Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations

Abstract

In this paper, a class of nonlinear two-dimensional (2D) integral equations of Volterra type, i.e. Volterra-Urysohn integral equations, is studied. Following the ideas of [24], and assuming that the kernels of the integral equation are Lipschitz functions with respect to the dependent variable, the existence and uniqueness of a solution to the integral equation is shown by a technique based on the Picard iterative method. Then, the Euler and trapezoidal discretization methods are used to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. It is proved that the solution of the Euler method has first order convergence to the exact solution of the integral equation while the solution of the trapezoidal method has quadratic convergence. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Some numerical examples are given which confirm our theoretical results.