Numerical solution of a class of quasilinear hyperbolic equations by reduction to the wave equation

An iterative method of solving quasilinear hyperbolic equations of the type $$ - [p_1 (u_x ,x,t)]_x + [p_2 (u_t ,x,t)]_t = f(x,t)$$ in the domain (0, 1)×(0,T) is proposed. For each given initial-boundary-value problem of this type with boundary conditions of the first kind (second kind), a conjugate problem of the same type that has boundary conditions of the second kind (first kind) is defined. From the relations connecting the solutions of a pair of conjugate problems, a series of wave equations is created. The method proposed consists in calculating the solutions of the wave equations of this series. Theoretical proof of the convergence of the solutions of the wave equations to the solutions of the conjugate quasilinear problems is left as an open question. However, numerical results are presented to demonstrate that, under favorable circumstances, the solutions of the wave equations do converge to the solutions of the conjugate quasilinear problems.     

Unsteady mixed convection from an isothermal circular cylinder

Summary Combined unsteady convection from an isothermal horizontal cylinder in a stream flowing vertically upwards has been investigated. Numerical solutions of the unsteady boundary-layer equations have been obtained at any station along the cylinder using the series truncation method. Solutions which are valid near the front and near stagnation points have been obtained using standard finite-difference methods. A series solution in powers of time has been obtained with which the numerical solutions has been checked.

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Unstetige, gemischte Konvektion um einen isothermen Kreiszylinder

Zusammenfassung Es wird eine kombinierte, unstetige Konvektion eines isothermen, horizontalen Zylinders in einer vertikal nach oben gerichteten Strömung untersucht. Numerische Lösungen der unstetigen Grenzschichtgleichungen werden an jeder Stelle längs des Zylinders durch die Verwendung der Reihenabbruchsmethode erhalten.Nahe der Vorderseite und nahe bei den Staupunkten gültige Lösungen werden durch Verwendung üblicher Methoden der finiten Differenzen erhalten. Mit Hilfe einer Reihenlösung in Potenzen der Zeit wird das numerische Ergebnis überprüft.

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Notation a radius of the cylinder - g acceleration of gravity - G _r Grashof number =g|T|a ³/v ² - Q heat transfer - R _e Reynolds number =U ₀ a/v - T temperature of fluid in the boundary layer - T ₀ temperature of the ambient fluid - T ₁ temperature of the cylinder - T temperature difference=T ₁–T ₀ - t time - U ₀ free stream - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r /R _e ² - coefficient of expansion - coefficient of kinematic viscosity - _w skin friction With 7 Figures     

Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter

Let the space curveL be defined implicitly by the (n, n+1) nonlinear systemH(u)=0. A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH. Moreover, a linear system of dimensionn+1 with 4 different right hand sides has to be solved per step. Under suitable conditions the method is shown to converge locally withQ-order two if a certain discretization stepsize is appropriately chosen. Two numerical examples confirm the theoretical results.     

MARKAL — Ein LP-Modell für die Internationale Energieagentur

Zusammenfassung MARKAL ist ein LP-Modell der Energieversorgung, mit dem für die Internationale Energieagentur Rechnungen zur langfristigen Technologiepolitik zur Öleinsparung durchgeführt werden. Als Mehrperiodenmodell deckt es einen Planungszeitraum von 40 Jahren ab und umfaßt alle wesentlichen Sektoren und Umwandlungsprozesse der Energiewirtschaft. Optimierungsrechnungen, die die Energieversorgung der wesentlichen westlichen Industriestaaten umfassen, zeigen unter anderem auf, daß die Chancen einer substantiellen Öleinsparung skeptisch zu beurteilen sind.

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MARKAL is a linear programming model of the energy supply used by the International Energy Agency for a long term technology evaluation to save oil. As a multi-period model, it covers a planning period of 40 years for all important sectors and technical processes within the energy economy. The model applications to the energy supplies of most industrialized western countries justify a certain scepticism about the chances for a substantial oil saving in the future.

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Diese Arbeit wurde als Vortrag auf der Sitzung der DGOR-Arbeitsgruppe Praxis der linearen Optimierung (PRALINE) unter dem Generalthema Energieplanung am 18. 1. 1980 in Jülich präsentiert.     

Estimating errors of numerical approximation for periodic analytic functions

We give a simple method based on Cauchy's integral formula for estimating the errors of numerical approximation for periodic analytic functions. We then obtain error estimates for the quadrature formulas of Chawla and Ramakrishnan [1] for the numerical evaluation of the Cauchy principal value integral $$I\left( {f;a} \right) = \int\limits_0^{2\pi } {f(x)\cot \left( {\left( {x - a} \right)/2} \right)dx,} $$ and for the quadrature formula of Garrick [2] for the evaluation ofI (f;o), Based on these error estimates, we are led to conclude that for the evaluation ofI (f;o), Garrick's formula has a better error estimate than the formula of Chawla and Ramakrishnan with the same number of function evaluations. Finally, we extend Garrick's formula for the evaluation ofI (f;a) for arbitrarya∈[0,2π); the extended formula has, for alla, the same error estimate as Garrick's formula. While, for a=x_j, the extended formula is identical with the quadrature formula of Wittich [3], fora≠x _j, the extended formula is much better in that it uses only half the number of function evaluations of Wittich's formula for the same accuracy.     

