Ordinary Dierential Equations Research Articles

Parallelization in time for thermo‐viscoplastic problems in extrusion of aluminium

AbstractThe ParaReal algorithm (C.R. Acad. Sci. Paris 2001; 332:1–6) is a parallel approach for solving numerically systems of ordinary differential equations by exploiting parallelism across the steps of the numerical integrator. The method performs well for dissipative problems and problems of fluid–structure interaction (Int. J. Numer. Methods Engng 2003; 58:1397–1434). We consider here a convergence analysis for the method and we report the performance achieved from the parallelization of a Stokes/Navier–Stokes code via the ParaReal algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

様々な半径のリムを持つ車輪の安定性について

This paper describes the dynamics and impact model of a wheel with various radius rim. The dynamics is expressed by a rst order linear ordinary dierential equation with respect to the absolute orientation of the wheel, and an analytic solution is derived. Poincare map is also derived analytically. Stability and basin of attraction (BoA) of the Poincare map are discussed. Finally, the analysis is validated through some numerical simulations. As a result, the rim radius aects the stability and broadens its BoA. The analysis helps understanding of not only a geometric tracking control but also many underactuated control methods for bipeds.

Applications of Prüfer transformations in the theory of ordinary differential equations

This article is a review article on the use of Prufer Transformations techniques in proving classical theorems from the theory of Ordinary Dierential Equations. We consider self-adjoint second order linear dierential equations of the form Lx = (p(t)x 0 (t)) 0 + g(t)x(t) = 0, t 2 (a,b). (?) We use Prufer transformation techniques (which are a gener- alization of Poincare phase-plane analysis) to obtain some of the main theorems of the classical theory of linear dieren- tial equations. First we prove theorems from the Oscillation Theory (Sturm Comparison theorem and Disconjugacy theo- rems). Furthermore we study the asymptotic behavior of the equation (?) when t ! 1 and we obtain necessary and suf- ficient conditions in order to have bounded solutions for (?). Finally, we consider a certain type of regular Sturm-Liouville eigenvalue problems with boundary conditions and we study their spectrum via Prufer transformations.

Convexifiable functions in integral calculus

A function is said to be convexiable if it becomes convex after adding to it a strictly convex quadratic term. In this paper we extend some of the basic integral properties of convex functions to Lipschitz continuously dieren tiable functions on real line. In particular, we give estimates of the mean value, a nonstandard form of Jensen's inequality, and an explicit representation of analytic functions. It is also outlined how one can use convexication to study ordinary dieren tial equations.

Topological properties of attainability sets of linear systems

Linear systems are best studied in control theory. They are both of applied interest in themselves and of theoretical importance. Indeed, the local investigation of a nonlinear control system by linearization results in a linear system whose behavior largely characterizes the local behavior of the original nonlinear system. The structure and properties of attainability sets play an important role in the analysis of control systems. In the present paper, we consider topological properties of attainability sets of linear systems. Consider an object whose motion is governed by the linear system of ordinary dierential equations _ x = A(t)x +B(t)u(t) ;t 2 [0;T]; (1) with the initial condition x(0) = 0: (2)

Multiscale Couplings In Prototype Hybrid Deterministic/Stochastic Systems: Part I, Deterministic Closures

We introduce and study a class of model prototype hybrid systems comprised of a microscopic stochastic surface process modeling adsorption/desorption and/or surface di.usion of particles coupled to an ordinary di.erential equation (ODE) displaying bifurcations excited by a critical noise parameter. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic models. We obtain deterministic mesoscopic models from the hybrid system by employing two methods: stochastic averaging principle and mean field closures. In this paper we focus on the case where phase transitions do not occur in the stochastic system. In the averaging principle case a faster stochastic mechanism is assumed compared to the ODE relaxation and a local equilibrium is induced with respect to the Gibbs measure on the lattice system. Under these circumstances remarkable agreement is observed between the hybrid system and the averaged system predictions. We exhibit several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory. As expected fluctuation driven rare events do occur in several parameter regimes which could not possibly be captured by the deterministic averaging principle equation.

