Improved Finite Difference Results for the Caputo Time-Fractional Diffusion Equation
Abstract
We begin with a treatment of the Caputo time-fractional diffusion equation, by using the Laplace transform, to obtain a Volterra intego-differential equation where we may examine the weakly singular nature of this convolution kernel.We examine this new equation and utilize a numerical scheme that is derived in parallel to the L1-method for the time variable and a usual fourth order approximation in the spatial variable. The main method derived in this paper has a rate of convergence of O(k 2 + h 4 ) for u(x, t) ∈ C 6 (Ω)×C 2 [0, T], which improves previous estimates by a factor of k α. We also present an alternative method for a first order approximation in time, which allows us to relax our regularity assumption to u(x, t) ∈ C 6 (Ω) × C 1 [0, T], while exhibiting order of convergence slightly less than O(k 1+α) in time. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.
Improved Finite Difference Results for the Caputo Time-Fractional Diffusion Equation
We begin with a treatment of the Caputo time-fractional diffusion equation, by using the Laplace transform, to obtain a Volterra intego-differential equation where we may examine the weakly singular nature of this convolution kernel.We examine this new equation and utilize a numerical scheme that is derived in parallel to the L1-method for the time variable and a usual fourth order approximation in the spatial variable. The main method derived in this paper has a rate of convergence of O(k 2 + h 4 ) for u(x, t) ∈ C 6 (Ω)×C 2 [0, T], which improves previous estimates by a factor of k α. We also present an alternative method for a first order approximation in time, which allows us to relax our regularity assumption to u(x, t) ∈ C 6 (Ω) × C 1 [0, T], while exhibiting order of convergence slightly less than O(k 1+α) in time. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.
