IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v190y2021icp678-690.html
   My bibliography  Save this article

A novel spectral Galerkin/Petrov–Galerkin algorithm for the multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth solutions

Author

Listed:
  • Hafez, Ramy M.
  • Zaky, Mahmoud A.
  • Hendy, Ahmed S.

Abstract

The usual classical polynomials-based spectral Galerkin and Petrov–Galerkin methods enjoy high-order accuracy for problems with smooth solutions. However, their accuracy and fidelity can be deteriorated when the solutions exhibit weakly singular behaviors and this issue becomes much more severe for polynomial-based spectral methods. The eigenfunctions of the Sturm–Liouville problems of fractional order serve as basis functions for constructing efficient spectral approximations for fractional differential models with nonsmooth solutions. In this paper, the Petrov–Galerkin spectral method is adopted to deal with the initial singularity in the temporal direction in which the first kind Jacobi poly-fractonomials are utilized as temporal trial functions and the second kind Jacobi poly-fractonomials as temporal test functions. Along the spatial direction, the Galerkin spectral method is adopted for the first time to deal with the boundary singularity in the spatial direction in which weighted Jacobi functions are utilized as bases in multi-dimensions. Various numerical experiments are provided to demonstrate the performance of the proposed schemes.

Suggested Citation

  • Hafez, Ramy M. & Zaky, Mahmoud A. & Hendy, Ahmed S., 2021. "A novel spectral Galerkin/Petrov–Galerkin algorithm for the multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 678-690.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:678-690
    DOI: 10.1016/j.matcom.2021.06.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421002275
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.06.004?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Li, Dongfang & Zhang, Chengjian, 2020. "Long time numerical behaviors of fractional pantograph equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 244-257.
    2. Cheng, Xiujun & Duan, Jinqiao & Li, Dongfang, 2019. "A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 452-464.
    3. Wang, Xiuping & Gao, Fuzheng & Liu, Yang & Sun, Zhengjia, 2020. "A Weak Galerkin Finite Element Method for High Dimensional Time-fractional Diffusion Equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    2. Biswas, Chetna & Singh, Anup & Chopra, Manish & Das, Subir, 2023. "Study of fractional-order reaction-advection-diffusion equation using neural network method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 15-27.
    3. Faheem, Mo & Khan, Arshad & Raza, Akmal, 2022. "A high resolution Hermite wavelet technique for solving space–time-fractional partial differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 588-609.
    4. Tang, Bo & Mao, Wenting & Zeng, Zhankuan, 2024. "A priori and a posteriori error estimates of a space–time Petrov–Galerkin spectral method for time-fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 559-572.
    5. Fardi, M. & Zaky, M.A. & Hendy, A.S., 2023. "Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 614-635.

    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. She, Mianfu & Li, Dongfang & Sun, Hai-wei, 2022. "A transformed L1 method for solving the multi-term time-fractional diffusion problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 584-606.
    2. Zhao, Jingjun & Zhang, Yanming & Xu, Yang, 2020. "Implicit Runge-Kutta and spectral Galerkin methods for the two-dimensional nonlinear Riesz space fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    3. Ahmed Z. Amin & Mahmoud A. Zaky & Ahmed S. Hendy & Ishak Hashim & Ahmed Aldraiweesh, 2022. "High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay," Mathematics, MDPI, vol. 10(17), pages 1-20, August.
    4. Li, Dongfang & Zhang, Chengjian, 2020. "Long time numerical behaviors of fractional pantograph equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 244-257.
    5. Wu, Longbin & Ma, Qiang & Ding, Xiaohua, 2021. "Energy-preserving scheme for the nonlinear fractional Klein–Gordon Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1110-1129.
    6. Hashemi, M.S. & Atangana, A. & Hajikhah, S., 2020. "Solving fractional pantograph delay equations by an effective computational method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 295-305.
    7. Sun, Lu-Yao & Fang, Zhi-Wei & Lei, Siu-Long & Sun, Hai-Wei & Zhang, Jia-Li, 2022. "A fast algorithm for two-dimensional distributed-order time-space fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    8. Ghosh, Uttam & Pal, Swadesh & Banerjee, Malay, 2021. "Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    9. Li, Yuyu & Wang, Tongke & Gao, Guang-hua, 2023. "The asymptotic solutions of two-term linear fractional differential equations via Laplace transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 394-412.
    10. Zhao, Jingjun & Li, Yu & Xu, Yang, 2019. "An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 124-138.
    11. Li, Lili & Zhao, Dan & She, Mianfu & Chen, Xiaoli, 2022. "On high order numerical schemes for fractional differential equations by block-by-block approach," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    12. Qin, Hongyu & Wu, Fengyan, 2019. "Several effective algorithms for nonlinear time fractional models," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    13. Abbaszadeh, Mostafa & Dehghan, Mehdi, 2021. "Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    14. Abdelkawy, M.A. & Alyami, S.A., 2021. "Legendre-Chebyshev spectral collocation method for two-dimensional nonlinear reaction-diffusion equation with Riesz space-fractional," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    15. Yao, Zichen & Yang, Zhanwen & Gao, Jianfang, 2023. "Unconditional stability analysis of Grünwald Letnikov method for fractional-order delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    16. Hosseininia, M. & Heydari, M.H., 2019. "Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction-diffusion equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 400-407.
    17. Sarita Nandal & Mahmoud A. Zaky & Rob H. De Staelen & Ahmed S. Hendy, 2021. "Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay," Mathematics, MDPI, vol. 9(23), pages 1-15, November.

    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:matcom:v:190:y:2021:i:c:p:678-690. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.