A FAST ALGORITHM FOR THE NUMERICAL SOLUTION OF AN INTEGRAL EQUATION WITH LOGARITHMIC KERNEL

Katharina Flemming and Peter Junghanns

Abstract: We describe an algorithm for the numerical solution of an integral equation of the form $$ -\frac1{\pi}\int_{-1}^1\left[(y-x)\ln|y-x|-h(x,y)\right]\frac{u(y) dy}{\sqrt{1-y^2}}=f(x),\quad-1<x<1, $$ which is based on a collocation-quadrature 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 ill-posed problem in (the best possible choice of) a pair of non-periodic Sobolev-like spaces. The present paper presents the technique, how to overcome this peculiarity in the investigation of the fast algorithm.

Keywords: first kind integral equation, ill-posed problem, collocation method, quadrature method

Classification (MSC2000): 65R20; 45B05, 45E99

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