scholarly journals On the Zeroes and the Critical Points of a Solution of a Second Order Half-Linear Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Critical Point ◽  
Lower Bounds ◽  
Critical Points ◽  
Upper Bounds ◽  
Second Order ◽  
Lyapunov Inequalities ◽  
Piecewise Continuous ◽  
Linear Differential

This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second-order half-linear differential equation(p(x)Φ(y'))'+q(x)Φ(y)=0, withp(x)andq(x)piecewise continuous andp(x)>0,Φ(t)=|t|r-2tandrbeing real such thatr>1. It also compares between them in several examples. Lower bounds (i.e., Lyapunov inequalities) for such a distance are also provided and compared with other methods.

scholarly journals The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Critical Point ◽  
Critical Points ◽  
Upper Bounds ◽  
Second Order ◽  
Distribution Of Zeroes ◽  
Linear Differential

This paper reuses an idea first devised by Kwong to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second order half-linear differential equation(p(x)Φ(y'))'+q(x)Φ(y)=0, withp(x),q(x)>0,Φ(t)=|t|r-2t, andrreal such thatr>1. It also compares it with other methods developed by the authors.

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 Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

Sign in / Sign up

Export Citation Format

Share Document