Evolution of near-inertial waves

1995 ◽  
Vol 301 ◽  
pp. 269-294 ◽  
Author(s):  
R. C. Kloosterziel ◽  
P. Müller
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Fluid Layer ◽  
Normal Modes ◽  
Buoyancy Frequency ◽  
Infinite Domain ◽  
Initial Value ◽  
Flux Boundary Condition

The three-dimensional evolution of near-inertial internal gravity waves is investigated for the case of a laterally unbounded fluid layer of constant finite depth. A general Green's function formulation is derived which can be used to solve initial value problems or study the effect of forcing. The Green's function is expanded in vertical normal modes, and is very singular. Convolutions with finite-sized initial conditions lead however to well-behaved solutions. Expansions in similarity solutions of the diffusion equation are shown to be an alternative for finding exact solutions to initial value problems, with respect to one normal mode. For the case of constant buoyancy frequency normal modes expansions are shown to be equivalent to expansions in an alternative series of which the first term is the response on the infinite domain, all the others being corrections to account for the no-flux boundary condition on the upper and lower boundaries.

scholarly journals A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra’s Population Growth Model

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Majid Tavassoli Kajani ◽  
Mohammad Maleki ◽  
Adem Kılıçman
Keyword(s):  
Population Growth ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Initial Value ◽  
Step Size ◽  
Integral Term ◽  
Multiple Step

A new shifted Legendre-Gauss collocation method is proposed for the solution of Volterra’s model for population growth of a species in a closed system. Volterra’s model is a nonlinear integrodifferential equation on a semi-infinite domain, where the integral term represents the effects of toxin. In this method, by choosing a step size, the original problem is replaced with a sequence of initial value problems in subintervals. The obtained initial value problems are then step by step reduced to systems of algebraic equations using collocation. The initial conditions for each step are obtained from the approximated solution at its previous step. It is shown that the accuracy can be improved by either increasing the collocation points or decreasing the step size. The method seems easy to implement and computationally attractive. Numerical findings demonstrate the applicability and high accuracy of the proposed method.

scholarly journals Finding symmetries by incorporating initial conditions as side conditions

2008 ◽  
Vol 19 (6) ◽  
pp. 701-715 ◽  
Author(s):  
JOANNA GOARD
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Initial Condition ◽  
Initial Value ◽  
Side Condition ◽  
Symmetry Generator ◽  
Infinitesimal Symmetry

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

Etude numérique des modes et fréquences propres d'une cavité à l'aide de la fonction de Green

1979 ◽  
Vol 57 (2) ◽  
pp. 208-217
Author(s):  
Jacques A. Imbeau ◽  
Byron T. Darling
Keyword(s):  
Point Source ◽  
Green’S Function ◽  
Green's Function ◽  
Normal Modes ◽  
Preceding Paper ◽  
Double Precision ◽  
Prolate Spheroidal ◽  
Normal Functions ◽  
Normal Frequency ◽  
High Degree

We apply the methods developed in our preceding paper (J. A. Imbeau and B. T. Darling. Can. J. Phys. 57, 190(1979)) for calculating the Green's function of a cavity to obtain the normal modes and normal frequencies of the cavity. As the frequency of the driving point source approaches that of a normal frequency the response (Green's function) of the cavity becomes infinite, and the form of the Green's function is dominated by the normal mode. There is also a 180° reversal of phase in passing through a resonance. In this way, we are able to calculate the normal frequencies of prolate spheroidal cavities to the full precision employed in the calculations (16 significant digits for double precision of the IBM-370). The Green's functions and the normal functions are also obtainable to a high degree of precision, except in the immediate vicinity of the surface of the cavity where they suffer a well-known discontinuity.

Perturbation dynamics in viscous channel flows

1997 ◽  
Vol 339 ◽  
pp. 55-75 ◽  
Author(s):  
W. O. CRIMINALE ◽  
T. L. JACKSON ◽  
D. G. LASSEIGNE ◽  
R. D. JOSLIN
Keyword(s):  
Initial Conditions ◽  
Normal Modes ◽  
Travelling Wave ◽  
Channel Flows ◽  
Transient Growth ◽  
Growth Data ◽  
Initial Value ◽  
Simulation Code ◽  
Temporal Behaviour ◽  
Fourier Decomposition

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where travelling wave normal modes are assumed as solutions, this work offers a means whereby arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behaviour, including both early-time transients and the long-time asymptotics, can be determined for any initial small-amplitude three-dimensional disturbance. The bases for the theoretical analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability criteria that are known for such flows can be reproduced. Also, optimal initial conditions measured in terms of the normalized energy growth can be determined in a straightforward manner and such optimal conditions clearly reflect transient growth data that are easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number, it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes when used without regard to a particular initial-value problem. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists (e.g. the Blasius boundary layer). Finally, results from the temporal theory are compared with the equivalent transient test case in the spatially evolving problem with the spatial results having been obtained using both a temporally and spatially accurate direct numerical simulation code.

scholarly journals A Green’s Function Approach for Dynamic Stress Analysis of Spherical Shell under the Isotropic Impact Load

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Jincheng Lv ◽  
Shike Zhang ◽  
Xinsheng Yuan
Keyword(s):  
Stress Distribution ◽  
Spherical Shell ◽  
Green’S Function ◽  
Green's Function ◽  
Dynamic Stress ◽  
Initial Conditions ◽  
Impact Load ◽  
Static Solution ◽  
Function Approach ◽  
Dynamic Stress Distribution

A Green’s function approach is developed for the analytic solution of thick-walled spherical shell under an isotropic impact load, which involves building Green’s function of this problem by using the appropriate boundary conditions of thick-walled spherical shell. This method can be used to analyze displacement distribution and dynamic stress distribution of the thick-walled spherical shell. The advantages of this method are able(1)to avoid the superposition process of quasi-static solution and free vibration solution during decomposition of dynamic general solution of dynamics,(2)to well adapt for various initial conditions, and(3)to conveniently analyze the dynamic stress distribution using numerical calculation. Finally, a special case is performed to verify that the proposed Green’s function method is able to accurately analyze the dynamic stress distribution of thick-walled spherical shell under an isotropic impact load.

GORKOV'S ANSATZ IN SUPERCONDUCTIVITY THEORY

1965 ◽  
Vol 43 (12) ◽  
pp. 2142-2149 ◽  
Author(s):  
A. J. Coleman ◽  
S. Pruski
Keyword(s):  
Density Matrix ◽  
Green’S Function ◽  
Green's Function ◽  
Order Parameter ◽  
Green’S Functions ◽  
Second Order ◽  
Bcs Theory ◽  
Initial Value ◽  
Ginzburg Landau ◽  
Superconductivity Theory

By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.

scholarly journals Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

2021 ◽  
Vol 24 (4) ◽  
pp. 32-39
Author(s):  
Hussein M. Sagban ◽  
◽  
Fadhel S. Fadhel ◽  
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

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