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

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  • 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
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    References listed on IDEAS

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    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.
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    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. 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.
    4. 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).
    5. Chakraborty, Samiran & Nelakanti, Gnaneshwar, 2023. "Superconvergence of system of Volterra integral equations by spectral approximation method," Applied Mathematics and Computation, Elsevier, vol. 441(C).

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