Non-existence criteria for Laurent polynomial first integrals

In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial rst integrals for a general nonlinear systems of ordinary dieren tial equations _ x = f(x); x 2 R n with f(0) = 0. We show that if the eigenvalues of the Jacobi matrix of the vector eld f(x) are Z-independent, then the system has no nontrivial Laurent polynomial

Method of the quasilinearization for nonlinear impulsive differential equations with linear boundary conditions

The method of quasilinearization for nonlinear impulsive dieren tial equations with linear boundary conditions is studied. The boundary conditions include periodic boundary conditions. It is proved that the convergence is quadratic. AMS ( MOS ) Subject Classic ations: 34A37, 34E05. In this paper a boundary value problem (BVP) for impulsive dieren tial equations with a family of linear two point boundary conditions is studied. An existence theo- rem is proved. An algorithm, based on methods of quasilinearization, for constructing successive approximations of the solution of the considered problem is given. The quadratic convergence of the iterates is proved. The obtained results are general- izations of the known results for initial value problems as well as boundary value problems for ordinary dieren tial equations and impulsive dieren tial equations. The method of quasilinearization has recently been studied and extended exten- sively. It is generating a rich history beginning with the works by Bellman and Kalaba (1). Lakshmikantham and Vatsala, and many co-authors have extensively developed the method and have applied the method to a wide range of problems. We refer the reader to the recent work by Lakshmikantham and Vatsala (9) and the extensive bibliography found there. The method has been applied to two-point boundary value problems for ordinary dieren tial equations and we refer the reader to the papers, (2, 3, 4, 8, 10, 11, 12), for example. Likewise impulsive equations have been generating a rich history. We refer the reader to the monograph by Lakshmikantham, Bainov, and Simeonov (6) for a thor- ough introduction to the material and an introduction to the literature. Methods of quasilinearization have been applied to impulsive dieren tial equations with various initial or boundary conditions. We refer the reader to (9) for references and we refer the reader to (2, 3, 13) in our bibliography. In this paper, we consider a family of

Evanescent solutions for linear ordinary differential equations

The problem of existence of the solutions for ordinary dieren tial equations vanishing at 1 is considered.

Persistence of attractors for one-step discretization of ordinary differential equations

We consider numerical one-step approximations of ordinary dierential equations and present two results on the persistence of attractors appearing in the numerical system. First, we show that the upper limit of a sequence of numerical attractors for a sequence of vanishing time step is an attractor for the approximated system if and only if for all these time steps the numerical one-step schemes admit attracting sets which approximate this upper limit set and attract with a uniform rate. Second, we show that if these numerical attractors themselves attract with a uniformly rate, then they converge to some set if and only if this set is an attractor for the approximated system. In this case, we can also give an estimate for the rate of convergence depending on the rate of attraction and on the order of the numerical scheme. AMS Classication: 65L20, 65L06, 34D45, 34E10

Spatial homogenization and internal layers in a reaction-diffusion system

For a system of reaction-diusion equations of activator-inhibitor type, we show that solutions undergo at least three stages of dynamical behaviour when the activator diuses slowly and reacts fast, and the inhibitor diuses fast. In the first stage, the inhibitor quickly decays to its spatial average (spatial homogenization of the inhibitor). In the second stage, the activator develops internal layers (formation of internal layers). In the third stage, the layers move according to a certain motion law (motion of interfaces) which is described by a system of ordinary dierential equations on finite time intervals. Asymptotic behaviour of the solutions of the interface equation is also analyzed. To describe the behaviour of the solutions of the reaction-diusion equations after the last interface equation becomes powerless, another type of interface equation is proposed.

