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In this work, we build a new numerical method to approximate the solution of Volterra’s nonlinear integro-differential equation. This method needs fewer conditions to converge, compared to the direct Nytröm method. Numerical tests show its efficiency. This new method is more practical and compatible with the Volterra nonlinear integro-differential equation.

In this article, our study deals with the existence and the uniqueness of the solution of a second degree integro-differential nonlinear Volterra equation with a weakly singular kernel, i.e., the solution depends on its speed (first derivative) and its acceleration (second derivative); whereas using Nyström method and product integration method with piecewise projection, we approximate this solution. Keywords Volterra equation • Integro-differential • Fixed point • Nonlinear equation • Product integration method Mathematics Subject Classification 45D05 • 45G05 • 45J99 • 45E99 • 65R20 Recently sps1 (2020) published a physical model explaining seismic phenomenon in deterministic manner. They managed to construct numerical approximations with the reality. In their model, they proved that the seismic function which models the above-mentioned phenomena is the unique solution of the Volterra nonlinear integro-differential equation of the form: Communicated by Hui Liang.

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