标题： Mittag-Leffler函数参数的单调性及亚扩散方程分数指数的确定 显示英文标题

标题： Monotonicity in the parameter of the Mittag-Leffler function and determining the fractional exponent of the subdiffusion equation

摘要： 本文证明了Mittag-Leffler函数$E_\rho(-t^\rho)$和$t^{\rho -1}E_{\rho,\rho}(-t^\rho)$关于参数$\rho$的严格单调性。随后，我们将这些结果应用于求解子扩散方程中确定分数阶导数的逆问题，其中，可用的测量值在时空的某一点给出。特别地，我们找到了该领域先前已知工作中缺失的条件。此外，所得结果适用于比先前考虑的更广泛的子扩散方程。给出了一个由Sh.A. Alimov构造的初边值问题的例子，对于该问题，所考虑的逆问题有唯一解。我们还指出了Mittag-Leffler函数的单调性在求解其他一些确定分数阶导数的逆问题中的应用。 显示英文摘要

摘要： In this paper, we prove the strict monotonicity in the parameter $\rho$ of the Mittag-Leffler functions $E_\rho(-t^\rho)$ and $t^{\rho -1}E_{\rho,\rho}(-t^\rho)$. Then, these results are applied to solve the inverse problem of determining the order of the fractional derivative in subdiffusion equations, where the available measurement is given at one point in space-time. In particular, we find the missing conditions in the previously known work in this area. Moreover, the obtained results are valid for a wider class of subdiffusion equations than those considered previously. An example of an initial boundary value problem constructed by Sh.A. Alimov is given, for which the inverse problem under consideration has a unique solution. We also point out the application of the monotonicity of the Mittag-Leffler functions to solving some other inverse problems of determining the order of a fractional derivative.