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Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds

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  • Zhao, Dazhi
  • Luo, Maokang

Abstract

Fractional derivative is a widely accepted theory to describe physical phenomena and processes with memory effect that is defined in the form of convolution with power kernel. Due to the shortcomings of power law distribution, some derivatives with other kernels are proposed, including Caputo–Fabrizio derivative, Atangana–Baleanu derivative and so on. In this paper, in order to provide some flexible and more appropriate tools which can better describe cases of the dynamics with memory effects or of nonlocal phenomena, we derive the definition of general fractional derivatives with memory effects named GC derivative and GRL derivative from some basic principles. We demonstrate that the mathematical expression of Gurtin–Pipkin theory of heat conduction with finite wave speeds is a special example of GC/GRL derivative.

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  • Zhao, Dazhi & Luo, Maokang, 2019. "Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 531-544.
    • Handle: RePEc:eee:apmaco:v:346:y:2019:i:c:p:531-544
    DOI: 10.1016/j.amc.2018.10.037
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    References listed on IDEAS

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    1. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "A new derivative with normal distribution kernel: Theory, methods and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 476(C), pages 1-14.
    2. Zhao, Dazhi & Pan, Xueqin & Luo, Maokang, 2018. "A new framework for multivariate general conformable fractional calculus and potential applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 271-280.
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    4. Sun, HongGuang & Hao, Xiaoxiao & Zhang, Yong & Baleanu, Dumitru, 2017. "Relaxation and diffusion models with non-singular kernels," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 590-596.
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    Cited by:

    1. Zhao, Dazhi & Luo, Maokang, 2019. "Supplementary remark to ‘Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds’ [Applied Mathem," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 175-176.
    2. Fernandez, Arran & Özarslan, Mehmet Ali & Baleanu, Dumitru, 2019. "On fractional calculus with general analytic kernels," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 248-265.
    3. Isah, Sunday Simon & Fernandez, Arran & Özarslan, Mehmet Ali, 2023. "On bivariate fractional calculus with general univariate analytic kernels," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    4. Shiri, Babak & Baleanu, Dumitru, 2023. "All linear fractional derivatives with power functions’ convolution kernel and interpolation properties," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
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    8. Tian, Yan & He, Guitian & Liu, Zhibin & Zhong, Linfeng & Yang, Xinping & Stanley, H. Eugene & Tu, Zhe, 2021. "The impact of memory effect on resonance behavior in a fractional oscillator with small time delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    9. Muhammad Samraiz & Ahsan Mehmood & Saima Naheed & Gauhar Rahman & Artion Kashuri & Kamsing Nonlaopon, 2022. "On Novel Fractional Operators Involving the Multivariate Mittag–Leffler Function," Mathematics, MDPI, vol. 10(21), pages 1-19, October.
    10. Zhao, Dazhi & Yu, Guozhu & Tian, Yan, 2020. "Recursive formulae for the analytic solution of the nonlinear spatial conformable fractional evolution equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
    11. Wei, Dongmei & Liu, Hailing & Li, Yongmei & Wan, Linchun & Qin, Sujuan & Wen, Qiaoyan & Gao, Fei, 2024. "Non-Markovian dynamics of time-fractional open quantum systems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).

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