A sinc quadrature method for the double-layer integral equation in planar domains with corners
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 are obtained. In the third part of the paper the Dirichlet problem for the Laplace equation \[ \Delta u= 0 \] is reduced to a Mellin type integral equation.
scientific article; zbMATH DE number 1240966