On the solution of Fredholm-Volterra integral equation with discontinuous kernel in time.

Abstract

The existence of a unique solution of Fredholm-Volterra integral equation (F-VIE) under certain conditions, are discussed and proved. The Fredholm integral term (FIT)  is considered in position with continuous kernel, while the Volterra integral term (VIT) in time with singular kernel. Using a numerical method, the F-VIE is transformed to a linear system of Volterra integral equations (LSVIEs). Then after using Toeplitz matrix method (TMM), we have a linear algebraic system (LAS). Finally, two applications are given, numerical results are obtained, and the error, in each case, is calculated.