Central part interpolation schemes for integral equations with singularities

Abstract

Two high order methods are constructed and analyzed for a class of Fredholm integral equations of the second kind with kernels that may have weak boundary and diagonal singularities. The proposed methods are based on improving the boundary behavior of the exact solution with the help of a change of variables, and on central part interpolation by polynomial splines on the uniform grid. A detailed error analysis for the proposed numerical schemes is given. This includes, in particular, error bounds under various types of assumptions on the equation, and shows that the proposed central part collocation approach has accuracy and numerical stability advantages compared with standard piecewise polynomial collocation methods, including the collocation at Chebyshev knots.