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 $$Kx \equiv a(t)x(t) + \frac{{b(t)}}{{\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{x(\tau )d\tau }}{{\tau - t}} + \frac{1}{{2\pi i}}\smallint _{\left| \tau \right| = 1} \frac{{h|t,\tau )x(\tau )}}{{\left| {\tau - t} \right|^\delta }}d\tau = f(t)$$ by a modification of the method of minimal residuals.