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  • 1
    Digitale Medien
    Digitale Medien
    A differential equation approach to nonlinear programming (1980)
    Springer
    Mathematical programming 18 (1980), S. 155-168 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Constrained Optimization ; Differential Equation ; Global Solution ; Nonlinear Programming
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract A new method is presented for finding a local optimum of the equality constrained nonlinear programming problem. A nonlinear autonomous system is introduced as the base of the theory instead of usual approaches. The relation between critical points and local optima of the original optimization problem is proved. Asymptotic stability of the critical points is also proved. A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points. Some numerical results are given to show the efficiency of the method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01588311
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  • 2
    Digitale Medien
    Digitale Medien
    On the computational utility of posynomial geometric programming solution methods (1982)
    Springer
    Mathematical programming 22 (1982), S. 163-201 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Geometric Programming ; Code Comparisons ; Numerical Testing ; Nonlinear Programming
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract Ten codes or code variants were used to solve the five equivalent posynomial GP problem formulations. Four of these codes were general NLP codes; six were specialized GP codes. A total of forty-two test problems was solved with up to twenty randomly generated starting points per problem. The convex primal formulation is shown to be intrinsically easiest to solve. The general purpose GRG code called OPT appears to be the most efficient code for GP problem solution. The reputed superiority of the specialized GP codes GGP and GPKTC appears to be largely due to the fact that these codes solve the convex primal formulation. The dual approaches are only likely to be competitive for small degree of difficulty, tightly constrained problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01581036
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  • 3
    Digitale Medien
    Digitale Medien
    A globally convergent constrained quasi-Newton method with an augmented lagrangian type penalty function (1982)
    Springer
    Mathematical programming 23 (1982), S. 75-86 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Constrained Optimization ; Global Convergence ; Nonlinear Programming ; Penalty Function ; Quasi-Newton Method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract The recently proposed quasi-Newton method for constrained optimization has very attractive local convergence properties. To force global convergnce of the method, a descent method which uses Zangwill's penalty function and an exact line search has been proposed by Han. In this paper a new method which adopts a differentiable penalty function and an approximate line is presented. The proposed penalty function has the form of the augmented Lagrangian function. An algorithm for updating parameters which appear in the penalty function is described. Global convergence of the given method is proved.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01583780
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  • 4
    Digitale Medien
    Digitale Medien
    A smoothing-out technique for min—max optimization (1980)
    Springer
    Mathematical programming 19 (1980), S. 61-77 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Optimization ; Nonlinear Programming ; Unconstrained Optimization ; Nondifferentiable Optimization ; Min—Max Problems
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract In this paper, we suggest approximations for smoothing out the kinks caused by the presence of “max” or “min” operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR n . Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01581628
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  • 5
    Digitale Medien
    Digitale Medien
    Accelerated convergence for the Powell/Hestenes multiplier method (1980)
    Springer
    Mathematical programming 18 (1980), S. 197-214 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Nonlinear Programming ; Constrained Optimization ; Augmented Lagrangian Methods ; Multiplier Methods
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract It is known that augmented Lagrangian or multiplier methods for solving constrained optimization problems can be interpreted as techniques for maximizing an augmented dual functionD c(λ). For a constantc sufficiently large, by considering maximizing the augmented dual functionD c(λ) with respect toλ, it is shown that the Newton iteration forλ based on maximizingD c(λ) can be decomposed into taking a Powell/Hestenes iteration followed by a Newton-like correction. Superimposed on the original Powell/Hestenes method, a simple acceleration technique is devised to make use of information from the previous iteration. For problems with only one constraint, the acceleration technique is equivalent to replacing the second (Newton-like) part of the decomposition by a finite difference approximation. Numerical results are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01588314
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  • 6
    Digitale Medien
    Digitale Medien
    On optimality conditions for structured families of nonlinear programming problems (1982)
    Springer
    Mathematical programming 22 (1982), S. 82-92 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Generic Programming ; Optimality Conditions ; Nonlinear Programming
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract Optimality conditions for families of nonlinear programming problems inR n are studied from a generic point of view. The objective function and some of the constraints are assumed to depend on a parameter, while others are held fixed. Techniques of differential topology are used to show that under suitable conditions, certain strong second-order conditions are necessary for optimality except possibly for parameter values lying in a negligible set.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01581027
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  • 7
    Digitale Medien
    Digitale Medien
    Comparative computer results of a new complementary pivot algorithm for solving equality and inequality constrained optimization problems (1980)
    Springer
    Mathematical programming 18 (1980), S. 169-185 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Nonlinear Programming ; Equality Constraints ; Fixed Points ; Complementary Pivoting Algorithms
