Approximate solution of the fractional advection-dispersion equation

In this paper, we consider practical numerical method to solve a space-time fractional advection-dispersion equation with variable coefficients on a finite domain. The equation is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative, and the first-order and second-order space derivatives by the Riemann-Liouville fractional derivative, respectively. Here, a new method for solving this equation is proposed in the reproducing kernel space. The representation of solution is given by the form of series and the n-term approximation solution is obtained by truncating the series. The method is easy to implement and the numerical results show the accuracy of the method.