Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.