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On discrete tempered fractional calculus and its application

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Abstract

Discrete tempered fractional calculus, as a fresh generalization of discrete fractional calculus, takes both the advantages of discrete fractional calculus and its tempered fractional counterpart, which has great potential value in the interpretation of special physical phenomena. However, its essential characteristics are far from clarifying. Meanwhile, the primary goal of this study is to enrich the vital characteristics of discrete tempered fractional calculus in the presence of required domains for diverse scenarios. First, the reciprocal properties of the tempered fractional summation and difference operators, as well as the discrete version of Taylor’s formula, are established. The formulae of the discrete Laplace transform for discrete tempered fractional calculus are also determined with the aid of discrete convolution. Then, the well-posedness of tempered fractional difference equation is investigated, along with the exploration on the corresponding Volterra summation equation, and the existence, uniqueness and continuous dependence as well. In the aspect of application, the criterion of Ulam–Hyers stability of tempered fractional difference equation on finite time interval is obtained in light of a novel discrete Gronwall inequality. Besides, all examples are showed to verify the effectiveness of the main results.

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Acknowledgements

Projects supported by the National Natural Science Foundation of China (Grant No. 11902108) and the Fundamental Research Funds for the Central Universities of China (Grant No. JZ2021HGTB0125). The authors are grateful to the anonymous referees for careful reading of this manuscript and valuable comments. And the authors would like to thank the help from the editors too.

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  1. School of Mathematics, Hefei University of Technology, Hefei, 230601, China

    Li Ma & Dahong Fan

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  1. Li Ma
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Correspondence to Li Ma.

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Ma, L., Fan, D. On discrete tempered fractional calculus and its application. Fract Calc Appl Anal 26, 1384–1420 (2023). https://doi.org/10.1007/s13540-023-00163-2

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