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

  • Copyright

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} }
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