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Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states

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Abstract

The fractional Fokker–Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., \(\mathbf {13}\), 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be replaced with fractional Laplacian. This paper focuses on solving the two-state Fokker–Planck system with fractional Laplacian. We first provide a priori estimate for this system under different regularity assumptions on the initial data. Then we use \(L_1\) scheme to discretize the time fractional derivatives and finite element method to approximate the fractional Laplacian operators. Furthermore, we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions. Finally, the effectiveness of the designed scheme is verified by one- and two-dimensional numerical experiments.

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References

  1. Adams, A., Fournier, J.F.: Sobolev Spaces. Academic Press, Cambridge (2003)

    MATH  Google Scholar 

  2. Acosta, G., Bersetche, F.M., Borthagaray, J.P.: A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian. Comput. Math. Appl. 74, 784–816 (2017)

    Article  MathSciNet  Google Scholar 

  3. Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55, 472–495 (2017)

    Article  MathSciNet  Google Scholar 

  4. Acosta, G., Borthagaray, J.P., Bruno, O., Maas, M.: Regularity theory and high order numerical methods for the (1D)-fractional Laplacian. Math. Comp. 87, 1821–1857 (2017)

    Article  MathSciNet  Google Scholar 

  5. Acosta, G., Bersetche, F.M., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. Fract. Calc. Appl. Anal. 22, 767–794 (2019)

    Article  MathSciNet  Google Scholar 

  6. Chen, S., Shen, J., Wang, L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comp. 85, 1603–1638 (2015)

    Article  MathSciNet  Google Scholar 

  7. Barkai, E.: Fractional Fokker–Planck equation, solution, and application. Phys. Rev. E 63, 046118 (2001)

    Article  Google Scholar 

  8. Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker–Planck equation. Phys. Rev. E 61, 132–138 (2000)

    Article  MathSciNet  Google Scholar 

  9. Bonito, A., Borthagaray, J.P., Nochetto, R.H., Otrola, E., Salgado, A.J.: Numerical methods for fractional diffusion. Comput. Vis. Sci. 19, 19–46 (2018)

    Article  MathSciNet  Google Scholar 

  10. Bazhlekova, E., Jin, B.T., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh–Stokes problem for a generalized second-grade fluid. Numer. Math. 131, 1–31 (2015)

    Article  MathSciNet  Google Scholar 

  11. Deng, W.H., Hou, R., Wang, W.L., Xu, P.B.: Modeling Anomalous Diffusion: From Statistics to Mathematics. World Scientific, Singapore (2020)

    Book  Google Scholar 

  12. Deng, W.H., Zhang, Z.J.: High Accuracy Algorithm for the Differential Equations Governing Anomalous Diffusion. World Scientific, Singapore (2019)

    Book  Google Scholar 

  13. Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)

    Article  MathSciNet  Google Scholar 

  14. Deng, W.H.: Numerical algorithm for the time fractional Fokker–Planck equation. J. Comput. Phys. 227, 1510–1522 (2007)

    Article  MathSciNet  Google Scholar 

  15. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  16. Elliott, C.M., Larsson, S.: Error estimates with smooth and nonsmooth data for a finite element method for the Cahn–Hilliard equation. Math. Comp. 58, 603–630 (1992)

    Article  MathSciNet  Google Scholar 

  17. Flajolet, P.: Singularity analysis and asymptotics of Bernoulli sums. Theoret. Comput. Sci. 215, 371–381 (1999)

    Article  MathSciNet  Google Scholar 

  18. Fujita, H., Suzuki, T.: Evolution problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Hand book of Numerical Analysis, vol. II, North-Holland (1991)

  19. Grubb, G.: Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu \)-transmission pseudodifferential operators. Adv. Math. 268, 478–528 (2015)

    Article  MathSciNet  Google Scholar 

  20. Heinsalu, E., Patriarca, M., Goychuk, I., Schmid, G., Hänggi, P.: Fractional Fokker–Planck dynamics: numerical algorithm and simulations. Phys. Rev. E 73, 046133 (2006)

    Article  Google Scholar 

  21. Jin, B.T., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Jin, B.T., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)

    Article  MathSciNet  Google Scholar 

  23. Klafter, J., Sokolov, I.M.: Anomalous diffusion spreads its wings. Phys. World 18, 29–32 (2005)

    Article  Google Scholar 

  24. Lewin, L.: Polylogarithms and Associated Functions. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  25. Li, B.J., Luo, H., Xie, X.P.: Analysis of the L1 scheme for fractional wave equations with nonsmooth data. (2019). arXiv:1908.09145 [math]

  26. Lin, Y.M., Xu, C.J.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  Google Scholar 

  27. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1974)

    MATH  Google Scholar 

  28. Lubich, C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52, 129–145 (1988)

    Article  MathSciNet  Google Scholar 

  29. Lubich, C.: Convolution quadrature and discretized operational calculus II. Numer. Math. 52, 413–425 (1988)

    Article  MathSciNet  Google Scholar 

  30. Lubich, C., Sloan, I.H., Thome, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65, 1–17 (1996)

    Article  MathSciNet  Google Scholar 

  31. Meerschaert, M.M., Scheffler, H.-P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249–261 (2006)

    Article  MathSciNet  Google Scholar 

  32. Nie, D.X., Sun, J., Deng, W.H.: Numerical scheme for the Fokker–Planck equations describing anomalous diffusions with two internal states. J. Sci. Comput. 83, 33 (2020)

    Article  MathSciNet  Google Scholar 

  33. Podlubny, I.: Fractional Differential Equations. Academic Press, Cambridge (1999)

    MATH  Google Scholar 

  34. Sun, J., Nie, D.X., Deng, W.H.: Fast algorithms for convolution quadrature of Riemann–Liouville fractional derivative. Appl. Numer. Math. 145, 384–410 (2019)

    Article  MathSciNet  Google Scholar 

  35. Xu, P.B., Deng, W.H.: Fractional compound Poisson processes with multiple internal states. Math. Model. Nat. Phenom. 13, 10 (2018)

    Article  MathSciNet  Google Scholar 

  36. Xu, P.B., Deng, W.H.: Lévy walk with multiple internal states. J. Stat. Phys. 173, 1598–1613 (2018)

    Article  MathSciNet  Google Scholar 

  37. Zhang, Z.Q.: Error estimates of spectral Galerkin methods for a linear fractional reaction–diffusion equation. J. Sci. Comput. 78, 1087–1110 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank Buyang Li for the discussions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Daxin Nie, Jing Sun & Weihua Deng

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  1. Daxin Nie
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  2. Jing Sun
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  3. Weihua Deng
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Correspondence to Weihua Deng.

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This work was supported by the National Natural Science Foundation of China under Grant Nos. 11671182 and 12071195, and the AI and Big Data Funds under Grant No. 2019620005000775.

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Nie, D., Sun, J. & Deng, W. Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states. Numer. Math. 146, 481–511 (2020). https://doi.org/10.1007/s00211-020-01148-6

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  • DOI: https://doi.org/10.1007/s00211-020-01148-6

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