Research Article | Open Access
Image Theory for Neumann Functions in the Prolate Spheroidal Geometry
Interior and exterior Neumann functions for the Laplace operator are derived in terms of prolate spheroidal harmonics with the homogeneous, constant, and nonconstant inhomogeneous boundary conditions. For the interior Neumann functions, an image system is developed to consist of a point image, a line image extending from the point image to infinity along the radial coordinate curve, and a symmetric surface image on the confocal prolate spheroid that passes through the point image. On the other hand, for the exterior Neumann functions, an image system is developed to consist of a point image, a focal line image of uniform density, another line image extending from the point image to the focal line along the radial coordinate curve, and also a symmetric surface image on the confocal prolate spheroid that passes through the point image.
Let be a smooth closed surface in , with being its interior and its exterior, respectively. An interior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential :where is the Laplace operator, is a fixed point in the open domain , is the Dirac delta function, is the outward normal derivative on the surface , and is a given function specified on the surface satisfying the constraintThis constraint follows by applying the divergence theorem to (1a). If we further demand the normal derivative to be constant on , then we arrive at probably the most common boundary condition used in developing a Neumann-Green’s function :where stands for the total surface area of .
The solutions of a Neumann problem when they exist, have the following integral representation (see , p.286-287):where stands for the weighted mean value of the solution on ; namely, , which can be chosen to be zero to simplify the equation. All other solutions to (4a) and (4b) can be obtained by adding an arbitrary constant to this solution.
Likely, an exterior Neumann function for the Laplace operator is the solution of the following boundary value problem for the potential : where now is a given point in the open domain .
Neumann functions are analogous to Green’s functions for Dirichlet problems, so they are often also called Green’s function for the Neumann problem or Green’s function of the second kind. While Dirichlet-Green’s functions are generally used for electrostatic problems where the potential is specified on bounding surfaces, Neumann-Green’s functions are often useful for finding temperature distributions where the bounding surfaces have specified currents. Similar to Dirichlet-Green’s functions, a Neumann-Green function can also be decomposed into a singular and a regular part as where is a harmonic function that secures the satisfaction of the boundary conditions. In what follows, is also called the reflected part of the Neumann function.
An image system for a Neumann function is a system of fictitious sources inside the complement of the fundamental domain that produces a potential equal to the reflected part of the Neumann function. These fictitious sources are commonly called images because they are located not in the real domain of interest for the problem but in its complement. In general, such an image system is not unique. There are all types of images, from isolated point images to continuous distributions of images on lines, curves, and surfaces to combinations of these images. While image theories for Dirichlet-Green’s functions have been studied quite extensively in the literature [3–11], much less work has been done with image theories for Neumann-Green’s functions.
In their interesting article , Dassios and Sten studied the Neumann function with the constant boundary condition and the corresponding image system in spherical and ellipsoidal geometry. For example, they have found that, in the spherical case, an image system for the exterior spherical Neumann function may consist of (i) a point image at the origin with strength , (ii) a point image at the conventional Kelvin image point with strength , where is the radius of the sphere, and (iii) a uniform line image with strength on the line segment between the origin and the Kelvin image point . On the other hand, an image system for the interior spherical Neumann function may consist of (i) a point image at with strength and (ii) a line image that extends from radially to infinity with field-point dependent charge density.
In the present work, we shall study Neumann functions and their image systems for the Laplace operator in the prolate spheroidal geometry. Although in theory the case to be considered here is a special case of that studied in , the authors still believe it is much beneficial to study it separately because the Neumann functions and their image systems in the ellipsoidal geometry have to be constructed using ellipsoidal harmonics, while those in the prolate spheroidal geometry can be constructed using prolate spheroidal harmonics, but the ellipsoidal harmonics are much more complicated to handle than the spheroidal ones. In addition, it seems that  contains several mistakes of omission and commission . The present paper is organized as follows. Section 2 provides a brief introduction to the prolate spheroidal coordinates and to the prolate spheroidal harmonics. Then, interior and exterior Neumann functions and their image systems for the Laplace operator in the prolate spheroidal geometry are developed in Sections 3 and 4, respectively. Finally, some concluding remarks are given in Section 5.
2. Elements of Prolate Spheroidal Harmonics
Prolate spheroid is defined bywhere . Here, the focal symmetry axis of the spheroid is aligned with the -axis, and the interfocal distance is with ; see Figure 1.
