High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.     

Phragmen-Lindelöf type results for second order quasilinear parabolic equations in ℝ2

In this paper Phragmen-Lindelöf type growth-decay estimates are derived for solutions of initial-boundary value problems associated with a class of quasilinear parabolic equations defined on a semi-infinite strip in ². The particular problems considered are ones in which homogeneous initial data and homogeneous Dirichlet conditions on the long sides of the strip are prescribed.This research was carried out while the first author held a visiting appointment at Cornell University and was partially supported by NSF grant #DMS-9100876.     

Stability of a family of multi-time-level difference schemes for the advection equation

Summary For the linear advection equation we consider explicit multi-time-level schemes of highest order which are one step in space direction only. If a stencil involvesk time steps we show that it is stable in theL ₂-sense for Courant numbers in the interval (0, 1/k). Since the order is 2k–1 one can use these schemes for high order discretization of the boundary conditions in hyperbolic initial value problems.Part of this work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which has been supported by the DFG     

Heat equation with degeneration in Holder and Slobodetskii spaces

We study initial-boundary value problems for the heat equation in which heat conductivity ²(x) may depend on the space variablex ⁺; the nonnegative function(x) is allowed to tend to infinity (respectively, zero) asx + (respectively,x +0). We prove that these problems are well posed and examine the smoothness of solutions. It is shown that criteria for smoothness of the solutions can be stated in terms of certain functionals, namely, the Hölder constant (for Hölder spaces) and the generalized Hölder constant (for Slobodetskii spaces).Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 189–203, August, 1995.     

On the analytic solutions of the monoenergetic neutron transport equation for slab geometry

The classical initial value problem for the monoenergetic neutron transport equation in slab geometry is solved, using the power series method. A general formula for analytic solutions of this equation is presented. It is shown that a polynomial solution exists only forc=1 and is linear inx and . Other analytic solutions are given in a closed form. The Taylor series expansion method is compared with the spherical harmonic approach.

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Zusammenfassung Das klassische Eigenwertproblem für die monoenergetische Neutronentransportgleichung in ebener Geometrie wird mit Hilfe der Taylorreihe gelöst und eine allgemeine Formel für analytische Lösungen dieser Gleichung angegeben. Es wird gezeigt, dass die Polynomiallösung nur fürc=1 exisitiert und linear ist in undx. Obwohl die analytischen Lösungen nur beschränkt praktische Anwendung haben zeigt sich, dass die abgebrochene Taylorreihe für praktische Probleme verwendbar ist. Rechnungen mit der abgebrochenen Taylorreihe wirden mitP _N -Rechnungen verglichen.

     

Iterative continuation and the solution of nonlinear two-point boundary value problems

In this paper we present a unified function theoretic approach for the numerical solution of a wide class of two-point boundary value problems. The approach generates a class of continuous analog iterative methods which are designed to overcome some of the essential difficulties encountered in the numerical treatment of two-point problems. It is shown that the methods produce convergent sequences of iterates in cases where the initial iterate (guess),x ₀, is far from the desired solution. The results of some numerical experiments using the methods on various boundary value problems are presented in a forthcoming paper.     

The approximate spectral projection method

In this paper we develop a general method for investigating the spectral asymptotics for various differential and pseudo-differential operators and their boundary value problems, and consider many of the problems posed when this method is applied to mathematical physics and mechanics. Among these problems are the Schrödinger operator with growing, decreasing and degenerating potential, the Dirac operator with decreasing potential, the quasi-classical spectral asymptotics for Schrödinger and Dirac operators, the linearized Navier-Stokes equation, the Maxwell system, the system of reactor kinetics, the eigenfrequency problems of shell theory, and so on. The method allows us to compute the principal term of the spectral asymptotics (and, in the case of Douglis-Nirenberg elliptic operators, also their following terms) with the remainder estimate close to that for the sharp remainder.     

KonvexeN-Stufen-Max-Min-Optimierung

Zusammenfassung Aus der Behandlung der Optimalen Skalierung von Analogrechenschaltungen ergab sich folgendes Problem: Zugrundegelegt wird als zulässiger Optimierungsbereich S₀ eine nichtleere, abgeschlossene, beschränkte und konvexe Teilmenge desR ⁿ . Unter allen Vektorenx S ₀ werden zunächst diejenigen gesucht, deren kleinste Komponente den inS ₀ größtmöglichen Wert hat. Ihre Menge sei mitS ₁ bezeichnet. Dann werden in einer zweiten Optimierungsstufe diejenigenx S ₁ gesucht, deren zweitkleinste Komponente den inS ₁ größtmöglichen Wert hat. Das so fort bis zurn-ten Stufe. Das Problem hat eine eindeutige Lösung, die sich, wie die Arbeit zeigt, rekursiv durch Lösen von einstufigen Max-Min-Optimierungsproblemen finden läßt. Es wird ein allgemeines Rechenverfahren angegeben. Sind die Nebenbedingungen linear, so können die auf den einzelnen Stufen zu lösenden Max-Min-Optimierungsprobleme auf Probleme der Linearen Optimierung zurückgeführt werden. Für diesen Fall wird ein ausgetestetes ALGOL-Programm angegeben.

