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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2018, Vol. 35 ›› Issue (6): 684-692.doi: 10.3969/j.issn.1005-3085.2018.06.008

• • 上一篇    下一篇

时间分数阶Fokker-Planck方程的Jacobi谱配置方法

周   琴1,   杨   银2   

  1. 1- 湖南涉外经济学院信息科学与工程学院，长沙  410205
    2- 湘潭大学科学工程计算与数值仿真湖南省重点实验室，湘潭  411105
  • 收稿日期:2017-01-15 接受日期:2018-01-04 出版日期:2018-12-15 发布日期:2019-02-15
  • 基金资助:
    国家自然科学基金(11671342)；湖南省自然科学基金(2018JJ2374)；湖南省教育厅重点项目(17A210).

Jacobi Collocation Method for Time Fractional Fokker-Planck Equations

ZHOU Qin1,   YANG Yin2   

  1. 1- School of Information Science and Engineering, Hunan International Economics University, Changsha 410205
    2- Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105
  • Received:2017-01-15 Accepted:2018-01-04 Online:2018-12-15 Published:2019-02-15
  • Supported by:
    The National Natural Science Foundation of China (11671342); the Natural Science Foundation of Hunan Province (2018JJ2374); the Scientific Research Fund of Hunan Provincial Education Department (17A210).

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摘要： 分数阶微分方程在工程、生物、金融等领域有广泛的应用．本文利用分数阶积分和微分公式的关系，针对一类带Dirichlet边值条件的时间分数阶Fokker-Planck方程，将其转化为与之等价的带有奇异核的积分微分方程，然后用高斯积分公式数值求解积分项，在时间和空间上都采用Jacobi谱配置法来离散求解积分微分方程．数值算例的结果表明，该方法是非常有效的，数值解具有谱精度，并且该方法容易推广到高维和非线性的情形．

关键词: Caputo分数阶导数, 时间分数阶Fokker-Planck方程, Jacobi谱配置法

Abstract: Fractional partial differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bioengineering and others. In this paper, we convert the time fractional Fokker-Planck equation into equivalent integral equations with singular kernel, then the model solution is discretized in time and space with a spectral expansion of the Lagrange interpolation polynomial. Numerical results demonstrate the spectral accuracy and efficiency of the collocation spectral method. The proposed technique is not only easy to implement, but also can be easily extended to multidimensional problems.

Key words: Caputo derivative, time-fractional Fokker-Planck equation, Jacobi collocation method

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