Time Fractional Advection - Diffusion Equation To Certain Definite Solution Of The Problem

The time fractional advection-dispersion equation is derived from the classical advection-dispersion equation whose 1-order time derivative is replaced by the time fractional order derivative ( 0 <α ≤ 1).The time fractional advection-dispersion equation is applied wildly in the realm of the natural science and the engineering. In the second section, the solution for the Cauchy problem of the time fractional advection-dispersion equation is considered by using Fourier transform and Laplace transform . A appropriate integral structure for the solution is represented What is more,the relationship between the solution for the Cauchy problem of the standard advection-dispersion equation and the probability density for water's Brown motion is attentioned in this paper. The solution for the Cauchy problem of the fractional advection-dispersion equation is proved to be the probability density for a certain motion which is different from the Brown motion. The mathematical expectation and the variance for the probability density are provided. The feature of the diffusion velocity is provided. In the third section, the solution for the mixed problem of the time fractional advection-dispersion equation is directly represented by using the method of separation of variables. Based on the proper function, a numerical solution for the mixed problem of the time fractional advection-dispersion equation is provided by using the spectral method The numerical schema is stable and the accuracy of the schema is estimated. In the fouth section, the numerical solution for the mixed problem of the time fractional advection-dispersion equation is provided by using the difference method.The difference schema is proved be unconditionally stable and convergent...