scholarly journals Identity-Type Functions and Polynomials

2007 ◽  
Vol 2007 ◽  
pp. 1-10
Author(s):  
M. Aslam Chaudhry ◽  
Asghar Qadir
Keyword(s):  
Differential Equation ◽  
Functional Equation ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Closed Form ◽  
Order Linear Differential Equation ◽  
Specific Function ◽  
Type Function ◽  
Form Representation ◽  
Linear Differential

For a noncommuting product of functions, similar to convolutions, an “identity-type function” leaving a specific function invariant is defined. It is evaluated for any choice of function on which it acts by solving a functional equation. A closed-form representation for the identity-type function of(1+t)−b(b>0)is obtained, which is a solution of a second-order linear differential equation with given boundary conditions. It yields orthogonal polynomials whose graphs are also given. The relevance for solution of boundary value problems by a series and convergence of the series are briefly discussed.

Lower bounds for the spectrum of a second order linear differential equation with a coefficient whose negative part is p-integrable

1984 ◽  
Vol 97 ◽  
pp. 105-107
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Differential Expression ◽  
Lower Bounds ◽  
Linear Operators ◽  
Order Linear Differential Equation ◽  
Negative Part ◽  
Order Linear ◽  
Linear Differential

SynopsisWe consider ihe differential expression M[y]: = −y″ + qy on [0, ∞) where q_∈ Lp [0, ∞) for some p ≧ 1. It is known that M, together with the boundary conditions y(0) = 0 or y′(0) = 0, defines linear operators on L2 [0, ∞). We obtain lower bounds for the spectra of these operators. Our bounds depend on the Lp norm of q_ and extend results of Everitt and Veling.

Optimal lower bounds for the spectrum of a second order linear differential equation with ap-integrable coefficient

1982 ◽  
Vol 92 (1-2) ◽  
pp. 95-101 ◽  
Author(s):  
E. J. M. Veling
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Differential Expression ◽  
Lower Bounds ◽  
The Self ◽  
Order Linear Differential Equation ◽  
Linear Differential ◽  
Adjoint Operators

SynopsisIn this note the differential expressionM[y] ≡ − y” + qy, q∈Lp(ℝ+) for some p ≧ l, is considered on [0,∞) together with the boundary condition either y(0) = 0 or y'(0) = 0. Lower bounds are given for the spectrum of the self-adjoint operatorsTgenerated by M[·] and these boundary conditions. The bounds depend on theLp-norm of the coefficientqand they improve results of Everitt and Eastham. The bounds are optimal.

scholarly journals The Distance between Points of a Solution of a Second Order Linear Differential Equation Satisfying General Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Linear Differential Equation ◽  
Minimum Distance ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Lower And Upper Bounds ◽  
Order Linear ◽  
Separated Boundary Conditions ◽  
Linear Differential

This paper presents a method to obtain lower and upper bounds for the minimum distance between pointsaandbof the solution of the second order linear differential equationy′′+q(x)y=0satisfying general separated boundary conditions of the typea11y(a)+a12y′(a)=0anda21y(b)+a22y′(b)=0. The method is based on the recursive application of a linear operator to certain functions, a recursive application that makes these bounds converge to the exact distance betweenaandbas the recursivity index grows. The method covers conjugacy and disfocality as particular cases.

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 A Comparison between the fourth order linear differential equation with its boundary value problem

2020 ◽  
Vol 99 (3) ◽  
pp. 18-25
Author(s):  
Karwan H.F. Jwamer ◽  
◽  
Rando R.Q. Rasul ◽  
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Fourth Order ◽  
Upper Bound ◽  
Boundary Value ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential

In this paper, we study a fourth order linear differential equation. We found an upper bound for the solutions of this differential equation and also, we prove that all the solutions are in L4(0, ∞). By comparing these results we obtain that all the eigenfunction of the boundary value problem generated by this differential equation are bounded and in L4(0, ∞).

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