A New Implicit Finite Difference Method for Solving Time Fractional Diffusion Equation

Document Type: Review Article

Author

Islamic Azad University,khomain Branch

Abstract

In this paper, a time fractional diffusion equation on a finite domain is con-
sidered. The time fractional diffusion equation is obtained from the standard
diffusion equation by replacing the first order time derivative by a fractional
derivative of order 0 < a< 1 (in the Riemann-Liovill or Caputo sence). In
equation that we consider the time fractional derivative is in the Caputo sense.
We propose a new finite difference method for solving time fractional diffu-
sion equation. In our method firstly, we transform the Caputo derivative into
Riemann-Liovill derivative. The stability and convergence of this method are
investigated by a Fourier analysis. We show that this method is uncondition-
ally stable and convergent with the convergence order O( 2+h2), where t and
h are time and space steps respectively. Finally, a numerical example is given
that confirms our theoretical analysis and the behavior of error is examined
to verify the order of convergence.

Keywords