In this paper, the prolate spheroidal coordinates are defined through where is the radial variable, is the angular variable, and is the azimuthal variable, respectively. The prolate spheroid (8) is called the reference prolate spheroid of this coordinate system. The coordinate surface of constant is a prolate spheroid confocal to the prolate spheroid (8), and such a confocal prolate spheroid, denoted by , can be written in the Cartesian coordinates asIn particular, with is the prolate spheroid (8), and corresponds to the focal line connecting the two focal points, respectively. The coordinate surfaces of constant are the two sheets of a hyperboloid of revolution with focal points , and in particular , , and correspond to the positive -axis beyond the focal point , the negative -axis beyond the focal point , and the -plane, respectively. The coordinate surface of constant is a plane through the -axis at an angle to the -plane. The metric coefficients for the prolate spheroidal coordinates are given by
The interior prolate spheroidal harmonics are and , and the exterior prolate spheroidal harmonics are and , for and , where and are the associated Legendre functions of the first and second kinds, respectively. In particular, and are the Legendre functions of the first and second kinds, respectively. Accordingly, the surface prolate spheroidal harmonics are and , for and , which are orthogonal over a confocal prolate spheroid with respect to the following geometrical weighting function:In fact, we have the following orthogonality relation: where is the Kronecker delta, the differential surface element on isand the normalization constants are
The surface prolate spheroidal harmonics form a complete set of eigenfunctions over a prolate spheroid . Therefore, any smooth function defined over the prolate spheroid can be expanded in terms of the surface prolate spheroidal harmonics. Furthermore, if is even with respect to , then it can be expanded in terms of only the even surface prolate spheroidal harmonics; namely,with the coefficients given by
3. Interior Neumann Functions and Their Image Systems
Given a unit source located at point inside the prolate spheroid , that is, , we assume, without loss of any generality, that the source is in the -plane; that is, . We consider interior Neumann functions with several different Neumann boundary conditions.
3.1. Nonconstant Boundary Condition
The interior Neumann function considered in this case, denoted by , is the solution of the following boundary value problem: where the normal derivative on is set to be a constant multiple of the geometrical weighting function on . Compared to the boundary condition (3) which assumes a constant normal derivative on , the boundary condition (20b) not only sounds more physical but also leads to a simpler mathematical formulation for the Neumann function. In fact, is the average of on the prolate spheroid sinceThis identity also guarantees that such a Neumann function exists. In this case, the weighted mean value of a function on the surface is .
Due to the azimuthal symmetry of the system, the interior Neumann function can be expressed in terms of the even interior prolate spheroidal harmonics aswhere are constants that have to be determined in such a way so as to satisfy the boundary condition (20b). Using (18), we can rewrite in the neighborhood of the boundary , where , as
Since the surface normal derivative at any point on isthe boundary condition (20b) demands that the following equation holds at every point on :
Obviously, is the arbitrary constant of the solution since . Indeed, integrating (25) over and using the orthogonality relation (13a), (13b), and (13c), we obtainwhich is an identity that can be verified using
Next, for each and , multiplying both sides of (25) by , integrating it over , and using the orthogonality relation (13a), (13b), and (13c), we obtainfrom which we getHence, the interior Neumann function is
The arbitrary constant may be fixed by demanding one additional condition, say In this case, noting that the origin in the prolate spheroidal coordinates corresponds to the point and thatwe get
3.2. Constant Nonzero Boundary Condition
The interior Neumann function with a constant nonzero normal derivative on , denoted by , is defined as the solution of the differential equation (20a) with the boundary conditionwhere stands for the total surface area of . In this case, the term in (5) is the average value of the solution on ; namely, .
Clearly, still serves as the arbitrary constant of the interior Neumann function. Indeed, integrating (33) over and using (13a), (13b), and (13c), we still have identity (26). Next, for each and , multiplying both sides of (33) by , integrating it over , and using the orthogonality relation (13a), (13b), and (13c), we obtainThe right-hand side of (34), however, is no longer zero for all and . Instead, noting thatwe haveTherefore, when or when and is odd, we still have (28), but when and is even, we instead have
3.3. Homogeneous Boundary Condition
If we demand the homogeneous boundary condition on , then the interior Neumann function, denoted by , is defined as the solution of the following boundary value problem : where stands for the volume of . The extra term in the differential equation (38a) is needed to guarantee the existence of such a Neumann function, and in this case the solution to the Neumann problem (4a) and (4b) is still given by (5) but with the term representing the average value of the solution in the volume ; namely, .
Let be a function in defined asThen, it is easy to see that and thatTherefore, the Neumann function can be obtained by subtracting from the Neumann function ; namely, we have
3.4. Image Systems for Interior Neumann Functions
Let us now turn to the construction of image systems for the interior Neumann functions. We will focus ourselves on the interior Neumann function given by (29), but the basic idea and the key conclusions apply to other interior Neumann functions as well.
