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Numerical Scheme for the Fokker–Planck Equations Describing Anomalous Diffusions with Two Internal States

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Abstract

Recently, the fractional Fokker–Planck equations (FFPEs) with multiple internal states are built for the particles undergoing anomalous diffusion with different waiting time distributions for different internal states, which describe the distribution of positions of the particles (Xu and Deng in Math Model Nat Phenom 13:10, 2018). In this paper, we first develop the Sobolev regularity of the FFPEs with two internal states, including the homogeneous problem with smooth and nonsmooth initial values and the inhomogeneous problem with vanishing initial value, and then we design the numerical scheme for the system of fractional partial differential equations based on the finite element method for the space derivatives and convolution quadrature for the time fractional derivatives. The optimal error estimates of the scheme under the above three different conditions are provided for both space semidiscrete and fully discrete schemes. Finally, one- and two-dimensional numerical experiments are performed to confirm our theoretical analysis and the predicted convergence order.

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References

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

    Article  Google Scholar 

  2. 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 

  3. 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 

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

    Article  MathSciNet  Google Scholar 

  5. Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2009)

    Article  MathSciNet  Google Scholar 

  6. Deng, W.H., Li, B.Y., Qian, Z., Wang, H.: Time discretization of a tempered fractional Feynman–Kac equation with measure data. SIAM J. Numer. Anal. 56, 3249–3275 (2018)

    Article  MathSciNet  Google Scholar 

  7. Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)

    Article  MathSciNet  Google Scholar 

  8. Gunzburger, M., Li, B.Y., Wang, J.L.: Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise. Math. Comp. 88, 1715–1741 (2018)

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Jin, B.T., Lazarov, R., Zhou, Z.: Error estimates for a semidiscrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51, 445–466 (2013)

    Article  MathSciNet  Google Scholar 

  11. Jin, B.T., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of a finite element method for the space-fractional parabolic equation. SIAM J. Numer. Anal. 52, 2272–2294 (2014)

    Article  MathSciNet  Google Scholar 

  12. Jin, B.T., Lazarov, R., Pasciak, J., Zhou, Z.: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, 561–582 (2015)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. 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 

  15. Jin, B.T., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Engrgy 346, 332–358 (2019)

    Article  MathSciNet  Google Scholar 

  16. Jin, B.T., Li, B.Y., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39, A3129–A3152 (2017)

    Article  MathSciNet  Google Scholar 

  17. Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)

    Article  MathSciNet  Google Scholar 

  18. Li, C.P., Ding, H.F.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Model. 38, 3802–3821 (2014)

    Article  MathSciNet  Google Scholar 

  19. Li, W.L., Xu, D.: Finite central difference/finite element approximations for parabolic integro-differential equations. Computing 90, 89–111 (2010)

    Article  MathSciNet  Google Scholar 

  20. 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 

  21. Liu, F.W., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Lubich, C.: Convolution quadrature revisited. BIT 44, 503–514 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. 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 

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

    MATH  Google Scholar 

  28. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)

    MATH  Google Scholar 

  29. 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 

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

    Article  MathSciNet  Google Scholar 

  31. Zeng, F.H., Li, C.P., Liu, F.W., Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35, A2976–A3000 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Funding was provided by National Natural Science Foundation of China (Grant No. 11671182).

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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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Nie, D., Sun, J. & Deng, W. Numerical Scheme for the Fokker–Planck Equations Describing Anomalous Diffusions with Two Internal States. J Sci Comput 83, 33 (2020). https://doi.org/10.1007/s10915-020-01218-9

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  • DOI: https://doi.org/10.1007/s10915-020-01218-9

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