A Two-Point Collocation Method for the Numerical Solution of One-Dimensional Hypersingular Integral Equations on Nonuniform Partitions

A quadrature formula has been constructed for calculating a hypersingular integral over asegment that uses the endpoints of the segment partition intervals as nodes of piecewise constantinterpolation of the integral density, as well as specially selected collocation points. A distinctivefeature of the proposed quadrature formula is the ability to calculate the integral of functions thatsuffer finitely many discontinuities of the first kind on the integration interval. On the basis of thequadrature formula constructed, a numerical scheme is developed for solving the characteristichypersingular integral equation on a nonregular grid. An estimate of the rate of convergence ofapproximate solutions to exact ones is proved in the class of piecewise Hölder functions.