## Abstract

This is the second of a pair of papers describing the use of boundary tracing for boundary value problems. In the preceding article, the theory of the technique was explained and it was shown how it enables one to use known exact solutions of partial differential equations to generate new solutions. Here, we illustrate the use of the technique by applying it to three equations of practical significance: Helmholtz's equation, Poisson's equation and the nonlinear constant mean curvature equation. A variety of new solutions are obtained and the potential of the technique for further application outlined.

## 1. Introduction

The boundary tracing technique is a procedure by which it is possible to use known exact solutions of partial differential equations to generate new solutions of boundary value problems. The procedure is conceptually easy: given a solution of a partial differential system satisfying a prescribed constraint on the boundary of the domain of interest, one simply deforms the boundary in such a way that the constraint remains fulfilled. This method thereby generates new domains and so leads to new exact results. Our thesis is that this technique can sometimes lead to solutions of both practical and theoretical interest and the intention here is to provide some evidence for this assertion. This is the last of a short series on the technique: in the first (Anderson *et al*. 2006), we obtained new exact solutions of the Laplace–Young equation which is of importance in capillarity. The solutions obtained there are pertinent to our understanding of the effect of corners and surface roughness on the height rise of liquids in a container; issues of particular concern in industrial dip-coating processes. In the forerunner and companion of the present paper, Anderson *et al*. (2007), hereafter referred to as I, generic properties of the tracing technique were discussed and examples were chosen to best illustrate these characteristics. There we showed how properties of the method follow from geometric considerations which, in turn, enable potential pitfalls in the application of the technique to be exposed.

In the present work the aim is to exhibit the power and flexibility of the technique by examination of three simple second-order partial differential equation applications. Exact results are obtained for Helmholtz's equation, Poisson's equation and the nonlinear constant mean curvature (CMC) equation. It is important to appreciate that a thorough systematic treatment of any of these equations would be long and turgid and our aim is not to produce a comprehensive list of solutions; instead the hope is that our discussion will illustrate how boundary tracing can be of use in problems of real interest. Many more solutions have been catalogued in the first authors' thesis (Anderson 2002) and the reader is directed there for further details.

The remainder of the paper is organized as follows. In §§2–4, we examine solutions for the three application areas in turn and conclude in §5 with a review of the results that have been achieved to date together with some final remarks.

## 2. Helmholtz's equation: a corner rounding problem

The boundary tracing technique originally arose as a product of an attempt to solve a so-called corner rounding problem. It is well known that the free surface of a liquid in a vessel rises sharply in the neighbourhood of an acute-angled corner in the containing vessel. Since perfect corners rarely occur in practice, the question arises as to the effect induced by some smoothing of the corner. The height rise of the surface is governed by the Laplace–Young equation with contact conditions holding around the boundary, so the matter can be resolved by solving the Laplace–Young equation in a slightly rounded wedge. Helmholtz's equation is a small-amplitude limit of the Laplace–Young equation and so a related problem is to determine the effect of corner rounding on the solution to the Helmholtz equation in the quarter-plane *x*, *y*≥0 subject to the linearized contact boundary condition. This problem, which is both much simpler and of greater general interest than the Laplace–Young problem, will be discussed here. Explicitly, this involves solving(2.1)where ** n** is the external normal to the rounded domain

*Ω*, see figure 1.

When *Ω* is the quarter-space *x*≥0, *y*≥0, the exact solution of equation (2.1) is simply(2.2)If the sharp corner at the origin were to be rounded off slightly, it would seem that a solution ought to follow from some kind of elementary perturbation method. However, attempts to obtain a solution in this way failed and it was this difficulty that ultimately resulted in the formulation of the method expounded in I and amplified here. The obstacle to generating the desired rounded domain solution is a consequence of the fact that the quarter-plane is not the only domain for which *η*=*η*^{*} is the solution; in fact there is a smooth rounded domain with exactly the same *η*=*η*^{*}. To see this, we note that the directional derivative of *η*^{*} along a line from the origin O at an angle of *π*/4 (so bisecting *Ω*) varies in magnitude from a value at O to zero as the distance from O tends to infinity. The conclusion is that there must be some location P along this line at which ∇*η*^{*}.** n**=1; in fact, P is

*x*=

*y*=(1/2)ln 2. This location can be used as a starting point for tracing a complete smooth curve along which the contact condition is satisfied; if

*y*(

*x*) defines the boundary then the contact condition (2.1) gives(2.3)By substituting for

*η*=

*η*

^{*}from equation (2.2) and requiring

*y*=

*x*at

*x*=(1/2)ln 2, we can determine the required smooth symmetric solution. The equation cannot be integrated exactly but it is a simple matter to show that the required curve exists and a good approximation can be found numerically. We thus have isolated at least one possible corner rounding solution.

