一类时间分数阶偏微分方程的解

黄凤辉; 郭柏灵

doi:10.3879/j.issn.1000-0887.2010.07.003

一类时间分数阶偏微分方程的解

doi: 10.3879/j.issn.1000-0887.2010.07.003

黄凤辉¹,
郭柏灵²

1.
华南理工大学理学院数学系,广州 510641;2.北京应用物理与计算数学研究所,北京 100088

基金项目: 国家教育部高等学校博士点基金新教师基金资助项目(20070561040);广东省自然科学基金资助项目(07300823);华南理工大学中央高校基本科研业务费专项资金(2009ZM0050)的资助

详细信息

作者简介:
黄凤辉(1974- ),女,江西人,副教授,博士(E-mail:huangfh@scu.tedu.cn);郭柏灵,中国科学院院士,研究员,博士生导师(联系人.E-mail:gb@lmail.iapcm.ac.cn).

中图分类号: O175.2
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出版历程
- 收稿日期: 1900-01-01
- 修回日期: 2010-05-25
- 刊出日期: 2010-07-15

General Solution for a Class of Time Fractional Partial Differential Equation

HUANG Feng-hui¹,
GUO Bo-ling²

1.
Department of Mathematics, School of Sciences, South China University of Technology, Guangzhou 510641, P. R. China;

摘要

摘要: 考虑一类时间分数阶偏微分方程，该方程包含几种特殊情况：时间分数阶扩散方程、时间分数阶反应-扩散方程、时间分数阶对流-扩散方程以及它们各自相对应的整数阶偏微分方程．通过Laplace-Fourier变换及其逆变换，该方程在空间全平面和半平面内的基本解可以求出，但其表达式则是通过适当的变形来求．另外，对于有限域上的初边值问题，则可由Sine(Cosine)-Laplace变换导出该方程的一种级数形式的解，并通过两个数值例子来说明该方法的有效性．
- 分数阶微分方程 /
- Caputo分数阶导数 /
- Green函数 /
- Laplace变换 /
- Fourier变换 /
- Sine(Cosine)变换
Abstract: A class of tmie fractional partial differential equation, including time fractional diffusion equation, tmie fractional reaction-diffusion equation, time fractional advection-diffusione-quation and their corresponding in teger-order partial differential equations, was considered. The fundam ental solutions for the Cauchy problem in a whole-space domain and signaling problem in a hal-fspace domain were obtained by using Fourier-Laplace trans forms and their inverse transforms. The appropriate structures for the Green functions were provided. On the other hand, the solutions in the form of a series for the in itial and boundary value p rob lem s in a bounded-space domain were derived by the Sine-Laplace or Cosine-Laplace transforms. Two examples were presented to show the application of the present technique.
- fractional differentia equation /
- Caputo fractional derivative /
- Green function /
- Laplace transform /
- Fourier transform /
- Sine(Cosine) transform

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参考文献(8)

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