Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior
Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior
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DOI：
10.1088/1361-6420/acab7a
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发表时间：
2022-06
期刊：
影响因子：
2.1
通讯作者：
Yikan Liu;Masahiro Yamamoto
中科院分区：
文献类型：
--
作者：
Yikan Liu;Masahiro Yamamoto
时间分数阶发展方程与抛物型方程最大的区别在于，其解的长时间或短时间行为取决于分数阶数，分数阶数的唯一性已在文献中得到了广泛的研究。与现有的结果不同，本文仅用解在单个空间点t = 0附近的渐近展开式的主项证明了阶数和参数的唯一性（对于后者可达一个乘数）.此外，我们发现了观测数据重合的未知初值或源项的特殊条件。作为副产品，我们甚至可以通过相同的数据得出初始值或源项的唯一性。证明依赖于进行拉普拉斯变换后的渐近展开式和广义本征函数的完备性。
As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t = 0 at a single spatial point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.