There are two other sets of traced solutions that can be obtained by integrating equation (2.3) away from points lying on one or other of the coordinate axes. By combining curves from these two families one can generate a plethora of domains compatible with the solution *η*^{*} and the flux condition (2.1), although they all have corners and so are unacceptable in our particular original context. Nevertheless, they are interesting in their own right, for it would not be anticipated that asymmetric non-smooth domains could admit symmetric solutions. The fact that all such domains produce indistinguishable results over the regions in common is perhaps even more of a surprise, although we have seen that this appears to be the rule rather than the exception.

Are there other smooth boundaries for which *η*=*η*^{*} is the requisite solution? To address this issue we need to refer to the theoretical results developed in I. The terminal curve is given by (∇*η*)^{2}=1, which for the prescribed *η* gives exp(−2*x*)+exp(−2*y*)=1. This curve passes through the point P found above and asymptotes to the axes. The viable domain lies under this curve. Evidently the *η*-contours touch the terminal curve at P and intersect it at all other points so that P is a critical terminal point in the language of I. All other points on the terminal curve are ordinary and thus do not generate realistic boundaries. The curvatures of the terminal curve at P and the *η*-contour are and so that *κ*_{Φ}>*κ*_{η} and P is a saddle on the tracing manifold. Thus, there are two smooth boundary curves through P with curvatures satisfying (using results obtained in I), so that *κ*≈0.437 or −1.144. The *η*-contour through P lies between these two traced curves and while the *κ*≈0.437 curve is smooth at P, it intersects the axes at an angle leading to a cornered region. Hence, there is just one curve that rounds the corner corresponding to *η*=*η*^{*}.

Of course to understand the effect of rounding on the solution behaviour it is not enough to have a single special result. In the above analysis, we have restricted our attention to domains generated using the solution *η*=*η*^{*}. By adding a small multiple of the radially symmetric point source solution to Helmholtz's equation which vanishes at infinity, we can obtain a whole range of solutions(2.4)where *K*_{0} is the Bessel function of order zero. This form satisfies the contact condition along the axes and can be used to generate suitably rounded domains. (The source solution is singular at the origin, which is, however, outside the rounded domains of interest). It can be shown that for each *ϵ* (within a restricted range) one can construct a smooth boundary that rounds the corner. The associated traced boundaries are displayed in figure 2. Of course there is no need to restrict attention to the radially symmetric solutions. By appending an arbitrary combination of solutions satisfying homogeneous Neumann solutions on the quarter domainone can generate domains with arbitrary rounding and obtain usefully explicit analytical results for the effect of natural rounding modes on, for example, the height rise of liquid in a corner.

Boundary rounding issues are of general interest in applied mathematics where sharp-angled domains are often used to model slightly rounded structures. The above work displays how boundary tracing may aid understanding of rounding effects. Equally, tracing can be used in the modelling of roughness and an example of this is given by Anderson *et al*. (2006) in the context of the Laplace–Young equation. Of course the fact that the new traced boundaries correspond to exact analytical results solutions is a bonus, both in the above work and in general.

