Ordinary differential equations and systems with time-dependent discontinuity sets

2004 ◽  
Vol 134 (4) ◽  
pp. 617-637 ◽  
Author(s):  
J. Ángel Cid ◽  
Rodrigo L. Pouso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Existence Results ◽  
Time Dependent ◽  
Nonlinear Part ◽  
First Order ◽  
Discontinuity Sets

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

scholarly journals Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. P. Markakis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Closed Form Solution ◽  
First Integrals ◽  
Form Solution ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Abel Equations

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

2008 ◽  
Vol 465 (2102) ◽  
pp. 585-608 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Nonlinear Odes ◽  
First Order ◽  
Open Questions ◽  
Motion Time

Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle–Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as time-dependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka–Volterra system and three-dimensional Rössler system as well as other examples including non-autonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

SOLVING OPERATOR DIFFERENTIAL EQUATIONS IN TERMS OF THE WEYL-ORDERED POLYNOMIALS

2002 ◽  
Vol 17 (30) ◽  
pp. 2009-2017 ◽  
Author(s):  
ZENG-BING CHEN ◽  
HUAI-XIN LU ◽  
JUN LI
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Systematic Approach ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Adiabatic Invariants ◽  
Formal Aspect ◽  
First Order ◽  
Heisenberg Equations ◽  
Heisenberg Equations Of Motion

A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.

ORDINARY DIFFERENTIAL EQUATIONS, FIRST INTEGRALS, CONFORMAL INVARIANTS, BOSE OPERATORS, COHERENT STATES AND ENTANGLEMENT

2008 ◽  
Vol 05 (07) ◽  
pp. 1057-1063
Author(s):  
WILLI-HANS STEEB
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Coherent States ◽  
Autonomous Systems ◽  
First Integrals ◽  
Conformal Invariants ◽  
First Order ◽  
Bose Operators ◽  
Space Description

Autonomous systems of first order ordinary differential equations can be embedded into a Hilbert space description by using Bose operators and Glauber coherent states. The first integrals and conformal invariants of the autonomous system can be represented by states in a Hilbert space. Thus the embedding in a Hilbert space allows us to study entanglement of the states. We apply it here to first integrals and conformal invariants which are expressed as states.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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