More specifically, we wish to build an image system that represents the following part of the interior Neumann function :As pointed out earlier, such an image system is not unique. By reasoning as Dassios and Sten did in the case of the ellipsoidal geometry , here we consider an image system consisting of a point image with strength at some exterior point (since the source is in the -plane, naturally the only point image should be in the -plane as well) as a continuous one-dimensional line image extending from the point to infinity along the radial coordinate curve with densitywhere is some constant and is some continuous function in , and as a continuous two-dimensional surface image on the confocal prolate spheroid with densitywhere are constants to be determined later. By reasoning as we did in the case of Dirichlet-Green’s function , the vanishing of the term in implies that the total strength (or in electrostatics, the total surface “charge”) on is zero, and the vanishing of the terms in implies that the distribution on is symmetric with its centroid at the origin. Such an image system is graphically illustrated in Figure 2.
As pointed out by Dassios , the existence of the continuous one-dimensional distribution of images in the proposed image system is characteristic of the Neumann boundary condition, which in fact was shown 70 years ago by Weiss who studied image systems through applications of Kelvin’s transformation in electricity, magnetism, and hydrodynamics [17, 18]. Dassios further provided an intuitive explanation of the one-dimensional distribution of images reflecting the physics of the underlying problem, that is, the Neumann boundary condition. Since the derivative is the limit of the difference of the solution between two points and the image interpretation of such difference demands a sequence of point images with gaps that are proportional to the distance between the two points, it then becomes clear that the set of point images evolves into a continuous curve as the two points approach each other in order to define the derivative. On the other hand, the existence of the continuous two-dimensional distribution of images over a closed surface in the image system is a direct consequence of the three-dimensional character of the spheroidal or ellipsoidal geometry as it compares with the essentially one-dimensional character of the spherical geometry in which, as mentioned in Section 1, point and line images are enough to represent interior Neumann functions.
The potential in the interior of the spheroid generated by this image system iswhere , , and is the differential length element on ; that is, . Therefore, we have
First, comparing the monopole term in both sides of (49), we havefrom which we get
The function in is something one can choose as long as the corresponding improper integrals in (49) all exist. For instance, we can chooseIn this case, the fact that the improper integral in (51) exists can be verified straightforwardly and the fact that the improper integrals in (49) for all exist is due to as . In fact, we have
Since the point image lies on the confocal prolate spheroid , we obtainwhich can be rewritten asEquation (58) provides us with a nonlinear algebraic equation for the radial coordinate of the point image . Unfortunately, this equation does not necessarily have a solution in . However, if we choose the constant as one satisfyingwhere is a constant such that , then (58) has a solution in . Indeed, define an auxiliary function asFirst, it can be demonstrated that is continuous on . Then, note that, for , as , we have and thus . Next, it is easy to see that . Therefore, (58) must have a solution in . In addition, we note that the constant should be chosen in such a way so as to guarantee that the point image is close to the boundary if the source is close to the boundary. For example, we can choose for some constant .
We were unable to prove the uniqueness of such a solution, but numerical investigations have suggested that the solution of (58) in should be unique. Figure 3 shows the graph of for the case of , (so ), and with . In this case, the unique solution is about . In general, Newton’s method can be used to solve (58) for . Once is found, we can use (55a) and (55b) to calculate and and consequently , and we can use (51) to calculate the strength of the point image, respectively.
4. Exterior Neumann Functions and Their Image Systems
Given a unit source located at point outside the prolate spheroid , so . Again, we assume the source is in the -plane so . We shall consider exterior Neumann functions with several different boundary conditions as well.
4.1. Nonconstant Boundary Condition
The exterior Neumann function considered in this case, denoted by , is the solution of the following boundary value problem:
Due to the azimuthal symmetry of the system, the exterior Neumann function can be expressed in terms of the even exterior prolate spheroidal harmonics aswhere are constants that have to be determined by the boundary condition (63b). Using (18), we can rewrite in the neighborhood of the boundary , where , asThe boundary condition (63b) demands that the following equation holds at every point on :
4.2. Constant Nonzero Boundary Condition
The exterior Neumann function with a constant nonzero normal derivative on , denoted by , is defined as the solution of the differential equation (63a) with the boundary conditions (63c) andIn this case, the exterior Neumann function is still given by (64), but now the boundary condition (72) demands that the following equation holds at every point on :
Integrating (73) over and using the orthogonality relation (13a), (13b), and (13c), we obtainTherefore, we still have . Next, for each and , multiplying both sides of (73) by , integrating it over , and using (13a), (13b), and (13c), we obtain