## 3. Poisson's equation

Poisson's equation arises in steady state heat conduction and field problems. As an illustration as to how boundary tracing applies to Poisson's equation problems, suppose that after appropriate scaling we are left with(3.1)and let us impose constant Neumann conditions around the boundary so that(3.2)we shall assume *K* and *c* are constants for simplicity. A straightforward application of the divergence theorem proves that a solution of this problem exists if and only if(3.3)a global flux conservation requirement. The function *η*=(1/4)*K*(*x*^{2}+*y*^{2}) is a radially symmetric solution of this problem when the boundary is the circle given by *x*^{2}+*y*^{2}=4*c*^{2}/*K*^{2}. The terminal curve is given by |∇*η*|=|*c*|, which for the prescribed *η* coincides with ∂*Ω*. The feasible region for tracing purposes is external to ∂*Ω* and, since the terminal curve is also an *η*-contour, all points on the terminal curve are critical terminal points. In radial coordinates for traced boundaries of the form *R*=*R*(*θ*) and *η*≡*η*(*r*), the boundary condition (3.2) becomeswhich for *η*=*Kr*^{2}/4 integrates to givefor arbitrary constant phase *ϕ*. Thus the traced curves are straight lines tangential to the circle as sketched in figure 3. The terminal curve is also a viable boundary, so we can construct a variety of domains by patching together suitable combinations of the traced straight line boundaries and portions of . Just a few of the possibilities are shown in figure 4 which gives examples of polygonal, lenticular and teardrop-shaped domains. Thus, for example, *η*=(1/4)*K*(*x*^{2}+*y*^{2}) is the solution to Poisson's equation with ∇*η*.** n**=

*c*within polygonal domains with all sides tangential to the circle ; a quite remarkable result. Through scaling we can find the solution (when it exists) to equation (3.1) in any regular polygon with constant Neumann boundary conditions. For example, with a triangle (which is not necessarily equilateral), if the in-circle of the triangle has radius

*r*

_{i}and centre (

*x*

_{i},

*y*

_{i}), then the desired solution within the triangle is , where the constants

*K*and

*c*must satisfy

*Kr*

_{i}=2

*c*. This solution in a triangle is well known to fluid dynamicists; it is associated with the flow of a viscous fluid through a pipe of triangular cross-section. Perhaps less familiar is the tracing result that the radial solution of Poisson's equation in three dimensions is also the solution in three-dimensional polygonal domains constructed using planes touching a sphere. Of course this includes all the regular polyhedra as particular cases.

Boundary tracing can be used to prove some interesting, if theoretical, exact results. As a simple illustration, it is recalled that solutions of Poisson's equation (3.1) must be such that the global flux condition (3.3) is satisfied on the initial domain, and hence, by extension, this relation will be satisfied for any domain constructed using tracing. This constraint is an area–perimeter relationship in two dimensions or a volume–surface area result in three dimensions. Thus, we have the following result:

*Given a polygon with all sides* *(or their extensions)* *tangential to a circle of radius R*, *the ratio of the perimeter* *P* *to the area A is A*/*P*=*R*/2; *that is the same ratio as for the inscribed circle*.

Using scaling with *η*=*Kr*^{2}/4 we see that this function satisfies equation (3.1) inside a polygon with ∇*η*.** n**=

*KR*/2 on the boundary. Then the divergence theorem gives

*K*|

*Ω*|=

*c*∂

*Ω*and so |

*Ω*|/∂

*Ω*=

*R*/2 as required. ▪

This result is well known to geometers but the radial solution also works for asymmetric polygonal domains and for domains constructed from combinations of circles and tangential lines, as well as for *N*-dimensional polytopes, so results of the above type also apply for these other shapes. Thus, for example, given an *N*-dimensional polytope with all edges tangential to a given *N*-sphere of radius *r*, the ratio of its *N*-volume *V*_{N} to its *N*-surface area *S*_{N} is just *r*/*N*.

Of course a large number of additional area/perimeter or volume/area results can be generated using tracing in association with other Poisson equation solutions but we do not explore this avenue any further here. Instead, we will briefly indicate a range of other traced domains that can be obtained using other Poisson's equation solutions.

