Volume 11, Issue 4
On the Numerical Solution of Logarithmic Boundary Integral Equations Arising in Laplace's Equations Based on the Meshless Local Discrete Collocation Method

Pouria Assari & Mehdi Dehghan

Adv. Appl. Math. Mech., 11 (2019), pp. 807-837.

Published online: 2019-06

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  • Abstract

In this article, we investigate the construction of a meshless  local  discrete collection method suitable for solving a class of boundary integral equations of the second kind with logarithmic singular kernels. These types of boundary integral equations can be deduced from boundary value problems of Laplace's equations with linear Robin boundary conditions. The numerical solution presented in the current paper is obtained by moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The logarithm-like singular integrals appeared in the method are computed via a particular nonuniform Gauss-Legendre quadrature rule. Since the offered scheme is based on the use of scattered points spread on the solution domain and does not require any background meshes, it can be identified as a meshless local discrete collocation (MLDC) method. We also obtain the error bound and the convergence rate of the presented method. The new technique is simple, efficient and flexible for most classes of boundary integral equations. The convergence accuracy of the new technique is examined over four integral equations on various domains and obtained results confirm the theoretical error estimates.

  • Keywords

Laplace's equation, boundary integral equation, logarithmic singular kernel, discrete collocation method, moving least squares (MLS) method, error analysis.

  • AMS Subject Headings

45A05, 41A25, 65D10, 45E99

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COPYRIGHT: © Global Science Press

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@Article{AAMM-11-807, author = {}, title = {On the Numerical Solution of Logarithmic Boundary Integral Equations Arising in Laplace's Equations Based on the Meshless Local Discrete Collocation Method}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {4}, pages = {807--837}, abstract = {

In this article, we investigate the construction of a meshless  local  discrete collection method suitable for solving a class of boundary integral equations of the second kind with logarithmic singular kernels. These types of boundary integral equations can be deduced from boundary value problems of Laplace's equations with linear Robin boundary conditions. The numerical solution presented in the current paper is obtained by moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The logarithm-like singular integrals appeared in the method are computed via a particular nonuniform Gauss-Legendre quadrature rule. Since the offered scheme is based on the use of scattered points spread on the solution domain and does not require any background meshes, it can be identified as a meshless local discrete collocation (MLDC) method. We also obtain the error bound and the convergence rate of the presented method. The new technique is simple, efficient and flexible for most classes of boundary integral equations. The convergence accuracy of the new technique is examined over four integral equations on various domains and obtained results confirm the theoretical error estimates.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0050}, url = {http://global-sci.org/intro/article_detail/aamm/13190.html} }
TY - JOUR T1 - On the Numerical Solution of Logarithmic Boundary Integral Equations Arising in Laplace's Equations Based on the Meshless Local Discrete Collocation Method JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 807 EP - 837 PY - 2019 DA - 2019/06 SN - 11 DO - http://dor.org/10.4208/aamm.OA-2018-0050 UR - https://global-sci.org/intro/article_detail/aamm/13190.html KW - Laplace's equation, boundary integral equation, logarithmic singular kernel, discrete collocation method, moving least squares (MLS) method, error analysis. AB -

In this article, we investigate the construction of a meshless  local  discrete collection method suitable for solving a class of boundary integral equations of the second kind with logarithmic singular kernels. These types of boundary integral equations can be deduced from boundary value problems of Laplace's equations with linear Robin boundary conditions. The numerical solution presented in the current paper is obtained by moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The logarithm-like singular integrals appeared in the method are computed via a particular nonuniform Gauss-Legendre quadrature rule. Since the offered scheme is based on the use of scattered points spread on the solution domain and does not require any background meshes, it can be identified as a meshless local discrete collocation (MLDC) method. We also obtain the error bound and the convergence rate of the presented method. The new technique is simple, efficient and flexible for most classes of boundary integral equations. The convergence accuracy of the new technique is examined over four integral equations on various domains and obtained results confirm the theoretical error estimates.

Pouria Assari & Mehdi Dehghan. (2019). On the Numerical Solution of Logarithmic Boundary Integral Equations Arising in Laplace's Equations Based on the Meshless Local Discrete Collocation Method. Advances in Applied Mathematics and Mechanics. 11 (4). 807-837. doi:10.4208/aamm.OA-2018-0050
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