In this paper, under certain conditions, we prove the existence of a unique solution of Volterra-Fredholm integral equation of the second kind with singular kernel in the space    using Banach fixed point theorem. Then, using a numerical method, we have a system of Fredholm integral equations. Therefore, some important theorems must be discussed. Moreover, Toeplitz matrix method is used, to obtain a nonlinear algebraic system, where the solution can be discussed. The estimate error of the numerical method is established and computed.

Volterra-Fredholm integral equation (V-FIE), nonlinear algebraic system (NAS), singular kernel, Toeplitz matrix method.