In this paper, under certain conditions, we prove the existence of a unique solution of mixed nonlinear integral equation of the second kind (MNIESK) with singular kernel in the space    using Banach fixed point theorem. Then, using two numerical methods, we have a nonlinear system of Fredholm integral equations (NLSFIEs). Moreover, we use Toeplitz matrix method and Product Nystrom method to obtain a nonlinear algebraic system (NLAS) which can be solved numerically. The error estimate is discussed. Some examples are solved numerically.

mixed integral equation (MIE), nonlinear system of Fredholm integral equations (NLSFIEs), nonlinear algebraic system (NLAS), singular kernel, Toeplitz matrix method, product Nystrom method.