Abstract

Time-fractional initial-boundary value problems of the form $D_t^\alpha u-p \varDelta u +cu=f$ are considered, where $D_t^\alpha u$ is a Caputo fractional derivative of order $\alpha \in (0,1)$ and the spatial domain lies in $\mathbb{R}^d$ for some $d\in \{1,2,3\}$. As $\alpha \to 1^-$ we prove that the solution $u$ converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where $D_t^\alpha u$ is replaced by $\partial u/\partial t$. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as $\alpha \to 1^-$, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as $\alpha \to 1^-$.

中文翻译：

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时间分数初始边界值问题中的误差估计的爆破

摘要

考虑形式为$ D_t ^ \ alpha up \ varDelta u + cu = f $的时间分数初始边界值问题，其中$ D_t ^ \ alpha u $是阶数\\ alpha \ in（0的Caputo分数阶导数，1）$，而空间域位于\ {1,2,3 \} $中的$ d \ $ mathbb {R} ^ d $。当$ \ alpha \到1 ^-$时，我们证明了解决方案$ u $在时空域上均匀收敛到经典抛物线初边值问题的解决方案，其中替换了$ D_t ^ \ alpha u $按$ \ partial u / \ partial t $。然而，正如我们所证明的那样，大多数对此时间分数问题进行数值方法的严格分析都有误差界限，其误差范围从$ \ alpha \到1 ^-$。我们表明，在某些情况下，可以修改这些分析以获取不会因为$ \ alpha \ to 1 ^-$爆炸的鲁棒错误界限。