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract This paper reports the development of a new algorithm for solving the general constrained optimization problem (that of optimizing an objective function subject to both equality and inequality constraints). The approach is based on the complementary pivoting algorithms which have been developed to solve certain classes of fixed point problems. The specific approach is to use the equality constraints to solve for some variables in terms of the remaining ones thus enabling one to eliminate the equality constraints altogether. The result, under certain circumstances, is an optimization problem which may be transformed into a fixed point problem in such a way that a complementary pivoting code may be used to search for a solution. Seventeen test problems have been solved by this method and the results are compared against those obtained from GRG (Generalized Reduced Gradient method). The results of the tests indicate that the fixed point approach is robust (all 17 problems were solved by this method where as GRG solved 16). As to the computer times, the fixed point code proved to be as fast or faster than GRG on the lower dimensional problems; however, as the dimension increased, the trend reversed and on a 40 dimensional problem GRG was approximately 11 times faster. The conclusion from these tests is that when the dimension of the original problem can be reduced sufficiently by the equality constraints, the fixed point approach appears to be more effective than GRG.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01588312
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  • 8
    Digitale Medien
    Digitale Medien
    Nonlinear programming via an exact penalty function: Global analysis (1982)
    Springer
    Mathematical programming 24 (1982), S. 137-161 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Nonlinear Programming ; Exact Penalty Methods ; Successive Quadratic Programming
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract In this paper we motivate and describe an algorithm to solve the nonlinear programming problem. The method is based on an exact penalty function and possesses both global and superlinear convergence properties. We establish the global qualities here (the superlinear nature is proven in [7]). The numerical implementation techniques are briefly discussed and preliminary numerical results are given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01585101
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  • 9
    Digitale Medien
    Digitale Medien
    Approximation of time-dependent free boundaries (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 179-199 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS) ; Primary 65N30 ; Secondary 35R35 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Two problems are considered in the paper: the first of them is connected with elliptic variational inequalities and consists in developing a moving obstacle algorithm for approximating the unknown free boundary; the other problem is linked with numerical solution of the Stefan problem, which is formulated in the similar way as in the elliptic case. Some computational aspects are also discussed in the paper.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400538
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  • 10
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    Digitale Medien
    Finite elements for parabolic equations backwards in time (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 207-227 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65M30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Several regularization methods for parabolic equations backwards in time together with the usual additional constraints for their solution are considered. The error of the regularization is estimated from above and below. For a “boundary value problem in time”-method, finite elements as well as a time discretization are introduced and the error with respect to the regularized solution is estimated, thus giving an overall error of the discrete regularized problem. The algorithm is tested in simple numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400540
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  • 11
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    Digitale Medien
    Adams methods for neutral functional differential equations (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 221-230 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a “non-smooth” solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01408695
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  • 12
    Digitale Medien
    Digitale Medien
    Improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximations (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 309-324 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65L10 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems. Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor ofε n+1 can be obtained (where ɛ is the “small” parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01407865
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  • 13
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    Digitale Medien
    Finite difference methods and their convergence for a class of singular two point boundary value problems (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 341-350 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L10 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x α y′)′=f(x,y), y(0)=A, y(1)=B, 0〈α〈1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all α∈(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM 1 based on just one evaluation off. For uniform mesh we obtain two methodsM 2 andM 3 each based on three evaluations off. For α=0,M 1 andM 2 both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h 2)-convergence established and illustrated by numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01407867
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  • 14
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    Digitale Medien
    On the numerical analysis of the Von Karman equations: Mixed finite element approximation and continuation techniques (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 371-404 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65 N 30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The purpose of this paper is to study the approximation of the Von Karman equations by the mixed finite element scheme of Miyoshi and to follow the solutions arcs at a neighbourhood of the first eigenvalue of the linearized problem. This last problem is solved by a continuation method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01407870
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  • 15
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    Digitale Medien
    Maximale Konvergenzordnung bei der numerischen Lösung von Anfangswertproblemen mit Splines (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 449-463 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65L65 ; CR: 5.17 ; AMS (MOS): 65L65 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Beschreibung / Inhaltsverzeichnis: Summary In [10] a general procedureV is presented to obtain spline approximations by collocation for the solutions of initial value problems for first order ordinary differential equations. In this paper the attainable order of convergence with respect to the maximum norm is characterized in dependence of the parameters involved inV; in particular the appropriate choice of the collocation points is considered.
