In this paper, we consider the Space-Time Fractional
Advection-Dispersion equation on a finite domain with variable
coefficients. Fractional Advection- Dispersion equation as a model for
transporting heterogeneous subsurface media as one approach to the
modeling of the generally non-Fickian behavior of transport. We use a
semi-analytical method as Reproducing kernel Method to solve the
Space-Time Fractional Advection-Dispersion equation so that we can get
better approximate solutions than the methods with which this problem
has been solved. The main obstacle to solve this problem is the
existence of a Gram-Schmidt orthogonalization process in the general
form of the reproducing kernel method, which is very time-consuming. So,
we introduce the Improved Reproducing Kernel Method, which is a
different implementation for the general form of the reproducing kernel
method. In this method, the Gram-Schmidt orthogonalization process is
eliminated to significantly reduce the CPU-time. Also, the present
method increases the accuracy of approximate solutions.