The solution of a class of free boundary problems

Mellin transform and Wiener-Hopf techniques are used to solve several free boundary problems occuring in groundwater flow. This is a novel application of a technique not usually associated with free boundary problems; here it is advocated as a powerful alternative to methods relying on conformal mappings. In particular analytical methods can now be applied to classes of problems whose explicit solution is otherwise unobtainable. Relatively simple integral representations are also provided for a class of problems that otherwise require complicated analysis involving hypergeometric functions.

The problems are formulated in an intermediate complex plane, and the Mellin transform is used to solve the problems directly in this plane. Alternative methods require a hodograph plane to be formulated; this is then conformally mapped to the intermediate plane. If the conformal mapping can be done then the full solution is determined.

As a by-product of this analysis the conformal mappings of some curvilinear polygons to a half plane are found. To perform such mappings directly for non-trivial problems involving more than three singular points requires the solution of Fuchsian differential equations that involve accessory parameters and unknown points. Apart from degenerate cases there appears to be no constructive method of obtaining solutions to such Fuchsian differential equations, as a result few practical problems whose representation in the hodograph plane involves curvilinear quadrangles have been solved. This paper has the aim of developing and demonstrating a simple, coherent strategy for tackling the free boundary problems that lead to a class of these more complicated situations and this substantially extends the range of problems that can be treated. The method presented sidesteps the difficulties associated with the free point and accessory parameter, and the solution for a problem where the hodograph plane is a curvilinear quadrangle with one vertex angle of 2 pi is found; in the physical plane this corresponds to a point of inflection on the free surface. Hence it is possible to conformally map a class of curvilinear quadrangles to an upper half plane using the analytic methods developed here and this resolves an open question in conformal mapping theory.

Two illustrative examples are also solved, one involving three singular points and this provides a comparison with previous techniques, and the other illustrates the occurrence of a point of inflection in a relatively simple situation. The alternative method advocated here is compared to the other analytical techniques available; these are briefly reviewed to place the Mellin transform, Wiener-Hopf approach in context.

Proceedings of the Royal Society A, 453, pp 1-24, 1997

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