A generalized moment representation of two free boundary problems (Q1093345)

This paper presents an alternate formulation of free boundary value problems for a time-dependent region in the x,y-plane in which Laplace's equation holds. Any harmonic function may be expressed in terms of any appropriate sequence of linearly independent solutions of Laplace's equation. \textit{S. Richardson} in his work on Hele Shaw flows [J. Fluid Mech. 56, 609-618 (1972; Zbl 0256.76024), and 102, 263-278 (1981; Zbl 0451.76015)] has given the coefficients in this expression with the help of the (time-dependent) moments of the (time-dependent) region, these are the domain integrals of the powers \((x+iy)^ n.\) Generalizing this approach (by another choice of the infinite set of harmonic functions than the set of powers) the authors treat two problems, first a moving-boundary value problem arising in electrochemical machining, and second the classical problem of steady water waves on the surface of an irrotational flow over a horizontal bed. Satisfying the boundary condition with the help of the mentioned set of generalized moments an infinite set of integral equations is found. Truncation of this infinite set and the use of a quadrature formula in the evaluation of the integrals bring about a highly nonlinear system of algebraic equations which is solved by a generalized Newton method. Some numerical results are given.