Image solutions for boundary value problems without sources

doi:10.1016/j.amc.2010.02.048

Applied Mathematics and Computation

Volume 216, Issue 5, 1 May 2010, Pages 1453-1468

https://doi.org/10.1016/j.amc.2010.02.048 Get rights and content

Abstract

In this paper, we employ the image method to solve boundary value problems in domains containing circular or spherical shaped boundaries free of sources. two and threeD problems as well as symmetric and anti-symmetric cases are considered. By treating the image method as a special case of method of fundamental solutions, only at most four unknown strengths, distributed at the center, two locations of frozen images and one free constant, need to be determined. Besides, the optimal locations of sources are determined. For the symmetric and anti-symmetric cases, only two coefficients are required to match the two boundary conditions. The convergence rate versus number of image group is numerically performed. The differences of the image solutions between 2D and 3D problems are addressed. It is found that the 2D solution in terms of the bipolar coordinates is mathematically equivalent to that of the simplest MFS with only two sources and one free constant. Finally, several examples are demonstrated to see the validity of the image method for boundary value problems.

Introduction

The image method is a popular approach in the theoretical physics [1] and has commonly been used in multidisciplines such as electro-magnetics, acoustics and optics. When solving problems by using the Green’s functions for a bounded domain, the reflection is described by one or successive image sources, and the position and sign of the image sources is chosen so that the boundary conditions can be satisfied [2]. Green’s function for a part of domain bounded by planes, circles or spherical surface in terms of the corresponding fundamental solution in the full space can be found in the literature [3]. In certain cases, it is possible to obtain the exact solution for a concentrated source in a domain through superimposing the infinite plane or infinite space solution for the given source and its image sources. Although the scope of this method is limited for special geometry, it yields a great deal of insight into the solution when it works [4], [5]. As a result of the aforementioned consideration, many theoretical studies concerning the Green’s function in circular and spherical boundaries have appeared in the literature. For example, Green’s function for plane boundaries has been investigated [6]. The image method was employed to solve edge dislocation in an anisotropic film-substrate system [7] and dielectric plate [8]. Chen et al. [9], [10] solved Green’s functions of annulus or concentric spheres by using the image method. It is found that almost all the related works on the image method deal with the problem with a true source in the domain. Although Cheng’s book [11] has employed the image method to solve the boundary value problems (BVPs) of an infinite space with two spherical boundaries, the frozen image locations were not found to be the focuses of the bispherical coordinates. However, we may wonder whether the image method may work for BVPs without sources in the domain. Bispherical and bipolar coordinates were always used to derive the analytical solutions for problems containing boundaries of two spheres or circles [12], respectively. The BVPs of eccentric annulus were solved in a unified way of conformal mapping [13]. Problems with several circular boundaries were solved by using the null-field BIEM [14].

In this paper, we will illustrate several examples to demonstrate the possible use of image method in solving 2D and 3D BVPs without sources. Symmetric, anti-symmetric and eccentric cases are considered. Based on the singularities distributed outside the domain for the image method, it can be seen as a special method of fundamental solutions (MFS) with optimal locations and strengths of sources. To verify our image idea, analytical solutions by using the bipolar and bispherical coordinates are used to check the accuracy of our results. Besides, numerical results using the conventional MFS and null-field BIEM are also given for comparison. An infinite space with two spherical cavities as well as an infinite plane with two circular holes are both considered. Besides, an eccentric sphere is also given. Also, the static result for a limiting case of two-spheres radiation to simulate Laplace problems is provided for comparison.

Section snippets

3D BVP

The problem of an infinite space with two spherical cavities is shown in Fig. 1(a) and the governing equation is $\nabla^{2} u (x) = 0, x \in D,$ where $\nabla^{2}$ is the Laplacian, u(x) is the potential function and D is the domain of interest. For a two-spheres case, the boundary conditions are $u (x) = V_{1}, x \in B_{1},$ $u (x) = V_{2}, x \in B_{2},$ where $B_{1}$ and $B_{2}$ are left and right spherical boundaries with constant boundary data of $V_{1}$ and $V_{2}$ , respectively. In this case, the analytical solution [12] was derived in terms of the bispherical coordinates

Case 1: an infinite space with two spherical cavities subject to symmetric boundary conditions (symmetric problem of $V_{1} = V_{2} = V = 1$

In the first case, the problem sketch for an infinite space with two spherical cavities is shown in Fig. 4. The centers of two cavities are set at (0, −2.5, 0) and (0, 2.5, 0), and the radii are both 1. By matching the boundary conditions, the analytical solution [12] can be simplified by using Eq. (4) as given below: $u (ξ, η) = \sqrt{2 \cosh η - 2 \cos ξ} \sum_{n = 0}^{\infty} [V \frac{\cosh (n + \frac{1}{2}) η}{\cosh (n + \frac{1}{2}) η_{0}}] e^{- (n + 1 / 2) η_{0}} P_{n} (\cos ξ) .$ By matching the boundary conditions, all the unknown coefficients in Eqs. (4), (5), (6), $q^{s} (N), c_{1}^{s} (N)$ and $c_{2}^{s} (N)$ , can

Conclusions

In this paper, five solutions for the 2 and 3D BVPs were obtained by using the image method. For the 3D case, we have found the strengths of the two initial sources at the two centers that can be determined in advance to satisfy its own boundary condition. The strengths of successive images are then calculated and their values become smaller and smaller. The final strengths of frozen images approach zero for sufficiently large number of successive images. However, the finding in the 3D case can

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Image solutions for boundary value problems without sources

Abstract

Introduction

Section snippets

3D BVP

Case 1: an infinite space with two spherical cavities subject to symmetric boundary conditions (symmetric problem of V1=V2=V=1

Conclusions

J. Sound Vib.

Eng. Anal. Bound. Elem.

Method of Theoretical Physics. Part I

Application of Green’s Functions in Science and Engineering

Progress and prospects in the theory of linear wave propagation

SIAM J. Numer. Anal.

The scope of the image method

Commun. Pure Appl. Math.

Image methods for constructing Green’s functions and eigenfunctions for domains with plane boundaries

J. Math. Phys.

Solutions for edge dislocation in anisotropic film-substrate system by the image method

Math. Mech. Solids: MMS.

Case 1: an infinite space with two spherical cavities subject to symmetric boundary conditions (symmetric problem of $V_{1} = V_{2} = V = 1$