Order Differential Problem Research Articles

Parameter estimation in exponentially fitted hybrid methods for second order differential problems

In this work we deal with exponentially fitted methods for the numerical solution of second order ordinary differential equations, whose solutions are known to show a prominent exponential behaviour depending on the value of an unknown parameter to be suitably determined. The knowledge of an estimation to the unknown parameter is needed in order to apply the numerical method, since its coefficients depend on the value of the parameter. We present a strategy for the practical estimation of the parameter, which is also tested on some selected problems.

D-branes with Lorentzian signature in the Nappi-Witten model

Lorentzian signature D-branes of all dimensions for the Nappi-Witten string are constructed. This is done by rewriting the gluing condition $J_+=FJ_-$ for the model chiral currents on the brane as a well posed first order differential problem and by solving it for Lie algebra isometries $F$ other than Lie algebra automorphisms. By construction, these D-branes are not twined conjugacy classes. Metrically degenerate D-branes are also obtained.

Transversality conditions for infinite horizon optimality: Higher order differential problems

Michel [P. Michel, Some clarifications on the transversality condition, Econometrica 58 (1990) 705–723] and Ekeland and Scheinkman [I. Ekeland, J.A., Scheinkman, Transversality conditions for some infinite horizon discrete time optimization problems, Mathematics of Operations Research 11 (1986) 216–229] presented the transversality condition for the first-order differential problems max x ∫ 0 ∞ v ( x ( t ) , x ̇ ( t ) , t ) d t , which may have unbounded objective functions. Kamihigashi [T. Kamihigashi, Necessity of transversality conditions for infinite horizon problems, Econometrica 69 (2001) 995–1012] showed a generalization of their transversality condition that does not assume concavity. Using the variational approach, this paper deals with higher order differential problems: max x ∫ 0 ∞ v ( x ( t ) , x ̇ ( t ) , x ̈ ( t ) , … , x ( n ) ( t ) , t ) d t . We derive two conditions: Euler’s condition and the transversality condition, for such problems in a simple manner. They are imperative for solving variational problems. Furthermore, two assumptions are necessary to induce the two conditions. We construct a counterexample in which the transversality condition is not satisfied without the two assumptions.

Gegenbauer Tau Methods With and Without Spurious Eigenvalues

It is proven that a class of Gegenbauer tau approximations to a fourth order differential eigenvalue problem of a hydrodynamic type provides real, negative, and distinct eigenvalues, as is the case for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and “inviscid” Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results show that the results are sharp: positive or complex eigenvalues arise outside of this class. The widely used modified tau approach is proved to be equivalent to the Galerkin method.

On the boundary operator of a fourth order differential problem

We study an unilateral elliptic fourth order problem and establish some uniqueness results. The expression of the boundary operation is also given

DEVELOPMENT OF APPROXIMATE METHODS FOR THE ANALYSIS OF PATCH DAMPING DESIGN CONCEPTS

This paper develops three approximate methods for the analysis of patch damping designs. Undamped natural frequencies and modal loss factors are calculated using the Rayleigh energy method and modal strain energy technique, respectively, without explicitly solving high order differential equations or complex eigenvalue problems. Approximate Method I is developed for sandwich beams assuming that damped mode shapes are given by the Euler beam eigenfunctions. The superposition principal is then used to accommodate any arbitrary mode shape, which may be obtained from modal experiments or the finite element method. In Method II, the formulation is further simplified with the assumption of a very compliant viscoelastic core. Finally, Method III considers a compact patch problem. The modal loss factor is then expressed as a product of terms related to material properties, layer thickness, patch size and patch performance. Approximate Methods II and III are also extended to rectangular plates. Formulations are verified by conducting analogous modal measurements and by comparing predictions with those obtained using the Rayleigh–Ritz method (without making any of the above mentioned assumptions). Several example cases are presented to demonstrate the validity and utility of approximate methods for patch damping design concepts.

Lateral Distortional Buckling: Predicting Elastic Critical Stress

The problem of restrained lateral beam buckling is being treated in this paper. The restraint represents complete lateral and arbitrary continuous rotational restraint of one flange, the latter is represented by the spring stiffness C0. A simple theory, in which the problem is transformed into an ordinary fourth order differential eigenvalue problem is presented and design‐tables are given. It refines a method previously developed by Svensson. The results obtained are compared with exact results for C0=0 (the classical theory) and C0=∞ the latter based on results given in the published literature. For C0=0 a maximum deviation of 1% and for C0= deviations in the range of 1% to 4% were found. The results also confirm that a method proposed by other researchers yields similar good results. It is concluded that the method developed in the present paper, due to its generality and simplicity, provides an accurate and quick method for solving complex lateral distortional buckling problems.

Boundary Conditions in Chebyshev and Legendre Methods

We discuss two different ways of treating non-Dirichlet boundary conditions in Chebyshev and Legendre collocation methods for second order differential problems. An error analysis is provided. The effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated.

Differential eigenvalue problems in which the parameter appears nonlinearly

Several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly. The equations are the result of the discretization of differential eigenvalue problems using a finite Chebyshev series. Two global methods are considered which determine the spectrum of eigenvalues without an initial estimate. A local iteration scheme with cubic convergence is presented. Calculations are performed for a model second order differential problem and the Orr-Sommerfeld problem for plane Poiseuille flow.

General nth order differential spectral problem: General structure of the integrable equations, nonuniqueness of recursion operator, and gauge invariance: B. G. Konopelchenko and V. G. Dubrovsky. Institute of Nuclear Physics, 630090, Novosibirsk 90, U.S.S.R.

General nth order differential spectral problem: General structure of the integrable equations, nonuniqueness of recursion operator, and gauge invariance: B. G. Konopelchenko and V. G. Dubrovsky. Institute of Nuclear Physics, 630090, Novosibirsk 90, U.S.S.R.

A uniqueness theorem for epd type equations in general spaces

A uniqueness theorem for solutions of Euler-Poisson-Darboux type equations in general spaces is proved which settles a question open for a number of years and gives a technique applicable to uniqueness questions for other second order abstract differential problems

Note sur la stabilité séculaire des étoiles

After some general remarks on secular stability and stellar evolution, the problem is discussed by means of the small perturbation method. Apart from recovering very simply the usual criteria of dynamical and vibrational stability, this leads, for secular stability, to a fourth order differential problem which seems more tractable than previous formulations. In the case of purely homologous motions which however do not correspond, in general, to a possible solution of the problem, one recovers the usual criterion.