时间空间分数阶对流-弥散方程的基本解

The Fundamental Solution of the Time-Space Fractional Advection-Dispersion Equation

考虑两类时间空间分数阶对流-弥散方程,它们是由传统的对流-弥散方程推广而来(时间一阶导数用μ∈(0,1]阶Caputo导数代替,空间一阶、二阶导数分别用α∈(0,1]和β∈(1,2]阶Riesz或Caputo导数代替).它们的Cauchy问题的基本解可以通过Laplace-Fourier变换得出,其表达式可以通过适当的变形求得,并证明了其空间概率密度的性质.

In this paper, we discuss two kinds of the time-space fractional advection-dispersion equa- tions. They are generalizations of the classical advection-dispersion equations,in which the first-order time derivatives are replaced with Caputo derivatives of order/zE （0,1] ,while the first- and two-order space de- rivatives are replaced with Reise or Caputo derivatives of order a E （0,1] and fie （1,2]. By the Laplace- Fourier transformation,we derive the fundamental solutions for theirs Cauchy problems. The appropriate structures for the Green functions are provided. We further prove that the Green functions are the spatial probability density functions.