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The time fractional diffusion equation and the advection-dispersion equation

Published online by Cambridge University Press:  17 February 2009

F. Huang  and
F. Liu
F. Huang
Affiliation:
Department of Mathematics, Xiamen University, Xiamen 361005, China; e-mail: [email protected]. School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China; e-mail: [email protected].
F. Liu
Affiliation:
Department of Mathematics, Xiamen University, Xiamen 361005, China; e-mail: [email protected]. School of Mathematical Sciences, Queensland University of Technology, Qld 4001, Australia; e-mail: [email protected].
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Abstract

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The time fractional diffusion equation with appropriate initial and boundary conditions in an n-dimensional whole-space and half-space is considered. Its solution has been obtained in terms of Green functions by Schneider and Wyss. For the problem in whole-space, an explicit representation of the Green functions can also be obtained. However, an explicit representation of the Green functions for the problem in half-space is difficult to determine, except in the special cases α = 1 with arbitrary n, or n = 1 with arbitrary α. In this paper, we solve these problems. By investigating the explicit relationship between the Green functions of the problem with initial conditions in whole-space and that of the same problem with initial and boundary conditions in half-space, an explicit expression for the Green functions corresponding to the latter can be derived in terms of Fox functions. We also extend some results of Liu, Anh, Turner and Zhuang concerning the advection-dispersion equation and obtain its solution in half-space and in a bounded space domain.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 3 , January 2005 , pp. 317 - 330
Copyright
Copyright © Australian Mathematical Society 2005

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