Skip to main content
SpringerLink
Log in

Semi-implicit Hermite–Galerkin Spectral Method for Distributed-Order Fractional-in-Space Nonlinear Reaction–Diffusion Equations in Multidimensional Unbounded Domains

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we construct an efficient Hermite–Galerkin spectral method for the nonlinear reaction–diffusion equations with distributed-order fractional Laplacian in multidimensional unbounded domains. By applying Gauss–Legendre quadrature rule for the distributed integral term, we first approximate the original distributed-order fractional problem by the multi-term fractional-in-space differential equation. Applying Hermite–Galerkin spectral method in space and backward difference method in time, we establish semi-implicit fully discrete scheme. For two- and three-dimensional cases of the original fractional problem, the linear systems are solved by the preconditioned conjugate gradients method. The main advantage of our method is that the original fractional problem is directly solved in the unbounded domains, thus avoiding the errors introduced by the domain truncations. The stability analysis is rigourously established, which shows that our scheme is unconditionally stable under suitable assumption on the nonlinear term. Several numerical examples are presented to validate both stability and accuracy of the numerical method. The numerical results of the fractional Allen–Cahn, Gray–Scott, and Belousov–Zhabotinskii models show that our semi-implicit methods produce good numerical solutions.

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

Access this article

Log in via an institution

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
Fig. 9
Fig. 10

Similar content being viewed by others

Data Availability

This manuscript has no associated data.

References

  1. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)

    Google Scholar 

  2. Magin, R.: Fractional Calculus in Bioengineering. Begell House Publishers Inc., Connecticut (2006)

    Google Scholar 

  3. Zaslavsky, G.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Pipkin, A.: Lectures on Viscoelasticity Theory, 2nd edn. Springer-Verlag, New York (1986)

    MATH  Google Scholar 

  5. Zhao, T., Mao, Z., Karniadakis, G.: Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations. Comput. Methods Appl. Mech. Eng. 348, 377–395 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Caputo, M.: Elasticit‘ae dissipazione. Zanichelli Publisher, Bologna (1969)

    Google Scholar 

  7. Sokolov, I., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. Acta Phys. Polon. B 35, 1323–1341 (2004)

    Google Scholar 

  8. Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225, 96–104 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations. Phys. Rev. E 66, 046129 (2002)

    Google Scholar 

  10. Jiao, Z., Chen, Y., Podlubny, I.: Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives. Springer, London (2012)

    MATH  Google Scholar 

  11. Song, F., Xu, C., Karniadakis, G.: A fractional phase-field model for two-phase flows with tunable sharpness: algorithms and simulations. Comput. Methods Appl. Mech. Eng. 305, 376–404 (2016)

    MathSciNet  MATH  Google Scholar 

  12. Song, F., Xu, C.: Spectral direction splitting methods for two-dimensional space fractional diffusion equations. J. Comput. Phys. 299, 196–214 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Liu, L., Feng, L., Xu, Q., Chen, Y.: Anomalous diffusion in comb model subject to a novel distributed order time fractional Cattaneo–Christov flux. Appl. Math. Lett. 102, 106116 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Bu, W., Ji, L., Tang, Y., Zhou, J.: Space-time finite element method for the distributed-order time fractional reaction diffusion equations. Appl. Numer. Math. 152, 446–465 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Shi, Y., Liu, F., Zhao, Y., Wang, F., Turner, I.: An unstructured mesh finite element method for solving the multi-term time fractional and Riesz space distributed-order wave equation on an irregular convex domain. Appl. Math. Model. 73, 615–636 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Fan, W., Liu, F.: A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Appl. Math. Lett. 77, 114–121 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Jia, J., Wang, H.: A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains. Comput. Math. Appl. 75, 2031–2043 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Guo, S., Mei, L., Zhang, Z., Li, C., Li, M., Wang, Y.: A linearized finite difference/spectral-Galerkin scheme for three-dimensional distributed-order time-space fractional nonlinear reaction–diffusion-wave equation: numerical simulations of Gordon-type solitons. Comput. Phys. Commun. 252, 107144 (2020)

