Publications de l'Institut Mathématique, Nouvelle Série Vol. 96[110], pp. 143–157 (2014) 

A FAST ALGORITHM FOR THE NUMERICAL SOLUTION OF AN INTEGRAL EQUATION WITH LOGARITHMIC KERNELKatharina Flemming and Peter JunghannsFakultät für Mathematik, Technische Universität, Chemnitz, GermanyAbstract: We describe an algorithm for the numerical solution of an integral equation of the form $$ \frac1{\pi}\int_{1}^1\left[(yx)\lnyxh(x,y)\right]\frac{u(y) dy}{\sqrt{1y^2}}=f(x),\quad1<x<1, $$ which is based on a collocationquadrature method and which has the same convergence rate as this method, but only $O(n\log n)$ complexity. This integral equation turns out to be an illposed problem in (the best possible choice of) a pair of nonperiodic Sobolevlike spaces. The present paper presents the technique, how to overcome this peculiarity in the investigation of the fast algorithm. Keywords: first kind integral equation, illposed problem, collocation method, quadrature method Classification (MSC2000): 65R20; 45B05, 45E99 Full text of the article: (for faster download, first choose a mirror)
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