Peter Junghanns

Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution

Cauchy singular integral equations on an interval are studied in weighted spaces of continuous functions. Convergence rates for a general method for their numerical solution are proved in weighted uniform norms. Applications of these results to the collocation method and the quadrature method are given.

A fast algorithm for Prandtl's integro-differential equation

Collocation and quadrature methods for singular integro-differential equations of Prandtls type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given.

Numerical analysis for one-dimensional Cauchy singular integral equations

The paper presents a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods. The aim of the paper is to show the main ideas and approaches, such as the concept of transforming the question of the stability of an operator sequence into an invertibility problem in a certain Banach algebra or the concept of certain scales of weighted Besov spaces to prove convergence rates of the sequence of the approximate solutions. Moreover, computational aspects, in particular the construction of fast algorithms, are discussed.

A collocation method for nonlinear Cauchy singular integral equations

The application of a collocation method with respect to the Chebyshev nodes of second kind together with a Newton iteration to a class of nonlinear Cauchy singular integral equations is discussed. The investigation of the convergence of the Newton method is based on the stability of the respective collocation method applied to linear Cauchy singular integral equations, which is proved by using Banach algebra techniques. Numerical results are presented.

Weighted Uniform Convergence of the Quadrature Method for Cauchy Singular Integral Equations

Collocation and quadrature methods for Cauchy singular integral equations on an interval with variable coefficients are studied. Convergence rates are proved in weighted uniform and uniform norms.

Uniform convergence of a fast algorithm for Cauchy singular integral equations

Abstract Error estimates for a fast algorithm applied to a Cauchy singular integral equation are proved in weighted uniform norms. This algorithm is essentially based on the application of discrete sine transformations. Numerical results are presented.

On the Numerical Solution of a Nonlinear Integral Equation of Prandtl’s Type

We discuss solvability properties of a nonlinear hypersingular integral equation of Prandtl’s type using monotonicity arguments together with different collocation iteration schemes for the numerical solution of such equations.

On polynomial collocation for Cauchy singular integral equations with fixed singularities

In this paper we consider a polynomial collocation method for the numerical solution of Cauchy singular integral equations with fixed singularities over the interval, where the fixed singularities are supposed to be of Mellin convolution type. For the stability and convergence of this method in weightedL2 spaces, we derive necessary and sufficient conditions.

Numerical analysis of Newton projection methods for nonlinear singular integral equations

Abstract Numerical methods of Newton iteration type for the solution of two classes of nonlinear Cauchy singular integral equations are studied. The existence of appropriate initial approximations, the solvability of the linearized discrete equations, and the weighted L2-convergence of the solutions of the discrete equations to a solution of the original problem are shown.

Newton methods for a class of nonlinear hypersingular integral equations

Different iterative schemes based on collocation methods have been well studied and widely applied to the numerical solution of nonlinear hypersingular integral equations (Capobianco et al. 2005). In this paper we apply Newton’s method and its modified version to solve the equations obtained by applying a collocation method to a nonlinear hypersingular integral equation of Prandtl’s type. The corresponding convergence results are derived in suitable Sobolev spaces. Some numerical tests are also presented to validate the theoretical results.