Image solutions for boundary value problems without sources

Chen, Jeng‐Tzong; Shieh, Hung-Chih; Lee, Ying‐Te; Lee, Jia-Wei

doi:10.1016/j.amc.2010.02.048

Cited by 7 publications

(4 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A successful 0307-904X/$ -see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2010.04.022 extension to 3D case can be found in the paper [8]. The optimal location of the MFS is also found by using the image method.…”

Section: Introductionmentioning

confidence: 96%

A study on the method of fundamental solutions using an image concept

Chen¹,

Shieh²,

Tsai³

et al. 2010

Applied Mathematical Modelling

Self Cite

View full text Add to dashboard Cite

a b s t r a c tIn this paper, both analytical and semi-analytical solutions for Green's functions are obtained by using the image method which can be seen as a special case of method of fundamental solutions (MFS). The image method is employed to solve the Green's function for the annular, eccentric and half-plane Laplace problems. In addition, an analytical solution is derived for the fixed-free annular case. For the half-plane problem with a circular hole and an eccentric annulus, semi-analytical solutions are both obtained by using the image concept after determining the strengths of two frozen image points and a free constant by matching boundary conditions. It is found that two frozen images terminated at the two focuses in the bipolar coordinates for the problems with two circular boundaries. A boundary value problem of an eccentric annulus without sources is also considered. Error distribution is plotted after comparing with the analytical solution derived by Lebedev et al. using the bipolar coordinates. The optimal locations for the source distribution in the MFS are also examined by using the image concept. It is observed that we should locate singularities on the two focuses to obtain better results in the MFS. Besides, whether the free constant is required or not in the MFS is also studied. The results are compared well with the analytical solutions.

show abstract

Section: Introductionmentioning

confidence: 96%

A study on the method of fundamental solutions using an image concept

Chen¹,

Shieh²,

Tsai³

et al. 2010

Applied Mathematical Modelling

Self Cite

View full text Add to dashboard Cite

show abstract

“…since ∇ 2 (1/r) = 0 for r = 0, it is easy to check that f 2 (r) is a solution for Laplace's equation in the region b < r < a that satisfies the boundary conditions (2) that in our case read f 2 (r 2 ) = f 2 (a) = 1 and f 2 (r 1 ) = f 2 (b) = 0 from Eq. (33) we can also obtain f 1 (r)…”

Section: Bhf's For the Case Of Two Concentric Spheresmentioning

confidence: 94%

Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries

Morales

Díaz

Herrera

2015

Journal of Electrostatics

View full text Add to dashboard Cite

We find solutions of Laplace's equation with specific boundary conditions (in which such solutions take either the value zero or unity in each surface) using a generic curvilinear system of coordinates. Such purely geometrical solutions (that we shall call Basic Harmonic Functions BHF's) are utilized to obtain a more general class of solutions for Laplace's equation, in which the functions take arbitrary constant values on the boundaries. On the other hand, the BHF's are also used to obtain the capacitance of many electrostatic configurations of conductors. This method of finding solutions of Laplace's equation and capacitances with multiple symmetries is particularly simple, owing to the fact that the method of separation of variables becomes much simpler under the boundary conditions that lead to the BHF's. Examples of application in complex symmetries are given. Then, configurations of succesive embedding of conductors are also examined. In addition, expressions for electric fields between two conductors and charge densities on their surfaces are obtained in terms of generalized curvilinear coordinates. It worths remarking that it is plausible to extrapolate the present method to other linear homogeneous differential equations.

show abstract

“…Solutions have also been proposed to deal with multilayer systems, and they include the matrix method [1], the thermal quadrupole method [3], the thin layer method [4], and methods based on the definition of potentials [5][6][7]. Chen et al have described the use of image method to solve 2D and 3D problems in unbounded and half-space domains containing circular or spherical shaped boundaries [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

Simões

Tadeu

Simões

2016

Journal of Applied Mathematics

View full text Add to dashboard Cite

This paper presents a set of fully analytical solutions, together with explicit expressions, in the time and frequency domain for the heat conduction response of homogeneous unbounded and of bounded rectangular spaces (three-, two-, and one-dimensional spaces) subjected to point, line, and plane heat diffusion sources. Particular attention is given to the case of spatially sinusoidal, harmonic line sources. In the literature this problem is often referred to as the two-and-a-half-dimensionalfundamental solutionor 2.5D Green’s functions. These equations are very useful for formulating three-dimensional thermodynamic problems by means of integral transforms methods and/or boundary elements. The image source technique is used to build up different geometries such as half-spaces, corners, rectangular pipes, and parallelepiped boxes. The final expressions are verified here by applying the equations to problems for which the solution is known analytically in the time domain.

show abstract

Image solutions for boundary value problems without sources

Cited by 7 publications

References 12 publications

A study on the method of fundamental solutions using an image concept

A study on the method of fundamental solutions using an image concept

Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries

Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

Contact Info

Product

Resources

About