Abstract
While the approximate solutions of one-dimensional nonlinear Volterra-Fredholm integral equations with smooth kernels are now well understood, no systematic studies of the numerical solutions of their multi-dimensional counterparts exist. In this paper, we provide an efficient numerical approach for the multi-dimensional nonlinear Volterra-Fredholm integral equations based on the multi-variate Legendre-collocation approach. Spectral collocation methods for multi-dimensional nonlinear integral equations are known to cause major difficulties from a convergence analysis point of view. Consequently, rigorous error estimates are provided in the weighted Sobolev space showing the exponential decay of the numerical errors. The existence and uniqueness of the numerical solution are established. Numerical experiments are provided to support the theoretical convergence analysis. The results indicate that our spectral collocation method is more flexible with better accuracy than the existing ones.