TY - JOUR
T1 - On the Degenerate Scale of an Infinite Plane Containing Two Unequal Circles
AU - Chen , Jeng-Tzong
AU - Kuo , Shyh-Rong
AU - Lien , Kuen-Ting
AU - Huang , Yi-Ling
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 - <p style="text-align: justify;">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&#39;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.</p>