Taiwanese Journal of Mathematics

The Spectral Method for Long-time Behavior of a Fractional Power Dissipative System

Hong Lu and Mingji Zhang

Full-text: Open access

PDF File (419 KB)

Abstract

In this paper, we consider the fractional complex Ginzburg-Landau equation in two spatial dimensions with the dissipative effect given by a fractional Laplacian. The periodic initial value problem of the fractional complex Ginzburg-Landau equation is discretized fully by Galerkin-Fourier spectral method, and the dynamical behaviors of the discrete system are studied. The existence and convergence of global attractors of the discrete system are obtained by a priori estimates and error estimates of the discrete solution. The numerical stability and convergence of the discrete scheme are proved.

Article information

Source
Taiwanese J. Math., Volume 22, Number 2 (2018), 453-483.

Dates
Received: 26 October 2016
Revised: 24 August 2017
Accepted: 12 September 2017
First available in Project Euclid: 14 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1507946430

Digital Object Identifier
doi:10.11650/tjm/170902

Mathematical Reviews number (MathSciNet)
MR3780728

Zentralblatt MATH identifier
06965381

Subjects
Primary: 65M12: Stability and convergence of numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N35: Spectral, collocation and related methods

Keywords
fractional Ginzburg-Landau equation global attractor numerical stability spectral method

Citation

Lu, Hong; Zhang, Mingji. The Spectral Method for Long-time Behavior of a Fractional Power Dissipative System. Taiwanese J. Math. 22 (2018), no. 2, 453--483. doi:10.11650/tjm/170902. https://projecteuclid.org/euclid.twjm/1507946430


