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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Manuscripta mathematica 26 (1978), S. 259-291 
    ISSN: 1432-1785
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let ψ ∈ Co [O,1] be positive; we show under what conditions there exists a ψ ∈ CO[O,1] such that the following RDI holds: ❘Mu(x)❘ ≤ ψ(x) (0≤x≤1) ⇒ ❘u(x)❘ ≤ ψ (0≤x≤1). In particular we provide a numerical procedure to calculate ψ. RDI are used to obtain error estimations and to solve related nonlinear problems. The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no “global splitting” the concept of a “local splitting” is introduced.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 27 (1976), S. 111-119 
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For a differential operatorL as defined in (2.1) we consider the eigenvalue problemL u=λu and describe a method to obtain a pointwise bound Ψ≧|u−v| wherev denotes the Rayleigh-Ritz approximation to an exact eigenfunctionu. The upper bound Ψ is continuous, but only piecewise differentiable and calculated by solving certain (inverse-positive!) auxiliary problems. Our method uses well-known estimations for Ψ(t i) at a finite number of pointst i∈[a, b] to calculate an upper, bound in the whole interval [a, b].
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 30 (1978), S. 93-101 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L10 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider non-inverse positive differential operators of the second order and show that error estimates can be obtained through methods of inverse positivity alone. This means a significant simplification if the operator has only one negative eigenvalue. In addition we show how to obtain Range-Domain implications for operators with mixed boundary conditions.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 25 (1976), S. 201-214 
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Φ≦Mu≦Ψ⇒−ϕ≦u≦ψ and an algorithm to construct these functions ϕ, ψ, Φ, Ψ. This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea 2 and for given functions Φ=Ψ we require ϕ=ψ∈C 0[0, 1] ∩C 2([0, 1]−T) whereT is some finite set), (Mψ) (t)≧Ψ(t), (t∈[0, 1]−T) and certain additional conditions for eacht∈T. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds-ϕ and ψ.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 57 (1990), S. 393-403 
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H15, 15A18, 15A22 ; CR: G.1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The computation of the smallest positive eigenvalueμ * of a quadratic λ-matrix is used to determine the stability of structures. In addition to existence results we derive two efficient and reliable methods to calculateμ *. Both methods are based on shift techniques which are discussed thoroughly with respect to convergence.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 105 (1991), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We derive a mathematical model which can be used to determine the evolution of the temperature in a Hot Dry Rock system. The essential feature of this approach is given by the fact that the solution is available by analytical methods. Therefore, long-term predictions of the evolution of the temperature as well as a detailed study concerning the influence of critical parameters become available. Although this approach is based on simplifying hypotheses, it provides a good overview of the evolution of the temperature in a Hot Dry Rock system which can be used for efficiency calculations.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Zeitschrift für angewandte Mathematik und Physik 31 (1980), S. 267-276 
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Summary In this article we discuss the buckling imperfection sensitivity of a column resting on a nonlinear elastic foundation. Usula methods require a special knowledge about the form of the imperfection. Here we use Range-Domain Implication to provide a pointwise bound for the solutions of the corresponding boundary problem if a pointwise bound for the imperfection may be assumed.
    Notes: Zusammenfassung Am Beispiel einer nichtlinearen Randwertaufgabe vierter Ordnung wird mit Hilfe von Einschließungsaussagen gezeigt, welchen Einfluß Störterme, die sich beispielsweise aus Materialunvollkommenheiten ergeben, auf die Lösung ausüben. Im allgemeinen sind diese Terme nicht exakt bekannt. Deshalb wurden bei solchen Problemen bisher für die Störterme plausible Ansätze verwendet. Hier werden wir statt dessen mit Hilfe von Einschließungsaussagen aus einer punktweisen Abschätzung der Störterme eine punktweise Abschätzung der Lösung gewinnen.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 141-150 
    ISSN: 1070-5325
    Keywords: Invariant manifolds ; Error estimates ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Many processes in the sciences and in engineering are modelled by dynamical systems and - in discretized version - by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 9
    Title: Numerical methods for bifurcation problems; 70
    Contributer: Küpper, Tassilo , Mittelmann, Hans D. , Weber, Helmut
    Publisher: Basel u.a. :Birkhäuser,
    Year of publication: 1984
    Pages: 584 S.
    Series Statement: Internationale Schriftenreihe zur numerischen Mathematik 70
    Type of Medium: Book
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  • 10
    Title: Bifurcation: analysis, algorithms, applications; Vol. 79
    Contributer: Küpper, Tassilo , Seydel, Rüdiger , Troger, Hans
    Publisher: Basel u.a. :Birkhäuser,
    Year of publication: 1987
    Pages: 359 S.
    Series Statement: International series of numerical mathematics Vol. 79
    Type of Medium: Book
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