Legendre polynomials approximation method for solving Volterra integral equations of the first kind with discontinuous kernels

The Volterra integral equations with discontinuous kernels have an essential role in the theory of evolving dynamical systems in economics, ecology and energetics. A new numerical scheme for solving Volterra integral equations of the first kind with piecewise continuous kernels presented in this paper. This type of equation has been little studied so far and the use of operational matrices for this kind of equation is a new and cost- efficient technique. Shifted Legendre orthogonal polynomials are used to solve Volterra integral equations with piecewise continuous kernels by transforming the main equation in to a system of linear algebraic equations. The error estimation is presented by using bounded operator and some numerical examples are illustrated to show the accuracy and efficiency of the presented method.