TRANSFORMATION OF AN AXIALSYMMETRIC DISK PROBLEM FOR THE HELMHOLTZ-EQUATION INTO AN ORDINARY DIFFERENTIAL-EQUATION

The problem of solving the three-dimensional Helmholtz equation in the exterior of a circular disk is considered where radially symmetric Di richlet data on the disk are assumed to be prescribed. This problem fo r example arises in the scattering of plane (sound) waves at an infini te plane screen with a circular aperture if the direction of the incid ent wave is normal to the screen, as well as in the process of diffusi on through a circular hole. By applying the factorization technique de veloped in [N. GORENFLO, M. WERNER, Solution of a finite convolution e quation with a Hankel kernel by matrix factorization, SIAM J. Math. An al., 28 (1997), pp. 434-451] to the disk problem an equivalent ordinar y differential equation is derived, whose solution leads directly to t he solution of the disk problem. This differential equation belongs to a class of ordinary differential equations which are of higher comple xity than the standard ordinary differential equations of mathematical physics. The examination of this new class of differential equations therefore is motivated.