The form of the spectral functions associated with a class of Sturm–Liouville equations with integrable coefficient

1987 ◽  
Vol 105 (1) ◽  
pp. 215-227 ◽  
Author(s):  
B.J. Harris
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Spectral Function ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Spectral Functions ◽  
Order Linear ◽  
Sturm Liouville ◽  
Linear Differential ◽  
Image Position

SynopsisWe consider the second order, linear differential equationin the case where, roughly, q ∊ L1 [0, ∞). We devise a representation for the spectral function, τ(t), associated with (*) which is valid for t sufficiently large. Our results are the best possible.

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

Stability and Response of Systems Satisfying a Second-Order Linear Differential Equation with Time-Dependent Coefficients

1957 ◽  
Vol 8 (1) ◽  
pp. 78-86
Author(s):  
A. W. Babister
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Linear Differential Equation ◽  
Second Order ◽  
Time Dependent ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Oscillatory Nature ◽  
Linear Differential ◽  
Image Position

SummaryThe differential equation considered iswhere all the a’s and b’s are real constants.The nature of the solution is investigated in the neighbourhood of the singular point and the conditions are found for logarithmic terms to be absent.The conditions for stability for large values of τ are determined; the system is stable ifare all positive for large values of τ.The form of the response is considered and its oscillatory (or non-oscillatory) nature investigated. The Sonin-Polya theorem is used to determine simple inequalities which must hold between the coefficients of the differential equation in any interval for the relative maxima of | x | to form an increasing or decreasing sequence in that interval.

The asymptotic form of the Titchmarsh–Weyl m-function associated with a second order differential equation with locally integrable coefficient

1986 ◽  
Vol 102 (3-4) ◽  
pp. 243-251 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equationin the case where the only restriction on the real-valued function q is

An exact method for the calculation of certain Titchmarsh-Weyl m-functions

1987 ◽  
Vol 106 (1-2) ◽  
pp. 137-142 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Second Order ◽  
Exact Method ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe consider the second order, linear, differential equationwhere q is real-valued. In the case we calculate the Titchmarsh-Weyl m-function associated with (*).

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