Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

A spectral order method for solving the nonlinear fourth-order time-fractional problem

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, the nonlinear fourth-order time-fractional equation is considered by the combination of the Sinc-Galerkin method and the double exponential (DE) transformation. The backward Euler method is employed for the first order time derivative and the L1 formula is applied to the discretization of the Caputo time fractional derivative. For the spatial approximation, the DE Sinc-Galerkin method is employed. Using \(2N+1\) Sinc points, the convergence rate of the DE Sinc-Galerkin scheme is \(O(\exp (-CN/\log (N))\). Three numerical experiments are implemented to show the feasibility and high efficiency of the proposed scheme. Besides, the numerical results demonstrate that the Sinc-Galerkin scheme can achieve expected convergence order for the problems with singular points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Abbaszadeh, M., Dehghan, M.: Direct meshless local Petrov-Galerkin (DMLPG) method for time-fractional fourth-order reaction-diffusion problem on complex domains. Comput. Math. Appl. 79, 876–888 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babaei, A., Moghaddam, B.P., Banihashemi, S., Machado, J.A.T.: Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations. Commun. Nonlinear Sci. Numer. Simul. 82, 104985 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baleanu, D., Machado, J., Luo, A.: Fractional Dynamics and Control. Springer Science & Business Media, Berlin (2011)

    Google Scholar 

  4. Du, Y., Liu, Y., Li, H., He, S.: Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation. J. Comput. Phys. 344, 108–126 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fakhar-Izadi, F.: Fully Petrov-Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation. Eng. Comput. (2020). https://doi.org/10.1007/s00366-020-00968-2

    Article  Google Scholar 

  6. Fei, M., Huang, C.: Galerkin-Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation. Int. J. Comput. Math. 97, 1183–1196 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gaudreau, P., Slevinsky, R., Safouhi, H.: The double exponential Sinc collocation method for singular Sturm-Liouville problems. J. Math. Phys. 57, 043505 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guo, L., Wang, Z., Vong, S.: Fully discrete local discontinuous Galerkin methods for some time-fractional fourth-order problems. Int. J. Comput. Math. 10, 1665–1682 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu, X., Zhang, L.: A new implicit compact difference scheme for the fourth-order fractional diffusion-wave system. Int. J. Comput. Math. 91, 2215–2231 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ji, C., Sun, Z., Hao, Z.: Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first dirichlet boundary conditions. J. Sci. Comput. 66, 1148–1174 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, Y., Du, Y., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70, 573–591 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, Y., Fang, Z., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Mark, R.J., Hall, M.W.: Differentegral interpolation from a bandlimited signal’s samples. IEEE Trans. Acoust. Speech Signal Process. 29, 872–877 (1981)

    Article  MATH  Google Scholar 

  14. Marshall, A.W., Olkin, I.: Inequalities: theory of majorization and its applications. Academic Press, New York (1979)

    MATH  Google Scholar 

  15. Morlet, A.C.: Convergence of the Sinc method for a fourth-order ordinary differential equation with an application. SIAM J. Numer. Anal. 32, 1475–1503 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nurmuhammad, A., Muhammad, M., Mori, M., Sugihara, M.: Double exponential transformation in the Sinc-collocation method for a boundary value problem with fourth-order ordinary differential equation. J. Comput. Appl. Math. 182, 32–50 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Okayama, T., Matsuo, T., Sugihara, M.: Approximate formulae for fractional derivatives by means of Sinc methods. J. Concr. Appl. Math 8, 470–488 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Oldham, K.B., Spainer, J.: The Fractional Calculus. Academic Press, New York (1974)

    Google Scholar 

  19. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  20. Qiu, W., Xu, D., Guo, J.: The Crank-Nicolson-type Sinc-Galerkin method for the fourth-order partial integro-differential equation with a weakly singular kernel. Appl. Numer. Math. 159, 239–258 (2021). https://doi.org/10.1016/j.apnum.2020.09.011

    Article  MathSciNet  MATH  Google Scholar 

  21. Qiu, W., Xu, D., Guo, J.: Numerical solution of the fourth-order partial integro-differential equation with multi-term kernels by the Sinc-collocation method based on the double exponential transformation. Appl. Math. Comput. 392, 125693 (2021). https://doi.org/10.1016/j.amc.2020.125693

    Article  MathSciNet  MATH  Google Scholar 

  22. Rashidinia, J., Nabati, M.: Sinc-Galerkin and Sinc-collocation methods in the solution of nonlinear two-point boundary value problems. Comput. Appl. Math. 32, 315–330 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Smith, R.C., Bogar, G.A., Bowers, K.L., Lund, J.: The Sinc-Galerkin method for fourth-order differential equations. SIAM J. Numer. Anal. 3, 760–788 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stenger, F.: Approximations via Whittaker’s cardinal function. J. Approx. Theory 17, 222–240 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  25. Stenger, F.: A"Sinc-Galerkin" method of solution of boundary value problems. Math. Comput. 33, 85–109 (1979)

    MathSciNet  MATH  Google Scholar 

  26. Sugihara, M.: Optimality of the double exponential formula-functional analysis approach. Numer. Math. 75, 379–395 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sugihara, M.: Near optimality of the Sinc approximation. Math. Comput. 72, 767–786 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci. 9, 721–741 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tanaka, K.I., Sugihara, M., Murota, K.: Function classes for successful DE-Sinc approximations. Math. Comput. 78, 1553–1571 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tariq, H., Akram, G.: Quintic spline technique for time fractional fourth-order partial differential equation, Numer. Meth. Part. Differ. Equ. 33, 445–466 (2017)

    MATH  Google Scholar 

  32. Wei, L., He, Y.: Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl. Math. Model. 38, 1511–1522 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, X., Zhang, H., Xu, D.: Orthogonal spline collocation method for the fourth-order diffusion system. Comput. Math. Appl. 9, 3172–3185 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, H., Yang, X., Xu, D.: A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation. Numer. Algorithms 80, 849–877 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, H., Yang, X., Xu, D.: An efficient spline collocation method for a nonlinear fourth-order reaction subdiffusion equation. J. Sci. Comput. 85, 1–18 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Science Foundation of China (No. 12071127; No. 12171147 ), the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20210469) and the Construct Program of the Key Discipline in Hunan (No. 19K056), Performance Computing and Stochastic Information Processing (Ministry of Education of China).

Author information

Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education),School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Jing Guo, Qing Pan, Da Xu & Wenlin Qiu

Authors
  1. Jing Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Da Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wenlin Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing Guo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, J., Pan, Q., Xu, D. et al. A spectral order method for solving the nonlinear fourth-order time-fractional problem. J. Appl. Math. Comput. 68, 4645–4667 (2022). https://doi.org/10.1007/s12190-022-01719-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-022-01719-w

Keywords

Search

Navigation