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基于组合神经网络的时间分数阶扩散方程计算方法

王江 陈文

王江, 陈文. 基于组合神经网络的时间分数阶扩散方程计算方法[J]. 应用数学和力学, 2019, 40(7): 741-750. doi: 10.21656/1000-0887.390288
引用本文: 王江, 陈文. 基于组合神经网络的时间分数阶扩散方程计算方法[J]. 应用数学和力学, 2019, 40(7): 741-750. doi: 10.21656/1000-0887.390288
WANG Jiang, CHEN Wen. A Combined Artificial Neural Network Method for Solving Time Fractional Diffusion Equations[J]. Applied Mathematics and Mechanics, 2019, 40(7): 741-750. doi: 10.21656/1000-0887.390288
Citation: WANG Jiang, CHEN Wen. A Combined Artificial Neural Network Method for Solving Time Fractional Diffusion Equations[J]. Applied Mathematics and Mechanics, 2019, 40(7): 741-750. doi: 10.21656/1000-0887.390288

基于组合神经网络的时间分数阶扩散方程计算方法

doi: 10.21656/1000-0887.390288
基金项目: 111引智计划(B12032);中央高校基本科研业务费(2017B01114)
详细信息
    作者简介:

    王江(1993—),男,硕士生(E-mail: 1150904137@qq.com);陈文(1967—2018),男,教授,博士,博士生导师(通讯作者. E-mail: chenwen@hhu.edu.cn).

  • 中图分类号: O232; O172

A Combined Artificial Neural Network Method for Solving Time Fractional Diffusion Equations

  • 摘要: 该文首次采用一种组合神经网络的方法,求解了一维时间分数阶扩散方程.组合神经网络是由径向基函数(RBF)神经网络与幂激励前向神经网络相结合所构造出的一种新型网络结构.首先,利用该网络结构构造出符合时间分数阶扩散方程条件的数值求解格式,同时设置误差函数,使原问题转化为求解误差函数极小值问题;然后,结合神经网络模型中的梯度下降学习算法进行循环迭代,从而获得神经网络的最优权值以及各项最优参数,最终得到问题的数值解.数值算例验证了该方法的可行性、有效性和数值精度.该文工作为时间分数阶扩散方程的求解开辟了一条新的途径.
  • [1] DIETHELM K, FORD N J. Analysis of fractional differential equations[J]. Journal of Mathematical Analysis and Applications,2002,265(2): 229-248.
    [2] CHEN W. A speculative study of 2/3-order fractional Laplacian modeling of turbulence: some thoughts and conjectures[J]. Chaos,2006,16(2): 023126. DOI: 10.1063/1.2208452.
    [3] BENSON D A, WHEATCRAFT S W, MEERSCHAERT M M. Application of a fractional advection-dispersion equation[J]. Water Resources Research,2000,36(6): 1403-1412.
    [4] DE ANDRADE M F, LENZI E K, EVANGELISTA L R, et al. Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces[J]. Physics Letters A,2005,347(4/6): 160-169.
    [5] SHEN S, LIU F, ANH V, et al. Detailed analysis of a conservative difference approximation for the time fractional diffusion equation[J]. Journal of Applied Mathematics and Computing,2006,22(3): 1-19.
    [6] CHEN W, YE L, SUN H. Fractional diffusion equations by the Kansa method[J]. Computers & Mathematics With Applications,2010,59(5): 1614-1620.
    [7] GORENFLO R, MAINARDI F, MORETTI D, et al. Time fractional diffusion: a discrete random walk approach[J]. Nonlinear Dynamics,2002,29(1/4): 129-143.
    [8] JAFARIAN A, MOKHTARPOUR M, BALEANU D. Artificial neural network approach for a class of fractional ordinary differential equation[J]. Neural Computing and Applications,2016,28(4): 765-773.
    [9] BASHEER I A, HAJMEER M. Artificial neural networks: fundamentals, computing, design, and application[J]. Journal of Microbiological Methods,2000,43(1): 3-31.
    [10] HANSS M. Applied Fuzzy Arithmetic: an Introduction With Engineering Applications [M]. Springer, 2010.
    [11] 陈文, 孙洪广, 李西成. 力学与工程问题的分数阶导数建模[M]. 北京: 科学出版社, 2010.(CHEN Wen, SUN Hongguang, LI Xicheng. Fractional Derivative Modeling in Mechanical and Engineering Problems [M]. Beijing: Science Press, 2010.(in Chinese))
    [12] NUSEIR A S, AL-HASOON A. Power series solutions for nonlinear systems of partial differential equations[J]. Journal of Applied Mathematics,2017,6(104/105): 5147-5159.
    [13] KURULAY M, BAYRAM M. A novel power series method for solving second order partial differential equations[J]. European Journal of Pure & Applied Mathematics,2009,2(2): 268-277.
    [14] HASSOUN M H. Fundamentals of Artificial Neural Networks [M]. MIT Press, 1995.
    [15] ZHANG J, ZHANG X, YANG B. An approximation scheme for the time fractional convection-diffusion equation[J]. Applied Mathematics and Computation,2018,335: 305-312.
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  • 被引次数: 0
出版历程
  • 收稿日期:  2018-11-16
  • 修回日期:  2018-11-20
  • 刊出日期:  2019-07-01

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