On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term

Abstract

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.

Block-by-block method for solving non-linear Volterra integral equation of the first kind

“…The numerical treatment that starts with the discretization of the equation its consequence is a non-linear algebraic system. It is necessary to solve this system by Newton's method and the best choice of initial point is u 0 = f (0) (see Ghiat et al 2021;Ghiat and Guebbai 2018;Ghiat et al 2020;Touati et al 2019;Deepa et al 2022;Linz 1987).…”

“…Also, we mention some of the models that have emerged recently, such as Fredholm and Volterra integro-differential equations, Fredholm integral equations and Volterra–Fredholm equations (Ghiat et al. 2020 ; Ghiat and Guebbai 2018 ; Ghiat et al. 2021 ; Touati et al.…”

“…There are different types of Volterra integral equations, like the non-linear weakly singular equation of the second kind which is analyzed in Micula (1862). Also, we mention some of the models that have emerged recently, such as Fredholm and Volterra integro-differential equations, Fredholm integral equations and Volterra-Fredholm equations (Ghiat et al 2020;Ghiat and Guebbai 2018;Ghiat et al 2021;Touati et al 2019). But the important question is how to search for a solution to these equations?…”

Application of Adomian polynomials for solving nonlinear integro-differential equations

“…Many of these equations have already been studied in earlier papers on related topics. Among them, the equations with a weakly singular kernel in the network studies [10,21], in the non-linear Fredholm form [4,31], in the COVID-19 researches [29], in the non-linear Volterra-Fredholm form [28,32], in the non-linear Volterra form [16,17,30], in the linear Fredholm form [2,19,27], and others.…”

Solving linear Fredholm integro-differential equation by Nyström method

“…However, this method was applied in the first state to approximate the solution of an integral equation and its theoretical framework is well given by Atkinson and Han in [13]. Recently, many others have relied on this method such as H. Guebbai et al in [14,15], H. Zhou and Q. Wang [16], S. Segni et al [17], M. Ghiat et al [18], S. Salah et al [19], Lemita and Guebbai [20]: to give an estimation solution of the non-linear integro-differential Volterra equation with a continuous kernel, or a weakly singular kernel [21,22] or to approximate a solution of a Fredholm linear system.…”