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.
7501088
Zur numerischen Behandlung gestörter Verzweigungsprobleme bei gewöhnlichen Differentialgleichungen
ZusammenfassungWir geben eine Möglichkeit der numerischen Behandlung von gestörten Verzweigungsproblemen bei Randwertaufgaben gewöhnlicher Differentialgleichungen mit Standardverfahren an. Dies gelingt mit Hilfe von Umparametrisierungen nach Störungsparametern und durch Aufstellung geeigneter regulärer Randwertaufgaben. Ein numerisches Beispiel — ein spezielles Stabknickungsproblem —belegt die Wirksamkeit unserer Methode.AbstractWe present a method for the numerical solution of perturbed bifurcation problems of boundary-value problems for ordinary differential equations by standard techniques. This is achieved by reparametrisation by perturbation parameters and definition of appropriate boundary-value problems which have isolated solutions. The efficiency of the method is demonstrated by a numerical example —a special rod buckling problem.
117077618
A Singular Multi-Grid Iteration Method for Bifurcation Problems
We propose an efficient technique for the numerical computation of bifurcating branches of solutions of large sparse systems of nonlinear, parameter-dependent equations. The algorithm consists of a nested iteration procedure employing a multi-grid method for singular problems. The basic iteration scheme is related to the Lyapounov-Schmidt method and is widely used for proving the existence of bifurcating solutions. We present numerical examples which confirm the efficiency of the algorithm.
123759806
Shooting Methods for Bifurcation Problems in Ordinary Differential Equations
We consider multiple shooting methods for bifurcation problems involving boundary value problems for ordinary differential equations. The case of bifurcation from a simple eigenvalue is treated as well as the solution of perturbed bifurcation problems. The original problem is discretizised via shooting techniques. This yields a finite-dimensional bifurcation problem which is solved by a special iteration scheme, having its origin in the theory of Lyapunov and Schmidt. A numerical example demonstrates that our algorithm workes well.
122551767
Numerical computation of the fourier transform using Laguerre functions and the Fast Fourier Transform
SummaryIn this paper we propose a numerical technique for the computation of Fourier transforms. It uses a bilateral expansion of the unknown transformed function with respect to Laguarre functions. The expansion coefficients are obtained via trigonometric interpolation and may be computed very efficiently by means of the Fast Fourier Transform. The convergence of the algorithm is analyzed and numerical results are presented which confirm that it works well.
120892864
Zur Verzweigung bei einfachen Eigenwerten
Let W and Y be real Banach spaces and F:W → Y be a smooth mapping. Let xɛW be a solution of F(z)=0, Bifurcation theory deals with the problem of determining the structure of the zeros of F near x. We give a modification of a well-known theorem of Crandall and Rabinowitz which is useful for applications, i.e. for the computation of bifurcation points. A numerical example is included.
107642364
The software factory challenge
The Eureka Software Factory project (ESF) was set up by a group of European partners in 1987. This text presents ESF's main ideas and achievements, and relates how the concepts pioneered by the project have become part of a general movement in both the the industrial and academic domains.
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.
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].