    MathSciNet  Google Scholar 

  19. Kazmi, K., Khaliq, A.: An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions. Appl. Numer. Math. 147, 142–160 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Guo, S., Mei, L., Zhang, Z., Jiang, Y.: Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction–diffusion equation. Appl. Math. Lett. 85, 157–163 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Mao, Z., Shen, J.: Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput. 39, A1928–A1950 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Tang, T., Yuan, H., Zhou, T.: Hermite spectral collocation methods for fractional PDEs in unbounded domains. Commun. Comput. Phys. 24, 1143–1168 (2018)

    MathSciNet  Google Scholar 

  23. Khosravian-Arab, H., Dehghan, M., Eslahchi, M.R.: Fractional spectral and pseudo-spectral methods in unbounded domains: theory and applications. J. Comput. Phys. 338, 527–566 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Khosravian-Arab, H., Dehghan, M., Eslahchi, M.R.: Fractional Sturm–Liouville boundary value problems in unbounded domains: theory and applications. J. Comput. Phys. 299, 526–560 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Gao, G., Sun, Z., Zhang, Y.: A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys. 231, 2865–2879 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Gao, G., Sun, Z.: The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain. J. Comput. Phys. 236, 443–460 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Brunner, H., Han, H., Yin, D.: Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain. J. Comput. Phys. 276, 541–562 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Ma, H., Zhao, T.: A stabilized Hermite spectral method for second-order differential equations in unbounded domains. Numer. Methods Part. Differ. Equ. 23, 968–983 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Tang, T.: The Hermite spectral method for Gauss-type function. SIAM J. Sci. Comput. 14, 594–606 (1993)

    MathSciNet  MATH  Google Scholar 

  30. Guo, B., Wang, L., Wang, Z.: Generalized Laguerre interpolation and pseudospectral method for unbounded domains. SIAM J. Numer. Anal. 43, 2567–2589 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Guo, S., Mei, L., Zhang, Z., Chen, J., He, Y., Li, Y.: Finite difference/Hermite-Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction–diffusion equation in unbounded domains. Appl. Math. Model. 70, 246–263 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M., Ainsworth, M., EmKarniadakis, G.: What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009 (2020)

    MathSciNet  Google Scholar 

  33. Song, F., Xu, C., Karniadakis, G.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39, A1320–A1344 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  35. Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. In: Springer Series in Computational Mathematics, vol. 41, Springer, Heidelberg ( 2011)

  36. Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2004)

    MATH  Google Scholar 

  37. Maroni, P., da Rocha, Z.: Connection coeffcients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation. Numer. Algorithms 47, 291–314 (2008)

    MathSciNet  MATH  Google Scholar 

  38. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (2008)

    MATH  Google Scholar 

  39. Zhao, X., Sun, Z., Hao, Z.: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation. SIAM J. Sci. Comput. 36, A2865–A2886 (2014)

    MATH  Google Scholar 

  40. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Google Scholar 

  41. Lam, L., Prost, J.: Solitons in Liquid Crystals. Springer, New York (1992)

    Google Scholar 

  42. Gray, P., Scott, S.K.: Sustained oscillations and other exotic patterns of behavior in isothermal reactions. J. Phys. Chem. 89, 22–32 (1985)

    Google Scholar 

  43. Belousov, B.: A periodic reaction and its mechanism. Ref. Radiat. Med. Medgiz 1, 145–160 (1959)

    Google Scholar 

Download references

Acknowledgements

The authors express their sincere thanks to the referees for their valuable comments which led to an improved version. The project is supported by NSF of China (11501441, 11601207, 11771348), Science Challenge Project of China (TZ2016002), NSF of ShaanXi Province (2020JQ-008), and Shanghai Sailing Program (16YF1404000).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Shimin Guo & Liquan Mei

  2. School of Sciences, Xi’an University of Technology, Xi’an, 710054, China

    Can Li

  3. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Zhengqiang Zhang

  4. School of Computer Engineering and Science, Shanghai University, Shanghai, 200444, China

    Ying Li

Authors
  1. Shimin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liquan Mei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Can Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhengqiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ying Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shimin 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, S., Mei, L., Li, C. et al. Semi-implicit Hermite–Galerkin Spectral Method for Distributed-Order Fractional-in-Space Nonlinear Reaction–Diffusion Equations in Multidimensional Unbounded Domains. J Sci Comput 85, 15 (2020). https://doi.org/10.1007/s10915-020-01320-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01320-y

Keywords

Search

Navigation