116988118
Efficient computation of stable bifurcating branches of nonlinear eigenvalue problems
123856837
Numerische Behandlung von Verzweigungsproblemen bei Gewöhnlichen Randwertaufgaben
We consider the numerical solution of bifurcation problems for ordinary differential equations in the case of simple eigenvalues. First we treat the case of bifurcation from the trivial solution. We also give a new method for the numerical treatment of secondary bifurcation. It consists in the accurate determination of the bifurcation point for which a convex unrestricted minimization problem is derived, and the computation of both branches by conventional methods after transforming the problem.
120338090
Numerical solution of a class of nonlinear boundary value problems for analytic functions
SummaryWe analyse a numerical method for solving a nonlinear parameter-dependent boundary value problem for an analytic function on an annulus. The analytic function to be determined is expanded into its Laurent series. For the expansion coefficients we obtain an operator equation exhibiting bifurcation from a simple eigenvalue. We introduce a Galerkin approximation and analyse its convergence. A prominent problem falling into the class treated here is the computation of gravity waves of permanent type in a fluid. We present numerical examples for this case.ZusammenfassungWir betrachten ein numerisches Verfahren zur Lösung eines nichtlinearen, parameterabhängigen Randwertproblems für analytische Funktionen auf einem Kreisring. Durch Laurent-Entwicklung erhalten wir für die Koeffizienten eine Operatorgleichung, die Verzweigung aufweist. Wir beweisen Konvergenz für das Galerkin-Verfahren. Als numerisches Beispiel untersuchen wir das Problem permanenter Wasserwellen unter dem Einfluß der Schwerkraft.
123517753
On the numerical approximation of secondary bifurcation problems
We discuss stable numerical methods for the approximation of the solutions of a nonlinear parameter - dependent equation near a non-trivial bifurcation point. The problem of finding the bifurcation point is reformulated as a well-posed equation of higher dimension. The nearby branches can be calculated in a stable manner after applying a certain transformation having its origin in the Lyapunov — Schmidt theory. We also treat the perturbed bifurcation problem and present numerical results.
117989535
Numerical Methods for Bifurcation Problems — A Survey and Classification
The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory.
123580420
Multi-Grid Solution of Bifurcation Problems
A continuation strategy is developed for a method of inverse iteration type to compute solutions of nonlinear eigenvalue problems. The solution curves are parametrized by a norm of the solution. Thus continuation near turning points does not represent any difficulties. A multi-grid version of the algorithm is presented and successfully applied to solve problems on general two-dimensional domains. Numerical results including a comparison with a more standard approach are also given for the continuation strategy proposed for the basic algorithm.
118567473
Compact convex sets in non-locally-convex linear spaces
In the theorems about compact convex sets it is usually assumed that the compact convex set is contained in a locally convex linear space. Examples for such theorems are the Schauder-Tychonoff fixed point theorem and the Krein-Milman theorem.The problem whether the Krein-Milman theorem remains true also in non locally convex linear spaces, was solved by J.W.Roberts; he gave examples for absolutely convex compact sets without extreme points ([R 76], [R 77], [Ro 84]).It is, however, still an open problem, whether the Schauder-Tychonoff theorem remains true in non locally convex linear spaces [M 81, Problem 54 of Schauder]. A sufficient condition fora compact convex set to have e.g. the fixed point property or to have extreme points is that it can be affinely embedded in a Hausdorff locally convex linear space.This observation, which was the starting point for [JOT 76] and [R 76], was left out of account in various newer publications about Schauder-Tychonoff's fixed point theorem in non locally convex spaces, e.g. [H 84].
12124100
EULER: a generalization of ALGOL, and its formal definition: Part II
In this section the algorithmic language EULER is described first informally and then formally by its syntax and semantics. An attempt has been made to generalize and extend some of the concepts of ALGOL, thus creating a language which is simpler and yet more flexible than ALGOL 60. A second objective in developing this language was to show that a useful programming language which can be processed with reasonable efficiency can be defined in rigorous formality.
16938020
EULER: a generalization of ALGOL and it formal definition: Part 1
In this section the algorithmic language EULER is described first informally and then formally by its syntax and semantics. An attempt has been made to generalize and extend some of the concepts of ALGOL, thus creating a language which is simpler and yet more flexible than ALGOL 60. A second objective in developing this language was to show that a useful programming language which can be processed with reasonable efficiency can be defined in rigorous formality.
814779
A microprogrammed implementation of EULER on IBM system/360 model 30
An experimental processing system for the algorithmic ~anguage EULER has been implemented in microprogramming on an IBM System/360 Model 30 using a second Read-Only Storage unit. The system consists of a mlcroprogrammed compiler and a microprogrammed String Language Interpreter, and of an I/O control program written in 360 machine language. The system is described and results are given in terms of microprogram and main storage space required and compiler and interpreter performance obtained. The role of microprogramming is stressed, which opens a new dimension in the processing of interpretive code. The structure and content of a higher level language can be matched by an appropriate interpretive language which can be executed efficiently by microprograms on existing computer hardware.