    Notizen: Zusammenfassung In [10] ist ein allgemeines VerfahrenV beschrieben, das die Lösungen von Anfangswertproblemen bei gewöhnlichen Differentialgleichungen erster Ordnung durch Splines approximiert. Die Konstruktion der Splines erfolgt hierbei mittels Kollokation. In dieser Arbeit wird die maximal erreichbare Konvergenzordnung vonV bezüglich der Maximumnorm in Abhängigkeit aller Parameter vonV charakterisiert, insbesondere wird auf die geeignete Wahl der Kollokationsknoten eingegangen.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01407875
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  • 16
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    Digitale Medien
    Fast Givens rotations for orthogonal similarity transformations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 47-56 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F30 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459074
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  • 17
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    Digitale Medien
    Exact solution of linear equations usingP-adic expansions (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 137-141 
    Details
    ISSN: 0945-3245
    Schlagwort(e): (MR 1980) AMS(MOS) ; 65F05, 15A06, 10M10, 10A30 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector $$\bar x$$ such that $$A\bar x \equiv b$$ (modp m ) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation $$\bar x$$ . For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n 3(logn)2) which is better than methods previously used.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459082
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  • 18
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    Digitale Medien
    Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 169-177 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400537
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  • 19
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    Digitale Medien
    Error bounds for computed eigenvalues and eigenvectors. II (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 201-206 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F05, 65F15, 65H10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper, motivated by Symm-Wilkinson's paper [5], we describe a method which finds the rigorous error bounds for a computed eigenvalue λ(0) and a computed eigenvectorx (0) of any matrix A. The assumption in a previous paper [6] that λ(0),x (0) andA are real is not necessary in this paper. In connection with this method, Symm-Wilkinson's procedure is discussed, too.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400539
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  • 20
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    Digitale Medien
    Discretization of multiparameter eigenvalue problems (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 319-328 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65J05, 65L15 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396449
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  • 21
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    The finite element method for parabolic equations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 373-406 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30, 65M20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper [3] for time-independent space meshes. In the context of the model problem studied there, results are given relating the effectiveness of the error estimator to properties of the solution, space meshes, and manner in which the meshes change. A procedure based upon this theory is presented for the adaptive construction of time-dependent meshes. The results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396452
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  • 22
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    Digitale Medien
    Second-order conditions for an exact penalty function (1980)
    Springer
    Mathematical programming 19 (1980), S. 178-185 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Nonlinear Programming ; Exact Penalty Function ; Constrained Optimization ; Piecewise Differentiable
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract In this paper we give first- and second-order conditions to characterize a local minimizer of an exact penalty function. The form of this characterization gives support to the claim that the exact penalty function and the nonlinear programming problem are closely related. In addition, we demonstrate that there exist arguments for the penalty function from which there are no descent directions even though these points are not minimizers.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01581639
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  • 23
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    A polynomially bounded algorithm for a singly constrained quadratic program (1980)
    Springer
    Mathematical programming 18 (1980), S. 338-343 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Nonlinear Programming ; Convex Programming ; Quadratic Programming ; Singly Constrained Quadratic Program
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract This paper presents a characterization of the solutions of a singly constrained quadratic program. This characterization is then used in the development of a polynomially bounded algorithm for this class of problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01588328
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  • 24
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    Digitale Medien
    Nonlinear programming via an exact penalty function: Asymptotic analysis (1982)
    Springer
    Mathematical programming 24 (1982), S. 123-136 
    Details
    ISSN: 1436-4646
    Schlagwort(e): Nonlinear Programming ; Exact Penalty Methods ; Successive Quadratic Programming
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract In this paper we consider the final stage of a ‘global’ method to solve the nonlinear programming problem. We prove 2-step superlinear convergence. In the process of analyzing this asymptotic behavior, we compare our method (theoretically) to the popular successive quadratic programming approach.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01585100
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  • 25
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    Digitale Medien
    Reguläre Zerlegungen und Berechnung des Spektralradius nichtnegativer Matrizen (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 201-216 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary On the basis of a Rayleigh Quotient Iteration method in [10] and a Maximal Quotient Iteration method in [5, 8] two algorithms for solving special eigenvalue problems are developed. The characteristic properties of these methods lie in the application of iterative linear methods to solving systems of linear equations. The convergence properties are investigated. We apply the algorithms to the computation of the spectralradius of a nonnegative irreducible matrix.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396060
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  • 26
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    Digitale Medien
    On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 235-246 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396701
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  • 27
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    Digitale Medien
    Realistic error bounds for a simple eigenvalue and its associated eigenvector (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 113-126 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F15 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396310
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  • 28
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    Digitale Medien
    An approximation theorem for second-order evolution equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 127-142 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 34G05, 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos τ for τ≧0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396311
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  • 29
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    Digitale Medien
    Méthodes multipas pour des équations paraboliques non linéaires (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 143-162 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65M15, 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We study the error due to the discretization in time of a nonlinear parabolic problem by a multistep method. Error estimates are obtained if the method is of the orderp (p〉1) and stronglyA(Θ)-stable $$\left( {0〈 \Theta〈 \frac{\pi }{2}} \right)$$ . The method is also applied to the Navier-Stokes equations in two dimensions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396312
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  • 30
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    Digitale Medien
    Enlarging the domain of convergence for multiple shooting by the homotopy method (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 231-240 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L10 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The homotopy method is a frequently used technique in overcoming the local convergence nature of multiple shooting. In this paper sufficient conditions are given that guarantee the homotopy process to be feasible. The results are applicable to a class of two-point boundary value problems. Finally, the numerical solution of two practical problems arising in physiology is described.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396319
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  • 31
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    Digitale Medien
    Une méthode multipas implicite-explicite pour l'approximation des équations d'évolution paraboliques (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 257-276 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS) ; 65M10 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Beschreibung / Inhaltsverzeichnis: Résumé Considérons une équation d'évolution parabolique linéaire associée à un opérateur linéaireA(t) dépendant du tempst. Nous développons dans cet article une méthode de discrétisation, basée sur les méthodes linéaires à pas multiples, traitant de manière implicite une partie de l'opérateurA(t) indépendante du temps, l'autre partie est traitée de manière explicite. Nous étudions la stabilité et la convergence de cette méthode.
