| Summary: | This article generalizes the results which were obtained in the paper [1], written together with my scientific-adviser, doctor-habilitat, professor Zolotarevschi V. Theoretical foundation of the collocation method and of mechanical quadrature method for
singular integro-differential equations systems (SIDE) in the case when the equations are given on a closed contour satisfying some conditions of smoothness, without their reduction to the unit circle, is given below. Let $\Gamma $ be a smooth Jordan border limiting the one-spanned area $F^{+}$, containing a point $ t=0$, $ F^{-}= C \setminus \{ F^{+}\cup \Gamma \}$,
$C $ is a full complex plane. Let $z= \psi (w)-$ be a function, mapping comformally and single-valuedly the surface
$\Gamma_{0}=\{|w| >1 \} $ on $F^{-} $ so that
$ \psi (\infty )= \infty ,\psi^{ (\prime )}(\infty ) >0$.
We shall assume that the function $ z= \psi (w)$ has its second derivative, satisfying on $\Gamma_{0} $ the H\"older condition with some parameter $ \nu$ $(0 < \nu <1); $ the class of such contours is denoted by $C (2; \nu ) [2,p.23]. $
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