Difference derivative for an integro-differential nonlinear Volterra equation

In this article, we propose a new numerical approximation method to deal with the unique solution of the nonlinear integro-differential Volterra equation. We are interested in a very particular form of this equation, in which the derivative of the sought solution appears under the integral sign in a nonlinear manner. Our vision is based on two different approaches: We use the Nyström method to transform the integral into a finite sum using a numerical integration formula, then we use the numerical backward difference derivative method to approach the derivative of our solution. This collocation between two different methods, the first outcome of the numerical processing of integral equations and the second outcome of the numerical processing of differential equations, gives a new nonlinear system for approaching the solution of our equation. We show that the system has a unique solution and that this numerical solution converges perfectly to our solution. A section is dedicated to numerical tests, in which we show the effectiveness of our new vision compared to two methods based only on numerical integration.