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

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.

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The Green’s function method in the problem of sound diffraction by an elastic shell of noncanonical shape

Acoustical Physics ◽

10.1134/s1063771014060062 ◽

2014 ◽

Vol 60 (6) ◽

pp. 617-623 ◽

Cited By ~ 2

Author(s):

S. L. Ilmenkov ◽

A. A. Kleshchev ◽

A. S. Klimenkov

Keyword(s):

Green’S Function ◽

Green's Function ◽

Function Method ◽

Elastic Shell ◽

Green’S Function Method ◽

Sound Diffraction

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On the spectra of SchroÌˆdinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients

10.32469/10355/4136 ◽

2005 ◽

Author(s):

◽

Vladimir Batchenko

Keyword(s):

Green’S Function ◽

Green's Function ◽

Complex Plane ◽

Mean Value ◽

Qualitative Behavior ◽

Conditional Stability ◽

One Dimensional ◽

Jacobi Operators ◽

The Mean ◽

Complex Valued

In this thesis we characterize the spectrum of one-dimensional SchroÌˆdinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of SchroÌˆdinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the SchroÌˆdinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.

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

EPJ Web of Conferences ◽

10.1051/epjconf/201818203005 ◽

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.

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The Spectral Matrix and Green's Function for Singular Self-Adjoint Boundary Value Problems

Canadian Journal of Mathematics ◽

10.4153/cjm-1954-019-4 ◽

1954 ◽

Vol 6 ◽

pp. 169-185 ◽

Cited By ~ 8

Author(s):

E. A. Coddington

Keyword(s):

Differential Operator ◽

Green’S Function ◽

Boundary Value Problems ◽

Green's Function ◽

Boundary Value ◽

Ordinary Differential Operator ◽

Real Interval ◽

Spectral Matrix ◽

Image Position ◽

Complex Valued

Let L denote the formal ordinary differential operator,where we assume the pk are complex-valued functions with n-k continuous derivatives on an open real interval a < x < b (a = — ∞, b = + ∞, or both may occur), p0(x) ≠ 0 on a < x < b, and L coincides with its Lagrange adjoint L+ given by.

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Green's function for the lossy wave equation

Revista Brasileira de Ensino de Física ◽

10.1590/s1806-11172008000100003 ◽

2008 ◽

Vol 30 (1) ◽

pp. 1302.1-1302.5 ◽

Cited By ~ 1

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.

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