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A numerical scheme for solving a class of logarithmic integral equations arisen from two-dimensional Helmholtz equations using local thin plate splines

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  • Assari, Pouria
  • Asadi-Mehregan, Fatemeh
  • Cuomo, Salvatore

Abstract

This paper presents a numerical method for solving logarithmic Fredholm integral equations which occur as a reformulation of two-dimensional Helmholtz equations over the unit circle with the Robin boundary conditions. The method approximates the solution utilizing the discrete collocation method based on the locally supported thin plate splines as a type of free shape parameter radial basis functions. The local thin plate splines establish an efficient and stable technique to estimate an unknown function by a small set of nodes instead of all points over the solution domain. To compute logarithm-like singular integrals appeared in the method, we use a particular nonuniform Gauss–Legendre quadrature rule. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error estimate of the proposed method is presented. Numerical results are included to show the validity and efficiency of the new technique. These results also confirm that the proposed method uses much less computer memory in comparison with the method established on the globally supported thin plate splines. Moreover, it seems that the algorithm of the presented approach is attractive and easy to implement on computers.

Suggested Citation

  • Assari, Pouria & Asadi-Mehregan, Fatemeh & Cuomo, Salvatore, 2019. "A numerical scheme for solving a class of logarithmic integral equations arisen from two-dimensional Helmholtz equations using local thin plate splines," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 157-172.
    • Handle: RePEc:eee:apmaco:v:356:y:2019:i:c:p:157-172
    DOI: 10.1016/j.amc.2019.03.042
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    References listed on IDEAS

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    1. Yao, Guangming & Duo, Jia & Chen, C.S. & Shen, L.H., 2015. "Implicit local radial basis function interpolations based on function values," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 91-102.
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    Cited by:

    1. 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.
    2. 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).

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