Applications of Fuchsian differential equations to free boundary problems

In a couple of recent papers free boundary problems and a class of conformal mappings involving curvilinear quadrilaterals were analysed primarily using transform methods. In both the free boundary and conformal mapping problems there is an underlying relation with Fuchsian differential equations and an alternative, conceptually simpler, solution technique for these problems would use solutions of these Fuchsian equations. Thus it was conjectured in those papers that a class of Fuchsian differential equations, commonly known as Heun's equation, has in some special, but relatively important, cases degenerate solutions; these involve hypergeometric functions. This complementary solution method is, in many cases, more convenient tl-lan the transform approach. The purpose of this paper is to explore the connections with the Fuchsian equations directly, and use the solutions to solve free boundary problems. The specific examples treated here come from a quasi-steady approximation to solidification problems and are not without interest in their own right. There are few analytical solutions for solidification problems in geometries of practical interest, and the solutions found here should be of use in that regard.