On the Expression of Exterior Neumann Problems with Infinite Boundaries as a Limit of Problems with Finite Boundaries

摘要

It is proved that the solutions of Neumann problems for the exterior of spheres (x sub 1 + b)(2) + (x sub 2)(2) + (x sub 3)(2) = b(2) where x is contained in a Euclidean three-dimensional space, R(3) converge to the solution of the exterior problem for the half-space x sub 1 or = 0 provided that the boundary ata converge in a certain sense. The method requires that there be some dissipation, which can be arbitrarily small. Fredholm integral equations are set up for the boundary data, and these are solved for large b by means of Neumann series. Convergence is proved by making use of estimates, in terms of b, of the terms of the series (which involve singular integrals). The procedure of expressing problems with infinite boundaries as a limit of problems with finite boundaries makes it possible to implement an effective numerical procedure for determining, for example, the entire near-field of a baffled piston.