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  • 1
    Electronic Resource
    Electronic Resource
    Zur numerischen Bestimmung periodischer Lösungen bei nichtlinearen Schwingungen (1952)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 3 (1952), S. 193-205 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary In order to solve a periodic problem, the solution of its boundary value problem is needed. We ask for a method with a high order of accuracy and with nonlinear equations containing only a few unknowns. In this case the use of ordinary finite differences is not recommended. The “Mehrstellenverfahren” uses terms (9) and is always applicable, but gives a nonlinear system with a large number of unknowns. The method described in this paper is applicable for equations (12) and uses the terms (13).A v anda v are evaluated from the fact that the remaining terms of Taylor's expansion are of an order as high as possible. Equations (15), (16), (17), and (18) are examples of such formulas. Some numerical examples are also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02008825
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  • 2
    Electronic Resource
    Electronic Resource
    Rohrströmung mit schwachem Drall (1954)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 5 (1954), S. 95-110 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary The steady laminar flow at high Reynolds' numbers in a straight tube of circular cross-section is investigated under the condition that swirl is present. Given any axially symmetrical distribution of the rotational velocities, the flow downstream is calculated in the special case that the rotational velocities are small compared to the axial velocity. The problem is reduced to an eigen-value problem of an ordinary linear differential equation of second order, the theory of which is developed in detail. Numerical values for the five lowest eigen-values and the corresponding eigen-functions are given. As an example, the decay of an initially rigid rotation is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01600770
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  • 3
    Electronic Resource
    Electronic Resource
    Historische Anmerkung zum Aufsatz: Einige Anwendungen funktional-analytischer Methoden in der praktischen Analysis (1954)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 5 (1954), S. 276-276 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01600336
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  • 4
    Electronic Resource
    Electronic Resource
    Fehlerabschätzungen bei Randwertaufgaben partieller Differentialgleichungen mit unendlichem Grundgebiet (1958)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 9 (1958), S. 118-128 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary Types of boundary value problems of partial differential equations for infinite domains are discussed which can easily be transformed in such a manner as to allow estimations of error (for approximate solutions) similar to the boundary maximum principle. First, second und third boundary value problems for the outer domain of linear elliptic and certain linear and nonlinear parabolic differential equations are examined. For elliptic differential equations one of the results is that the secound boundary value problem for more than two dimensions can be included. The estimates of the paper can thus be applied to problems of flow around some object, not in the case of two but of three dimensions. This is in a certain sense a counterpart to the conformal mappings method which is successful for two but not for three dimensions. Numerical examples show that estimations of error can easily be carried out.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01600629
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  • 5
    Electronic Resource
    Electronic Resource
    Einige Anwendungen funktionalanalytischer Methoden in der praktischen Analysis (1953)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 4 (1953), S. 327-357 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary In this collection of results a formulation of the basic theorem of iteration methods is given, which is particularly suitable for practical applications since it permits testing the presumptions in a straight forward way. A uniform existence and error estimation theory results, which is valid for many various types of ordinary and partial differential equations. The well-known criteria for iteration methods for linear equations are automatically included and are extended to apply to systems of non-linear equations. Initial and boundary value problems for ordinary and partial differential equations are also treated. For numerical calculations and especially for non-linear equations, it is rather important to chose a suitable norm. In problems of a monotone nature, a check on the accuracy may be achieved by the simulataneous application of the theorems on iteration methods and on error estimation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02074980
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  • 6
    Electronic Resource
    Electronic Resource
    Strukturen geometrischer Ornamente (1988)
    Springer
    Journal of geometry 31 (1988), S. 42-64 
    Details
    ISSN: 1420-8997
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider double-periodic tilings of the whole plane from the view of graph theory, not with respect to symmetry groups. We suppose that the graph is planar and connected and that the fundamental domain contains a finite number of vertices and edges. We assign to every tiling a tableau. There exists a fundamental formula connecting the number of all numbers of the tableau with the sum of the reciprocals of all these numbers and with the number p of lines in the tableau; the formula is proved even if multiple edges or loops occur. By this way we get a graph-theoretic classification of the tilings. We introduce families F of tilings and their ranks. The family F={k1,k2...,ks} (with k1〉k2〉...〉ks〉O) is the set of all tilings, the tableau of which contains all the numbers kj and no others. The smallest number p of lines (which occur for the tilings of the family F) is the rank of F and has special geometric interest. Some open questions are mentioned at the end.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01222384
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  • 7
    Electronic Resource
    Electronic Resource
    Monotone Operatoren bei Anfangswertaufgaben von Differentialgleichungen (1960)
    Springer
    Mathematische Zeitschrift 74 (1960), S. 158-166 
    Details
    ISSN: 1432-1823
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01180480
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  • 8
    Electronic Resource
    Electronic Resource
    Über die Instabilität beim Verfahren der zentralen Differenzen für Differentialgleichungen zweiter Ordnung (1953)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 4 (1953), S. 153-154 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary In the numerical solution of an ordinary differential equation by step by step methods it may be that the computational solution is instable and has after a certain number of integration steps of the lengthh nothing to do with the solution of the differential equation. For the method of central differences for the differential equationy″=f(x, y, y′) figure 3 gives in the (A, B)-plane (withA=h 2 f y andB=h f y′ ) a general view of the risk of instability; in the region I there is no risk for instability on account of the method of central differences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02067579
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  • 9
    Book
    Book
    Optimierungsaufgaben; 15 (1966)
    Berlin u.a. :Springer,
    Details
    Title: Optimierungsaufgaben; 15
    Author: Collatz, Lothar
    Contributer: Wetterling, Wolfgang
    Publisher: Berlin u.a. :Springer,
    Year of publication: 1966
    Pages: 181 S.
    Series Statement: Heidelberger Taschenbücher 15
    Type of Medium: Book
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  • 10
    Book
    Book
    Optimization problems : Optimierungsaufgaben, engl.; Vol. 17 (1975)
    Berlin u.a. :Springer,
    Details
    Title: Optimization problems : Optimierungsaufgaben, engl.; Vol. 17
    Author: Collatz, Lothar
    Contributer: Wetterling, Wolfgang
    Publisher: Berlin u.a. :Springer,
    Year of publication: 1975
    Pages: 356 S.
    Series Statement: Applied mathematical sciences Vol. 17
    Type of Medium: Book
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