The approximate solution of singular integro-differential equations systems on smooth contours in spaces Lp

• Main Author: Iu. Caraus
• Format: Article
• Language: English
• Published: Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova 1997-08-01
• Series: Computer Science Journal of Moldova
• Online Access: http://www.math.md/nrofdownloads.php?file=/files/csjm/v5-n2/v5-n2-(pp198-209).pdf

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]. $