IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v392y2021ics0096300320306718.html
   My bibliography  Save this article

Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation

Author

Listed:
  • Abbaszadeh, Mostafa
  • Dehghan, Mehdi

Abstract

In the Galerkin weak form technique based on various kernels that they do not have δ-Kronecker property, in order to apply the essential boundary condition, there are two straight strategies that one of them is the Lagrange multiplier method and another one is the penalty method. In the penalty method the main boundary value problem (BVP) is converted to a new BVP with Robin boundary condition. So, we obtain a new BVP that it must be solved. The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. For this aim, we select fractional modified distributed-order anomalous sub-diffusion equation. At the first stage, we propose a second-order difference scheme for the temporal variable. The convergence and stability analysis for the time-discrete scheme are proposed. At the second stage, we derive the full-discrete scheme based on the Galerkin weak form and shape functions of reproducing kernel particle method (RKPM) as the mentioned shape functions do not have the δ-Kronecker property. Furthermore, it is shown that when the penalty parameter goes to infinity then the solutions of BVP with Robin boundary condition are convergent to the solutions of BVP based on the essential boundary condition. The proposed examples verify that the present error estimate is true.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:392:y:2021:i:c:s0096300320306718
    DOI: 10.1016/j.amc.2020.125718
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300320306718
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2020.125718?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. 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.
    2. Wu, Guo-Cheng & Baleanu, Dumitru & Deng, Zhen-Guo & Zeng, Sheng-Da, 2015. "Lattice fractional diffusion equation in terms of a Riesz–Caputo difference," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 335-339.
    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. Sun, Fengxin & Wang, Jufeng & Xu, Ying, 2024. "An improved stabilized element-free Galerkin method for solving steady Stokes flow problems," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    2. Saffarian, Marziyeh & Mohebbi, Akbar, 2021. "Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    3. Mahmoud A. Zaky & Ahmed S. Hendy & Rob H. De Staelen, 2021. "Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System," Mathematics, MDPI, vol. 9(2), pages 1-22, January.

    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. 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).
    2. Zeng, Shengda & Baleanu, Dumitru & Bai, Yunru & Wu, Guocheng, 2017. "Fractional differential equations of Caputo–Katugampola type and numerical solutions," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 549-554.
    3. 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.
    4. Wu, Guo-Cheng & Baleanu, Dumitru & Luo, Wei-Hua, 2017. "Lyapunov functions for Riemann–Liouville-like fractional difference equations," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 228-236.
    5. 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).
    6. 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.
    7. 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.
    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. Liu, Xiping & Jia, Mei, 2019. "Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 230-242.
    10. Huang, Lan-Lan & Baleanu, Dumitru & Mo, Zhi-Wen & Wu, Guo-Cheng, 2018. "Fractional discrete-time diffusion equation with uncertainty: Applications of fuzzy discrete fractional calculus," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 508(C), pages 166-175.
    11. 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.
    12. Baleanu, Dumitru & Wu, Guo–Cheng & Zeng, Sheng–Da, 2017. "Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 99-105.
    13. 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.
    14. Qin, Hongyu & Wu, Fengyan, 2019. "Several effective algorithms for nonlinear time fractional models," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.

    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:apmaco:v:392:y:2021:i:c:s0096300320306718. 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: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.