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

Abstract

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.