Fast solvers of generalized arifoil equation of index -1 (Q2784829)

The author considers the generalized airfoil equation NEWLINE\[NEWLINE(Bv)(x) :=\int_{-1}^{1}\left(\frac{1}{\pi}\frac{1}{x-y}+b_1(x,y)\log|x-y|+ b_2(x,y)\right)v(y) dy=g(x).NEWLINE\]NEWLINE The kernel functions \(b_1\) and \(b_2\) are assumed to be smooth. In this case, \(B\) represents a linear continuous Fredholm operator in different weighted spaces \(L_\sigma^2(-1, 1)\); the index of \(B\) depends on the weight \(\sigma(x)\). In the present paper, \(\sigma(x)=1/\sqrt{1-x^2}\); then the index of \(B\in\mathcal L(L_\sigma^2(-1, 1))\) is \(-1\). With the cosine transformation the above integral equation is reduced to the 1-periodic integral equation which is discretized by a fully discrete version of the trigonometric collocation method. The conjugate gradient method is applied to solve the discretized problem. The approximate solution appears to be of optimal accuracy in a scale of Sobolev norms, and the \(N\) parameters of the approximate solution can be determined by \( O(N\log N)\) arithmetical operations.