On perturbations of a translationally-invariant differential equation

1988 ◽  
Vol 110 (1-2) ◽  
pp. 1-25 ◽  
Author(s):  
R.J. Magnus
Keyword(s):  
Differential Equation ◽  
Perturbation Parameter ◽  
Dimensional Euclidean Space ◽  
Existence Of A Solution ◽  
Smooth Perturbation ◽  
Invariant Differential ◽  
Smooth Dependence ◽  
Space Conditions ◽  
Perturbed Equation ◽  
Unperturbed Solution

SynopsisWe study certain perturbations of the differential equation Δu − u + up = 0 on all of n-dimensional Euclidean space. Conditions are obtained which ensure the existence of a solution to the perturbed equation near a given solution to the unperturbed equation. We have to overcome degeneracy of the unperturbed solution and lack of smooth dependence on the perturbation parameter. An abstract version of the argument is sketched in a functional-analytic setting related toequivariant bifurcation theory. We consider also a smooth perturbation with several parameters and study the singularities of the mapping which maps each solution to its associated parameters.

scholarly journals On partial fractional Sturm–Liouville equation and inclusion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zohreh Zeinalabedini Charandabi ◽  
Hakimeh Mohammadi ◽  
Shahram Rezapour ◽  
Hashem Parvaneh Masiha
Keyword(s):  
Differential Equation ◽  
Liouville Equation ◽  
Existence Of Solutions ◽  
Liouville Problem ◽  
Existence Of A Solution ◽  
Contractive Mappings ◽  
Sturm Liouville

AbstractThe Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence of a solution for partial fractional Sturm–Liouville equation by using the α-ψ-contractive mappings. Also, we give an illustrative example. By using the α-ψ-multifunctions, we prove the existence of solutions for inclusion version of the partial fractional Sturm–Liouville problem. Finally by providing another example and some figures, we try to illustrate the related inclusion result.

scholarly journals Asymptotic Comparison of the Solutions of Linear Time-Delay Systems with Point and Distributed Lags with Those of Their Limiting Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-37 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Linear Time ◽  
Type Theorem ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Functional Differential ◽  
Autonomous Differential Equations ◽  
Perturbed Equation

This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.

ODE observers for DAE systems

2018 ◽  
Vol 36 (4) ◽  
pp. 1375-1393 ◽  
Author(s):  
Thomas Berger ◽  
Timo Reis
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Linear Time ◽  
Observer Design ◽  
Simple Criterion ◽  
Linear Time Invariant ◽  
Ode System ◽  
Dae Systems ◽  
Invariant Differential ◽  
Time Invariant

Abstract We consider linear time-invariant differential-algebraic systems which are not necessarily regular. The following question is addressed: when does an (asymptotic) observer which is realized by an ordinary differential equation (ODE) system exist? In our main result we characterize the existence of such observers by means of a simple criterion on the system matrices. To be specific, we show that an ODE observer exists if, and only if, the completely controllable part of the system is impulse observable. Extending the observer design from earlier works we provide a procedure for the construction of (asymptotic) ODE observers.

scholarly journals Solution Of Singularly Perturbed Delay Differential Equations Using Liouville Green Transformation

2021 ◽  
Vol 12 (4) ◽  
pp. 325-335
Author(s):  
M. Adilaxmi , Et. al.
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Perturbation Problem ◽  
Delay Differential ◽  
Singularly Perturbation ◽  
The Given

This paper envisages the use of Liouville Green Transformation to find the solution of singularly perturbed delay differential equations. First, using Taylor series, the given singularly perturbed delay differential equation is approximated by an asymptotically equivalent singularly perturbation problem. Then the Liouville Green Transformation is applied to get the solution. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and perturbation parameter.

QUANTUM STOCHASTIC ANALYSIS VIA WHITE NOISE OPERATORS IN WEIGHTED FOCK SPACE

2002 ◽  
Vol 14 (03) ◽  
pp. 241-272 ◽  
Author(s):  
DONG MYUNG CHUNG ◽  
UN CIG JI ◽  
NOBUAKI OBATA
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
White Noise ◽  
Fock Space ◽  
Existence Of A Solution ◽  
Quantum Stochastic Differential Equation ◽  
Unique Existence ◽  
Regularity Properties ◽  
Weighted Fock Space ◽  
White Noises

White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

scholarly journals Reducibility for a Class of Almost-Periodic Differential Equations with Degenerate Equilibrium Point under Small Almost-Periodic Perturbations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Wenhua Qiu ◽  
Jianguo Si
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Equilibrium Point ◽  
Almost Periodic Solution ◽  
Time Dependent ◽  
Almost Periodic ◽  
Degenerate Equilibrium ◽  
Periodic Transformation ◽  
Perturbed Equation ◽  
Kam Method

This paper focuses on almost-periodic time-dependent perturbations of an almost-periodic differential equation near the degenerate equilibrium point. Using the KAM method, the perturbed equation can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation. Hence, for the equation we can obtain a small almost-periodic solution.

scholarly journals Some results of differentiability for the solution of an integral equations system

2014 ◽  
Vol 30 (2) ◽  
pp. 147-159
Author(s):  
MARIA DOBRITOIU ◽  
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Initial Data ◽  
Integral Equations ◽  
Generalized Contraction ◽  
System Of Integral Equations ◽  
Contraction Theorem ◽  
The Fixed Point Theorem ◽  
Smooth Dependence

Using the fixed point theorem given by [Rus, I. A., A Fiber generalized contraction theorem and applications, Mathematica, 41(64) (1999), No. 1, 85–90] and an idea of [Sotomayor, J., Smooth dependence of solution of differential equation on initial data: a simple proof, Bol. Soc. Brasil., 4 (1973), No. 1, 55–59] we establish some conditions of differentiability of the solution for the following system of integral equations: ... and such we obtain two theorems of differentiability. Finally, two examples are given.

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