### (a) Other radial solutions

So far we have used boundary tracing on the radially symmetric solution of Poisson's equation (3.1) which is a quadratic in *r*. Of course there are additional radial solutions so we may ask what additional domains may be possible if the more general radial solution(3.4)is used as our starting point. If we keep the constant flux condition ∇*η*.** n**=

*c*, then exact solutions of the form (3.4) can be obtained in ring-shaped domains. The possible terminal curves are radially symmetric and given bywhich has real solutions for

*r*when

*c*

^{2}>2

*KA*given by the positive values ofThere thus may be zero, one or two terminal curves with the associated viable regions changing depending on the relative sizes of

*A*and

*K*. Such terminal curves, if they exist, are critical terminal curves, so a large range of possible domains with or without holes can be constructed. If we define

*A*

^{*}≡

*c*

^{2}/2

*K*, then there are five cases to consider:

*A*<0,

*A*=0, 0<

*A*<

*A*

^{*},

*A*=

*A*

^{*}and

*A*>

*A*

^{*}. When

*A*<0 or 0<

*A*<

*A*

^{*}there are both inner and outer viable regions, and as

*A*→0 from either above or below we retrieve the polynomial solution for

*η*discussed previously as the inner viable region shrinks to nothing. As

*A*→

*A*

^{*}from below, the gap between the two domains reduces and for

*A*>

*A*

^{*}the viable domain becomes the whole plane. The behaviours of the five cases have been calculated numerically and some sample domains are sketched in figure 5. Thus there are various combinations of polygons, circular and heart-shaped outer boundaries with or without holes of various shapes.

### (b) Channel solutions

The simple polynomial solution *η*=*y*^{2}/2 satisfies the Poisson's equation (3.1) with *K*=1 and the flux condition (3.2) in the channel |*y*|<*c*. Traced solutions are given by(3.5)for arbitrary *C*. By patching together such solutions, one can generate solutions in channels with one or more spikes on either or both channel sides; one example is displayed in figure 6.

### (c) Robin solutions

If a body is heated internally and this heat is expelled from its surface into the surrounding environment, a standard task is to determine the temperature rise expected within the body. Often a Robin condition of the formis used to model the heat loss from the surface. Evidently, the temperature rise will be strongly dependent on the body shape and under normal circumstances numerical procedures are necessary to obtain results for the effect of shape on the body temperature. However, there are exact one-dimensional solutions available for a channel and using tracing one can again greatly extend the range of exact solutions. Thus, using the simple channel solution *η*=*Ky*^{2}/2+*A* as a starting point, one can use tracing to derive exact solutions to Poisson's equation subject to Robin conditions in channels with kinks or with changing width. Using the same solution, external domains containing finite or infinite rounded obstructions can be constructed. Some representative domains are displayed in figure 7 and, again, more examples can be found in Anderson (2002).

The above examples of course represent a small sample of the domains that can be generated using very simple Poisson solutions. Again, it should be emphasized that all the above results are exact. Evidently, the range of domains can be further extended by superposition so that it may be possible to shape the domains to fit the circumstance of interest. One possible application area for the above results is in the area of microchip design, where high temperatures can be problematic.

## 4. The constant mean curvature equation

It is appropriate to illustrate boundary tracing by considering at least one nonlinear example. Surfaces of CMC appear in many contexts including gas dynamics, soap bubbles and micro-gravity capillarity. The equation determining a surface *η*(*x*, *y*) of CMC *κ* is(4.1)which is the simplified form of the Laplace–Young equation when gravity is ignored, surface tension is *σ* and the excess pressure *p* across the surface *η* is non-zero. This equation has been the subject of extensive research, both theoretical and experimental, and a comprehensive survey of relevant work has been compiled by Finn (1999). When *κ*=0, the CMC equation is referred to as the minimal surface equation which is of importance in optimization. In the surface tension context the contact (or constant angle) condition(4.2)around the boundary ∂*Ω* of the domain is often appropriate, and the existence and stability of available solutions is often the issue. In a finite domain, an application of the divergence theorem givesas a necessary constraint for solutions. Whereas the Laplace–Young equation admits just two exact solutions (before tracing), many such solutions have been generated for the CMC equation by a variety of means; see, for example, Wente (1986) or Eells (1987). This surfeit of exact solutions makes the CMC equation a prime candidate for the application of tracing.

