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A new framework for multivariate general conformable fractional calculus and potential applications

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  • Zhao, Dazhi
  • Pan, Xueqin
  • Luo, Maokang

Abstract

Some applications of fractional partial differential equations (FPDEs) based on conformable fractional derivative (CFD) are discussed recently. As an extension of CFD — power kernel is generalized to other normalized probability distribution kernels, general conformable fractional derivative (GCFD) provides a framework to explain the physical meaning of CFD and describe which phenomena or processes GCFD/CFD are suitable to model. We will establish the multivariate theory of GCFD in this paper. Compared with traditional hereditary fractional calculus like Riemann–Liouville type or Caputo type, the unified theory of GCFD vector calculus has well forms with simplicity and elegance. As applications, fractional Maxwell’s equations of GCFD are given to describe electromagnetic fields of media that demonstrate inhomogeneous, complicated and local properties, and some fractional parabolic equations are presented as natural extensions of some nonlinear parabolic equations that cover several important famous equations.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:510:y:2018:i:c:p:271-280
    DOI: 10.1016/j.physa.2018.06.070
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    References listed on IDEAS

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    1. Eslami, Mostafa, 2016. "Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 141-148.
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    Cited by:

    1. Xiao, Guanli & Wang, JinRong & O’Regan, Donal, 2020. "Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    2. 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.
    3. 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).
    4. Abd-Allah Hyder & Ahmed H. Soliman & Clemente Cesarano & M. A. Barakat, 2021. "Solving Schrödinger–Hirota Equation in a Stochastic Environment and Utilizing Generalized Derivatives of the Conformable Type," Mathematics, MDPI, vol. 9(21), pages 1-16, October.
    5. Abdeljawad, Thabet & Al-Mdallal, Qasem M. & Jarad, Fahd, 2019. "Fractional logistic models in the frame of fractional operators generated by conformable derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 119(C), pages 94-101.
    6. Yang, Yang & Wang, Xiuqin, 2022. "A novel modified conformable fractional grey time-delay model for power generation prediction," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).

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