On the solution arising in two-cylinders electrostatics

Jeng-Tzong Chen; Shing-Kai Kao; Yen-Ting Chou; Wei-Chen Tai; Jeng-Tzong Chen; Shing-Kai Kao; Yen-Ting Chou; Wei-Chen Tai

doi:10.3934/mbe.2023439

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 6: 10007-10026. doi: 10.3934/mbe.2023439

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On the solution arising in two-cylinders electrostatics

1.
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
2.
Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
3.
Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
4.
Department of Civil Engineering, National Cheng Kung University, Tainan 701401, Taiwan
5.
Department of Civil Engineering, National Taiwan University, Taipei 106216, Taiwan

Academic Editor: Feliz Manuel Barrão Minhós

Received: 11 November 2022 Revised: 10 February 2023 Accepted: 13 February 2023 Published: 27 March 2023

The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is investigated by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel of the bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. The uniqueness of solution, infinite solution, and no solution are examined therein. A single cylinder (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) at infinity in the BIE is addressed. Ordinary and degenerate scales in the BIE are both discussed. Furthermore, the solution space represented by the BIE is explained after comparing it with the general solution. The present finding is compared to those of Darevski ^[2] and Lekner ^[4] for identity.
- boundary value problem,
- bipolar coordinates,
- degenerate scale,
- degenerate kernel,
- null-field boundary integral equation
Citation: Jeng-Tzong Chen, Shing-Kai Kao, Yen-Ting Chou, Wei-Chen Tai. On the solution arising in two-cylinders electrostatics[J]. Mathematical Biosciences and Engineering, 2023, 20(6): 10007-10026. doi: 10.3934/mbe.2023439

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Abstract

The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is investigated by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel of the bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. The uniqueness of solution, infinite solution, and no solution are examined therein. A single cylinder (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) at infinity in the BIE is addressed. Ordinary and degenerate scales in the BIE are both discussed. Furthermore, the solution space represented by the BIE is explained after comparing it with the general solution. The present finding is compared to those of Darevski ^[2] and Lekner ^[4] for identity.

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