On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term
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On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term Mourad Ghiat1 · Hamza Guebbai1
· Muhammet Kurulay1 · Sami Segni1,2
Received: 20 April 2020 / Revised: 8 June 2020 / Accepted: 20 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
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
1 Introduction 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.
B
Hamza Guebbai [email protected] ; [email protected] Mourad Ghiat [email protected] ; [email protected] Muhammet Kurulay [email protected] ; [email protected] Sami Segni [email protected] ; [email protected]
1
Université 8 Mai 1945 Guelma Laboratoire de Mathématiques Appliquées et de Modélisation, BP 401, 24000 Guelma, Algeria
2
Yildiz Teknik Universitesi, Istanbul, Turkey 0123456789().: V,-vol
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∀t ∈ [0, T ] , u(t) =
t
(t − s) K t, s, u(s), u (s) ds + f (t),
0
where u is the seismic function to be found in C 1 (0, T ) and f is deduced from the physical model such as f ∈ C 1 (0, T ). This kind of integro-differential equation in which the solution derivative appears under the integral sign in a nonlinear way caught most mathematicians attention in both analytical and numerical directions, since the physical models that give rise to these equations play a large role in the understanding of different natural phenomena, such as: the evolution of forests, cellular behavior, mechanical phenomena, and repetitive movement. The peculiarity of these equations lies in the fact that they manage to explain the evolution of the phenomenon when this latter is related to its speed in a nonlinear manner. In addition to that, because of its integration form, it takes into consideration the behavior in full of this evolution. In this article, we are interested in a new generalization of this equation. Let us assume that the solution depends in a nonlinear way on its velocity (first derivative) and its acceleration (second derivative) appearing under the integral. The problem is presented in the following form: For a given g
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