Variation formulas of solution for a delay differential equation taking into account delay perturbation and the continuous initial condition

2011 ◽  
Vol 18 (2) ◽  
pp. 345-364
Author(s):  
Tamaz Tadumadze
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Delay Differential Equation ◽  
Initial Function ◽  
Initial Moment ◽  
Initial Condition ◽  
Constant Delay ◽  
Non Linear ◽  
Linear Differential ◽  
Delay Differential

Abstract Variation formulas of solution are proved for a non-linear differential equation with constant delay. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

scholarly journals Oscillation of the even - order delay linear differential equation

2015 ◽  
Vol 31 (1) ◽  
pp. 69-76
Author(s):  
J. DZURINA ◽  
◽  
B. BACULIKOVA ◽  
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Necessary And Sufficient Conditions ◽  
Even Order ◽  
Linear Differential ◽  
Delay Differential ◽  
Necessary And Sufficient

In the paper we offer criteria for oscillation of the even order delay differential equation y(n)(t) + p(t)y(ct) = 0 We provide detail analysis of the properties of this equation, we offer necessary and sufficient conditions for oscillation of studied equation and we fulfill the gap in the oscillation theory.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramterνoccurs, but theν= 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factorε(ν) is required, so that the aspect ratioεof a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion inεgives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

scholarly journals A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1332 ◽  
Author(s):  
Ábel Garab
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Continuous Functions ◽  
Oscillation Criterion ◽  
Variable Delay ◽  
All Solutions ◽  
Linear Differential ◽  
Delay Differential ◽  
Additive Form ◽  
Condition B

We consider linear differential equations with variable delay of the form x ′ ( t ) + p ( t ) x ( t − τ ( t ) ) = 0 , t ≥ t 0 , where p : [ t 0 , ∞ ) → [ 0 , ∞ ) and τ : [ t 0 , ∞ ) → ( 0 , ∞ ) are continuous functions, such that t − τ ( t ) → ∞ (as t → ∞ ). It is well-known that, for the oscillation of all solutions, it is necessary that B : = lim sup t → ∞ A ( t ) ≥ 1 e holds , where A : = ( t ) ∫ t − τ ( t ) t p ( s ) d s . Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and τ , condition B > 1 / e implies that all solutions of the above delay differential equation are oscillatory.

CORRELATION FUNCTIONS OF DISORDER OPERATORS IN MASSIVE FREE THEORIES

2004 ◽  
Vol 19 (supp02) ◽  
pp. 126-133
Author(s):  
G. DELFINO ◽  
P. GRINZA ◽  
P. MOSCONI ◽  
G. MUSSARDO
Keyword(s):  
Differential Equation ◽  
Form Factor ◽  
Linear Differential Equation ◽  
Correlation Functions ◽  
Functional Equations ◽  
Matrix Elements ◽  
Fermion Fields ◽  
Non Linear ◽  
Point Correlation ◽  
Linear Differential

A unified analysis of the disorder operators for ghosts, complex boson and fermion fields is presented. Matrix elements on the asymptotic states of these operators can be exactly computed by solving the Form Factor functional equations. The two–point correlation functions of the disorder operators depend only on the statistics and can be expressed in terms of a solution of a non–linear differential equation of Painleve' type.

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