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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
1. | Unité de recherche : Multifractales et Ondelettes, Faculté des Sciences de Monastir, Av. de l'environnement 5019 Monastir, Tunisia |
References:
[1] |
C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.
doi: 10.1016/0022-0396(89)90176-9. |
[2] |
J. L. Bona, M. Chen and J.-C Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
[3] |
J. L. Bona, F. Demengel and K. Promislow, Fourier splitting and dissipation of nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 477-502.
doi: 10.1017/S0308210500021478. |
[4] |
M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with a nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst., 27 (2010), 1473-1492.
doi: 10.3934/dcds.2010.27.1473. |
[5] |
M. Chen and O. Goubet, Long-time asymptotic behavior of 2D dissipative boussinesq system, Discrete Contin. Dyn. Syst., 17 (2007), 509-528.
doi: 10.3934/dcds.2007.17.509. |
[6] |
S. Dumont and J.-B Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, 10 (2013), 333-349. |
[7] |
D. Dutykh, Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids, 28 (2009), 430-443.
doi: 10.1016/j.euromechflu.2008.11.003. |
[8] |
D. Dutykh and F. Dias, Viscous potential free surface flows in a fluid layer of finite depth, C.R.A.S, Série I, 345 (2007), 113-118.
doi: 10.1016/j.crma.2007.06.007. |
[9] |
, tm., ().
|
[10] |
A. C Galucio, J.-F Deü and F. Dubois, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605.
doi: 10.1177/1077546307087427. |
[11] |
A. C Galucio, J.-F Deü, S. Mengué and F. Dubois, An adaptation of the Gear scheme for fractional derivatives, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6073-6085.
doi: 10.1016/j.cma.2005.10.013. |
[12] |
O. Goubet and G. Warnault, Decay of solutions to a linear viscous asymptotic model for waterwaves, Chinese Ann. Math. Ser. B, 31 (2010), 841-854.
doi: 10.1007/s11401-010-0615-2. |
[13] |
N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884, Springer-Verlag, Berlin, 2006.
doi: 10.1007/b133345. |
[14] |
T. Kakutani and M. Matsuuchi, Effect of viscosity on long gravity waves, J. Phys. Soc. Japan, 39 (1975), 237-246.
doi: 10.1143/JPSJ.39.237. |
[15] |
P. Liu and A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech., 520 (2004), 83-92.
doi: 10.1017/S0022112004001806. |
show all references
References:
[1] |
C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49.
doi: 10.1016/0022-0396(89)90176-9. |
[2] |
J. L. Bona, M. Chen and J.-C Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318.
doi: 10.1007/s00332-002-0466-4. |
[3] |
J. L. Bona, F. Demengel and K. Promislow, Fourier splitting and dissipation of nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 477-502.
doi: 10.1017/S0308210500021478. |
[4] |
M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with a nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst., 27 (2010), 1473-1492.
doi: 10.3934/dcds.2010.27.1473. |
[5] |
M. Chen and O. Goubet, Long-time asymptotic behavior of 2D dissipative boussinesq system, Discrete Contin. Dyn. Syst., 17 (2007), 509-528.
doi: 10.3934/dcds.2007.17.509. |
[6] |
S. Dumont and J.-B Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, 10 (2013), 333-349. |
[7] |
D. Dutykh, Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids, 28 (2009), 430-443.
doi: 10.1016/j.euromechflu.2008.11.003. |
[8] |
D. Dutykh and F. Dias, Viscous potential free surface flows in a fluid layer of finite depth, C.R.A.S, Série I, 345 (2007), 113-118.
doi: 10.1016/j.crma.2007.06.007. |
[9] |
, tm., ().
|
[10] |
A. C Galucio, J.-F Deü and F. Dubois, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605.
doi: 10.1177/1077546307087427. |
[11] |
A. C Galucio, J.-F Deü, S. Mengué and F. Dubois, An adaptation of the Gear scheme for fractional derivatives, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6073-6085.
doi: 10.1016/j.cma.2005.10.013. |
[12] |
O. Goubet and G. Warnault, Decay of solutions to a linear viscous asymptotic model for waterwaves, Chinese Ann. Math. Ser. B, 31 (2010), 841-854.
doi: 10.1007/s11401-010-0615-2. |
[13] |
N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884, Springer-Verlag, Berlin, 2006.
doi: 10.1007/b133345. |
[14] |
T. Kakutani and M. Matsuuchi, Effect of viscosity on long gravity waves, J. Phys. Soc. Japan, 39 (1975), 237-246.
doi: 10.1143/JPSJ.39.237. |
[15] |
P. Liu and A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech., 520 (2004), 83-92.
doi: 10.1017/S0022112004001806. |
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