Transformation of an axialsymmetric disk problem for the Helmholtz equation into an ordinary differential equation
type: Item
instance of: scholarly article
title: Transformation of an axialsymmetric disk problem for the Helmholtz equation into an ordinary differential equation
zbMATH Open document ID: 0915.45005
DOI: 10.1007/BF01196517
author: Norbert Gorenflo
published in: Integral Equations and Operator Theory
publication date: 1999-06-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 45H05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 1216472
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: axialsymmetric disk problem
zbMATH Keywords: factorization technique
zbMATH Keywords: integral equation
Wikidata QID: Q115393923
reviewed by: Johannes Elschner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1216472