APPROXIMATE SOLUTION OF SECOND KIND INTEGRAL EQUATIONS ON INFINITE CYLINDRICAL SURFACES

摘要

The paper considers second kind integral equations of the form phi(x) = g(x) + integral(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 Ic 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 --> infinity. 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. [References: 12]