一类时间分数阶偏微分方程的解  

General Solution for a Class of Time Fractional Partial Differential Equation

作　　者：黄凤辉[1] 郭柏灵[2] 

机构地区：[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

出　　处：《应用数学和力学》2010年第7期781-790,共10页Applied Mathematics and Mechanics

基　　金：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

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.A class of time fractional partial differential equation,including time fractional diffusion equation,time fractional reaction-diffusion equation,time fractional advection-diffusion equation and their corresponding integer-order partial differential equations,was considered.The fundamental solutions for the Cauchy problem in a whole-space domain and signaling problem in a half-space domain were obtained by using Fourier-Laplace transforms 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 initial and boundary value problems 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.

关 键 词：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 

分 类 号：O175.2[理学—数学]