An efficient computational method for nonlinear mixed Volterra–Fredholm integral equations

In this study, an efficient hybrid method based on the trapezoidal rule formula and block-by-block scheme is developed to solve a class of nonlinear mixed Volterra–Fredholm integral equations. In these problems, the Volterra and Fredholm kernels are dependent on the time and space variables, respectively. The proposed algorithm uses the trapezoidal rule formula to approximate the Fredholm kernel and block-by-block technique to approximate the Volterra kernel. This strategy is able to simultaneously find several approximations to solve the problem under consideration at a set of nodes without asking for starting values. The convergence analysis of the proposed procedure is discussed, and the validity and efficiency of this scheme are illustrated by solving some numerical examples.