Victor D. Didenko

Approximation methods for singular integral equations with conjugation on curves with corners

Approximation methods for singular integral operators with continuous coefficients and conjugation as well as for the double-layer potential operator on curves with corners are investigated with respect to their stability. The methods under consideration include several kinds of quadrature rules and collocation and qualocation methods. The approach is based on the thorough use of Banach algebra techniques and local principles. Complete necessary and sufficient stability conditions are derived.

Stability of the Nyström Method for the Sherman-Lauricella Equation

The stability of the Nystrom method for the Sherman-Lauricella equation on piecewise smooth closed simple contour

On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

\Gamma

Approximation methods for the Muskhelishvili equation on smooth curves

is studied. It is shown that in the space

On the Spectral Radius of Convolution Dilation Operators

L_2

Mellin convolution operators in Bessel potential spaces

the method is stable if and only if certain operators associated with the corner points of

Approximate foveated images and reconstruction of their uniform pre-images

\Gamma

Spectral radii of refinement and subdivision operators

are invertible. If

Spline Approximation Methods for the Biharmonic Dirichlet Problem on Non-Smooth Domains

\Gamma

On Real and Complex Spectra in some Real C*-Algebras and Applications

does not have corner points, the method is always stable. Numerical experiments show the transformation of solutions when the unit circle is continuously transformed into the unit square, and then into various rhombuses. Examples also show an excellent convergence of the method.