### (a) Channel solutions

If liquid is constrained between two plates separated by a distance 2*d* then under equilibrium conditions the shape of the free surface of the liquid can be obtained by solving the CMC equation (4.1) on −*d*<*y*<*d* with the contact condition (4.2) satisfied on *y*=±*d*. The solution is given by(4.3)and *H* determined by volume constraints; that is the surface is cylindrical with the radius *R* determined by the contact angle and plate spacing. The excess pressure required to sustain this profile given by *p*=*Rσ*. Tracing ideas allow us to determine the solution if the bounding plates are not parallel but instead have some indentations. If we assume a boundary of the form *y*=*f*(*x*), then the contact boundary condition (4.2) becomeswhich can be integrated to give(4.4)where *x*_{0} is an arbitrary constant. Sample solutions are displayed in figure 8*a*. The terminal curves *y*=±*d* correspond to the original plate locations and the viable domains for tracing purposes are the strips *d*≤|*y*|<*R* which lie outside the original domain. Since the terminal curves are also *η*-contours they form critical terminal curves along which the above-traced solutions match smoothly (at *x*_{0}). The wall solutions and the above-traced solutions can be patched together to generate a large number of possible boundaries; one such solution is displayed in figure 8*b*. Recall that all such boundaries share the same solution so that the above indentations on either or both plates have no effect on the surface shape; the surface remains cylindrical with radius *R* within the original domain. The example displayed in figure 8*b* is particularly interesting because corners can have a dramatic effect on the surface shape both locally and globally, and in fact the very existence of solutions is threatened by the presence of such corners. However, corners constructed using the traced curves above have no effect whatsoever! It is interesting to note that there is a limit to the size of corner angle that can be constructed using the above-traced curves: this occurs when the corner P is located at *y*=*R*, see figure 8*b*; the associated (maximum) corner angle is *α*=*π*/2−*γ*. Concus & Finn (1996) have shown that no CMC solution can exist for *α* greater than this value (a result which mirrors similar restrictions in the case of the Laplace–Young equation), so the results are consistent. Of further interest are rough domains that can be constructed using traced solutions. Interestingly enough, the surface shape within the channel is again not affected by roughness generated by traced curves generated corrugations, a situation that has been previously noted in the Laplace–Young context (Anderson *et al*. 2006).

One of the principal features of tracing is that the procedure specifically seeks out boundaries that do not alter the solution within the original domain, so the technique is tailor-made to seek out possible changes in the boundary that will not cause changes in the behaviour of the surface; an issue of major concern in surface tension work.

### (b) Spherical solutions

The lower half of a sphere of radius *R* is a solution to the constant curvature problem when *κ*=2/*R*. In terms of radial coordinates the solution is given by , the associated terminal curve is given by *r*=*R* cos *γ* and the viable domain lies in the annulus *R* cos *γ*≤*r*≤*R*. The contact condition (4.2) integrates to give for the traced curves *r*(*θ*)so the traced curves are straight lines tangential to the circle *r*=*R* cos *γ*. Thus we have the familiar situation in which the terminal curve satisfies the boundary conditions and polygonal domains can be constructed akin in those found in §2 for Poisson's equation. Suppose we attempt to piece together parts of traced boundaries derived from the spherical solution. If in the composite there is a corner with included angle 2*α* then the boundary condition implies that solutions can only exist if the half-angle . Extending the argument to a polygonal-shaped container reveals that a spherical-cap solution of the CMC equation exists if there are concentric circles of radii *R*_{1} and *R*_{2} such that the sides of the polygon are all tangential to the inner circle and the vertices all lie inside the larger one. In this case, the contact angle on the edge of the container satisfies cos *γ*=*R*_{1}/*R*_{2}. As a simple application of these ideas, equilateral triangles with side length *L* will admit spherical cap solutions when *γ*<*π*/3 and then the spherical cap has a radius . Examples of solutions are presented in figure 9.

## 5. Concluding remarks

In this paper we have fleshed out some of the ideas propounded in I to show how boundary tracing methods can be used in three application areas. We have generated some indicative new results, but the scope of the boundary tracing technique is so wide that little more than a quick sketch of some of the possibilities was feasible. We have previously noted that many more solutions and applications can be found in Anderson (2002) and the three examples chosen here represent only a small selection of results that have been extracted to date.

