Averaging for ordinary differential equations and functional differential equations

2007 ◽  
pp. 286-305 ◽  
Author(s):  
Tewfik Sari
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Functional Differential Equations ◽  
Functional Differential

scholarly journals Approximation methods and the generalised Fuller index for semi-flows in Banach spaces

1986 ◽  
Vol 29 (3) ◽  
pp. 299-308 ◽  
Author(s):  
A. J. B. Potter
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Banach Spaces ◽  
Functional Differential Equations ◽  
Periodic Points ◽  
Approximation Methods ◽  
Functional Differential ◽  
Linear Differential ◽  
Fuller Index

In [3] Fuller introduced an index (now called the Fuller index) in order to study periodic solutions of ordinary differential equations. The objective of this paper is to give a simple generalisation of the Fuller index which can be used to study periodic points of flows in Banach spaces. We do not claim any significant breakthrough but merely suggest that the simplistic approach, presented here, might prove useful for the study of non-linear differential equations. We show our results can be used to study functional differential equations.

scholarly journals Oscillation for Certain Nonlinear Neutral Partial Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Quanwen Lin ◽  
Rongkun Zhuang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Smooth Boundary ◽  
Functional Differential Equations ◽  
Second Order ◽  
Piecewise Smooth Boundary ◽  
Linear Ordinary Differential Equations ◽  
Partial Functional Differential Equations ◽  
Functional Differential

We present some new oscillation criteria for second-order neutral partial functional differential equations of the form(∂/∂t){p(t)(∂/∂t)[u(x,t)+∑i=1lλi(t)u(x,t-τi)]}=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(x,t)f(u(x,t))-∑j=1mqj(x,t)fj(u(x,t-σj)),(x,t)∈Ω×R+≡G, whereΩis a bounded domain in the EuclideanN-spaceRNwith a piecewise smooth boundary∂ΩandΔis the Laplacian inRN. Our results improve some known results and show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.

scholarly journals Ultimate Boundedness of the Systems Governed by Stochastic Differential Equations

1972 ◽  
Vol 47 ◽  
pp. 111-144 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Functional Differential ◽  
The Stability

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

scholarly journals Distributional and entire solutions of ordinary differential and functional differential equations

1983 ◽  
Vol 6 (2) ◽  
pp. 243-269 ◽  
Author(s):  
S. M. Shah ◽  
Joseph Wiener
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Linear Equations ◽  
Functional Differential Equations ◽  
Generalized Function ◽  
Entire Solutions ◽  
Polynomial Coefficients ◽  
Functional Differential

A brief survey of recent results on distributional and entire solutions of ordinary differential equations (ODE) and functional differential equations (FDE) is given. Emphasis is made on linear equations with polynomial coefficients. Some work on generalized-function solutions of integral equations is also mentioned.

RUNGE–KUTTA METHODS FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 15 (08) ◽  
pp. 1203-1251 ◽  
Author(s):  
STEFANO MASET ◽  
LUCIO TORELLI ◽  
ROSSANA VERMIGLIO
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Polynomial Function ◽  
Functional Differential Equations ◽  
Order Conditions ◽  
Polynomial Functions ◽  
Retarded Functional Differential Equations ◽  
Runge Kutta ◽  
Functional Differential ◽  
Order Four

We introduce Runge–Kutta (RK) methods for Retarded Functional Differential Equations (RFDEs). With respect to RK methods (A, b, c) for Ordinary Differential Equations the weights vector b ∈ ℝs and the coefficients matrix A ∈ ℝs×s are replaced by ℝs-valued and ℝs×s-valued polynomial functions b(·) and A(·) respectively. Such methods for RFDEs are different from Continuous RK (CRK) methods where only the weights vector is replaced by a polynomial function. We develop order conditions and construct explicit methods up to the convergence order four.

Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points

1995 ◽  
Vol 125 (4) ◽  
pp. 759-776 ◽  
Author(s):  
Teresa Faria ◽  
Luis T. Magalhães
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Equilibrium Points ◽  
Functional Differential Equations ◽  
Finite Dimensional ◽  
Retarded Functional Differential Equations ◽  
Functional Differential ◽  
Necessary And Sufficient

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

Equivalence of the MTS Method and CMR Method for Differential Equations Associated with Semisimple Singularity

2014 ◽  
Vol 24 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Pei Yu ◽  
Yuting Ding ◽  
Weihua Jiang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Differential Equations ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Delay Differential

In this paper, the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method is proved for computing the normal forms of ordinary differential equations and delay differential equations. The delay equations considered include general delay differential equations (DDE), neutral functional differential equations (NFDE) (or neutral delay differential equations (NDDE)), and partial functional differential equations (PFDE). The delays involved in these equations can be discrete or distributed. Particular attention is focused on dynamics associated with the semisimple singularity, and both the MTS and CMR methods are applied to compute the normal forms near the semisimple singular point. For the ordinary differential equations (ODE), we show that the two methods are equivalent up to any order in computing the normal forms; while for the differential equations with delays, we obtain the conditions under which the normal forms, derived by using the MTS and CMR methods, are identical up to third order. Different types of practical examples with delays are presented to demonstrate the application of the theoretical results, associated with Hopf, Hopf-zero and double-Hopf singularities.

Linear Functional-Differential Equations in a Banach Algebra*

1978 ◽  
Vol 21 (4) ◽  
pp. 435-439 ◽  
Author(s):  
W. J. Fitzpatrick ◽  
L. J. Grimm
Keyword(s):  
Functional Analysis ◽  
Differential Equations ◽  
Banach Algebra ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Linear Functional ◽  
Banach Algebras ◽  
Functional Differential Equations ◽  
Differential Systems ◽  
Functional Differential

The theory of analytic differential systems in Banach algebras has been investigated by E. Hille and others, see for instance Chapter 6 in [4].In this paper we show how a projection method used by W. A. Harris, Jr., Y. Sibuya, and L. Weinberg [3] can be applied to study a class of functional differential equations in this setting. The method, based on functional analysis, had been used extensively by L. Cesari [1] in similar forms for boundary value problems, and by J. K. Hale, S. Bancroft, and D. Sweet [2]. We also obtain as corollaries several results for ordinary differential equations in Banach algebras which were proved in a different way by Hille.

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