On the approximate solution of a class of linear singular integral equations. (Russian) Zbl 0571.65111

Galimyanov, A. F.

Konstr. Teor. Funkts. Funkts. Anal. 4, 3-9 (1983).

The following linear singular equation is studied \[ (1)\quad a(t)x(t)+(b(t)/\pi i)\int_{\gamma}x(\tau)d\tau /(\tau -t)\ln ((t- \tau)/c)=f(t),\quad t\in \gamma, \] where a(t), b(t), f(t) are continuous functions and \(\gamma\) is the unit circle in the complex plane, \(c>2\). The numerical computation of the solution x of equation (1) is based on the collocation method using a polynomial expression for x on the basis of Hölder spaces \(H_{\beta}\) \((0<\beta \leq 1)\). An estimation of the error by numerical computation of the integral \(Bx(t)=(1/\pi i)\int_{\gamma}x(\tau)d\tau /(\tau -t)\ln ((\tau -t)/2)\) is given, too.

Reviewer: J.Kofroň

MSC:

65R20	Numerical methods for integral equations
45E05	Integral equations with kernels of Cauchy type

Keywords:

error estimates; collocation method; Hölder spaces