Numerical Solution of Volterra Integral Equations of the First Kind with Piecewise Continuous Kernel

Integral equations are in the core of many mathematical models in physics, economics and ecology. Volterra integral equations of the first kind with jump discontinuous kernels play important role in such models and they are considered in this article. Regularization method and sufficient conditions are derived for existence and uniqueness of the solution of such integral equations. An efficient numerical method based on the mid-rectangular quadrature rule is proposed for these equations with jump discontinuous kernels. The accuracy of proposed numerical method is $\mathcal{O}(N^{-1})$. The model examples demonstrate efficiency of proposed method: errors, two mesh differences and orders of convergent.