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Generalized fractional calculus on time scales based on the generalized Laplace transform

Author

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  • Li, Xin
  • Ma, Weiyuan
  • Bao, Xionggai

Abstract

This paper aims to develop definitions and properties about the generalized Laplace transform, fractional integral and derivative on time scales. On the basis of the α−derivative and generalized exponential function, the Laplace transform is extended to the generalized case with respect to another function on time scales. Then, the generalized fractional integral and derivative on time scales are defined by employing the inverse generalized Laplace transform to unify a variety of definitions about continuous and discrete fractional calculus. Moreover, some significant theorems are derived to further increase the availability of these proposed operators. Finally, two examples with different kernel functions are given to verify the feasibility of the theoretical results.

Suggested Citation

  • Li, Xin & Ma, Weiyuan & Bao, Xionggai, 2024. "Generalized fractional calculus on time scales based on the generalized Laplace transform," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    • Handle: RePEc:eee:chsofr:v:180:y:2024:i:c:s0960077924001504
    DOI: 10.1016/j.chaos.2024.114599
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    References listed on IDEAS

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    1. Ting-Ting Song & Guo-Cheng Wu & Jia-Li Wei, 2022. "Hadamard Fractional Calculus On Time Scales," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(07), pages 1-14, November.
    2. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    3. Ismail, G.M. & Abdl-Rahim, H.R. & Abdel-Aty, A. & Kharabsheh, R. & Alharbi, W. & Abdel-Aty, M., 2020. "An analytical solution for fractional oscillator in a resisting medium," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
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