Approximation der optimalen Vorhersage der Ausgabe stochastischer Quellen

This paper discusses the prediction of the output of a stochastic sourceQ having positive transition function. If the sourceQ has two output alphabets,X andY, then there exists such a stochastic automaton \(\mathfrak{A}(V_{opt} )\) with input alphabetX, output alphabetY, and a countable set of states, that realizes the optimal predictionV _opt fromX toY. Here it will be shown thatV _opt can be approximated with any precision by a predictionV which is realized by a stochastic automaton with finite set of states.     

Mass appraisal of residential apartments: An application of Random forest for valuation and a CART-based approach for model diagnostics

To the best knowledge of authors, the use of Random forest as a potential technique for residential estate mass appraisal has been attempted for the first time. In the empirical study using data on residential apartments the method performed better than such techniques as CHAID, CART, KNN, multiple regression analysis, Artificial Neural Networks (MLP and RBF) and Boosted Trees. An approach for automatic detection of segments where a model significantly underperforms and for detecting segments with systematically under- or overestimated prediction is introduced. This segmentational approach is applicable to various expert systems including, but not limited to, those used for the mass appraisal.     

Experimente zum Strömungswiderstand in gekrümmten,rotierenden Kanälen mit quadratischem Querschnitt

Zusammenfassung Bei ausgebildeter Strömung wurde der Druckverlust in rotierenden, gekrümmten Kanalstrecken experimentell ermittelt. Speziell interssierte der Einfluß der Krümmung auf den Reibungswiderstand und im besonderen der der Rotation, die sowohl in Richtung der Durchflußgeschwindigkeit als auch entgegengesetzt dazu erfolgte. Die strömungsspezifischen Parameter wie Reynolds-und Taylor-Zahl erfuhren eine Änderung im Bereich 3·10²<Re<2·10⁵ und 0Ta<10⁵. Das Krümmungsverhältnis variierte zwischen 0,0496a/r _m 0,337. Aufgezeigt wird die Möglichkeit einer universellen Darstellung des Widerstandsbeiwertes für den nicht rotierenden Kanal bei einerseits laminarer und anderseits bei turbulenter Bewegung. Im rotierenden Kanal beobachtet man beträchtliche Druckverlusterhöhungen, deren Größe von der Rotationsrichtung abhängt. Bei großen Taylor-Zahlen wurde eine Unabhängigkeit des Strömungswiderstandes vom Krümmungsverhältnis festgestellt.

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Experimental investigations on the friction resistance in rotating curved channels with square cross-section

Summary For fully developed flow the pressure loss in rotating curved channels was investigated experimentally. The influence of the curvature on the friction coefficient and particularly on the rotation, which was realized in the direction of the through flow as well as in the opposite direction was of special interest. The specific parameters of the flow field, like the Reynolds-number and the Taylor-number were varied within the range of 3·10²<Re<2·10⁵ and 0Ta<10⁵. The curvature ratio were varied within 0,0496a/r _m 0,337. The possibility of an universal presentation of the friction coefficient for the nonrotating channel is shown for laminar and turbulent through flow. In the rotating channel a considerable pressure loss increase can be observed. The magnitude depends on the direction of rotation. In case of great Taylor-numbers the curvature ratio was found to be independent of the friction resistance.

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Bezeichnungen a Kantenlänge - mittlere Geschwindigkeit - l Kanallänge - r _m Krümmungsradius - Volumenstrom - _p Druckdifferenz - Widerstandsbeiwert - =57/Re Widerstandsbeiwert des geraden Kanals - dynamische Viskosität - Dichte - Winkelgeschwindigkeit - Dean-Zahl - Reynolds-Zahl - Taylor-Zahl Mit 6 Abbildungen     

On grammar forms with terminal context

Summary A context-free grammar form G defines a family of context-free grammars {ie397-01}(G). In this paper {ie397-02}(G) is extended to give the family of terminal context grammars {ie397-03}(G), which were introduced by Book [4]. It is shown that the families of languages defined by {ie397-04}(G) and {ie397-05}(G) are equal when G is an -free context-free grammar form. This generalizes the result of Book [4] for context-free grammars and underlines the usefulness of the notion of grammar forms in formal language theory.The work of the second author was carried out partially under a National Research Council of Canada Grant. No. A-7700 and partially under the auspices of the University of Karlsruhe     

On a direct algorithm for the generation of log-normal pseudo-random numbers

The random variable \(\left( {\prod {_{i = 1}^n {{X_i } \mathord{\left/ {\vphantom {{X_i } {X_{i + n} }}} \right. \kern-0em} {X_{i + n} }}} } \right)^{{1 \mathord{\left/ {\vphantom {1 {\sqrt {2n} }}} \right. \kern-0em} {\sqrt {2n} }}}\) is used to generate standard log-normal variables Λ(0, 1), where theX _i are independent uniform variables on [0, 1].