formula to approximate the integral. The advantage of the method is that it requires relatively few collocation points to obtain a relatively small error and does not require the calculation of integrals. Under certain sufficient conditions, the existence and uniqueness of the original equation are given. In addition, the existence and uniqueness of the solutions of the discrete equations are given using the theory of compact operators. The convergence analysis and error estimates of the method are also derived. Finally, several numerical examples are used to demonstrate its efficiency and accuracy.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper presents a method for solving the two-dimensional nonlinear Volterra-Fredholm integral equation using Lagrange interpolation and Legendre-Gauss quadrature formula. The method has the advantage of requiring few collocation points, resulting in small errors, and eliminates the need for integral calculation. It provides proofs for the existence and uniqueness of the original equation under certain conditions, as well as for the solutions of the discrete equations using compact operators theory. Furthermore, the convergence analysis and error estimates are derived, and numerical examples are provided to demonstrate its efficiency and accuracy.