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On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems

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  • Stanislav Yu. Lukashchuk

    (Department of High Performance Computing Technologies and Systems, Ufa State Aviation Technical University, 12 K. Marx Str., Ufa 450008, Russia)

Abstract

The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point.

Suggested Citation

  • Stanislav Yu. Lukashchuk, 2022. "On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems," Mathematics, MDPI, vol. 10(13), pages 1-17, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2319-:d:854423
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    References listed on IDEAS

    as
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    4. Tianhang Gong & Wei Feng & Songlin Zhao, 2022. "Symmetry Analysis and Conservation Laws for a Time-Fractional Generalized Porous Media Equation," Mathematics, MDPI, vol. 10(5), pages 1-21, February.
    5. Dorjgotov, Khongorzul & Ochiai, Hiroyuki & Zunderiya, Uuganbayar, 2018. "Lie symmetry analysis of a class of time fractional nonlinear evolution systems," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 105-117.
    6. Inc, Mustafa & Yusuf, Abdullahi & Isa Aliyu, Aliyu & Baleanu, Dumitru, 2018. "Time-fractional Cahn–Allen and time-fractional Klein–Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 493(C), pages 94-106.
    7. Saberi, Elaheh & Reza Hejazi, S., 2018. "Lie symmetry analysis, conservation laws and exact solutions of the time-fractional generalized Hirota–Satsuma coupled KdV system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 296-307.
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