Solutions of a non-linear differential equation. I

1967 ◽  
Vol 63 (3) ◽  
pp. 743-754 ◽  
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

We consider in this paper solutions of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

scholarly journals Existence of limit cycles for a class of autonomous systems

1983 ◽  
Vol 28 (3) ◽  
pp. 331-337
Author(s):  
Anthony Sofo
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Autonomous Systems ◽  
Non Linear ◽  
Stable Periodic ◽  
Linear Differential ◽  
Image Position

A proof is given for the existence of at least one stable periodic limit cycle solution for the polynomial non-linear differential equation of the formin some cases where the Levinson-Smith criteria are not directly applicable.

A Generalization of Picard's Method of Successive Approximation

1936 ◽  
Vol 32 (1) ◽  
pp. 86-95 ◽  
Author(s):  
H. O. Hirschfeld
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Successive Approximation ◽  
Linear Integral Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Linear Integral ◽  
Image Position

It is well known that the boundary value problem for the non-linear differential equationcan be reduced with help of a Green's function K (x, ξ) to a non-linear integral equation of the type

The existence of infinitely many solutions all bifurcating from λ = 0

1991 ◽  
Vol 118 (3-4) ◽  
pp. 295-303 ◽  
Author(s):  
Wolfgang Rother
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Critical Point ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe consider the non-linear differential equationand state conditions for the function q such that (*) has infinitely many distinct pairs of (weak) solutions such that holds for all k ∈ ℕ. The main tools are results from critical point theory developed by A. Ambrosetti and P. H. Rabinowitz [1].

Solutions of a non-linear differential equation. II

1968 ◽  
Vol 64 (1) ◽  
pp. 127-139
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position ◽  
Solution Behaviour

We consider in this paper the solution behaviour as s → 0 of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

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.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

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.

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 Associated Mathieu Functions

1922 ◽  
Vol 41 ◽  
pp. 94-99 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Periodic Solutions ◽  
Mathieu Functions ◽  
Linear Differential ◽  
Image Position

The periodic solutions of the linear differential equation,which reduce to Mathieu functions when v = 0 or 1, will be known as the associated Mathieu functions. The significance of this terminology will appear in the following section.

Linear differential equations with constant coefficients

2019 ◽  
Vol 103 (557) ◽  
pp. 257-264
Author(s):  
Bethany Fralick ◽  
Reginald Koo
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Auxiliary Equation ◽  
Real Solutions ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Complex Solutions ◽  
Image Position ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

We consider the second order homogeneous linear differential equation (H) $${ ay'' + by' + cy = 0 }$$ with real coefficients a, b, c, and a ≠ 0. The function y = emx is a solution if, and only if, m satisfies the auxiliary equation am2 + bm + c = 0. When the roots of this are the complex conjugates m = p ± iq, then y = e(p ± iq)x are complex solutions of (H). Nevertheless, real solutions are given by y = c1epx cos qx + c2epx sin qx.

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