Title: A second order difference scheme for time fractional diffusion equation with generalized memory kernel Show Chinese title

Title: 一种用于具有广义记忆核的分数阶扩散方程的二阶差分格式

Abstract: In the current work we build a difference analog of the Caputo fractional derivative with generalized memory kernel ($_\lambda$L2-1$_\sigma$ formula). The fundamental features of this difference operator are studied and on its ground some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid $L_2$ - norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems. Show Chinese abstract

Abstract: 在当前的工作中，我们构建了一个具有广义记忆核的Caputo分数阶导数的差分模拟（$_\lambda$L2-1$_\sigma$公式）。 该差分算子的基本特性得到了研究，并在此基础上设计了一些差分格式，这些格式可以为具有变系数的广义时间分数阶扩散方程提供二阶时间精度的近似。 我们证明了给定格式在网格$L_2$- 范数下的稳定性与收敛性，其收敛速度等于近似误差的阶数。 所获得的结果通过针对一些测试问题进行的数值计算得到了支持。