Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via three-dimensional block-pulse functions

We are concerned here with a three-dimensional nonlinear mixed Volterra–Fredholm integral equations of the second kind which include many key integral that appear in the theory of nonlinear parabolic boundary value problems. The existence of a unique solution will be proved. A new numerical method for solving these type of equations will be presented. The method is based upon three-dimensional block-pulse functions approximation. In addition convergence analysis of the method is discussed. Illustrative examples are included to demonstrate the validity and applicability of the technique.