A qualitative study and numerical simulations for a time-delayed dispersive equation
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A qualitative study and numerical simulations for a time-delayed dispersive equation Kaïs Ammari1
· Boumediène Chentouf2
· Nejib Smaoui2
Received: 13 August 2020 / Revised: 4 October 2020 / Accepted: 9 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided. Keywords Nonlinear dispersive equation · Time-delay · Stability · Numerical simulations Mathematics Subject Classification 35L05 · 35M10
1 Introduction The qualitative and numerical analysis of nonlinear dispersive equations has attracted the attention of a huge number of authors from various disciplines. This is due to the fact that such equations describe miscellaneous physical phenomena, such as surface water waves in shallow water [22,38], turbulent states in a distributed chemical reaction system and plane flame propagation [40,52], propagation of ion-acoustic waves in plasma, and pressure waves in liquid–gas bubble mixture [27,30,37,41,65–67].
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Boumediène Chentouf [email protected] Kaïs Ammari [email protected] Nejib Smaoui [email protected]
1
UR Analysis and Control of PDEs, UR 13ES64, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia
2
Department of Mathematics, Faculty of Science, Kuwait University, 13060 Safat, Kuwait
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K. Ammari et al.
It is worth noting that the nonlinearity in the equations governing the models mentioned above makes the mathematical problem more challenging, and its analysis often requires elaborate techniques. The situation is even more complicated when a timedelay occurs in the equation (see for instance [1,3–5,8–11,28,34,46] for other types of physical systems and [63,64] for a general class of fourth-order delay differential equations). One particular dispersive equation is the nonlinear partial differential equation (PDE) known in literature as the Korteweg–de Vries–Burgers (KdVB) equation in a bounded interval Ut (x, t) − ω Uxx (x, t) + U (x, t)Ux (x, t) + λ [Uxxx (x, t) + Ux (x, t) + b(x)U (x, t)] = 0, (x, t) ∈ Q. In the above equation and throughout this article, x is the space variable, t is the time, U is the amplitude of the wave, Q = (0, ν)×(0, +∞), where ν represents a length of the space variable. Additionally, ω and λ are positive physical parameters, while b(x) is a given nonnegative function. The above equation exhibits the properties of dispersion and dissipation, and has been widely used to describe a number of physical parameters such as unidimensional propagation of small waves
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