Posteriori error estimates for the nonlinear Volterra-Fredholm integral equations

We study the numerical approximation of the nonlinear Volterra-Fredholm integral equations by combining the discrete time collocation method [1] and the new formulation of Kumar and Sloan [2], which converts an integral equation of the conventional Hammerstein form into a conductive form for approximation by a collocation method. The intrinsic merit of this alternative formulation lies in its computational savings. Posteriori error estimates of the method for two typical nonlinearities (i.e., algebraic and exponential nonlinearity) are obtained. Some remarks on the generalization of the method to higher-dimensional cases are offered, and finally some numerical examples are given.