A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions-Reference-Cited by-同舟云学术

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Published:2024-05-06 Issue:2 Volume:4 Page:596-611
ISSN:2673-9909
Container-title:AppliedMath
language:en
Short-container-title:AppliedMath

Author:

Feng Chao-Kang¹^ORCID,Tang Jyh-Haw²^ORCID

Affiliation:

1. Department of Aerospace Engineering, Tamkng University, Tamsui, New Taipei City 251301, Taiwan

2. Department of Civil Engineering, Chung Yuan Christian University, Zhongli, Taoyuan City 320314, Taiwan

Abstract

The infinite series solution to the boundary-value problems of Laplace’s equation with discontinuous Dirichlet boundary conditions was found by using the basic method of separation of variables. The merit of this paper is that the closed-form solution, or the singular similarity solution in the semi-infinite strip domain and the first-quadrant domain, can be generated from the basic infinite series solution in the rectangular domain. Moreover, based on the superposition principle, the infinite series solution in the rectangular domain can be related to the singular similarity solution in the semi-infinite strip domain. It is proven that the analytical source-type singular behavior in the infinite series solution near certain singular points in the rectangular domain can be revealed from the singular similarity solution in the semi-infinite strip domain. By extending the boundary of the rectangular domain, the infinite series solution to Laplace’s equation in the first-quadrant domain can be derived to obtain the analytical singular similarity solution in a direct and much easier way than by using the methods of Fourier transform, images, and conformal mapping.

Publisher

MDPI AG

Link

https://www.mdpi.com/2673-9909/4/2/33/pdf

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