第一类Dirichlet边界下空间四阶时间多项变阶分数阶慢扩散方程初边值问题的差分方法

A Finite Difference Scheme for the Fourth-order Time Multi-termVariable-order Fractional Sub-diffusion Equations withthe First Dirichlet Boundary Value Conditions

研究求解第一类Dirichlet边界条件下空间四阶时间多项变阶分数阶慢扩散方程的差分方法.首先,应用降阶法将原方程转换为等价的低阶方程组.然后在特殊点处考虑此方程组,并对所得方程两端同时作用平均值算子.通过巧妙定义平均值算子,对边界条件进行处理,使所得差分格式全局上达到收敛阶O(τ^(2)+h^(4)),其中τ和h分别是时间步长和空间步长.借助离散能量分析方法,证明了所提格式的无条件稳定性和收敛性.最后,给出数值算例,进一步验证了上述结论.

The present work mainly focuses on the finite difference methods for solving space fourth-order and time multi-term variable-order fractional sub-diffusion equations with the first Dirichlet boundary value conditions.Using the method of order reduction,the original equation is converted into an equivalent lower-order system,then the system is considered at some special points.Next,an average operator is performed on both hand sides of the resultant equality.With some novel techniques to handle the first Dirichlet boundary value,the global convergence of the proposed difference scheme reaches O(τ^(2)+h^(4)),withτand h the temporal and spatial step size,respectively.By the discrete energy analysis method,the unconditional stability and convergence of the developed difference scheme are proved.A numerical example is used to further verify the above conclusions.