Approximate solution of second kind integral equations on infinite cylindrical surfaces

Peplow, A. T. and Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 (1995) Approximate solution of second kind integral equations on infinite cylindrical surfaces. SIAM Journal on Numerical Analysis, 32 (2). pp. 594-609. ISSN 0036-1429 doi: 10.1137/0732026

Abstract/Summary

The paper considers second kind integral equations of the form $\phi (x) = g(x) + \int_S {k(x,y)} \phi (y)ds(y)$ (abbreviated $\phi = g + K\phi $), in which S is an infinite cylindrical surface of arbitrary smooth cross section. The “truncated equation” (abbreviated $\phi _a = E_a g + K_a \phi _a $), obtained by replacing S by $S_a $, a closed bounded surface of class $C^2 $, 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 - K_a $ is invertible and $(I - K_a )^{ - 1} $ is uniformly bounded for all sufficiently large a, and that $\phi _a $ converges to $\phi $ in an appropriate sense as $a \to \infty $. 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 infinite 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.