Approximation of operators with reproducing nonlinearities

We consider an equation of the form Au+N(u)=u in a Hilbert space and assume that the nonlinearity N is reproducing relative to a known sequence of vectors. Under this assumption the Rayleigh-Ritz-Galerkin approximations lead to a simple class of nonlinear algebraic eigenvalue problems.In a general variational case we show that Lusternik-Schnirelmann critical values of Rayleigh-Ritz-Galerkin problems provide upper bounds to those of the original problem. Lower bounds are constructed in the case N(u)=B^*(Bu)³.The author would like to thank the European Research Office for their assistance in this research.     

A note on the effect of ground absorption in the measurement of aircraft noise

It is shown that results of available theoretical analyses of the far-field sound pressure due to a simple source above an absorbing plane are capable of simple interpretation, and that certain published experimental values of excess attenuation obtained from measurements near the ground are consistent with such results. The effects of changing various parameters are demonstrated and estimates made of the errors likely to be incurred if effects of ground absorption are ignored when determining normalized values of the noise of aircraft.     

A Class of Operator Equations in Nonlinear Mechanics

A nonlinear evolution equation in Hilbert space is introduced and is shown to govern a broad class of problems of nonlinear solid mechanics. The simplest oscillations of this equation are described by a “generalized Liénard equation.”

AMS(MOS) Classification Nos: 73D35, 73F15, 58D25, 35L35, 34G20, 34C15.     

Bifurcation of periodic solutions for wave equations with scalar nonlinearities

We treat wave equations with “scalar nonlinearities” and demonstrate the connection between the bifurcation theory of an associated system of Hammerstein integral equations and the existence of periodic solutions of the nonlinear wave equation.     

Criticality in a model for thermal ignition in three or more dimensions

An explicitly resolvable model, which was instroduced in a previous paper (see [1]), is used to obtain exact behaviour of its bifurcation curves. The model closely approximates the true Arrhenius law for a spherical vessel of reacting material undergoing an exothermic reaction in three or more dimensions. For a sequence of values of a parameter, which is the reciprocal of the dimensionless activation energy, the number of the solutions changes for certain values of the eigenparameter Further, there exist solutions for all then is non-zero.

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Zusammenfassung Mit Hilfe eines explizit lösbaren Modells, das in einer früheren Arbeit eingeführt wurde (siehe [1]), erhält man das exakte Verhalten der zugehörigen Verzweigungskurven. Das Modell approximiert gut das Arrheniussche Gesetz für exotherme Reaktionen in einem sphärischen Topf in drei oder mehr Dimensionen. Für eine Folge von Werten des Parameters, welches als Reziproke der dimensionslosen Aktivierungsenergie dient, ändert sich die Anzahl der Lösungen zum Eigenwert, der durch gesteuert wird. Weiter gibt es für alle mindestens eine Lösung, sofern 0 gilt.

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Supported in part by the Deutsche Forschungsgemeinschaft and in part by the Victoria University of Wellington Fellowship Committee.     

Bifurcation diagrams and periodic solutions to nonlinear equations

Summary We consider a class of simple examples and investigate the connection between the solutions of the nonlinear wave equation and the solutions of the corresponding bifurcation problem.

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Zusammenfassung An einfachen Klassen von Beispielen wird der Zusammenhang untersucht zwischen den periodischen Lösungen nichtlinearer Wellengleichungen und den Lösungen des zugehörigen Verzweigungsproblems.

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Research supported in part by the European Research Office of the United States Army.     

Methods for lower bounds to frequencies of continuous elastic systems

Zusammenfassung Es werden Verfahren zur Berechnung der unteren Grenzen des Frequenzgangs freier Schwingungen von kontinuierlichen linear-elastischen Systemen dargestellt. Diese Verfahren beruhen auf einer systematischen Verbesserung der durch explizite Lösung ähnlicher, jedoch einfacherer Schwingungsprobleme hergeleiteten unteren Grenzen. Die Allgemeinheit unserer Darstellung ermöglicht die Erfassung einer umfangreichen Klasse linearer Theorien. Die Anwendbarkeit dieser Verfahren wird an Beispielen aus der klassischen Plattentheorie erläutert, und es werden numerische Ergebnisse für vier Balkenprobleme angegeben.

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Sponsored in part by the Aeronautical Research Laboratories, OAR, through the European Office of Aerospace Research, United States Air Force.

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Sponsored in part by the Department of the Navy under contract NOw-62-0604-c with the Bureau of Naval Weapons.