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Summary The question of optimal scaling of analogue computer set-ups leads to the following problem: LetS ₀, the feasible region of optimization, be defined as a non-empty, closed, bounded, and convex subset ofR ⁿ . The first problem is to find those vectorsx S ₀ whose smallest component has the greatest possible value inS ₀. Let the set of these optimal vectors beS ₁. The second stage of the optimization process consists of finding thosex S ₁ whose second-smallest component has the greatest possible value inS ₁. Continue this process up to then-th stage. The paper shows that the optimization problem has a unique solution which can be found recursively by solving a certain number of one-stage-max-min-optimization problems. A general algorithm will be given. If the constraints of then-stage-max-min-optimization problem are linear, the one-stage-max-min-optimization problems mentioned above can be reduced to Linear Programming problems. For this case a tested ALGOL-programme will be given.

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Diese Arbeit ist Teil einer am Institut für Angewandte Mathematik der Technischen Hochschule München unter Anleitung von Herrn o. Prof. Dr. rer. nat. habil.J. Heinhold angefertigten Dissertation.Vorgel. v.:J. Heinhold.     

An initial value problem for a functional equation

We consider the functional equationf(A(x,y))=B(f(x),f(y)), whereA andB are averages. It is known that such a functional equation has exactly one continuous solution satisfying a given two-point condition. By analogy with the theory of differential equations, we may regard the functional equation, together with a two-point condition, as a boundary value problem. (Then each boundary value problem has a unique continuous solution.) If we replace the two-point condition with the specification of a value and derivative at just one point, we obtain an initial value problem.Consider the initial value problemsf(A(x,y))=B(f(x),f(y)),f(a)=s,f(a)=, obtained by fixinga ands and allowing to vary through the set of positive real numbers. The main result of this paper gives a necessary and sufficient condition for each of the initial value problems to have a unique continuous solution, under the hypothesis that at least one of the problems has a continuous solution. This is a partial answer to the problem of determining conditions which are sufficient for the existence of a unique continuous solution of a given initial value problem.     

An Algebraic Version of the Cantor-Bernstein-Schröder Theorem

The Cantor-Bernstein-Schröder theorem of the set theory was generalized by Sikorski and Tarski to -complete boolean algebras, and recently by several authors to other algebraic structures. In this paper we expose an abstract version which is applicable to algebras with an underlying lattice structure and such that the central elements of this lattice determine a direct decomposition of the algebra. Necessary and sufficient conditions for the validity of the Cantor-Bernstein-Schröder theorem for these algebras are given. These results are applied to obtain versions of the Cantor-Bernstein-Schröder theorem for -complete orthomodular lattices, Stone algebras, BL-algebras, MV-algebras, pseudo MV-algebras, ukasiewicz and Post algebras of order n.     

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term. Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated. Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.     

The value distribution of Hecke <Emphasis Type="Italic">L</Emphasis>-functions in the grössencharacter aspect

In this paper we investigate the value distribution of Hecke L-functions with parametrized grössencharacters. We prove the analogue of Bohrs result for the Riemann zeta function. Received: 12 March 2003     

A geometrical approach to bifurcation for nonlinear boundary value problems

Summary In this note we use a new averaging method, which was introduced in [2], to explain the geometrical behaviour of systems governed by nonlinear boundary value problems of the formy+g(y)=K sin(t),y(0)=y(/)=0. We show by numerical computations that global features of the solutions (such as the number of solutions, their magnitude, bifurcation behaviour, etc.) agree in both the original and averaged model. As an example, the pendulum equation is discussed in detail.

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Zusammenfassung In dieser Arbeit benutzen wir eine neue, in [2] eingeführte Mittelwertmethode, um das geometrische Verhalten nichtlinearer Randwertprobleme der Formy+g(y)=K sin(t),y(0)=y(/)=0. zu erklären. Wir belegen durch numerische Untersuchungen, daß globale Eigenschaften der Lösungen (wie z. B. die Anzahl der Lösungen, ihre Größenordnung, das Verzweigungsverhalten usw.) in der originalen und genäherten Gleichung übereinstimmen. Als Beispiel wird die Pendelgleichung ausführlich diskutiert.

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Supported by the Deutsche Forschungsgemeinschaft under grant No. BA 735/3-1     

Uniqueness in some higher order elliptic boundary value problems

Summary The uniqueness of the solution to two boundary value problems for the linear equation ³ u –a ² u +b u –cu =F and to two boundary value problems for the quasilinear differential equation ² u +w(u) =f are proved. The proofs follow as a consequence of maximum principles for a functional which is defined on solutions to the differential equation.

