Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip
type: Item
instance of: scholarly article
title: Explicit solution of a Dirichlet-Neumann wedge diffraction problem with a strip
zbMATH Open document ID: 1069.47073
author: L. P. Castro
author: Frank-Olme Speck
author: F. S. Teixeira
published in: Journal of Integral Equations and Applications
publication date: 2004-11-11T00:00:00Z
review text: Let \(\Omega_+\) indicate the first quadrant in \(\mathbb R^2\) and \(\Omega_-\) its complement, \(\Gamma_1:= \{(0,x_2) : x_2 > 0\}\), \(\Gamma_2:= \{(x_1,0) : 0 < x_1 < a\}\), with \(a \in \mathbb R^+\), \(k \in \mathbb C\) with \(\text{Im}(k) > 0\). The authors study the following boundary value problem: \[ \begin{cases} (\Delta + k^2) u = 0 & \text{ in } \Omega_\pm ,\\ u(0^+,x_2) = u(0^-,x_2) = f_+(x_2), & (0, x_2) \in \Gamma_1,\\ {\partial u \over \partial x_2}(x_1,0^+) = {\partial u \over \partial x_2}(x_1,0^-) = - g_a(x_1), & (x_1,0) \in \Gamma_2. \end{cases}\tag{1} \] Here \(f_+ \in H^{s-1/2}(\mathbb R^+)\), \(g_a \in H^{s-3/2}(]0, a[)\) for some \(s \in ]1/2, 3/2[\). A solution of (1) is (by definition) an element \(u \in L^2(\mathbb R^2)\) such that \(u_{\Omega^\pm} \in H^s(\Omega^\pm)\). The boundary conditions are intended in a generalized sense. Problem (1) models a rectangular wedge diffraction problem. The authors are able to prove a well-posedness result and to write down the solution explicitly in terms of inverse matrixes involving Wiener-Hopf and Wiener-Hopf-Hankel operators.
reviewed by: Davide Guidetti
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 2114498
zbMATH Keywords: boundary value problem
zbMATH Keywords: generalized boundary conditions
zbMATH Keywords: rectangular wedge diffraction problem
zbMATH Keywords: well-posedness
zbMATH Keywords: Wiener-Hopf operators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181074982
OpenAlex ID: W2013854771
cites work: Q4287397
cites work: Convolution operators and factorization of almost periodic matrix functions
cites work: Regularity properties and generalized inverses of delta-related operators
cites work: Relations between convolution type operators on intervals and on the half-line
cites work: Q4424398
cites work: Q3753635
cites work: Q2723193
cites work: On Fredholm Properties of a Class of Hankel Operators
cites work: FACTORIZATION OF ALMOST PERIODIC MATRIX-VALUED FUNCTIONS AND THE NOETHER THEORY FOR CERTAIN CLASSES OF EQUATIONS OF CONVOLUTION TYPE
cites work: Q4695311
cites work: Some results on the invertibility of Wiener-Hopf-Hankel operators
cites work: Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant
cites work: Two canonical wedge problems for the Helmholtz equation
cites work: A contribution to the quarter-plane problem in diffraction theory
cites work: Wiener-Hopf-Hankel operators for some wedge diffraction problems with mixed boundary conditions
cites work: Q3251655
cites work: Toeplitz operators with semialmost periodic symbols
cites work: Mixed boundary value problems of the type of Sommerfeld's half-plane problem
cites work: Q5607484
cites work: Diffraction by a rectangular wedge: Wiener-Hopf-Hankel formulation
cites work: Q4169289
cites work: Q3997078
DOI: 10.1216/JIEA/1181074982
description: scientific article; zbMATH DE number 2114498