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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 (5): 570-578.doi: 10.3969/j.issn.1005-3085.2018.05.008

• • 上一篇    下一篇

Legendre函数法求解分数阶偏微分方程的数值解

朱   帅1,3,   解加全2,   吴世跃3   

  1. 1- 山西大同大学工学院，山西  大同  037003
    2- 太原科技大学机械工程学院，山西  太原  030024
    3- 太原理工大学矿业工程学院，山西  太原  030024
  • 收稿日期:2016-10-09 接受日期:2017-03-06 出版日期:2018-10-15 发布日期:2018-12-15
  • 基金资助:
    国家科技支撑计划(2007BAK29B01)；山西省科技攻关项目(2007031120-02).

Legendre Function Method for Solving Fractional-order Partial Differential Equations

ZHU Shuai1,3,   XIE Jia-quan2,   WU Shi-yue3   

  1. 1- School of Technology, Shanxi Datong University, Datong, Shanxi 037003
    2- School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan, Shanxi 030024
    3- School of Mining Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024
  • Received:2016-10-09 Accepted:2017-03-06 Online:2018-10-15 Published:2018-12-15
  • Supported by:
    The National Science and Technology Support Program (2007BAK29B01); the Science and Technology Breakthrough Project of Shanxi Province (2007031120-02).

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摘要： 分数阶偏微分方程作为一类常见的微分方程用以描述工程等实际问题．较传统的解析方法而言，本文提出的数值算法在计算精度及计算效率上有更大的优势．借助分数阶Legendre函数对待求方程中的二元函数进行级数展开，并结合算子矩阵将待求方程转化为非线性代数方程组，然后通过数学软件求解该方程组，获得原方程的数值解．本文介绍的分数阶Legendre函数法能更精确的模拟工程问题中一些复杂的数学现象，而且在函数推导及构造上都比较简单，很小的级数展开就能达到满意的数值精度．最后给出的误差分析也验证了该方法的收敛性．

关键词: 分数阶Legendre函数, 算子矩阵, 分数阶偏微分方程, 数值解, Tau方法

Abstract: Fractional partial differential equations are regarded as one kind of common differ-ential equations to describe the engineering problems. Compared with other traditional methods, the proposed method has a great advantage on computational precision and efficiency. Legendre function method expands the dual functions as basis functions, and transforms the original equations into a system of algebra equations combined with the operational matrices. This paper introduces the method which can accurately stimulate the complex mathematical phenomenon of the engineering problems, and is relatively simple in the function construction and theoretical derivation. Finally, the error analysis is presented to verify the effectiveness and robust.

Key words: Legendre functions, operational matrix, fractional partial differential equations, numerical solution, Tau method

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