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A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem

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  • Zhang, Hui
  • Jiang, Xiaoyun
  • Yang, Xiu

Abstract

In the paper, we consider a time-space spectral method to get the numerical solution of time-space fractional Fokker–Planck initial-boundary value problem. The temporal discretization is constructed by Jacobi polynomials and the spatial discretization is composed by Legendre polynomials. Moreover, we present the stability and convergence analysis strictly. The main advantages of the provided method are spectrally accurate in time and space and high computational efficiency. In addition, we introduce the inverse problem based on the spectral form with high-order accuracy of the direct problem for the first time, the Levenberg–Marquardt (L–M) method is proposed to estimate the two fractional derivatives α and 2β. Some numerical results presented are consistent with the theoretical analysis.

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  • Zhang, Hui & Jiang, Xiaoyun & Yang, Xiu, 2018. "A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 302-318.
    • Handle: RePEc:eee:apmaco:v:320:y:2018:i:c:p:302-318
    DOI: 10.1016/j.amc.2017.09.040
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    References listed on IDEAS

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    1. Yu, Bo & Jiang, Xiaoyun & Wang, Chu, 2016. "Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 106-118.
    2. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    3. Gaviraghi, B. & Annunziato, M. & Borzì, A., 2017. "Analysis of splitting methods for solving a partial integro-differential Fokker–Planck equation," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 1-17.
    4. Chen, Yanping & Zhou, Jianwei, 2015. "Error estimates of spectral Legendre–Galerkin methods for the fourth-order equation in one dimension," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1217-1226.
    5. Wang, JinRong, 2015. "Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 315-323.
    6. Fan, Wenping & Jiang, Xiaoyun & Qi, Haitao, 2015. "Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 40-49.
    7. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.
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    Cited by:

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