Export citation

References

  • C. Bu, On the Cauchy problem for the $1+2$ complex Ginzburg-Landau equation, J. Austral. Math. Soc. Ser. B 36 (1994), no. 3, 313–324.
    Zentralblatt MATH: 0829.35119
    Digital Object Identifier: doi:10.1017/S0334270000010468
  • L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425–461.
    Zentralblatt MATH: 1148.35097
    Digital Object Identifier: doi:10.1007/s00222-007-0086-6
  • C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982), no. 157, 67–86.
    Zentralblatt MATH: 0567.41008
    Digital Object Identifier: doi:10.1090/S0025-5718-1982-0637287-3
    Mathematical Reviews (MathSciNet): MR637287
  • W. Deng, B. Li, W. Tian and P. Zhang, Boundary problems for the fractional and tempered fractional operators, arXiv:1702.03639.
    arXiv: 1702.03639
    Mathematical Reviews (MathSciNet): MR3747454
    Digital Object Identifier: doi:10.1137/17M1116222
  • C. R. Doering, J. D. Gibbon, D. D. Holm and B. Nicolaenko, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity 1 (1988), no. 2, 279–309.
    Zentralblatt MATH: 0655.58021
    Digital Object Identifier: doi:10.1088/0951-7715/1/2/001
  • C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of the complex Ginzburg-Landau equation, Phys. D 71 (1994), no. 3, 285–318.
    Zentralblatt MATH: 0810.35119
    Digital Object Identifier: doi:10.1016/0167-2789(94)90150-3
  • J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl. 344 (2008), no. 2, 1005–1017.
    Mathematical Reviews (MathSciNet): MR2426328
    Digital Object Identifier: doi:10.1016/j.jmaa.2008.03.061
    Zentralblatt MATH: 1140.81357
  • V. L. Ginzburg, On superconductivity and superfluidity (what I have and have not managed to do), as well as on the `physical minimum' at the beginning of the 21st century, Reviews of Mondern Physics 76 (2004), 981–998.
  • V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Russian in Zh. Eksp. Teor. Fiz. (ZhETF) 20 (1950), 1064–1082.
  • B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput. 204 (2008), no. 1, 468–477.
    Zentralblatt MATH: 1163.35483
  • B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differential Equations 36 (2011), no. 2, 247–255.
    Zentralblatt MATH: 1211.35268
  • B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl. 361 (2010), no. 1, 131–138.
    Zentralblatt MATH: 1179.35320
    Digital Object Identifier: doi:10.1016/j.jmaa.2009.09.009
    Mathematical Reviews (MathSciNet): MR2567288
  • H. Lu, S. Lü and Z. Feng, Asymptotic dynamics of 2D fractional complex Ginzburg-Landau equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013), no. 12, 1350202, 12 pp.
    Mathematical Reviews (MathSciNet): MR3158317
    Zentralblatt MATH: 1284.35456
    Digital Object Identifier: doi:10.1142/S0218127413502027
  • S. Lü, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numer. Math. J. Chinese Univ. 22 (2000), no. 1, 1–9.
  • S. Lü and Q. Lu, Asymptotic behavior of three-dimensional Ginzburg-Landau type equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 (2006), no. 2, 209–220.
    Zentralblatt MATH: 1091.35034
  • ––––, Exponential attractor for the 3D Ginzburg-Landau type equation, Nonlinear Anal. 67 (2007), no. 11, 3116–3135.
    Zentralblatt MATH: 1162.35014
    Digital Object Identifier: doi:10.1016/j.na.2006.10.005
  • V. G. Mazja, Sobolev Spaces, Springer-Verlag, 1985.
  • R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Solidi. B 133 (1986), no. 1, 425–430.
  • K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Phys. D 41 (1990), no. 2, 232–252.
    Zentralblatt MATH: 0696.35177
    Digital Object Identifier: doi:10.1016/0167-2789(90)90125-9
    Mathematical Reviews (MathSciNet): MR1049129
  • A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), no. 4, 753–764.
    Zentralblatt MATH: 0933.37029
    Digital Object Identifier: doi:10.1063/1.166272
  • S. Salsa, Optimal regularity in lower dimensional obstacle problems, in Subelliptic PDE's and Applications to Geometry and Finance, 217–226, Lect. Notes Semin. Interdiscip. Mat. 6, Semin. Interdiscip. Mat. (S.I.M.), Potenza, 2007.
    Zentralblatt MATH: 1137.35083
  • S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: Theory and applications, New York: Gordon and Breach Science, 1987.
  • J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal. 38 (1990), no. 4, 201–209.
    Zentralblatt MATH: 0684.65095
    Digital Object Identifier: doi:10.1080/00036819008839963
  • M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature 363 (1993), no. 6424, 31–37.
  • Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), no. 6, 1842–1864.
    Zentralblatt MATH: 1163.35019
    Digital Object Identifier: doi:10.1016/j.jfa.2009.01.020
    Mathematical Reviews (MathSciNet): MR2498561
  • V. E. Tarasov and G. M. Zaslavsky, Fractional Ginzburg-Landau equation for fractal media, Phys. A 354 (2005), 249–261.
  • ––––, Fractional dynamics of coupled oscillators with long-range interaction, Chaos 16 (2006), no. 2, 023110, 13 pp.
    Zentralblatt MATH: 1152.37345
  • R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences 68, Springer-Verlag, New York, 1997.
    Zentralblatt MATH: 0871.35001
  • H. Weitzner and G. M. Zaslavsky, Some applications of fractional equations, Commun. Nonlinear Sci. Numer. Simul. 8 (2003), no. 3-4, 273–281.
    Zentralblatt MATH: 1041.35073
    Digital Object Identifier: doi:10.1016/S1007-5704(03)00049-2
    Mathematical Reviews (MathSciNet): MR2007006
  • G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), no. 6, 461–580.
    Zentralblatt MATH: 0999.82053
    Digital Object Identifier: doi:10.1016/S0370-1573(02)00331-9
  • ––––, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005.
    Zentralblatt MATH: 1083.37002
  • G. M. Zaslavsky and M. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, Chaos 11 (2001), no. 2, 295–305.
    Zentralblatt MATH: 1080.37584
    Digital Object Identifier: doi:10.1063/1.1355358

Mathematical Society of the Republic of China

Taiwanese Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more