时间分数阶演化方程阶数恢复的稳定性
Stability in recovering the order of a time-fractional evolution equation
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中文总结 AI 辅助
针对时间分数阶演化方程，在未知时间因子但属有界容许类且t=0处有给定非零值的条件下，从一次标量观测恢复分数阶，证明一致小时间展开与Hölder稳定性估计，并揭示超临界微分情形。
中文摘要 AI 辅助
我们考虑具有形如$p(t)f$的源项的时间分数阶演化方程中分数阶的恢复问题，其中时间因子$p$也未知，但属于一个有界容许类，且在$t=0$处具有给定的非零值。从一次标量观测出发，我们证明了一个一致的小时间展开，并获得了分数阶的Hölder估计。更精确地说，如果固定导数阶$\gamma$低于每个容许分数阶，则稳定性指数为$\alpha_+/(2\alpha_+-\gamma)$；对于未微分的观测，这给出指数$1/2$。我们还表明，将数据微分到大于所有容许分数阶的阶是超临界的：当两个分数阶不同时，相应的导数在$t=0$附近无界。在附录中证明了两参数Mittag-Leffler函数（包括对角情形）的一致估计。
英文摘要
We consider the recovery of the fractional order in a time-fractional evolution equation with a source of the form $p(t)f$, where the temporal factor $p$ is also unknown but belongs to a bounded admissible class and has a prescribed nonzero value at $t=0$. From one scalar observation we prove a uniform small-time expansion and obtain a Hölder estimate for the fractional order. More precisely, if a fixed derivative order $γ$ lies below every admissible fractional order, then the stability exponent is $α_+/(2α_+-γ)$; for the undifferentiated observation this gives the exponent $1/2$. We also show that differentiating the data to an order larger than all admissible fractional orders is supercritical: the corresponding derivative is unbounded near $t=0$ whenever the two fractional orders are different. A uniform estimate for the two-parameter Mittag--Leffler function, including the diagonal case, is proved in the appendix.
发表机构
- V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences（乌兹别克斯坦科学院 V.I.罗曼诺夫斯基数学研究所）
- School of Engineering, Central Asian University（中亚大学工程学院）
- Graduate School of Mathematical Sciences, The University of Tokyo（东京大学数理科学研究科）
- Department of Mathematics, Faculty of Arts and Sciences, Zonguldak Bülent Ecevit University（宗古尔达克比伦特杰奇维特大学文理学院数学系）
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