Showing papers on "Integrating factor published in 1973"

Ordinary Differential Equations

Abstract: Basic Concepts.- Basic Theorems.- Linear Systems.- Proofs of the Main Theorems.- Differential Equations on Manifolds.

Differential Equations and Optimal Choice Problems

Abstract: Asymptotic forms for the optimal payoff and optimal stop rule for a generalized class of "secretary" problems are obtained by the analysis of a related family of ordinary differential equations.

Ordinary Differential Equations of Order > 1

Abstract: Many differential equations arising in engineering and the physical sciences, particularly in the study of vibratory or oscillatory phenomena, have the form (1) in which αn, αn-1,…, αl, αo and f are given functions of x. Such an equation is known as a linear ordinary differential equation of order n. In this chapter it will sometimes be convenient to write a linear equation in the abbreviated form (2) where L denotes a linear differential operator: The linearity property of L is expressed by the easily confirmed result where c1 and c2 are any constants and y and z any functions of x possessing derivatives up to the nth order.

Functional analysis and nonlinear differential equations

Abstract: where AT is a nonlinear operator in a real Hubert space S, and £ is a linear differential operator in S with preassigned linear homogeneous boundary conditions. The idea is to reduce the problem to a finite dimensional setting and this technique has been used by several authors. We use here a method due to Cesari [4]. This method has been extensively developed in the existence analysis of differential equations by Cesari, Hale, Locker, Mawhin and others. For a detailed bibliography one is referred to Cesari [5]. In this paper, by applying results from the theory of monotone operators, we show that, under suitable monotonicity hypotheses on N, the equation Ex = Nx can be solved. In the present short presentation we restrict ourselves to the simplest hypotheses on E9 N and S, even though the results obtained here hold under more general conditions.

On P-Adic Differential Equations. III. On P-Adically Bounded Solutions of Ordinary Linear Differential Equations with Rational Function Coefficients.

Abstract: For clarity of exposition, the underlying motivation is indicated at the end of this article under concluding Remarks 3, rather than at the beginning.

Observer theory for delayed differential equations

Abstract: A minimal order observer for a deterministic differential delay equation is proposed. A desirable feature of thi9 observer is both its stability property and its analytical complexity compare favourably with the minimal order observer for ordinary differential equations

On the Asymptotic Behavior of the Solutions of Some Third and Fourth Order Non- Autonomous Differential Equations

Abstract: where functions appeared in the equations are real valued. The dots indicate differentiation with respect to t and all solutions considered are assumed to be real. The problem is to give conditions to ensure that all solutions of (1.1), (1.2), (1.3) and (1.4) tend to zero as t— >oo. This problem has received a considerable amount of attention during the past twenty years, particulary when equations are autonomous. Many of these results are summarized in

Singularities of the cosmological model of the Bianchi IX type according to the qualitative theory of differential equations

Abstract: An exact definition and calculation of all singular points (in the sense of qualitative theory of differential equations) are presented for the Einstein equations in a homogeneous cosmological model of the Bianchi IX type, as well as their separatrices. This makes possible an exact statement and solution of the problem regarding the initial states of the Universe at early stages of evolution which are “typical” for the sign of time corresponding to expansion (in contrast to contraction, for which the typical states have been found by Belinskii, Lifshitz, and Khalatnikov and the analytically complex structure of the cosmological singularity has been elucidated). The initial typical states for Universe expansion indicated in the paper correspond asymptotically to power-law solutions with three types of time-factor asymptotics: that of the Friedman quasiisotropic type, that of the Taub type and a previously unknown type.

Fundamental theory of contingent differential equations in Banach space

Abstract: For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider the special case of a pointvalued or ordinary differential equation. 0. Introduction. Consider the contingent differential equation

Smooth dependence of solutions of differential equations on initial data: A simple proof

Abstract: A simple proof of smooth dependence of solutions of ordinary differential equations, with respect to initial conditions, is given.

Analytic wave front sets for solutions of linear differential equations of principal type

Abstract: The propagation of analyticity for solutions u of P(x, D)u = f is studied, in terms of wave front sets, for a large class of differential operators P = P(x, D) of principal type. In view of a theorem by L. Hó'rmander [9J, the results obtained imply rather precise results about the surjectivity of the mapping P: c°°(0) — c°°(n). Introduction. Let P = P(x, D) be a linear differential operator with C°°-coefficients in an open set ß C R\". When P is elliptic, then the classical regularity theorem for elliptic equations says that the distribution u is infinitely differentiable whenever Pu is and, if the coefficients of P are analytic, then u is analytic where Pu is analytic. The corresponding question, for more general operators, of describing the set of singularities of u when that of Pu is given has been much studied lately (see [10] and the references there). The introduction of the concept of wave front sets (see [7] and [14]) has added precision to the statements and has also simplified many proofs. For operators with real principal part P (x, D), such that the Hamilton field Hp = Z [(dPm(x,d/d{)d/dx. {dPm(x,&/dx.)d/d£.] m \\&jzn is nondegenerate when £ £ R\" = P\"\\{0j, and for some operators with complex coefficients, very complete results for the C -case are obtained by Duistermaat and Hó'rmander in [5] (see also [17]). Corresponding results, concerning analyticity, are proved in [l] (operators with constant coefficients), [9] and [11], under the assumption that P is real and d¿ P (x, £) = (dP (x, ¿¡)/d£., • • • ,