This article proposes a simple efficient direct method for solving Volterra integral equation of the first kind. By using block-pulse functions and their operational matrix of integration, first kind integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. Some examples are presented to illustrate efficiency and accuracy of the proposed method.

Abstract-In this paper a moment method simulation of electromagnetic scattering problem is presented. An effective numerical method for solving this problem based on the method of moments and using block-pulse basis functions is proposed. Some examples of engineering interest are included to illustrate the procedure. The scattering problem is treated in detail, and illustrative computations are given for some cases. This method can be generalized to apply to objects of arbitrary geometry and arbitrary material.

Abstract-A new and effective direct method to determine the numerical solution of specific nonlinear Volterra-Fredholm integral and integro-differential equations is proposed. The method is based on vector forms of block-pulse functions (BPFs). By using BPFs and its operational matrix of integration, an integral or integro-differential equation can be transformed to a nonlinear system of algebraic equations.Some numerical examples are provided to illustrate accuracy and computational efficiency of the method. Finally, the error evaluation of this method is presented. The benefits of this method are low cost of setting up the equations without applying any projection method such as Galerkin, collocation, . . . . Also, the nonlinear system of algebraic equations is sparse.

a b s t r a c tAn effective direct method to determine the numerical solution of the specific nonlinear Volterra-Fredholm integro-differential equations is proposed. The method is based on new vector forms for representation of triangular functions and its operational matrix. This approach needs no integration, so all calculations can be easily implemented. Some numerical examples are provided to illustrate the accuracy and computational efficiency of the method.

a b s t r a c tMost integral equations of the first kind are ill-posed, and obtaining their numerical solution often leads to solving a linear system of algebraic equations of a large condition number. So, solving this system is difficult or impossible. For numerically solving Volterra integral equation of the first kind an efficient expansion-iterative method based on the block-pulse functions is proposed. Using this method, solving the first kind integral equation reduces to solving a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to solve any linear system of algebraic equations. To show the convergence and stability of the method, some computable error bounds are obtained. Numerical examples are provided to illustrate that the method is practical and has good accuracy.