PROPERTIES A AND B OF nTH ORDER LINEAR DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT

Sucient conditions for the nth order linear dierential equation u (n) (t) + p(t)u(u(t)) = 0; n i 2; to have Property A or Property B are established in both the de- layed and the advanced cases. These conditions essentially improve many known results not only for dierential equations with deviating arguments but for ordinary dierential equations as well.

Laura and Petrarch: An Intriguing Case of Cyclical Love Dynamics

Three ordinary dierential equations are proposed to model the dynamics of love between Petrarch, a celebrated Italian poet of the 14th century, and Laura, a beautiful but married lady. The equations are nonlinear but can be studied through the singular perturbation approach if the inspiration of the poet is assumed to have very slow dynamics. In such a case, explicit conditions are found in the appeals of Laura and Petrarch and in their behavioral parameters that guarantee the existence of a globally stable slow-fast limit cycle. These conditions are consistent with the relatively clear portrait of the two personalities one gets while reading the poems addressed to Laura. On the basis of the scarce and only qualitative information available, the calibration of the parameters is also performed; the result is that the calibrated model shows that the poet's emotions followed for about 20 years a quite regular cyclical pattern ranging from the extremes of ecstasy to despair. All these ndings agree with the recent results of Frederic Jones, who, through a detailed stylistic and linguistic analysis of the poems inspired by Laura, has discovered Petrarch's emotional cycle in a fully independent way.

Relativistic Two-Body Problem: Existence and Uniqueness of Two-Sided Solutions to Functional-Differential Equations of Motion

We study a class of explicitly Poincare-invariant equations of motion (EMs) of two point bodies with a finite speed of propagation of interactions (combination of retarded and advanced ones) that may be considered as functional-diere ntial equations or dierential equations with deviating argument of a neutral type. Under conditions having a clear physical interpretation it is proved that there exist ordinary dierential equations with all weakly-relativistic solutions satisfying the initial EMs. The existence and uniqueness of two-sided solutions of initial EMs on the infinite time interval are investigated.

The Method of an Exact Linearization of n-order Ordinary Differential Equations

Necessary and sucient conditions are found that the n-order nonlinear and nonautonomous ordinary dierential equation could be transformed into a linear equation with constant coecients with the help, generally speaking, nonlocal transformation of dependent and independent variables. These conditions are expressed in termes of factorization through first order nonlinear dierential operators. Examples are considered also.

The Cauchy-Nicoletti problem with poles

The Cauchy-Nicoletti boundary value problem for a sys- tem of ordinary dierential equations with pole-type singularities is investigated. The conditions of the existence, uniqueness, and non- uniqueness of a solution in the class of continuously dierentiable functions are given. The classical Banach contraction principle is combined with a special transformation of the original problem.

Group Analysis of Nonlinear Heat-Conduction Problem for a Semi-Infinite Body

The transformation group theoretic approach is applied to present an analysis of the nonlinear unsteady heat conduction problem in a semi‐infinite body. The application of one‐parameter group reduces the number of independent variables by one, and consequently the governing partial dierential equation with the boundary and initial conditions to an ordinary dierential equation with the appropriate corresponding boundary conditions. The ordinary dierential equation is solved analytically for some special forms of the thermal parameters. The general analysis developed in this study corresponds to thermal parameters that has dierent forms with coordinates and time.

On Lie Reduction of the Navier-Stokes Equations

Lie reduction of the Navier-Stokes equations to systems of partial dierential equations in three and two independent variables and to ordinary dierential equations is described.

On non-Lie ansatzes and new exact solutions of the classical Yang-Mills equations

We ssuggest an eective method for reducing Yang-Mills equations to systems of ordinary dierential equations. With the use of this method, we construct wide families of new exact solutions of the Yang-Mills equations. Analysis of the solutions obtained shows that they correspond to conditional symmetry of the equations under study.

On the stability of solutions of linear boundary value problems for a system of ordinary differential equations

Linear boundary value problems for a system of ordinary dierential equations are considered. The stability of the solution with respect to small perturbations of coecients and boundary val- ues is investigated.