两类分数阶对流-扩散方程的有限差分方法

考虑两类分数阶偏微分方程,空间分数阶对流-扩散方程和时间-空间分数阶对流-扩散方程。基于移位的Grünwald公式,在第一类方程中,空间分数阶导数用加权平均有限差分法来近似,用特征值方法给出了稳定性分析,误差估计为O(τ+h);在第二类方程中,时间导数逼近用高阶近似,根据最大模估计方法证明了稳定性,其收敛阶为O(τ2-max{γ1,γ2}+h),这里γ1,γ2分别是方程中出现的两项Caputo时间分数阶导数的阶。数值实例验证了理论结果。

Finite difference methods for two kinds of fractional convection-diffusion equations

Two kinds of fractional partial differential equations are considered.One is a space-fractional convection-diffusion equation and the other is a time-space fractional convection-diffusion equation.Based on the shifted Grünwald formula,the weighted average finite difference method is used to approximate the spatial fractional derivatives in the first equation,and its stability is studied by eigenvalue analysis.The error estimate is O(τ+h).A high order approximation for the temporal derivative is used for the second equation.The stability is given by the technique of maximum norm analysis,with the convergence order O(τ2-max{γ1,γ2}+h),where γ1,γ2 are the orders of the two Caputo time fractional derivatives,respectively.Numerical examples are presented to demonstrate the theoretical results.