带有分数阶边界条件的一维分数阶扩散方程差分方法

Finite Difference Approximations for a One-dimensional Fractional Diffusion Equation with Dractional Boundary Conditions

对带有分数阶边界条件一维分数阶扩散方程进行了数值研究,分别利用移位的和标准的Grünwald-Letnikov分数阶算子对方程中Riemann-Liouville空间分数阶导数和分数阶边界条件中Riemann-Liouville空间分数阶导数进行了离散,在此基础上建立了一种隐式有限差分方法。然后分析了该方法的解的存在唯一性、相容性、稳定性和收敛性。最后通过数值实例验证了该方法的有效性。

A practical numerical method to solve a one-dimensional fractional diffusion equation with fractional boundary conditions was examined. By using shifting and standard Grünwald-Letnikov fractional order operator to discrete Riemann-Liouville fractional derivative in equation and fractional boundary conditions,an implicit finite difference method was proposed. Then the existence and uniqueness of solutions for the method were discussed. The stability,consistency and convergence of the method were established. Finally,a numerical experiment was used to prove the effectiveness of the proposed format.