In this paper, the properties of the hybrid Functions which consist of Block-Pulse Functions plus Legendre polynomials are presented. Then, integral equations are converted into an algebraic system by hybrid of general Block-Pulse Functions and the Legendre polynomials. In continue, approximate solutions of integral equations are derived, finally the numerical examples are included to demonstrate the validity and applicability of the algorithm.

In this work, a new simple direct method to solve nonlinear Fredholm-Volterra integral equations is presented. By using Block-Pulse (BP) Functions, their operational matrices and Taylor expansion a nonlinear Fredholm-Volterra integral equation converts to a nonlinear system. Some numerical examples illustrate accuracy and reliability of our solutions.

In this paper, an effective numerical method is introduced for the treatment of nonlinear two-dimensional Volterra integro-differential equations. Here, we use the so-called two-dimensional Block-Pulse Functions.First, the two-dimensional Block-Pulse operational matrix of integration and differentiation has been presented. Then, by using these matrices, the nonlinear two-dimensional Volterra integro-differential equation has been reduced to an algebraic system. Some numerical examples are presented to illustrate the effectiveness and accuracy of the method.