Hans-Jürgen Dobner

Bounds for the solution of hyperbolic problems

This article describes a newly developed and implemented method for computing guaranteed errorbounds for the solution of hyperbolic initial value problems. The basic concepts—modified fixed point theorems and approximated operators—allow an a posteriori error-estimation automatically. Therefore, no a priori knowledge of Lipschitz constants, monotonicity properties or additional error analysis is necessary.ZusammenfassungEs wird ein neu entwickeltes und implementiertes Verfahren zur Berechnung von garantierten Fehlerschranken bei hyperbolischen Anfangswert-problemen beschrieben. Die grundlegenden Konzepte—modifizierte Fixpunktsätze und angenäherte Operatoren—gestatten eine automatische a posteriori Fehlerabschätzung. Deshalb ist keine a priori Kenntnis von Lipschitzkonstanten, Monotonieeigenschaften oder zusätzliche Fehleranalyse erforderlich.

Verified computation of Lame´ functions with high accuracy

The Lamé functions are of considerable interest for boundary-value problems in electromagnetics and mechanics that involve ellipsoid geometry. Up to now efficient algorithms for the verified computation of these functions do not exist. In this article we provide a new effective scheme for the numerical treatment of Lamé functions of the first and second kind. It involves computing the Lamé functions with high accuracy combined with safe error estimates.

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lame functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.

A fractional integro-differential equation of volterra type

An analytical and numerical treatment of a fractional integro-differential equation has been considered recently by Boyadjiev et al. [1]. This paper deals with a fractional generalization of the Free Electron Laser (FEL) equation, and the solution is obtained by a method that combines the variation of parameters and successive approximations. The numerical values have been obtained employing the algebra system MAPLE V.

Bounds of high quality for first kind Volterra integral equations

E-Methods for solving linear Volterra integral equations of the first kind with smooth kernels are considered.E-Methods are a new type of numerical algorithms computing numerical approximations together with mathematically guaranteed close error bounds. The basic concepts from verification theory are sketched and such self-validating numerics derived. Computational experiments show the efficiency of these procedures being an advance in numerical methods.AbstractРассматриваютсяE-методы решения динейных интегральных уравнений Вольтерра первого рода с гладким ядром.E-методы представляют собой новый тип численных алгоритмов, позволяюяющих нолучить одновременно с численными приближениями математически гарантированные тесные гранипы погрешностей. В работе излагаются основные понятия теории верификации и ее прияожения в области самоверифицируюпщх вычислений. Численные эксперименты показывают эффективность этих новых вычислительных методов. mis© H.-J. Dobner, 1996

Verified Solution of the Integral Equations for the Two-Dimensional Dirichlet and Neumann Problem

Verified Solution of the Integral Equations for the Two-Dimensional Dirichlet and Neumann Problem. In this article selfvalidating numerical methods for including the Solution of the Dirichlet and Neumann problem in the plane are constructed. Here an additional error analysis to estimate roughly the quality of the computed Solution is obsolete. The so-called verification or E-methods compute a mathematically guaranteed enclosure for the true Solution of these problems.

Attacking a Conjecture in Mathematical Physics by Combining Methods of Computational Analysis and Scientific Computing

We consider a conjecture on the sum of eigenvalues of two integral operators arising in potential and scattering theory for the case that the underlying surface is a triaxial ellipsoid. This concerns computation of Lamé functions which are anyway of great interest in electromagnetics and mechanics. We provide a new effective scheme for the numerical treatment of these special functions. It involves computing the Lamé functions with high accuracy combined with safe error estimates.

On Kernel Inclusions

Kernel inclusions are used to derive a new and efficient solution method for Fredholm integral equations. Concepts from enclosure theory and interval analysis are combined and lead to effective error bounds for a broad class of kernels.

Computing guaranteed error bounds for problems in renewal theory

Equations of renewal type are linear Volterra integral equations having the special form