On the mixed nonlinear integro-differential equations with weakly singular kernel

In this paper, we study the solution existence and uniqueness of a nonlinear Volterra-Fredholm integro-differential equation with weakly singular kernel. The singular part appears in the Volterra integral when we derive the equation. To construct a numerical approximation of the solution we use numerical methods based on the approximation of the integral as the Nystrom method if the integrand is regular and the product integration method with piecewise linear approximation for the weakly singular case. The convergence order depends on the coefficient of the solution and the order of regularity of the regular kernel. Finally, we build numerical example to ensure the validity and applicability of our method.