    Notizen: Summary Let us consider a linear parabolic equation which is associated with a time dependent operatorA(t). In this paper, we present a method, which is founded on linear multistep methods, which discretize a time-independent part of the operatorA(t) in an implicit way, and the other part in an explicit way. We study stability and convergence for this method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396412
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  • 32
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    Digitale Medien
    Mixed finite elements in ℝ3 (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 315-341 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS) ; 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We present here some new families of non conforming finite elements in ℝ3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwell's equations and equations of elasticity.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396415
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  • 33
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    Digitale Medien
    A mixed finite element approximation of the Navier-Stokes equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 381-404 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399007
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  • 34
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    Digitale Medien
    Characterizations of dominant and dominated solutions of linear recursions (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 421-442 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65Q05, 65F99, 39A10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Dominated solutions of a linear recursion (i.e. solutions which are outgrown by other ones) cannot be computed in a stable way by forward recursion. We analyze this dominance phenomenon more closely and give practically significant characterizations for dominated and dominant solutions. For a dominated solution, in particular, this leads to a stable computational method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399009
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  • 35
    Digitale Medien
    Digitale Medien
    Stability of fast algorithms for matrix multiplication (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 63-72 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 15A63 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Non commutative fast algorithms to compute 2×2 matrix product are classified with regard to stability. An analysis of the rounding error propagation is presented for then×n matrix multiplication algorithms obtained by recursive partitioning.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395989
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  • 36
    Digitale Medien
    Digitale Medien
    Least squares with a quadratic constraint (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 291-307 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F20 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We present the theory of the linear least squares problem with a quadratic constraint. New theorems characterizing properties of the solutions are given. A numerical application is discussed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396656
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  • 37
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    Digitale Medien
    Matrices related to interpolatory quadratures (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 309-318 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F15 ; 65D30 ; CR: 5.14 ; 5.16
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary An explicit formula for the coefficients of interpolatory quadratures with knots of arbitrary multiplicity which uses the eigenvectors and principal vectors of certain matrices is derived. The construction and properties of such matrices are discussed, and applications for the evaluation of Gauss and generalized Gauss-Lobatto quadratures are indicated.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396657
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  • 38
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    Digitale Medien
    An approximation technique for the numerical solution of a Stefan problem (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 141-154 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 35 A 40 ; 35 K 05 ; 65 N 15 ; 65 K 05 ; 80 A 20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A nonlinear approximation technique for the numerical solution of certain free boundary problems is proposed. The method is shown for a degenerate one-dimensional Stefan problem. For this problem, an error estimate, which is independent of the used algorithm, is derived. Numerical examples are discussed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395813
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  • 39
    Digitale Medien
    Digitale Medien
    LU decompositions of generalized diagonally dominant matrices (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 57-69 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F05 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Using the simple vehicle ofM-matrices, the existence and stability ofLU decompositions of matricesA which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination onA are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsAx=0, arising in Markov queuing networks, input-output models in economics and compartmental systems, whereA or −A is an irreducible, singularM-matrix. This paper extends earlier work by Funderlic and Plemmons and by Varga and Cai.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459075
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  • 40
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    Digitale Medien
    Computing theCS decomposition of a partitioned orthonormal matrix (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 297-306 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F25 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396447
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  • 41
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    Digitale Medien
    On improving the absolute stability of local extrapolation (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 329-337 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396450
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  • 42
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    Digitale Medien
    The finite element method for parabolic equations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 339-371 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; 65M20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396451
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  • 43
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    Digitale Medien
    Die mehrstellenformeln für den Laplaceoperator (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 171-187 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called “Mehrstellenformeln’ or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396058
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  • 44
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    Digitale Medien
    An algorithm for a linear diophantine equation and a problem of Frobenius (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 349-352 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 10B05 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We solve the diophantine equation $$\sum\limits_{j = 1}^n {a_j x_j } = L$$ for nonnegative variablesx j , wherea j andL are positive integers. We characterize both the values ofL that lead to solutions and those that do not lead to solutions. We solve the Frobenius problem of finding the largest value ofL for which no solution exists.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01403673
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  • 45
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    Digitale Medien
    A generalization of the Stein-Rosenberg theorem to Banach spaces (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 403-409 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS):65F 10, 47B 55 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In the first part of this note we prove a generalization of the Stein-Rosenberg theorem; the context is that of real Banach spaces with a normal reproducing cone and the operators involved are positive and completely continuous. Our generalization of the Stein-Rosenberg theorem improves the modern version of it as stated by F. Robert in [5, §2]. In the second part, we discuss briefly how our results are related to other versions of the Stein-Rosenberg theorem. In the last section we describe a situation to which the results in the first part can be applied.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01403677
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  • 46
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    Digitale Medien
    Effective computation of periodic orbits and bifurcation diagrams in delay equations (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 457-467 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05, 65Q05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01403681
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  • 47
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    Digitale Medien
    A finite element method for the Tricomi problem (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 13-20 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): primary 65M05 ; 65M10 ; 65M15 ; secondary: 35M05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We consider the numerical solution of the Tricomi problem. Using a weak formulation based on different spaces of test and trial functions, we construct a new Galerkin procedure for the Tricomi problem. Existence, uniqueness, and uniform stability of the approximate solution is proven, and a priori error bounds are given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396366
