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Integral Equations
Stan Richardson

Conformal Mapping

Many different methods exploiting integral equations have been developed to determine numerically the function giving the conformal map of the unit disk onto some given simply-connected domain; for an account of many of these, see Henrici [9]. As an example of an integral equation that is both nonlinear and singular, we consider the one introduced by Theodorsen for this purpose.

Let the unit disk reside in the plane of so its boundary is given by , and let the domain onto which we wish to map this disk reside in the plane of . We assume that the origin lies within this domain and that its boundary is given in polar form by some known function . The map we seek takes points on the boundary of the disk in the -plane (identified by the angle ) onto points on the boundary of the domain in the -plane (identified by the angle ). The problem reduces to finding the dependence between these two angles; once we have determined the function , the mapping itself can easily be constructed.

If we normalize the map by requiring that the origins in the two planes correspond, and that the map induces no rotation at the origin, the function satisfies Theodorsen's integral equation

The integral here is to be interpreted as a Cauchy principal value, there being a singularity at . (It is possible to transform this equation to a form that is technically nonsingular (see Kythe [10], for example), but the removable singularity that results involves the computation of values as the ratio of small quantities, even if one manages to avoid . It is generally preferable to keep the singularity and deal with it correctly, rather than disguise it and hope for the best.) We solve this equation using the iteration

Since an increase of in must yield an increase of in , a sensible approximate solution with which to begin the iteration is

This is also the exact solution if, and only if, the domain in the -plane is a circular disk centered on the origin.

As a first example, we will consider a domain bounded by a lemniscate of Booth (called an inverted ellipse by some authors because it is the inverse of an ellipse with respect to a circle), for which we know that the exact mapping function is given by

Here is a parameter satisfying . For the domain is a circular disk of unit radius, while as the domain degenerates to two circular discs of radius touching tangentially. We will consider the particular case

We can see how the mapping distorts the unit disk using PolarMap from the Graphics`ComplexMap` package that is already available to us.

We see that we have nonnegligible distortion, for the image domain is not convex. For this domain the function is

and we can calculate that the exact solution is

for ; the restriction on the range here is necessary because it is not the principal value of the inverse tangent function that is relevant outside this range. (Using the two-variable form ArcTan[Cos[Theta], p Sin[Theta]] instead gives a formula that is correct for , but the symmetry makes such an extension irrelevant here.) As before, we can introduce a plot to enable us to keep track of the error.

Our initial approximation is not very accurate.

We need another package to deal with the Cauchy principal value.

Because the domain onto which we are mapping is symmetric about both axes in the -plane, the function must be the sum of and a function that is an odd function of both and . It is natural to approximate this function using a linear sum of functions of the form , where must be an even integer. For this we use Fit with just 10 such functions, and therefore introduce

With this basis enforcing the correct symmetry, we need to compute values for the interpolation of approxsoln[Theta] only for , and should not compute for or because the symmetry already implies that we will have the correct values there. Such thinking leads us to define

Some preliminary trials now establish that we obtain satisfactory results with

At points in , the mapping function is expressed in terms of via the Cauchy integral formula, while on the relationship between and is more immediate. As an approximation to for we have

Arg[z] has been inserted into the range of integration here to cope with instances when is near 1 and the pole of the Cauchy integral is close to the contour of integration; Mathematica appreciates being told where problems are likely to occur!

We can test the error at various points using

We can now plot the absolute value of the error on the boundary to assess the accuracy of our approximation there.

We can also obtain an estimate of the overall accuracy in the interior by finding the maximum absolute value of the error at a random distribution of interior points. Because of the maximum modulus principle we, of course, expect the maximum error to occur on the boundary.

We give a second example that requires more ingenuity; the difficulties will mean that Mathematica takes a little longer, but it can still cope. Consider the square in the plane of defined by and . The exact mapping function is given by a Schwarz-Christoffel transformation and can be expressed in terms of elliptic integrals (see Kober [11], for example), but Mathematica handles the computations more efficiently if we use numerical integration and define the mapping function as follows.

Unaided, PolarMap does not deal with this example very effectively; it cuts off the corners of the square. We can improve matters by substituting a correct Line for the incorrect one.

For this example, is a function of period that is equal to for , and can therefore be defined by

As before, Theodorsen's integral equation is solved for the function , and the exact solution for is now

Again we begin with

To give a visual assessment of the accuracy we now use

The initial approximation is not very close.

Because of the symmetry, we can concentrate on the interval ; we know that is equal to plus an odd function of that must vanish at both and . Since these points correspond to the corners of the square, it is near them that difficulties arise, but a local analysis shows that has square root singularities and an unbounded derivative at these points. Thus, in spite of the periodicity, trigonometric interpolation is not appropriate for this example and we use Fit with a more apt set of basis functions for our representation. A suitable set is given by

To try to obtain a uniform distribution of errors we should, as before, concentrate the interpolation points near the ends of the interval, and therefore introduce a nonlinear scaling similar to one used before.

In fact, because of the extra difficulties we now have at the ends, we will push points even closer to them by using new[new[Theta]]. (If larger errors near the ends are not a problem, you can obtain smaller errors over the major part of the interval by just using new[Theta].) We use iterstep constructed much as before; it produces from the interpolation on an approxsoln[Theta] that is applicable for all values of .

Satisfactory results are now obtained with

Our approximation for the mapping is given by

This is much as before, but Mathematica now appreciates being warned to expect trouble with NIntegrate at points corresponding to the corners. We can test the error at various points as in the previous example. This gives a plot of the absolute value of the error on the boundary.

And here we obtain the maximum absolute value of the error at a random distribution of interior points.



     
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