Volterra-renewal integral equations: a combined simplified numerical approach

This paper presents a novel numerical approach for solving Volterra-renewal integral equations, which arise in various elds, including biology, engineering, and economics. Traditional treatment of the Volterra-renewal integral equations systems of this type utilises computationally large iterative algorithms. In order to tackle these limitations, we propose a hybrid method that is numerical in nature whereby an analytic and discretization techniques are used to obtain accurate and efficient solutions. Several test problems have been solved and the peculiarities of this method demonstrated, its possibility to solve a wide class of renewal-type problems is also presented. We have addressed a number of obstacles, such as managing nonlinearities and memory effects over extended periods of time, by using the Picard approach to provide a more accurate numerical approximation. We illustrate the benets of the proposed methods over closed-form solutions using numerical examples.