Monotonicity in the Parameter of the Mittag-Leffler Function and Determining the Fractional Exponent of the Subdiffusion Equation

In this paper, we establish the strict monotonic dependence on the parameter $$\rho$$ for the Mittag-Leffler functions $$E_{\rho}(-t^{\rho})$$ and $$t^{\rho-1}E_{\rho,\rho}(-t^{\rho})$$ . Using these monotonicity properties, we further address an inverse problem concerning the identification of the fractional derivative order 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. Furthermore, we emphasize the application of the monotonicity property of the Mittag-Leffler functions in addressing inverse problems concerned with identifying the order of a fractional derivative.

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