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  • 48
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    Digitale Medien
    Deferred corrections using uncentered end formulas (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 21-33 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65B05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The trapezoidal rule with deferred corrections using uncentered end formulas is shown to converge. While the proof technique is more specialized than the standard asymptotic expansion approach, it has some advantages. In addition to providing a more complete theoretical justification for current implementations of deferred corrections with the trapezoidal rule, the approach given here will hopefully apply for several other discretization methods.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396367
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  • 49
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    Digitale Medien
    Unconditionally stable explicit methods for parabolic equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 57-68 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; 65M20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396370
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  • 50
    Digitale Medien
    Digitale Medien
    On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 69-79 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the ‘usual’D−L−U, and (ii) of an hermitian positive definite matrixA intoD−L−U, whereD is hermitian positive definite andL=1/2(A−D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in ω, ω≧0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396371
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  • 51
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    Digitale Medien
    The rate of convergence of an approximate matrix factorization semi-direct method (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 237-251 
    Details
    ISSN: 0945-3245
    Schlagwort(e): CR: 5.14 ; 5.17 ; AMS(MOS): 65F10 ; 65N10
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The definition of acceleration parameters for the convergence of a sparseLU factorization semi-direct method is shown to be based on lower and upper bounds of the extreme eigevalues of the iteration matrix. Optimum values of these parameters are established when the eigenvalues of the iteration matrix are either real or complex. Estimates for the computational work required to reduce theL 2 norm of the error by a specified factor ɛ are also given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396653
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  • 52
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    Digitale Medien
    On the design of high order exponentially fitted formulae for the numerical integration of stiff systems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 253-266 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65LO5 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Single step exponentially fitted integration formulae of orders 4 and 6 are derived. The final approximation to the required solution is obtained via a linear combination of asymptotically less accurate solutions obtained using conventional implicit integration formulae. This linear combination is performed in such a way that the final integration formula, as well as having an increased order of accuracy, integrates a certain pre-determined ordinary differential equation exactly. The algorithms developed are illustrated by means of some numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396654
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  • 53
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    Digitale Medien
    The indicatrix method in free boundary problems (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 39-52 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): Primary 65L10 ; Secondary 35R35 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary An approximate method for nonlinear problems with functional constraints is considered, in which the constraint in the whole domain is replaced by the constraint on a manifold of lower dimension. The stability criterion is introduced, and convergence theorems are proved for the onedimensional problem. Numerical results for the elastic-plastic torsion problem are given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395807
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  • 54
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    Digitale Medien
    On fixed point linear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 53-59 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; 47B55 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary By means of successive partial substitutions it is possible to obtain new fixed point linear equations from old ones and it is interesting to determine how the spectral radius of the corresponding matrices varies. We prove that, when the original matrix is nonnegative, this variation is decreasing or increasing, depending on whether the original matrix has its spectral radius smaller or greater than 1. We answer in this way a question posed by F. Robert in [5].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395808
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  • 55
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    Digitale Medien
    Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 365-382 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396438
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  • 56
    Digitale Medien
    Digitale Medien
    A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 417-419 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required. The technique is also applicable to other computational methods which use the bisection process.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396441
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  • 57
    Digitale Medien
    Digitale Medien
    A note onC o Galerkin methods for two-point boundary problems (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 447-453 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS) 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary As is known [4]. theC o Galerkin solution of a two-point boundary problem using piecewise polynomial functions, hasO(h 2k ) convergence at the knots, wherek is the degree of the finite element space. Also, it can be proved [5] that at specific interior points, the Gauss-Legendre points the gradient hasO(h k+1) convergence, instead ofO(h k ). In this note, it is proved that on any segment there arek−1 interior points where the Galerkin solution is ofO(h k+2), one order better than the global order of convergence. These points are the Lobatto points.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396444
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  • 58
    Digitale Medien
    Digitale Medien
    On the stability of backward differentiation methods (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 467-471 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS) 65L20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The approximation of linear systemsy′=−A(t)y+b(t) by backward differentiation methods up to order 5 is considered. It is proved that the error does not increase if the real symmetric matrixA(t) is positive definite andA′(t) is negative semi-definite.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396446
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  • 59
    Digitale Medien
    Digitale Medien
    Approximation of Hopf bifurcation (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 15-37 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65 N 30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We make several assumptions on a nonlinear evolution problem, ensuring the existence of a Hopf bifurcation. Under a fairly general approximation condition, we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399309
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  • 60
    Digitale Medien
    Digitale Medien
    The numerical solutions of interface problems by infinite element method (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 39-50 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399310
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  • 61
    Digitale Medien
    Digitale Medien
    A contribution to the theory of condition (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 85-96 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS65F05 ; 65F 15 ; 65F 20 ; 65F 35 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper the use of the condition number of a problem, as defined by Rice in 1966, is discussed. For the eigenvalue, eigenvector, and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399313
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  • 62
    Digitale Medien
    Digitale Medien
    Exact bounds for the solution branches of nonlinear eigenvalue problems (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 29-40 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N25 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A method is constructed which yields a strip containing the full solution sets of nonlinear eigenvalue problems of the formu=λTu. The strip can be narrowed iteratively, and the method applies for both stable and unstable branches. Its high degree of accuracy is demonstrated by numerical examples. In particular, a lower bound is given for the critical value at which criticality is lost in the thermal ignition problem for the unit ball.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01463996
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  • 63
    Digitale Medien
    Digitale Medien
    Analysis of finite element methods for second order boundary value problems using mesh dependent norms (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 41-62 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01463997
