Transformation of an axialsymmetric disk problem for the Helmholtz equation into an ordinary differential equation (Q1269738)

The author considers the one-dimensional integral equation \[ \int^1_0 sg(s) \int^{2\pi}_0 \exp (ik(r^2 + s^2 - 2rs \cos \psi)^{1/2})/(r^2 + s^2 - 2rs \cos \psi)^{1/2} d\psi ds = f(r),\;r \in [0,1], \] which occurs in the problem of solving the 3D Helmholtz equation in the exterior of a circular disk where radially symmetric Dirichlet data are prescribed on the disk. To this equation, the factorization technique developed by \textit{N. Gorenflo} and \textit{M. Werner} [Solution of a finite convolution equation with a Hankel kernel by matrix factorization. SIAM J. Math. Anal. 28, No. 2, 434-451 (1997)] is applied. This leads to an equivalent ordinary differential equation, whose solution gives the representation of the solution of the disk problem.