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An inverse problem of determining the fractional order in the TFDE using the measurement at one space-time point

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Abstract

In this paper we are concerned with an inverse problem for determining the order of time fractional derivative in a time fractional diffusion equation (TFDE in short), where the available measurement is given at a single space-time point. The inverse problem is transformed to a nonlinear algebraic equation of the fractional order based on the solution to the forward problem. We prove that the algebraic equation possesses a unique solution by the strict monotonicity of the Mittag-Leffler function, and the inverse problem is of uniqueness. Numerical examples are presented to show the unique solvability of the inverse problem.

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Acknowledgements

The authors thank the anonymous referees and the editor for their valuable comments. The first author thanks the support of National Natural Science Foundation of China, under Grant No. 11871313, and Natural Science Foundation of Shandong Province, China, under Grant No. ZR2019MA021. The authors also give thanks to Prof. Yamamoto M. for his helpful instructions on the estimation of the eigenfunctions. The part of this manuscript was submitted as a pre-print in Arxiv: https://arxiv.org/abs/2111.13017.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, Shandong, China

    Gongsheng Li, Xianzheng Jia & Yi Zhang

  2. College of Science, Inner Mongolia University of Technology, Hohhot, 010100, Inner Mongolia, China

    Zhen Wang

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  1. Gongsheng Li
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  2. Zhen Wang
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  3. Xianzheng Jia
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Correspondence to Gongsheng Li.

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Li, G., Wang, Z., Jia, X. et al. An inverse problem of determining the fractional order in the TFDE using the measurement at one space-time point. Fract Calc Appl Anal 26, 1770–1785 (2023). https://doi.org/10.1007/s13540-023-00170-3

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  • DOI: https://doi.org/10.1007/s13540-023-00170-3

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