The approximate solution of singular integral equations

A computational scheme of collocation type is proposed for a singular linear integral equation with a power singularity in the regular integral and the justification is given. The results obtained are used to justify the approximate solution of the singular integral equation
$$K(x)\equiv a(t)x(t)+\frac{b(t)}{\pi i}\int_{|\tau|=1}\frac{x(\tau)d\tau}{\tau-t}+ \frac1{2\pi i}\int_{|\tau|=1}\frac{h(t,\tau)x(\tau)}{|\tau-t|^\delta}d\tau=f(t)$$
by a modification of the method of minimal residuals.