scholarly journals Cylindrical symmetry: II. The Green's function in 3+ 1 dimensional curved space

2018 ◽  
Vol 182 ◽  
pp. 03005
Author(s):  
Gopinath Kamath
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Lagrangian Density ◽  
Cylindrical Symmetry ◽  
Order Partial Differential Equation ◽  
Loop Order ◽  
Schwarzschild Solution ◽  
Partial Differential ◽  
Curved Space ◽  
Cylindrically Symmetric

An exact solution to the heat equation in curved space is a much sought after; this report presents a derivation wherein the cylindrical symmetry of the metric gμν in 3 + 1 dimensional curved space has a pivotal role. To elaborate, the spherically symmetric Schwarzschild solution is a staple of textbooks on general relativity; not so perhaps, the static but cylindrically symmetric ones, though they were obtained almost contemporaneously by H. Weyl, Ann. Phys. Lpz. 54, 117 (1917) and T. Levi-Civita, Atti Acc. Lincei Rend. 28, 101 (1919). A renewed interest in this subject in C.S. Trendafilova and S.A. Fulling, Eur.J.Phys. 32, 1663(2011) - to which the reader is referred to for more references - motivates this work, the first part of which (cf.Kamath, PoS (ICHEP2016) 791) reworked the Antonsen-Bormann idea - arXiv:hep-th/9608141v1 - that was originally intended to compute theheat kernel in curved space to determine - following D.McKeon and T.Sherry, Phys. Rev. D 35, 3584 (1987) - the zeta-function associated with the Lagrangian density for a massive real scalar field theory in 3 + 1 dimensional stationary curved space to one-loop order, the metric for which is cylindrically symmetric. Using the same Lagrangian density the second part reported here essentially revisits the second paper by Bormann and Antonsen - arXiv:hep 9608142v1 but relies on the formulation by the author in S. G. Kamath, AIP Conf.Proc.1246, 174 (2010) to obtain the Green's function directly by solving a sequence of first order partial differential equations that is preceded by a second order partial differential equation.

Green’s Function for a Closed, Infinite, Circular Cylindrical Elastic Shell

2005 ◽  
Vol 73 (2) ◽  
pp. 183-188
Author(s):  
J. G. Simmonds
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Circular Cylindrical Shell ◽  
Formal Solution ◽  
Elastic Shell ◽  
Constant Thickness ◽  
Order Partial Differential Equation ◽  
Relative Order ◽  
Outer Expansion ◽  
Complex Valued

An acceptable variant of the Koiter–Morley equations for an elastically isotropic circular cylindrical shell is replaced by a constant coefficient fourth-order partial differential equation for a complex-valued displacement-stress function. An approximate formal solution for the associated “free-space” Green’s function (i.e., the Green’s function for a closed, infinite shell) is derived using an inner and outer expansion. The point wise error in this solution is shown rigorously to be of relative order (h∕a)(1+h∕a∣x∣), where h is the constant thickness of the shell, a is the radius of the mid surface, and ax is distance along a generator of the mid surface.

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’.

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

scholarly journals Green's function for the lossy wave equation

2008 ◽  
Vol 30 (1) ◽  
pp. 1302.1-1302.5 ◽  
Author(s):  
R. Aleixo ◽  
E. Capelas de Oliveira
Keyword(s):  
Wave Equation ◽  
Integral Representation ◽  
Green’S Function ◽  
Bessel Function ◽  
Green's Function ◽  
Integral Transform ◽  
Dispersive Media ◽  
Order Partial Differential Equation ◽  
Fourier Cosine ◽  
Transient Electromagnetic

Using an integral representation for the first kind Hankel (Hankel-Bessel Integral Representation) function we obtain the so-called Basset formula, an integral representation for the second kind modified Bessel function. Using the Sonine-Bessel integral representation we obtain the Fourier cosine integral transform of the zero order Bessel function. As an application we present the calculation of the Green's function associated with a second-order partial differential equation, particularly a wave equation for a lossy two-dimensional medium. This application is associated with the transient electromagnetic field radiated by a pulsed source in the presence of dispersive media, which is of great importance in the theory of geophysical prospecting, lightning studies and development of pulsed antenna systems.

Lyapunov Stability of a Class of Distributed Parameter Systems

1971 ◽  
Vol 93 (2) ◽  
pp. 79-85
Author(s):  
A. Frank D’Souza
Keyword(s):  
Differential Operator ◽  
Green’S Function ◽  
Green's Function ◽  
Linear Differential Operator ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Partial Differential ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Function Matrix

The mathematical model of dynamical systems is represented as an initial-boundary value problem described by nonlinear vector-matrix valued partial differential equations. The linear partial differential operator associated with the nonlinear system is restricted to a time-invariant operator with domain dense in Hilbert space. New stability results reported in this paper show the existence of quadratic Lyapunov functions that yield both necessary and sufficient conditions for asymptotic stability of linear systems satisfying certain restrictions and the use of these forms for the stability investigation of a class of nonlinear systems. The proofs of the stability theorems employ the spectral representation of the Green’s function matrix of the associated linear differential operator. Therefore, in the earlier part of the paper well-known properties of linear operators are stated in order to express the Green’s function matrix in the form of spectral expansion.

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