There are two features of tracing that underpin the successful application of the technique. First, and perhaps the most surprising, is the extraordinary range of domain shapes that can be generated using a single solution by judicious patching together of available tracing curves. In linear problems, superposition of simple solutions can greatly extend the range of available domains to the extent that matching the base function to produce the required boundary becomes possible. The second feature that is important is exactness: all the results derived are precise and no approximations are made. On combining these two characteristics one has available an enormous range (essentially a continuum) of exact solutions even for nonlinear problems. As well as finding new solutions, the precision of boundary tracing sometimes enables one to derive theoretical and practical estimates on solutions of ‘difficult’ equations, such as the Laplace–Young or the CMC equation.

The range of domains generated by boundary tracing come as something of a surprise: we have produced asymmetric domains admitting symmetric solutions, domains with corners and with smoothed-out corners, domains with fine-scale structure, mesh-like domains and domains consisting of several unconnected parts. One drawback of boundary tracing is however the fact that one seems to have very limited control over the domain shapes themselves. Although patching and superposition (if available) do enable one to extend the domain range well beyond what might be expected, it would be preferable to have a less esoteric, but more predictable, range of traced curves to work with. One might expect domains generated using solutions of a particular partial differential equation to share some common characteristics, but, of course, we know perfectly well that the boundary condition can be equally influential as far as solution behaviour is concerned and so it is perhaps not surprising that results are not simple. Indeed, we have seen that elementary solutions can give rise to complex domains which change drastically with small changes in boundary condition parameters. Anderson (2002) has made a first attempt to predict what domains might be anticipated based on knowledge of the equation and boundary condition types, but much still needs to be done in this direction. In particular, a careful study of homogeneous boundary condition problems would be valuable. Results obtained using tracing are of intrinsic interest for inverse problems and, at the very least, an identification of the viable domain for basic solutions would enable one to separate out problems that arise owing to genuine non-uniqueness from those that are due to inaccuracy.

Our focus here has been on specific partial differential equations from a mathematical perspective. It should be recalled that the boundary tracing technique arose out of an attempt to answer a specific practical problem for industrial dip-coating, and it is our hope that the method might be useful for attacking various modelling problems in industrial and scientific applications. In such problems, the equations are often nonlinear and/or the geometry awkward, so that often one needs to resort to numerical simulations which can lead to inconclusive or unconvincing results. In such circumstances, similarity solutions may be available and boundary tracing can lead to useful answers—for example, we have already seen that the technique can be applied to corner and rough surface problems. Problems involving many boundaries with possibly intricate shapes may also be addressed as we have seen in I. A referee suggested that the technique might find application in computational fluid dynamics; examples include the evaluation of the performance of wings whose geometry or surface properties have been affected by wear and tear. Also in this context useful alternative grids may be generated to handle geometric or three-dimensional issues. While both here, and in I, we have concentrated on boundary tracing for exact solutions, the approach can also be implemented when the solution can only be obtained numerically. Often a numerical solution of an ordinary differential equation can be obtained very accurately and boundary tracing can then be applied to generate new results for the underlying partial differential equation. This procedure has been successfully used to generate solutions for the height rise of liquids in rough capillaries (Anderson 2002).

To conclude, it has to be accepted that the usefulness of any technique should be judged by the applicability of the results obtained. The theory expounded in I, combined with the particular applications outlined here, suggest that the boundary tracing technique is rather more than an oddity restricted to producing a few mathematical curiosities. At the very least, it is known that exact solutions of a partial differential boundary value problem are valuable as yardsticks against which the reliability of numerical methods can be assessed. Boundary tracing clearly has the potential for generating many solutions of physically important equations and one might expect that among these some will be of real significance. On the other hand, to claim that boundary tracing is potentially a revolutionary technique that deserves to be recognized as a standard method is premature. There is much that needs to be done to explore the approach; in particular, we identify the desirability of extension to both higher-order and higher-dimensional problems and the application of the technique to other equations of physical importance. In the meantime our hope is that the findings thus far have whetted the appetite and shown that boundary tracing may have a place in the toolbox of techniques available for studying boundary value problems.

## Acknowledgments

We are grateful to the referees of both I and II for their careful reading of the manuscripts and their suggestions for improvements to the work.

## Footnotes

↵† Present address: Rising Sun Research, 133 Gouger Street, Adelaide, South Australia 5000, Australia.

- Received January 23, 2007.
- Accepted May 4, 2007.

- © 2007 The Royal Society