IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i13p2319-d854423.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/13/2319/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/10/13/2319/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Sahadevan, R. & Prakash, P., 2017. "On Lie symmetry analysis and invariant subspace methods of coupled time fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 107-120.
    2. Jin-Fa Cheng & Yu-Ming Chu, 2012. "Fractional Difference Equations with Real Variable," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-24, December.
    3. 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.
    4. 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.
    5. Hashemi, M.S., 2015. "Group analysis and exact solutions of the time fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 141-149.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Khalique, Chaudry Masood & Motsepa, Tanki, 2018. "Lie symmetries, group-invariant solutions and conservation laws of the Vasicek pricing equation of mathematical finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 871-879.
    2. Zhang, Zhi-Yong & Li, Guo-Fang, 2020. "Lie symmetry analysis and exact solutions of the time-fractional biological population model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    3. Biswas, Swapan & Ghosh, Uttam & Raut, Santanu, 2023. "Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    4. Muhammad Bilal Khan & Aleksandr Rakhmangulov & Najla Aloraini & Muhammad Aslam Noor & Mohamed S. Soliman, 2023. "Generalized Harmonically Convex Fuzzy-Number-Valued Mappings and Fuzzy Riemann–Liouville Fractional Integral Inequalities," Mathematics, MDPI, vol. 11(3), pages 1-24, January.
    5. Yusuf, Abdullahi & Inc, Mustafa & Isa Aliyu, Aliyu & Baleanu, Dumitru, 2018. "Efficiency of the new fractional derivative with nonsingular Mittag-Leffler kernel to some nonlinear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 220-226.
    6. Mohammadizadeh, Fatemeh & Rashidi, Saeede & Hejazi, S. Reza, 2021. "Space–time fractional Klein-Gordon equation: Symmetry analysis, conservation laws and numerical approximations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 476-497.
    7. Berhe, Hailay Weldegiorgis & Qureshi, Sania & Shaikh, Asif Ali, 2020. "Deterministic modeling of dysentery diarrhea epidemic under fractional Caputo differential operator via real statistical analysis," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    8. Ali, Ahmad T. & Khater, Mostafa M.A. & Attia, Raghda A.M. & Abdel-Aty, Abdel-Haleem & Lu, Dianchen, 2020. "Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    9. Nass, Aminu M., 2019. "Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 370-380.
    10. Inc, Mustafa & Yusuf, Abdullahi & Aliyu, Aliyu Isa & Baleanu, Dumitru, 2018. "Investigation of the logarithmic-KdV equation involving Mittag-Leffler type kernel with Atangana–Baleanu derivative," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 520-531.
    11. 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.
    12. Tanwar, Dig Vijay, 2022. "Lie symmetry reductions and generalized exact solutions of Date–Jimbo–Kashiwara–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    13. Jeong, Darae & Li, Yibao & Choi, Yongho & Lee, Chaeyoung & Yang, Junxiang & Kim, Junseok, 2021. "A practical adaptive grid method for the Allen–Cahn equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 573(C).
    14. Zhang, Yi, 2019. "Lie symmetry and invariants for a generalized Birkhoffian system on time scales," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 306-312.
    15. Hashemizadeh, E. & Ebrahimzadeh, A., 2018. "An efficient numerical scheme to solve fractional diffusion-wave and fractional Klein–Gordon equations in fluid mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 1189-1203.
    16. Ávalos-Ruiz, L.F. & Zúñiga-Aguilar, C.J. & Gómez-Aguilar, J.F. & Escobar-Jiménez, R.F. & Romero-Ugalde, H.M., 2018. "FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 177-189.
    17. Hashemi, M.S., 2021. "A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    18. Bakhshandeh-Chamazkoti, Rohollah & Alipour, Mohsen, 2022. "Lie symmetries reduction and spectral methods on the fractional two-dimensional heat equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 97-107.
    19. Inc, Mustafa & Yusuf, Abdullahi & Aliyu, Aliyu Isa & Baleanu, Dumitru, 2018. "Lie symmetry analysis, explicit solutions and conservation laws for the space–time fractional nonlinear evolution equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 371-383.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2319-:d:854423. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.