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  • 1
    Electronic Resource
    Electronic Resource
    Local error estimation by doubling (1985)
    Springer
    Computing 34 (1985), S. 179-190 
    Details
    ISSN: 1436-5057
    Keywords: 65 L 05 ; Doubling ; Richardson extrapolation ; error estimates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Schrittverdopplung, oder Richardson-Extrapolation, ist ein allgemeines Prinzip zur Schätzung des lokalen Fehlers eines Einschrittverfahrens zur numerischen Lösung eines Anfangswertproblems für Systeme gewöhnlicher Differentialgleichungen. Es werden einige Beiträge zur Theorie dieses Vorgehens gemacht. Die Prinzipien zum Vergleich expliziter Runge-Kutta-Formeln werden betrachtet und durch eine ausgewogene Bewertung des Verdoppelns bei Formeln vierter Ordnung illustriert. Einige scheinbar widersprüchliche Vergleiche aus der Literatur werden aufgeklärt.
    Notes: Abstract Doubling, or Richardson extrapolation, is a general principle for the estimation of the local error made by a one-step method for the numerical solution of the initial value problem for a system of ordinary differential equations. Some contributions are made to the theory of doubling. Principles of comparing explicit Runge-Kutta formulas are reviewed and illustrated in a balanced appraisal of doubling in the context of fourth order formulas. Some apparently contradictory comparisons in the literature are explained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02259844
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  • 2
    Electronic Resource
    Electronic Resource
    Invariant imbedding and the calculation of eigenvalues for Sturm-Liouville systems (1969)
    Springer
    Computing 4 (1969), S. 10-23 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird eine neue Technik, basierend auf einer Methode, welche invariante Einbettung (Invariant Imbedding) mit derRicatti-Transformation kombiniert, dargestellt zum Zwecke der Berechnung der Eigenwerte von Differentialgleichungssystemen, welche vomSturm-Liouville-Typ sind. Eine sehr einfache numerische Prozedur ist entwickelt worden, die leicht zu programmieren ist und verläßliche Unterprogramme benutzt. Dieses Verfahren vermag eine umfangreiche Klasse von Problemen zu handhaben. Unter ihnen sind solche inbegriffen, bei denen der Eigenwert in einer nichtlinearen Form auftritt; auch Fälle, wo der Eigenwert in der Randbedingung vorkommt; und schließlich auch Gleichungen, die Singularitäten aufweisen. Es sei betont, daß die hier von uns behandelten numerischen Berechnungen im allgemeinen stabil sind und sehr genaue Ergebnisse hervorzubringen vermögen.
    Notes: Summary A new technique based upon an invariant imbedding-Ricatti transformation approach is presented for the calculation of eigenvalues ofSturm-Liouville type systems of differential equations. A very simple numerical procedure is developed which is easily programmed and which uses reliable subroutines. The method is capable of handling a large class of problems. Included among these are problems in which the eigenvalues appears in a non-linear fashion, cases in which the eigenvalue occurs in the boundary condition, and equations which have singularities. The numerical computations are generally well-conditioned and very accurate results were obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02236538
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  • 3
    Electronic Resource
    Electronic Resource
    Uniformly accurate Sturm-Liouville eigenvalues (1992)
    Springer
    Computing 47 (1992), S. 379-385 
    Details
    ISSN: 1436-5057
    Keywords: 65L05 ; Sturm-Liouville Problem ; Eigenvalue Approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Finite Differenzen und finite Elemente werden allgemein verwendet zur näherungsweisen Berechnung desk-ten Eigenwertes eines regulären Sturm-Liouville-Problemes durch denk-ten Eigenwert eines Matrizenproblemes der Dimensionn. Die Genauigkeit nimmt rapide ab mit zunehmendenk, wobei es fürk〉n keine Näherungen gibt. Viele neuere Arbeiten beschäfftigten sich damit, die Genauigkeit der Approximation mit wenig Rechenaufwand zu verbessern. Es wird gezeigt, daß es für viele Methoden möglich ist, mit vernachlässigbaren Rechenkosten und minimalen Komplikationen, Näherungen für allek zu erhalten, die gleichmäßig genau sind ink.
