Abstract
A theoretical foundation is proposed for some computing schemes of quadratures method for solving nonlinear singular integral equations (SIE) given on arbitrary closed smooth contour. Estimations of convergence rate of approximative solution are obtained in Holder spaces.
Highlights
In the case of more smooth contours of integration Γ, namely, contours from class C(2, v) the un improvable estimates of convergence rate of method are proved
A theoretical foundation is proposed for some computing schemes of quadratures method for solving nonlinear singular integral equations (SIE) given on arbitrary closed smooth contour
Estimations of convergence rate of approximative solution are obtained in Holder spaces
Summary
In the case of more smooth contours of integration Γ, namely, contours from class C(2, v) the un improvable estimates of convergence rate of method are proved. Convergence of quadratures methods for nonlinear SIE given on arbitrary closed smooth contours. Let Γ be a closed smooth contour (Mushelishvili, 1968) bounding a simple connected region F+ of the complex plane C containing the point t = 0. In the Banach space of functions Hβ(Γ) (Mushelishvili, 1968) satisfying on Γ the Holder condition with the exponent β (0 < β < 1) consider a nonlinear SIE. Where Φ[t; u; v](t ∈ Γ; |u|, |v| < ∞), h(t, τ; u) (t, τ ∈ Γ; |u| < ∞) and f (t) are known continuous functions of their arguments, the singular integral.
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