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  • 64
    Digitale Medien
    Digitale Medien
    Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil III (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 143-154 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L65 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In the first part [1] a general procedure is presented to obtain polynomial spline approximations for the solutions of initial value problems for ordinary differential equations; furthermore a divergence theorem is proved there. Sufficient conditions for convergence of the method are given in the second part [2]. The remaining case which has not been considered in [1] and [2] is treated in the present paper. In this special case the procedure is equivalent to an unstable two-step method with special initial values; nevertheless, convergence can be proved. Finally,A 0-stability of the method as well as the influence of rounding errors are investigated.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396056
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  • 65
    Digitale Medien
    Digitale Medien
    Iterative solutions of nonlinear initial value differential equations in Chebyshev series using Lie series (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 1-13 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 34A45, 34A50, 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper Lie series are presented in Chebyshev form and applied to the iterative solution of initial value problems in differential equations. The resulting method, though algebraically complicated, is of theoretical interest as a generalisation of Taylor series methods and iterative Chebyshev methods. The theory of the method is discussed and the solutions of some simple scalar equations are analysed to illustrate the behaviour of the process.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01463994
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  • 66
    Digitale Medien
    Digitale Medien
    A method of characteristics solving the initial-boundary value problem of a hyperbolic differential equation of second order (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 217-234 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65M25, 35L20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The ALGOL-procedure1 char2 presented in this paper can be applied to the initial or initial-boundary value problem of a quasilinear hyperbolic differential equation of second order. A method of characteristics is combined with extrapolation to the limit. Thus, the results are very accurate. The same accuracy can also be obtained if the initial values are only piecewise smooth.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396700
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  • 67
    Digitale Medien
    Digitale Medien
    Finite dimensional approximation of nonlinear problems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 1-25 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65 N 30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We begin in this paper the study of a general method of approximation of solutions of nonlinear equations in a Banach space. We prove here an abstract result concerning the approximation of branches of nonsingular solutions. The general theory is then applied to the study of the convergence of two mixed finite element methods for the Navier-Stokes and the von Kármán equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395985
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  • 68
    Digitale Medien
    Digitale Medien
    Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 33-52 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A modified variational formulation, recently introduced by Taylor, Beresford and Wilson for solving second order problems, using the nonconforming Wilson element is here analysed. It is shown that the Patch Test is satisfied and that stresses and displacements are respectively first and second order accurate for arbitrary quadrilateral meshes.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395987
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  • 69
    Digitale Medien
    Digitale Medien
    Numerical solution of steady-state porous flow free boundary problems (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 73-98 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS: 35R35, 65P05, 76S05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395990
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  • 70
    Digitale Medien
    Digitale Medien
    A finite element lumped mass scheme for solving eigenvalue problems of circular arches (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 267-290 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396655
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  • 71
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    Digitale Medien
    A note onB-stability of Runge-Kutta methods (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 319-331 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS) ; 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396658
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  • 72
    Digitale Medien
    Digitale Medien
    Finite dimensional approximation of nonlinear problems (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 1-30 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395805
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  • 73
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    Digitale Medien
    Inversion of tridiagonal matrices (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 333-345 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 15 A 09, 15-04 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist. Ukita's theorem is extended to characterize the class of matrices that have tridiagonal inverses.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396436
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  • 74
    Digitale Medien
    Digitale Medien
    Éléments finis mixtes incompressibles pour l'équation de Stokes dans ℝ3 (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 97-112 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We introduce some new families of finite element approximation for the stationary Stokes and Navier Stokes equations in a bounded domain in ℝ3. These elements can used tetahedrons or cubes. The approximation satisfie exactly the incompressibility condition.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399314
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  • 75
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    Digitale Medien
    On the convergence of the symmetric SOR method for matrices with red-black ordering (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 113-117 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We prove that if the matrixA has the structure which results from the so-called “red-black” ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0〈ω〈2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399315
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  • 76
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    Digitale Medien
    Zur Konvergenz eines Verfahrens von D.J. Evans zur iterativen Verbesserung einer Näherung für die inverse Matrix (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 163-173 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Recently D.J. Evans introduced an implicit matrix inversion process showing asymptotic behaviour which is superior to that of the well known Schulz-method. In this paper we give sufficient conditions for convergence, prove some error bounds and show that under certain conditions the iterates are converging monotonously.
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    URL: http://dx.doi.org/10.1007/BF01408691
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  • 77
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    Digitale Medien
    The effect of a thin inclusion of high rigidity in an elastic body (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 251-270 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: We study an elastic body comprising an inclusion whose thickness is 2∊ and for which Lamé's constants are λ∊ and μ. We are interested in the limit behaviour of the inclusion, when ∊ → 0 and μ∊ → ∊; that is when the inclusion becomes thiner and more rigid. Different behaviours are possible, according to the rate at which μ∊ converges to infinity; the limit inclusion may “vanish” if it is not very rigid, but also it may be entirely solid.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020302
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  • 78
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    Digitale Medien
    Nonselfadjoint operators in diffraction and scattering (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 327-346 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020307
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  • 79
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    Digitale Medien
    Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3 (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 1-14 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p 〉 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm rot }\upsilon {\rm = }\omega {\rm, div }\upsilon = 0,\upsilon \cdot \bar n||_{\partial \Omega } = 6 $$\end{document} where v - velocity vector, ω - vorticity vector and n is an unit outward vector normal to the boundary,(2) the existence of solutions of the following evolution problem for given velocity vector \documentclass{article}\pagestyle{empty}\begin{document}$$ \begin{array}{l} \omega _t + \upsilon ^\kappa \omega _x \kappa - \omega ^\kappa \upsilon _x \kappa = F \equiv {\rm rot }f \\ \omega |_{t = 0} = \omega _0,\omega |_{s1} = \eta \\ \end{array} $$\end{document}(3) the method of successive approximations, using solvability of problems (1) and (2).
