Representation of exact solution for the nonlinear Volterra–Fredholm integral equations

In this paper, the representation of the exact solution for the nonlinear Volterra–Fredholm integral equations will be obtained in the reproducing kernel space. The exact solution is given by the form of series. Its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained. The error of the approximate solution is monotone deceasing in the sense of ‖·‖W21[a,b]. The intrinsic merit of the method given in the paper lies in its speedy convergence. In addition, it must be pointed out that additional conditions as linearization or unjustified assumptions are not required for nonlinear cases.