A quadrature method for the hypersingular integral equation on an interval
The paper is concerned with quadrature methods for the approximate solution of the hypersingular integral equation \[ p.f.\;\int^1_0 {u(\tau)\over |\tau- t|^2} d\tau= f(t),\quad 0\leq t\leq 1.\tag{1} \] Before discretization the parametrization \[ \gamma(s)= v(s)^\alpha/(v(s)^\alpha+ v(1- s)^\alpha),\quad 0\leq s\leq 1,\;\alpha> 3/2 \] with a certain cubic polynomial \(v\) is applied to (1), leading to the transformed equation \[ p.f.\;\int^1_0 {\gamma'(s) \gamma'(\sigma)\over |\gamma(\sigma)- \gamma(s)|^2} u(\gamma(s)) d\sigma= f(\gamma(s)) \gamma'(s),\quad 0\leq s\leq 1,\tag{2} \] with a regular solution. Then (2) is discretized by using a simple quadrature rule on a uniform mesh, and results on stability and error estimates in the norm of the Sobolev space \(H^{1/2}\) are proved. Moreover, corresponding results are obtained for a quadrature method based on the cosine transformation \[ \gamma(s)= (1- \cos\pi s)/2,\qquad 0\leq s\leq 1. \]
scientific article; zbMATH DE number 851873
scientific article