IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v201y2025ip1s0960077925012044.html

Meshless scheme for solving backward higher-order time-fractional parabolic equations with an extremely long time span

Author

Listed:
  • Chang, Chih-Wen

Abstract

This study developed a regularization-free method for solving backward higher-order time-fractional parabolic equations with an extremely long time span, which are severely-ill-posed equations. In this method, two boundary conditions and a terminal time-wave function are used to establish the group symmetry. The constructed governing equation is then transformed into a linear combination of basis functions to acquire the complete solution. Furthermore, the method constructs an alternative symmetry group basis, which allows the governing equation to be transformed into a linear system and solved completely. When several numerical experiments are examined, we clearly show that the proposed approach can effectively handle the backward higher-order time-fractional parabolic equations. In addition, the proposed method remains robust even when the terminal data are extremely small and contaminated with significant noise, with the corresponding numerical results being highly accurate and stable. The proposed method has three advantages: (i) it does not require the estimation of additional data, (ii) it avoids the use of regularization parameters, and (iii) it does not involve any complex process.

Suggested Citation

  • Chang, Chih-Wen, 2025. "Meshless scheme for solving backward higher-order time-fractional parabolic equations with an extremely long time span," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).
    • Handle: RePEc:eee:chsofr:v:201:y:2025:i:p1:s0960077925012044
    DOI: 10.1016/j.chaos.2025.117191
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925012044
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.117191?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. Abdon Atangana & Necdet Bildik & S. C. Oukouomi Noutchie, 2014. "New Iteration Methods for Time-Fractional Modified Nonlinear Kawahara Equation," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-9, January.
    2. Zhang, Yadong & Feng, Minfu, 2023. "A local projection stabilization virtual element method for the time-fractional Burgers equation with high Reynolds numbers," Applied Mathematics and Computation, Elsevier, vol. 436(C).
    3. Peng, Xiangyi & Xu, Da & Qiu, Wenlin, 2023. "Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 702-726.
    4. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    5. Qiao, Li & Yang, Fan & Li, Xiaoxiao, 2024. "Simultaneous identification of the unknown source term and initial value for the time fractional diffusion equation with local and nonlocal operators," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    6. Li, Changpin & Li, Dongxia & Wang, Zhen, 2021. "L1/LDG method for the generalized time-fractional Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 357-378.
    7. Qiu, Wenlin & Chen, Hongbin & Zheng, Xuan, 2019. "An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 298-314.
    8. Zhang, Qifeng & Sun, Cuicui & Fang, Zhi-Wei & Sun, Hai-Wei, 2022. "Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    9. Abdon Atangana & Necdet Bildik & S. C. Oukouomi Noutchie, 2014. "New Iteration Methods for Time‐Fractional Modified Nonlinear Kawahara Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    10. Rashid Jan & Normy Norfiza Abdul Razak & Sultan Alyobi & Zaryab Khan & Kamyar Hosseini & Choonkil Park & Soheil Salahshour & Siriluk Paokanta, 2024. "Fractional Dynamics Of Chronic Lymphocytic Leukemia With The Effect Of Chemoimmunotherapy Treatment," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 32(02), pages 1-16.
    11. Djennadi, Smina & Shawagfeh, Nabil & Abu Arqub, Omar, 2021. "A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 150(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. Peng, Xiangyi & Xu, Da & Qiu, Wenlin, 2023. "Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 702-726.
    2. Daniel Olvera & Alex Elías-Zúñiga, 2014. "Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    3. Nattakorn Sukantamala & Supawan Nanta, 2021. "On Solitary Wave Solutions for the Camassa‐Holm and the Rosenau‐RLW‐Kawahara Equations with the Dual‐Power Law Nonlinearities," Abstract and Applied Analysis, John Wiley & Sons, vol. 2021(1).
    4. Zhang, Hao & Liu, Mengmeng & Guo, Tao & Xu, Da, 2024. "Three finite difference schemes for generalized nonlinear integro-differential equations with tempered singular kernel," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 1199-1217.
    5. Nihed Teniou, 2025. "A Modified Regularization Method for Inverse Problems of Nonhomogeneous Differential Operator Equation," Journal of Mathematics, John Wiley & Sons, vol. 2025(1).
    6. Bukhari, Ayaz Hussain & Raja, Muhammad Asif Zahoor & Shoaib, Muhammad & Kiani, Adiqa Kausar, 2022. "Fractional order Lorenz based physics informed SARFIMA-NARX model to monitor and mitigate megacities air pollution," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    7. Polwang, Apipoom & Poochinapan, Kanyuta & Wongsaijai, Ben, 2025. "Numerical simulation of wave flow : Integrating the BBM-KdV equation using compact difference schemes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 236(C), pages 70-89.
    8. M. Asha & T. P. Surekha, 2023. "Development of OFDM technique for underwater communication in system on chip," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 14(3), pages 977-988, June.
    9. Jun Lu & Lianpeng Shi & Chein-Shan Liu & C. S. Chen, 2022. "Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters," Mathematics, MDPI, vol. 10(13), pages 1-17, June.
    10. Cai, Min & Tong, Weiwei, 2026. "Nonuniform L1/spectral element algorithm for the time fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 239(C), pages 361-375.
    11. Michael A. Awuya & Gbenga O. Ojo & Nazim I. Mahmudov, 2022. "Solution of Space‐Time Fractional Differential Equations Using Aboodh Transform Iterative Method," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).
    12. Hamid, Muhammad & Usman, Muhammad & Haq, Rizwan Ul & Tian, Zhenfu, 2021. "A spectral approach to analyze the nonlinear oscillatory fractional-order differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    13. Karakoc, Seydi Battal Gazi & Saha, Asit & Sucu, Derya Yıldırım, 2023. "A collocation algorithm based on septic B-splines and bifurcation of traveling waves for Sawada–Kotera equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 12-27.
    14. Singh, Deeksha & Pandey, Rajesh K., 2026. "An efficient Newton-ADI scheme for 2D time-fractional reaction–diffusion equations with weak initial singularity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 438-450.
    15. Sahoo, S. & Ray, S. Saha, 2017. "Invariant analysis with conservation laws for the time fractional Drinfeld–Sokolov–Satsuma–Hirota equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 725-733.
    16. Yang, Fan & Li, Si-Ying & Li, Xiao-Xiao, 2025. "Inverse random source problem for the stochastic Caputo–Hadamard time-fractional diffusion equation driven by fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).
    17. Ju, Yuejuan & Yang, Jiye & Liu, Zhiyong & Xu, Qiuyan, 2023. "Meshfree methods for the variable-order fractional advection–diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 489-514.
    18. Can Liu & Xinming Zhang & Boying Wu, 2020. "Quasilinearized Semi-Orthogonal B-Spline Wavelet Method for Solving Multi-Term Non-Linear Fractional Order Equations," Mathematics, MDPI, vol. 8(9), pages 1-15, September.
    19. Zhang, Qifeng & Sun, Cuicui & Fang, Zhi-Wei & Sun, Hai-Wei, 2022. "Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    20. Chen, Hao & Nikan, Omid & Qiu, Wenlin & Avazzadeh, Zakieh, 2023. "Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 209(C), pages 248-266.

    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:201:y:2025:i:p1:s0960077925012044. 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.