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040102
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  • 80
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    Galerkin approximations for the equation Lu + Nu = f (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 286-290 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: Introducing the concept of a supercontinuous operator, we obtain a general convergence theorem for Galerkin approximations. Under the stronger assumption that N is a monotone operator with N(0) = 0, we show norm convergence of the unique Galerkin approximations to the unique solution.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040119
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  • 81
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    Biorthogonal sequences of solutions of the generalized feller equation (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 317-353 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: The generalized Feller equation is a linear, autonomous, parabolic equation of a positive space variable and a time variable. Its coefficients are power functions of the space variable, and they depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the standard heat equation. The main objective of the present paper is to establish series expansions of solutions of the generalized Feller equation in terms of the elements of two sequences of particular solutions. The elements of one of these sequences are particular initial condition solutions. The two sequences are biorthogonal. The main result is that a solution does have the desired expansion property if and only if it has the Huygens property in some neighborhood of the origin of the time variable.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040122
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  • 82
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    Ein gemischtes randwertproblem der kreiszylinderschale mit beliebigen ausschnitten (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 354-381 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: The problem of stress determination in the area of cut-outs in circular cylindrical shells at given loads is of great interest in industrial practice. This work deals with a mixed boundary value problem of a differential equation derived according to the theory of shallow shells. On part Ct1 of the boundary, the displacements are given, whereas the stresses are specified on the remaining part Ct2. Starting from the Betti-Maxwell principle and with the aid of the fundamental solutions for unit loads and unit displacements, integral representations can be derived for the displacement functions as well as the stress functions.The problem is then transformed into an equivalent system of Fredholm integral equations of the first kind with logarithmic kernels as the main part. As the integral equations together with the auxiliary conditions form a strongly elliptical system of pseudo-differential operators, the Galerkin method converges. Assuming that curves Ct1 and Ct1 do not have points of intersection and that the data are sufficiently regular, the required functions are approximated by cubic splines and, for simplicity's sake, the integral equation system is solved by approximation with a collocation method. In view of the complicated terms of the kernel functions, the kernels are split into a regular and a singular part, the regular part being in turn replaced by cubic splines. The remaining integrations are done numerically by means of Gaussian quadrature formulae. The applicability of the method is demonstrated with the example of a cylinder under internal pressure.
    Zusätzliches Material: 5 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040123
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  • 83
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    Zum 80. Geburtstag von Wolfgang Haack (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 451-453 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Zusätzliches Material: 1 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040129
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  • 84
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    Zur theorie rotierender und schwingender schaufelkränze in einer unterschallstromung durch einen ringkanalDie vorliegende Arbeit wurde von der Deutschen Forschungsgemeinschaft unter der Numraer Me 261/3 unterstützt. (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 454-479 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In this paper the three-dimensional perturbation flow induced by a rotating and oscillating blade row which operates in a subsonic flow in axial direction of an annular channel is studied. The velocity potential is reduced to the infinite Hilbert space vector of Fourier coefficients of an eigen-function expansion with respect to vanishing normal derivatives on both cylinder walls. These coefficients satisfy an infinite set of ordinary differential equations of second order after an application of a one-dimensional Fourier transform in axial direction.Several canonical two-part mixed boundary value problems are then investigated by reduction to “infinite two-by-two-Wiener-Hopf functional systems”. In case of strong factorizability of certain matrix-operator-valued functions on the line these systems may be solved explicitely. Criteria for the factorization are not given here.
    Zusätzliches Material: 5 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040130
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  • 85
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    Über die existenz und eindeutigkeit einer lösung für ein randwertproblem für eine gleichung vom gemischten typ in einem rechteck (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 510-528 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In the paper a boundary value problem is studied for the equation of mixed typek(y)uxx + uyy + r(x, y)u = f(x, y) in the rectangular domain {(x, y)| -1 〈 x 〈 +1, yc 〈 y 〈 yH} with yc 〈 0, yH 〉 0, k(y) = sign y|y|m, m 〉 0 (and more generally for a function k = k(y) with k(O) = 0, k(y)y 〉 O for y ≠ O). Specific for the stated problem is that no data are prescribed on the line {(x, yc), -1 〈 x 〈 +1}. It is proved that the formulated problem is well-posed in the sense that there is at most one quasi-regular solution and that a generalized solution exists. The energy-integral-(abc-)method is used to show uniqueness and to obtain an apriori estimate for the solution of the adjoint problem whence the existence statement follows.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040133
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  • 86
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    Rounding errors in numerical solutions of two linear equations in two unknownsDedicated to Professor Wolfgang Haack on the occasion of his eightieth birthday. (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 549-571 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: New condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.
