Approximate solution of second kind integral equations on infinite cylindrical surfaces

Abstract

The paper considers second kind integral equations of the form (abbreviated)φφK+=g, in which S is an infinite cylindrical surface of arbitrary smooth cross-section. The “truncated equation” (abbreviated )aaaaKEφφ+=g, obtained by replacing S by Sa, a closed bounded surface of class C2, the boundary of a section of the interior of S of length 2a, is also discussed. Conditions on k are obtained (in particular, implying that K commutes with the operation of translation in the direction of the cylinder axis) which ensure that I-K is invertible, that I - Ka is invertible and (I — Ka)-1 uniformly bounded for all sufficiently large a, and that aφ converges to φ in an appropriate sense as ∞→a. Uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained. A boundary integral equation, which models three-dimensional acoustic scattering from an infi- nite rigid cylinder, illustrates the application of the above results to prove existence of solution (of the integral equation and the corresponding boundary value problem) and convergence of a particular collocation method.