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Zusammenfassung Die Eindeutigkeit der Lösung zweier Randwertaufgaben für die lineare Gleichung ³ u –a ² u +b u –cu =F und zweier Randwertaufgaben für die quasilineare Differentialgleichung ² u +w(u) =f wird bewiesen. Der Beweis folgt aus einem Maximumprinzip für ein Funktional, das für die Lösungen der Gleichung definiert ist.

     

Lineare Ersatzprogramme für unscharfe Entscheidungsprobleme Zur Optimumbestimmung bei unscharfer Problembeschreibung

Zusammenfassung In der vorliegenden Arbeit werden lineare Programme als Ersatzformulierungen unscharfer Linearer Programmierungsprobleme vorgeschlagen. Dabei wird, abweichend vom gegenwärtigen Stand in der Diskussion der Fuzzy Sets-Theorie, auf die Minimumbildung für das logische Und verzichtet; sie wird durch eineadditive Verknüpfung ersetzt.Ausgehend von einer kurzen Erörterung der Bestandteile und einer Typologie des Entscheidungsprozesses werden spezielle Entscheidungsprobleme formuliert, welche mit den derzeit vorliegenden Algorithmen der linearen Programmierung gelöst werden können. Ein Beispiel wird dabei in jeweils modifizierter Form die Vorgehensweisen illustrieren. Es schließt sich eine detaillierte Begründung der hier vorgeschlagenen Methode an.

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Summary In this article we formulate Linear Programming Problems as substitutes of fuzzy decisions. In contrast toZadeh's suggestion to handle the logical and by applying the minimum-operator to the membership functions, we suggest toadd membership functions for combining fuzzy sets in the sense of the logical and.In the first sections we discuss some essentials of fuzzy decision problems and we design a typology for fuzzy decision problems.Accordingly, in succeeding sections, we formulate special decision problems that can be solved by existing algorithms. A numerical example will illustrate our approach through the entire text. The last sections contain a detailed justification for our way of solving fuzzy decision problems.

     

Single step methods for general second order singular initial value problems with spherical symmetry

A one parameter family of self-starting explicit Runge-Kutta-Nyström methods has been obtained for the solution of the general second order singular initial value problem with spherical symmetry of the formu+2r ^–1 u=f(r, u, u),u(0)=A, u(0)=0. The methods are exact foru(r)=r ^–1, l,r, r ²,r ³ andr ⁴.     

On the complexity of following the central path of linear programs by linear extrapolation II

A class of algorithms is proposed for solving linear programming problems (withm inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newton's method to compute points on the central pathx(r), r>0, and this allows to estimate the complexity, i.e. the total numberN = N(R, ) of steps needed to go from an initial pointx(R) to a final pointx(), R>>0, by an integral of the local weighted curvature of the (primal—dual) path. Here, the central curve is parametrized with the logarithmic penalty parameterr0. It is shown that for large classes of problems the complexity integral, i.e. the number of stepsN, is not greater than constm log(R/), where < 1/2 e.g. = 1/4 or = 3/8 (note that = 1/2 gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1/3.As a byproduct, many analytical and structural properties of the primal—dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameterr is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.On leave from the Institute of Mathematics, Eötvös University Budapest, H-1080 Budapest, Hungary.     

Initial value problems for a system of nonlinear equations on a half-line

The paper deals with initial value problems for the system of nonlinear equations on a half-line where ₁, ₁ are real constant connections.The first problem for the nonlinear Schrödinger equation (NLS) is obtained from the system (A) when . In this case, the boundary problem associated with this NLS does not have a discrete spectrum, and the solution of the NLS is uniquely found from the given initial condition. Further, we show two remarkable classes of matrix potentials, in which the inverse scattering problem associated with system (A) can be solved exactly. In the case where the given scattering data for (A) consist of only two simple poles, the exact soliton solution of system (A) is presented. This happens if and only if the time evolution of the scattering data for (A) obeys some time evolution equations. In this case, the initial value problem for the NLS which is obtained from (A) when is solved exactly.     

An Evolution Program for Non-Linear Transportation Problems

In this paper we describe main features of a Strongly Feasible Evolution Program (SFEP) designed to solve non-linear network flow problems. The program can handle non-linearities both in the constraints and in the objective function. The solutions procedure is based on a recombination operator in which all parents in a small mating pool have equal chance of contributing their genetic material to offspring. When offspring is created with better fitness value than that of the worst parent, the worst parent is discarded from the mating pool while the offspring is placed in it. The main contributions are in the massive parallel initialization procedure which creates only feasible solutions with simple heuristic rules that increase chances of creating solutions with good fitness values for the initial mating pool, and the gene therapy procedure which fixes defective genes ensuring that the offspring resulting from recombination is always feasible. Both procedures utilize the properties of network flows. The algorithm is capable of handling mixed integer problems with non-linearities in both constraints and the objective function. Tests were conducted on a number of previously published transportation problems with 49 and 100 decision variables, which constitute a subset of network flow problems. Convergence to equal or better solutions was achieved with often less than one tenth of the previous computational efforts.