New algorithms for solving nonlinear mixed integral equations

In this article, the existence and unique solution of the nonlinear Volterra-Fredholm integral equation (NVFIE) of the second kind is discussed. We also prove the solvability of the second kind of the NVFIE using the Banach fixed point theorem. Using quadrature method, the NVFIE leads to a system of nonlinear Fredholm integral equations (NFIEs). The existence and unique numerical solution of this system is discussed. Then, the modified Taylor's method was applied to transform the system of NFIEs into nonlinear algebraic systems (NAS). The existence and uniqueness of the nonlinear algebraic system's solution are discussed using Banach's fixed point theorem. Also, the stability of the modified error is presented. Some numerical examples are performed to show the efficiency and simplicity of the presented method, and all results are obtained using Wolfram Mathematica 11.