Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples

Romberg Type Cubature over Arbitrary Triangles

We develop an extrapolation algorithm for numerical integration over arbitary non-standard triangles in IR², which parallels the well-known univariate Romberg method. This is done by a suitable generalization of the trapezoidal rule over triangles, for which we can prove the existence of an asymptotic expansion. Our approach relies mainly on two ideas: The use of barycentric coordinates and the interpretation of the trapezoidal rule as the integral over an interpolating linear spline function. Since our method works for arbitrary triangles, it yields - via triangulation - a method for cubature over arbitrary, possibly non-convex, polygon regions in IR². Moreover, also numerical integration over convex polyhedra in IR d, d > 2 , can be accomplished without difficulties. Numerical examples show the stability and efficiency of the algorithm

Generalized extrapolation methods for solving nonlinear Fredholm integral equations

In this paper we develop a class of generalized extrapolation methods for numerical solution of nonlinear Fredholm integral equations of the second kind. The direct representation of this class allows us to simply discretize the nonlinear integral equations with smooth kernels. This approach enjoys several outstanding features of numerical methods such as: economized computational cost, the high order accuracy, direct implementation, discrretization on arbitrary nodes and applying the methods with positive weights. The comparison results demonstrate the superior results of the new class of methods versus the classical and recent approaches