IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v199y2025ip3s0960077925007519.html

Solutions of (1+1) and (m+1)-dimensional time-fractional delay PDEs with the Hilfer derivative: Separable and invariant subspace methods

Author

Listed:
  • Priyendhu, K.S.
  • Prakash, P.
  • Victor, Stéphane

Abstract

The main aim of this work is to systematically present two analytical approaches that are known as (i) the separable method and (ii) the invariant subspace method to solve the scalar and coupled time-delay linear and nonlinear time-fractional PDEs with the Hilfer arbitrary-order derivative. Also, this work investigates how to compute different possible types of exact solutions for the k-component coupled (m+1)-dimensional time-delay time-fractional PDEs with the Hilfer arbitrary-order derivative through the invariant subspace method together with and without the linear space variable transformation. More precisely, we show the effectiveness and usefulness of the separable and invariant subspace methods to obtain various types of variable separable forms of exact solutions for the scalar and k-component coupled (1+1)-dimensional time-delay linear and nonlinear time-fractional heat equations with the Hilfer arbitrary-order derivative. In addition, we explicitly illustrated the importance of the invariant subspace method together with and without the linear space variable transformation to compute the variable separable forms of exact solutions for the 2-component coupled (2+1)-dimensional time-delay nonlinear time-fractional diffusion convection reaction systems with the Hilfer arbitrary-order derivative subject to suitable initial and boundary conditions. From this study, we notice that the Euler-gamma, trigonometric, exponential, three-parameter Mittag-Leffler, and polynomial functions are involved in the derived exact solutions. Further, we provide the comparative study of the discussed methods along with illustrative examples in the appropriate places as well as with the existing literature wherever possible.

Suggested Citation

  • Priyendhu, K.S. & Prakash, P. & Victor, Stéphane, 2025. "Solutions of (1+1) and (m+1)-dimensional time-fractional delay PDEs with the Hilfer derivative: Separable and invariant subspace methods," Chaos, Solitons & Fractals, Elsevier, vol. 199(P3).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p3:s0960077925007519
    DOI: 10.1016/j.chaos.2025.116738
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925007519
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.116738?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    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. Bu, Weiping & Zheng, Xin, 2025. "Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 237(C), pages 70-85.
    3. Rui, Weiguo, 2018. "Idea of invariant subspace combined with elementary integral method for investigating exact solutions of time-fractional NPDEs," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 158-171.
    4. Prakash, P. & Priyendhu, K.S. & Lakshmanan, M., 2025. "Generalized separable solutions for (2+1) and (3+1)-dimensional m-component coupled nonlinear systems of PDEs under three different time-fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    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. Prakash, P. & Priyendhu, K.S. & Lakshmanan, M., 2025. "Generalized separable solutions for (2+1) and (3+1)-dimensional m-component coupled nonlinear systems of PDEs under three different time-fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 191(C).
    2. 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.
    3. Rui, Weiguo & Yang, Xinsong & Chen, Fen, 2022. "Method of variable separation for investigating exact solutions and dynamical properties of the time-fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 595(C).
    4. 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.
    5. Tanwar, Dig Vijay, 2022. "Lie symmetry reductions and generalized exact solutions of Date–Jimbo–Kashiwara–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    6. 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.
    7. 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.
    8. 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).

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:eee:chsofr:v:199:y:2025:i:p3:s0960077925007519. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.