On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation

1949 ◽  
Vol 1 (2) ◽  
pp. 191-198 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Arbitrary Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Partial Differential ◽  
Second Order Differential Equation ◽  
Image Position

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

Note on the Uniqueness of the Green'S Functions Associated With Certain Differential Equations

1950 ◽  
Vol 2 ◽  
pp. 314-325 ◽  
Author(s):  
D. B. Sears
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Order Differential Equation ◽  
Green’S Functions ◽  
Second Order ◽  
Second Order Differential Equation

Conditions to be imposed on q(x) which ensure the uniqueness of the Green's function associated with the linear second-order differential equation

On the Forced Lienard Equation

1969 ◽  
Vol 12 (1) ◽  
pp. 79-84 ◽  
Author(s):  
R.R. Stevens
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Liénard Equation ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Lienard Equation

We consider the second order differential equation(1)with the assumptions that(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.Also, throughout this paper, F(x) denotes an antiderivative of f(x).

On a Linear Partial Differential Equation of Hyperbolic Type

1922 ◽  
Vol 41 ◽  
pp. 76-81
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Order Partial Differential Equation ◽  
Sound Waves ◽  
Partial Differential ◽  
Method Of Solution ◽  
Image Position

Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equationwhere α, β, γ are functions of x and y.

scholarly journals Modelling and prediction of the peak-radiated sound in subsonic axisymmetric air jets using acoustic analogy-based asymptotic analysis

2019 ◽  
Vol 377 (2159) ◽  
pp. 20190073 ◽  
Author(s):  
Mohammed Z. Afsar ◽  
Adrian Sescu ◽  
Stewart J. Leib
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Asymptotic Analysis ◽  
Green’S Function ◽  
Green's Function ◽  
Exact Formula ◽  
Convolution Product ◽  
Acoustic Analogy ◽  
Mean Flow ◽  
Partial Differential

This paper uses asymptotic analysis within the generalized acoustic analogy formulation (Goldstein 2003 JFM 488 , 315–333. ( doi:10.1017/S0022112003004890 )) to develop a noise prediction model for the peak sound of axisymmetric round jets at subsonic acoustic Mach numbers (Ma). The analogy shows that the exact formula for the acoustic pressure is given by a convolution product of a propagator tensor (determined by the vector Green's function of the adjoint linearized Euler equations for a given jet mean flow) and a generalized source term representing the jet turbulence field. Using a low-frequency/small spread rate asymptotic expansion of the propagator, mean flow non-parallelism enters the lowest order Green's function solution via the streamwise component of the mean flow advection vector in a hyperbolic partial differential equation. We then address the predictive capability of the solution to this partial differential equation when used in the analogy through first-of-its-kind numerical calculations when an experimentally verified model of the turbulence source structure is used together with Reynolds-averaged Navier–Stokes solutions for the jet mean flow. Our noise predictions show a reasonable level of accuracy in the peak noise direction at Ma = 0.9, for Strouhal numbers up to about 0.6, and at Ma = 0.5 using modified source coefficients. Possible reasons for this are discussed. Moreover, the prediction range can be extended beyond unity Strouhal number by using an approximate composite asymptotic formula for the vector Green's function that reduces to the locally parallel flow limit at high frequencies. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.

scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

Periodic solutions for a second-order differential equation with indefinite weak singularity

2019 ◽  
Vol 149 (5) ◽  
pp. 1135-1152 ◽  
Author(s):  
José Godoy ◽  
Manuel Zamora
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Second Order ◽  
Constant Sign ◽  
Piecewise Constant ◽  
Weak Singularity ◽  
Second Order Differential Equation ◽  
Image Position

AbstractAs a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation $${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

On the stability of a second-order differential equation

1965 ◽  
Vol 5 (1) ◽  
pp. 8-16 ◽  
Author(s):  
D. E. Daykin ◽  
K. W. Chang
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
The Stability ◽  
Image Position ◽  
Familiar Form

In this note we discuss the stability at the origin of the solutions of the differential equation where a dot indicates a differentiation with respect to time, and α, β are real-valued functions of any arguments. We tacitly assume that α, β are such that solutions to (1) do in fact exist. Under the transformation equation (1) takes the equivalent familiar form .

The Existence of Continuable Solutions of a Second Order Differential Equation

1977 ◽  
Vol 29 (3) ◽  
pp. 472-479 ◽  
Author(s):  
G. J. Butler
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Real Line ◽  
Order Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
The Real ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Sign Restriction

A much-studied equation in recent years has been the second order nonlinear ordinary differential equationwhere q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

scholarly journals On the “elementary” solution of Laplace's Equation

1931 ◽  
Vol 2 (3) ◽  
pp. 135-139 ◽  
Author(s):  
H. S. Ruse
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Elementary Solution ◽  
General Linear ◽  
Laplace’S Equation ◽  
Partial Differential ◽  
Laplace's Equation ◽  
Image Position

Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.

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