November  2014, 19(9): 2837-2863. doi: 10.3934/dcdsb.2014.19.2837

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

Received  June 2013 Revised  March 2014 Published  September 2014

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.
Citation: Imen Manoubi. 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. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837
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.

[1]

Min Chen, S. Dumont, Louis Dupaigne, Olivier Goubet. Decay of solutions to a water wave model with a nonlocal viscous dispersive term. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1473-1492. doi: 10.3934/dcds.2010.27.1473

[2]

Yong Zhou. Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 525-532. doi: 10.3934/dcds.2006.14.525

[3]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267

[4]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

[5]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039

[6]

Moulay Rchid Sidi Ammi, Mostafa Tahiri, Delfim F. M. Torres. Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 621-637. doi: 10.3934/dcdss.2021155

[7]

Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021041

[8]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021

[9]

Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038

[10]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082

[11]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851

[12]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations and Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021

[13]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721

[14]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076

[15]

Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029

[16]

Zbigniew Gomolka, Boguslaw Twarog, Jacek Bartman. Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem. Conference Publications, 2011, 2011 (Special) : 505-514. doi: 10.3934/proc.2011.2011.505

[17]

Shakir Sh. Yusubov, Elimhan N. Mahmudov. Optimality conditions of singular controls for systems with Caputo fractional derivatives. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021182

[18]

Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004

[19]

Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2022002

[20]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]