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Leibniz-type rule of variable-order fractional derivative and application to build Lie symmetry framework

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  • Zhang, Zhi-Yong
  • Liu, Cheng-Bao

Abstract

In this paper, we first study some properties of the variable-order fractional derivative defined by the Caputo fractional derivative and particularly present a Leibniz-type rule, which makes the variable-order fractional derivative to be expressed as an infinite sum of integer-order derivatives. Then we use such properties to build a Lie symmetry framework for a class of scalar variable-order fractional partial differential equations and show that such type of equations has an elegant symmetry structure, which facilitates us to explore the symmetry properties. By means of the Lie symmetry framework and the symmetry structure, we perform a Lie symmetry classification of the variable-order fractional diffusion equations and construct the corresponding symmetry reductions and certain exact solutions.

Suggested Citation

  • Zhang, Zhi-Yong & Liu, Cheng-Bao, 2022. "Leibniz-type rule of variable-order fractional derivative and application to build Lie symmetry framework," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    • Handle: RePEc:eee:apmaco:v:430:y:2022:i:c:s0096300322003423
    DOI: 10.1016/j.amc.2022.127268
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    References listed on IDEAS

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    1. Li, Zheng & Wang, Hong & Xiao, Rui & Yang, Su, 2017. "A variable-order fractional differential equation model of shape memory polymers," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 473-485.
    2. Iskenderoglu, Gulistan & Kaya, Dogan, 2020. "Symmetry analysis of initial and boundary value problems for fractional differential equations in Caputo sense," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
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