Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative

Imen Manoubi

doi:10.3934/dcdsb.2014.19.2837

Article Contents

2014, Volume 19, Issue 9: 2837-2863. Doi: 10.3934/dcdsb.2014.19.2837

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Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative

Imen Manoubi^1,

1.
Unité de recherche : Multifractales et Ondelettes, Faculté des Sciences de Monastir, Av. de l'environnement 5019 Monastir

Received: May 31, 2013

Revised: February 28, 2014

Published: October 2014

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Abstract

In this paper, we study the water wave model with a nonlocal viscous term \begin{equation*} u_t + u_x + \beta u_{x x x} + \frac{\sqrt \nu}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds + u u_x = v u_{xx}, \end{equation*} where $\frac{1}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds $ is the Riemann-Liouville half derivative. We prove the well-posedness of the equation and we investigate theoretically and numerically the asymptotical behavior of the solutions. Also, we compare our theoretical and numerical results with those given in [4] for a similar equation.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q53.

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