A sinc quadrature method for the double-layer integral equation in planar domains with corners
type: Item
instance of: scholarly article
title: A sinc quadrature method for the double-layer integral equation in planar domains with corners
zbMATH Open document ID: 0916.65135
author: Rainer Kress
author: Ian H. Sloan
author: Frank Stenger
published in: Journal of Integral Equations and Applications
publication date: 1999-07-19T00:00:00Z
full work available at URL: CONT10-3.html
review text: Let \[ D_d= \{z= x+ iy; x\in R, | y|< d\}\subset\mathbb{C}. \] For \(d\leq \pi/2\), the function \[ w(z)= {1\over 1+ e^{-z}},\quad z\in\mathbb{C}, \] maps the strip \(D_d\) bijectively onto an eye-shaped domain \(E_d= \{w(z): z\in D_d\}\) centered around the interval \((0,1)\). Let \(S_{\alpha,d}\) be the space of all functions \(g\), which are holomorphic in \(E_d\), real-valued on \((0,1)\), and which satisfy \(| t(1- t)|^{1- \alpha}| g(t)|\leq\alpha\), for \(t\in E_d\). The first part of the paper is devoted to sinc quadrature rules for the calculation of the integral \(\int^1_0 g(t)dt\), where \(g\in S_{\alpha,d}\). A quadrature rule with error \(| E_{n,h}(g)|\leq Ce^{-\mu n^{1/2}}\| g\|_{S_{\alpha,d}}\), for some positive constants \(C\) and \(\mu\) depending on \(d\) and \(\alpha\) is constructed. In the second part of the paper the application of the sinc quadrature method to the approximate solution of a Mellin type integral equation \[ \varphi(t)- \int^1_0 K(t,\tau)[\varphi(\tau)- \varphi(0)]d\tau+ \gamma(t)\varphi(0)= f(t)\tag{1} \] is investigated. The kernel \(K\) is assumed to have period one with respect to \(t\) and be continuous for \(0\leq t\), \(\tau\leq 1\) with the exception of the four corners of the square \([0,1]\times [0,1]\). In these corners \(K\) has Mellin type singularities. Let \(K(t,\tau)= L(t,\tau)+ M(t,\tau)\), where \[ | L(t,\tau)|\leq {1\over \tau} k\Biggl({t\over\tau}\Biggr),\quad (t,\tau)\in Q,\quad t\leq 1/2, \] \[ | L(t,\tau)|\leq {1\over 1-\tau} k\Biggl({1- t\over 1-\tau}\Biggr),\quad (t,\tau)\in Q,\quad t\geq 1/2, \] \[ Q= \{(t,\tau)\in [0,1]\times [0,1]: 0< t+\tau< 2\}, \] \[ k(0)= 0,\quad \int^\infty_0 {k(s)\over s} ds< 1. \] For equation (1) conditions of existence and uniqueness are received. The quadrature method is used for an approximate solution of equation (1). In this method, the integral in (1) is approximated by the sinc quadrature rule. Conditions of solvability of the quadrature method and an error estimation ar…
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65N38
zbMATH DE Number: 1240966
zbMATH Keywords: double-layer integral equation
zbMATH Keywords: sinc quadrature rules
zbMATH Keywords: sinc quadrature method
zbMATH Keywords: Mellin type integral equation
zbMATH Keywords: error estimation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Laplace equation
reviewed by: Ilia V. Boikov
MaRDI profile type: Publication
OpenAlex ID: W1992563060
cites work: Q5635885
cites work: Q4207544
cites work: An algorithm for the approximate solution of integral equations of Mellin type
cites work: High-Order Methods for Linear Functionals of Solutions of Second Kind Integral Equations
cites work: A Nyström method for boundary integral equations on domains with a piecewise smooth boundary
cites work: Linear integral equations
cites work: A Nyström method for boundary integral equations in domains with corners
cites work: Q4003877
cites work: Q5607548
cites work: Numerical analysis for integral and related operator equations.
cites work: Q3139836
DOI: 10.1216/JIEA/1181074232
description: scientific article; zbMATH DE number 1240966