Volterra’s integral equation of the second kind, with discontinuous kernel

Abstract

In this equation the function K(w, t ) is called the kerllel; the desired function is u(z) . VVhen the functions K(w, {) and +(.z) are continuous there is no diSculty in finding a continuous 6(z) that shall satisfy equation (1). This general case of the equation (1), under the conditions solelythat +(z) be continuouswhen a-z-b and that X(x, t:) be continuous in the triangular region a-t:-z _ b, was first investigated by VOLTERRA, t who showed that there is one and only one continuous solution in the interval a _ z c b . His method applies without essential change to equations whose kernels are finite in the region a c t _ z _ b and have discontinuities, provided the discontinuities are regularly t distributed. Let us consider, however, a certain equation to which we are led by a hydrostatical problem. Suppose we are given a tube lying in a vertical plane along a curve of arbitrary shape, s t6(.z), where s is the distance along the curve and x the altitude. Let us fill this tube with a liquid of variable linear density v, and then regulate its height .z in the tube by allowing various amounts to flow through the bottom. Let us then legard zo as an analytic function of the depth in the liquid, i. e. zo zo ( Z t: ) . The average linear density is given by the formula