IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v350y2019icp249-265.html
   My bibliography  Save this article

A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations

Author

Listed:
  • Assari, Pouria
  • Dehghan, Mehdi

Abstract

This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.

Suggested Citation

  • Assari, Pouria & Dehghan, Mehdi, 2019. "A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 249-265.
    • Handle: RePEc:eee:apmaco:v:350:y:2019:i:c:p:249-265
    DOI: 10.1016/j.amc.2019.01.013
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300319300219
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2019.01.013?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. Assari, Pouria & Dehghan, Mehdi, 2017. "A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 424-444.
    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. Luo, Yidong, 2020. "Galerkin method with trigonometric basis on stable numerical differentiation," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    2. Torkaman, Soraya & Heydari, Mohammad & Loghmani, Ghasem Barid, 2023. "A combination of the quasilinearization method and linear barycentric rational interpolation to solve nonlinear multi-dimensional Volterra integral equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 366-397.
    3. Maurya, Rahul Kumar & Devi, Vinita & Srivastava, Nikhil & Singh, Vineet Kumar, 2020. "An efficient and stable Lagrangian matrix approach to Abel integral and integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 374(C).
    4. Chakraborty, Samiran & Nelakanti, Gnaneshwar, 2023. "Superconvergence of system of Volterra integral equations by spectral approximation method," Applied Mathematics and Computation, Elsevier, vol. 441(C).
    5. Parand, K. & Aghaei, A.A. & Jani, M. & Ghodsi, A., 2021. "A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 114-128.

    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. Kumhar, Raju & Kundu, Santimoy & Pandit, Deepak Kr. & Gupta, Shishir, 2020. "Green’s function and surface waves in a viscoelastic orthotropic FGM enforced by an impulsive point source," Applied Mathematics and Computation, Elsevier, vol. 382(C).
    2. Akbari, Tahereh & Esmaeilbeigi, Mohsen & Moazami, Davoud, 2024. "A stable meshless numerical scheme using hybrid kernels to solve linear Fredholm integral equations of the second kind and its applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 1-28.
    3. Xu, Fei & Huang, Qiumei, 2019. "An accurate a posteriori error estimator for semilinear Neumann problem and its applications," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    4. Pan, Yubin & Huang, Jin & Ma, Yanying, 2019. "Bernstein series solutions of multidimensional linear and nonlinear Volterra integral equations with fractional order weakly singular kernels," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 149-161.

    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:350:y:2019:i:c:p:249-265. 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.