带有分数阶边界条件的一维Riesz分数阶扩散方程差分方法

Finite difference approximations for one-dimensional Riesz fractional diffusion equation with fractional boundary condition

摘要:

本文对带有分数阶边界条件的一维Riesz分数阶扩散方程进行了数值研究.本文利用分数阶中心差分公式对方程中的Riemann-Liouville空间分数阶导数进行离散,并且利用标准的Grünwald-Letnikov分数阶算子对分数阶边界条件中的Riemann-Liouville空间分数阶导数进行离散,进而建立了一种隐式有限差分格式,然后讨论了该方法的解的存在唯一性，分析了该格式的相容性、稳定性和收敛性.最后， 本文通过数值实例验证了该方法的有效性.

Abstract:

In this paper, we examine a practical numerical method to solve a one-dimensional Riesz fractional diffusion equation with fractional boundary conditions. In order to propose an implicit finite difference method, we use the fractional centered derivative approach to approximate the Riesz fractional derivative and use the standard Grünwald-Letnikov fractional order operator to discrete the Riemann-Liouville fractional derivative in fractional boundary conditions. Then we discuss the existence and uniqueness of solution for the method. The stability, consistency and convergence of the method are also established. Finally, a numerical experiment is proposed to show the effectiveness of the method.