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

International Journal of Mathematical Modelling & Computations

مقاله 1، دوره 8، 1 (WINTER) - شماره پیاپی 29، فروردین 2018، صفحه 1-14   

نوع مقاله: Review Article

نویسنده

elham afshari^*

Islamic Azad University,khomain Branch

چکیده

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.

کلیدواژه‌ها

fractional derivative؛ finite difference method؛ Stability and convergence؛ Fourier analysis؛ time fractional diffusion equation