Advertisement

Numerical Algorithms

, Volume 82, Issue 2, pp 553–571 | Cite as

Numerical solutions to time-fractional stochastic partial differential equations

  • Guang-an ZouEmail author
Original Paper

Abstract

This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the anomalous diffusion in porous media with random effects with thermal memory. A standard finite element approximation is used in space as well as a spatial-temporal discretization which is achieved by a new algorithm in time direction. The strong convergence error estimates for both semidiscrete and fully discrete schemes in a semigroup framework are proved, which are based on the error estimates for the corresponding deterministic equations, and together with the optimal regularity results of mild solutions to the time-fractional SPDEs. Numerical results are finally reported to confirm our theoretical findings.

Keywords

Time-fractional derivative Stochastic heat-type equations Finite element method Error estimates 

Mathematics Subject Classification (2010)

65C30 65N30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank the referees very much for the constructive comments and valuable suggestions which would help us to improve the quality of the paper.

Funding information

This study is supported by the National Nature Science Foundation of China (Grant No. 11626085) and Key Scientifc Research Projects of Colleges and Universities in Henan Province, China (19A110002).

References

  1. 1.
    Chen, C., Hong, J., Ji, L.: Mean-square convergence of a symplectic local discontinuous Galerkin method applied to stochastic linear Schrödinger equation. IMA J. Numer. Anal. 37(2), 1041–1065 (2016)zbMATHGoogle Scholar
  2. 2.
    Chen, L., Hu, G., Hu, Y., Huang, J.: Space-time fractional diffusions in Gaussian noisy environment. Stochastics 89(1), 171–206 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z.Q., Kim, K.H., Kim, P.: Fractional time stochastic partial differential equations. Stoch. Process. Appl. 125(4), 1470–1499 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Deng, K., Deng, W.: Finite difference/predictor-corrector approximations for the space and time fractional Fokker-Planck equation. Appl. Math. Lett. 25(11), 1815–1821 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Feng, X., Li, Y., Zhang, Y.: Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise. SIAM J. Numer. Anal. 55(1), 194–216 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Foondun, M., Nane, E.: Asymptotic properties of some space-time fractional stochastic equations. Math. Z. 287, 493–519 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, Article ID 298628. (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jentzen, A., Kloeden, P.E.: The numerical approximation of stochastic partial differential equations. Milan J. Math. 77(1), 205–244 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235(11), 3285–3290 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  11. 11.
    Kruse, R.: Strong and Weak Approximation of Semilinear Stochastic Evolution Equations. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models. World Scientific, Singapore (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Mijena, J.B., Nane, E.: Space-time fractional stochastic partial differential equations. Stoch. Proc. Appl. 125(9), 3301–3326 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mijena, J.B., Nane, E.: Intermittence and space-time fractional stochastic partial differential equations. Potential Anal. 44(2), 295–312 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Oksendal, B.: Stochastic Differential Equations: an Introduction with Applications. Springer, Berlin (2013)zbMATHGoogle Scholar
  16. 16.
    Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007)zbMATHGoogle Scholar
  17. 17.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1984)zbMATHGoogle Scholar
  18. 18.
    Wang, H., Du, N.: A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation. J. Comput. Phys. 253(15), 50–63 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43(4), 1363–1384 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yao, R., Bo, L.: Discontinuous Galerkin method for elliptic stochastic partial differential equations on two and three dimensional spaces. Sci. China Math. 50(11), 1661–1672 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yoo, H.: Semi-discretization of stochastic partial differential equations on R 1 by a finite-difference method. Math. Comput. 69(230), 653–666 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976–A3000 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhai, S., Feng, X., He, Y.: An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation. J. Comput. Phys. 269(15), 138–155 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zou, G., Wang, B.: Solitary wave solutions for nonlinear fractional Schrödinger equation in Gaussian nonlocal media. Appl. Math. Lett. 88, 50–57 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zou, G., Lv, G., Wu, J.L.: On the regularity of weak solutions to space-time fractional stochastic heat equations. Statist. Probab. Lett. 139, 84–89 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yuste, S.B., Quintana-Murillo, J.: Fast, accurate and robust adaptive finite difference methods for fractional diffusion equations. Numer. Algorithms 71(1), 207–228 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan UniversityKaifengChina

Personalised recommendations