    Zusätzliches Material: 2 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040135
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  • 87
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    Asymptotic study of the macroscopic behaviour of a solid-fluid mixture (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 1-11 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In the framework of homogenization theory we study a mixture of an elastic solid and a viscous compressible fluid with periodic structure and its limit behaviour as the period tends to zero Existence, uniqueness and convergence theorems are given. The limit behaviour is viscoelastic.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020102
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  • 88
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    Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung ΔU + k2U = 0 (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 48-67 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k2U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L2(∂G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L2(∂G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G.The well-known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020106
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  • 89
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    A singular perturbation approach for a reaction diffusion system arising in nuclear reactor theory (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 91-107 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: The qualitative behavior of the solutions of a reaction-diffusion system arising in the theory of nuclear reactors is investigated by means of singular perturbation techniques, the small parameter ∊ representing the inverse of neutron velocity. At the lowest approximation for small ∊ the solutions exhibit oscillations about the unique strictly positive equilibrium, much as in the case of the associated lumped parameter system. A two-times representation for solutions of small amplitudes is also given.
    Zusätzliches Material: 2 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020109
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  • 90
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    On uniqueness of Neumann-Tricomi problem in R2 (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 168-177 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: We consider the equation of mixed type (k(y) ⋛ 0 whenever y ⋛ 0) in a region G which is bounded by the curves: A piecewise smooth curve Γ lying in the half-plane y 〉 0 which intersects the line y = 0 at the points A(-1, 0) and B(0, 0). For y 〈 0 by a piecewise smooth curve Γ through A which meets the characteristic of (1) issued from B at the point P and the curve Γ which consists of the portion PB of the characteristic through B. We obtain sufficient conditions for the uniqueness of the solution of the problem L[u] = f, dnu: = k(y)uxdy - uydx|γ0 = = Ψ(s) for a “general” function k(y), when r(x, y) is not necessarily zero and Γ1 is of a more general form then in the papers of V. P. Egorov [6], [7].
    Zusätzliches Material: 2 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020204
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  • 91
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    Numerical solution of Hopf bifurcation problems (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 178-190 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ∊. It has isolated solutions in the ∊-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.
    Zusätzliches Material: 1 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020205
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  • 92
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    Zur L∞-Konvergenz finiter Elemente bei parabolischen Differentialgleichungen (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 221-234 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: For a parabolic model problem it is shown that the convergence rate of higher order finite element approximation is quasioptimal in L∞. Moreover, the estimate does not depend on the interval [0, T]. The essential tool of the proof is a modified weighted norm technique.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020208
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  • 93
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    Scattering of elastic waves through a heterogeneous medium (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 271-287 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In this paper we study the elastic wave diffraction in R3 through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020303
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  • 94
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    Masthead (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980) 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020401
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  • 95
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    Über die Gaußsche Methode zur angenäherten Berechnung von IntegralenMeinem verehrten Lehrer, Herrn Prof. Dr. Cl. Müller, in Dankbarkeit zum 60. Geburtstag gewidmet. (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 397-409 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: The purpose of this paper is the application of Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020403
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  • 96
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    Zur numerischen Lösung gewöhnlicher Differential-gleichungen mit Splines in einem SonderfallHerrn Claus Müller zum 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 419-428 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020405
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  • 97
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    Some remarks on the horizontal line methodHerrn Claus Müller zum 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 471-479 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: In the following paper a methodological survey with respect to applications of the horizontal line method (Rothe's method) to a class of initial boundary value problems is given. By means of results from abstract perturbation theory, convergence results and error estimates are established for several special initial boundary value problems of mathematical physics.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020409
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  • 98
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    Elektrodynamik mit SpinHerrn Prof. Dr. Claus Müller zu seinem 60. Geburtstag am 20. 2. 1980 gewidmet (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 457-470 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: Atomistic equations of the electromagnetic field for a particle with spin are derived from a Lagrangian. These equations are consistent with the equations of motion for such a particle. The resulting phenomenological equations are the well-known equations of Maxwell for the electromagnetic field in matter. The atomistic field equations for a particle with spin and magnetic moment give a dipole field. This result and the corresponding quantum mechanics for a particle with spin are applied to compute the hyperfine structure of the hydrogen atom by perturbation theory.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020408
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  • 99
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    On the boundary value problem of the biharmonic operator on domains with angular corners (1980)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 2 (1980), S. 556-581 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.
    Zusätzliches Material: 1 Ill.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670020416
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  • 100
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    On complete recovery of geophysical data (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 33-73 
    Details
    ISSN: 0170-4214
    Schlagwort(e): Mathematics and Statistics ; Applied Mathematics
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Mathematik
    Notizen: One considers an elastic halfspace with depth (x1) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x2, x3)δ(t) for x1 = 0 are applied with displacement responses ui = gi(t, x2, x3) at x1 = 0 (i = 1, 2, 3). Let vi(t, x1) = ∫∫uidx2dx3 (i = 1, 2 is enough) and set w(t, x1) = ∫∫x2u1dx2dx3. One obtains a system of 3 differential equations for v1, v2, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled vi can then be solved to produce 2 functions A1 and A2 involving (Q, λ, μ) as functions of “bound” variables y1 and y2 containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x1. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1002/mma.1670040105
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