The time fractional diffusion equation and the advection-dispersion equation

Abstract

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 a = 1 with arbitrary n, or n = 1 with arbitrary a. 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.

Fractional high order methods for the nonlinear fractional ordinary differential equation

“…This topic has received a great 1 Email address :3425linran@sohu.com 2 Corresponding author. Email address :fwliu@xmu.edu.cn,f.liu@qut.edu.au deal of attention in the last decade [7,15,[22][23][24], especially in the fields of viscoelastic materials [1,10,11,27], electrochemical processes [8], dielectric polarization [28], colored noise [29], anomalous diffusion, signal processing [21], control theory [24], advection and dispersion of solutes in natural porous or fractured media [2,3] and chaos [20]. Djrbasjan et al [6] considered the Cauchy problem with multi-term fractional derivatives, and proved the Cauchy problem has a unique solution.…”

The Solution of Initial Boundary Value Problem with Time and Space-Fractional Diffusion Equation via a Novel Inner Product

“…There are various studies in literature supporting this conclusion [1][2][3][4][5][6][7][8]. By making use of Mittag-Leffler function, characteristic equations of fractional ODEs are solved and solutions of them are constructed efficiently.…”

Fundamental solution to the Cauchy problem for the time-fractional advection-diffusion equation

“…In the case of the fractional advection-diffusion equation, as a rule, different numerical methods have been used to find the solution: the implicit difference method based on the shifted Grünwald approximation [9] and the explicit difference method [10], the fractional variational method [11], the finite volume method [12], etc. In [13,14] the solution to one-dimensional time-fractional advection-diffusion equation was obtained in terms of the H-function.…”