Analysis and numerical approximation to time-fractional diffusion equation with a general time-dependent variable order
Analysis and numerical approximation to time-fractional diffusion equation with a general time-dependent variable order
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DOI：
10.1007/s11071-021-06353-y
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发表时间：
2021-05
影响因子：
5.6
通讯作者：
Xiangcheng Zheng;Hong Wang
中科院分区：
文献类型：
--
作者：
Xiangcheng Zheng;Hong Wang
变阶时间分数阶扩散方程（tFDES）通过Hurst指数来调节周围介质的分形维数变化，为描述颗粒在可变形非均质材料中的非均匀扩散输运提供了一种有竞争力的工具.分析了一般变阶分数次积分的映射性质，并在此基础上证明了相应的变阶tFDE模型在多维空间中的适定性和光滑性质.然后，我们推导和分析一个完全离散的有限元近似，在其中，我们开发了一种新的分解L-1离散系数证明的数值方案的最优阶误差估计的基础上，只有对数据的正则性假设。数值实验证实了理论研究结果。
Variable-order time-fractional diffusion equations (tFDEs), in which the variable fractional order accommodates the fractal dimension change of the surrounding medium via the Hurst index, provide a competitive instrument to describe anomalously diffusive transport of particles through deformable heterogeneous materials. We analyze the mapping properties of the fractional integral with a general time-dependent variable order, based on which we prove the well-posedness and smoothing properties of corresponding variable-order tFDE model in multiple space dimensions. We then derive and analyze a fully discretized finite element approximation, in which we develop a novel decomposition of L-1 discretization coefficients to prove an optimal-order error estimate of the numerical scheme based only on the regularity assumptions on the data. Numerical experiments are performed to substantiate the theoretical findings.