    Notes: Abstract Popular methods like finite differences and finite elements approximate thekth eigenvalue of a regular Sturm-Liouville problem by thekth eigenvalue of a matrix problem of dimensionn. The accuracy deteriorates rapidly ask increases, and there is no approximation at all fork〉n. Much recent work has been devoted to improving inexpensively the accuracy of the approximation with respect to increasingk. It is shown here that for many methods, it is possible with negligible cost and minimal complication to obtain approximations for allk that are uniformly accurate ink.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02320206
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  • 4
    Electronic Resource
    Electronic Resource
    Better extrapolation codes (1985)
    Springer
    Computing 35 (1985), S. 369-374 
    Details
    ISSN: 1436-5057
    Keywords: 65L05 ; Ordinary diffeential equations ; extrapolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird gezeigt, wie man den gängigen Codes, die Systeme von gewöhnlichen Differentialgleichungen durch Extrapolation mit variabler Ordnung lösen, eine Formel der Ordnung 1 hinzufügen kann, was das Verhalten der Codes in verschiedener Hinsicht verbessert.
    Notes: Abstract It is shown how to add a formula of order one to the orders available in the popular variable order codes that solve systems of ordinary differential equations by extrapolation. This improves the codes in several ways.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02240201
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  • 5
    Electronic Resource
    Electronic Resource
    A-stable block implicit one-step methods (1972)
    Springer
    BIT 12 (1972), S. 252-266 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of methods for solving the initial value problem for ordinary differential equations is studied. We developr-block implicit one-step methods which compute a block ofr new values simultaneously with each step of application. These methods are examined for the property ofA-stability. A sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizesr=1,2,..., 8 these methods areA-stable while those forr=9,10 are not. We constructA-stable formulas having arbitrarily high orders of accuracy, even stiffly (strongly)A-stable formulas.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932819
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  • 6
    Electronic Resource
    Electronic Resource
    Evaluation of implicit formulas for the solution of ODEs (1979)
    Springer
    BIT 19 (1979), S. 495-502 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Implicit formulas are quite popular for the solution of ODEs. They seem to be necessary for the solution of stiff problems. An acceptance test must be made of the approximate solution of the algebraic equations arising in the evaluation of an implicit formula. A reliable way to make this test is presented. It offers many practical advantages over current tests.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931266
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  • 7
    Electronic Resource
    Electronic Resource
    On a paper of roark and wing (1965)
    Springer
    Numerische Mathematik 7 (1965), S. 394-395 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01436252
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  • 8
    Electronic Resource
    Electronic Resource
    Note on a paper of Bailey and Waltman (1966)
    Springer
    Archive for rational mechanics and analysis 22 (1966), S. 308-309 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00285422
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  • 9
    Electronic Resource
    Electronic Resource
    Some nonlinear boundary value problems (1967)
    Springer
    Archive for rational mechanics and analysis 25 (1967), S. 123-134 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00281292
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  • 10
    Electronic Resource
    Electronic Resource
    Contraction mappings for nonlinear boundary value problems (1968)
    Springer
    Computing 3 (1968), S. 205-214 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Diese Arbeit behandelt schwach nichtlineare Randwertprobleme der Form $$\begin{gathered} y'' (t) + p (t) y'(t) + f(t,y (t)) = 0 \hfill \\ y(\alpha ) = A, y(b) = B \hfill \\ \end{gathered} $$ wobei Lösungsverfahren, die derPicard-Methode ähnlich sind, benützt werden. Einer Lösung vorgegebene Schranken werden verwendet, um die Konvergenz einer geeigneten Funktionenfolge zu erzeugen.
    Notes: Summary This paper treats mildly nonlinear boundary value problems of the form $$\begin{gathered} y'' (t) + p (t) y'(t) + f(t,y (t)) = 0 \hfill \\ y(\alpha ) = A, y(b) = B \hfill \\ \end{gathered} $$ byPicard-like methods.A priori bounds on a solution are used to generate contraction mappings on a suitable set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02277217
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