Volume 12, Issue 5
On the Degenerate Scale of an Infinite Plane Containing Two Unequal Circles

Jeng-Tzong Chen, Shyh-Rong Kuo, Kuen-Ting Lien & Yi-Ling Huang

Adv. Appl. Math. Mech., 12 (2020), pp. 1280-1300.

Published online: 2020-07

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  • Abstract

In this paper, we analytically derived the degenerate scale of an infinite plane containing two unequal circles and numerically implemented by using the BEM. We provide two methods to analytically derive the degenerate scale. One is using the degenerate kernel and the other is using the conformal mapping. The closed-form fundamental solution of the two-dimensional Laplace equation is expanded to the degenerate kernel form in order to analytically study the degenerate scale in the BIE. Moreover, we used the technique of the conformal mapping in order to analytically study the degenerate scale in the complex variables. Then, a boundary value problem can be transformed into a Green's function. Finally, we prove the equivalence of the two analytical formulas derived by using the degenerate kernel and by using the complex variables. They are also examined by using the BEM. Good agreement is made. Besides, the case of the two equal circles is just a special one of the present formula.

  • Keywords

Degenerate scale, degenerate kernel, complex variables, bipolar coordinates, conformal mapping.

  • AMS Subject Headings

45A05, 30B10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-12-1280, author = {Jeng-Tzong Chen , and Shyh-Rong Kuo , and Kuen-Ting Lien , and Yi-Ling Huang , }, title = {On the Degenerate Scale of an Infinite Plane Containing Two Unequal Circles}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2020}, volume = {12}, number = {5}, pages = {1280--1300}, abstract = {

In this paper, we analytically derived the degenerate scale of an infinite plane containing two unequal circles and numerically implemented by using the BEM. We provide two methods to analytically derive the degenerate scale. One is using the degenerate kernel and the other is using the conformal mapping. The closed-form fundamental solution of the two-dimensional Laplace equation is expanded to the degenerate kernel form in order to analytically study the degenerate scale in the BIE. Moreover, we used the technique of the conformal mapping in order to analytically study the degenerate scale in the complex variables. Then, a boundary value problem can be transformed into a Green's function. Finally, we prove the equivalence of the two analytical formulas derived by using the degenerate kernel and by using the complex variables. They are also examined by using the BEM. Good agreement is made. Besides, the case of the two equal circles is just a special one of the present formula.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0076}, url = {http://global-sci.org/intro/article_detail/aamm/17749.html} }
TY - JOUR T1 - On the Degenerate Scale of an Infinite Plane Containing Two Unequal Circles AU - Jeng-Tzong Chen , AU - Shyh-Rong Kuo , AU - Kuen-Ting Lien , AU - Yi-Ling Huang , JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1280 EP - 1300 PY - 2020 DA - 2020/07 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0076 UR - https://global-sci.org/intro/article_detail/aamm/17749.html KW - Degenerate scale, degenerate kernel, complex variables, bipolar coordinates, conformal mapping. AB -

In this paper, we analytically derived the degenerate scale of an infinite plane containing two unequal circles and numerically implemented by using the BEM. We provide two methods to analytically derive the degenerate scale. One is using the degenerate kernel and the other is using the conformal mapping. The closed-form fundamental solution of the two-dimensional Laplace equation is expanded to the degenerate kernel form in order to analytically study the degenerate scale in the BIE. Moreover, we used the technique of the conformal mapping in order to analytically study the degenerate scale in the complex variables. Then, a boundary value problem can be transformed into a Green's function. Finally, we prove the equivalence of the two analytical formulas derived by using the degenerate kernel and by using the complex variables. They are also examined by using the BEM. Good agreement is made. Besides, the case of the two equal circles is just a special one of the present formula.

Jeng-Tzong Chen, Shyh-Rong Kuo, Kuen-Ting Lien & Yi-Ling Huang. (2020). On the Degenerate Scale of an Infinite Plane Containing Two Unequal Circles. Advances in Applied Mathematics and Mechanics. 12 (5). 1280-1300. doi:10.4208/aamm.OA-2019-0076
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