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

محورهای موضوعی : فصلنامه ریاضی

elham afshari ¹ (Islamic Azad University,khomain Branch)

تاریخ دریافت : 1396/06/04 تاریخ پذیرش : 1397/07/17 تاریخ انتشار : 1396/10/11

کلید واژه: finite difference method, fractional derivative, Stability and convergence, Fourier analysis, time fractional diffusion equation,

چکیده مقاله :

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.