A Method for Solving Singular Integrodifferential Equations

Abstract

This paper describes a general technique for solving singular integrodifferential equations for a function f ( x ) (0≤ x <∞) which can be written as $u(x)=π-1∫0∞v(s)s-xds(0≤x&lt;∞)$ where u ( x ) and v ( x ) are linear forms in f ( x ) and its derivatives. The coefficients in these forms are complex constants. Such equations occur in many problems that involve solving Laplace's equation in the half-plane subject to ‘mixed’ boundary conditions. Here, though, to be specific, we motivate our study by considering the equations that occur when studying the motion of an interface separating two incompressible inviscid fluids which are of different densities and are flowing under gravity with different speeds. Upstream, the fluids are separated by a semi-infinite flat plate and the interface starts at the trailing edge of the plate. The pressure jump across the interface is assumed to be proportional to its curvature. Typically, for such problems, x denotes distance along the interface and f ( x ) denotes the Fourier transform with respect to time of its displacement normal to the plane of the plate. (The dependence of u , v , and f on the transform variable is not explicitly stated.) The main aim of the paper is to describe a simple procedure that quickly yields a representation for the general solution to a wide class of singular integrodifferential equations. This representation is more useful, and is more elegant, than that obtained by a straightforward application of the techniques described in Muskhelishvili's (1946) book. (The argument showing that the different representations are equivalent is also given.) The use of the Cauchy relations, which are needed to obtain the integrodifferential equations, requires that v ( x )←0 as x ←∞. However, some interfaces are unstable and, according to linear theory, v grows without bound with increasing x . One of the major achievements of this study is to show how the analyses should be modified in order to describe such spatially growing instabilities.