The Exact Solution,Berry's Phase and Coherent State for the Driven Generalized Time-Dependent Harmonic Oscillator

The Lewis'invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for thirr system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adiabatic limit. The relation between the exact Lewis'phase and the corresponding classical phase angle L'discusrred.     

Direct analytical solutions to non-uniform beam problems

The direct analytical solution to the vibration of non-uniform beams with and without discontinuities and with various boundary conditions is presented. Results are compared to results from the exact solution for certain cases where the exact solution has been obtained. It is shown that the direct solution converges to the exact solution, in fact, with “indefinite accuracy” just as Hamilton stated that it would.     

广义含时谐振子的精确解和Berry相因数

本文利用Lewis-Riesenfeld的量子理论,求出广义含时谐振子的精确解。研究了此精确解的绝热渐近极限,并求出广义谐振子在量子绝热情形的Berry相因数。进而利用此精确解构造了广义含时谐振子的相干态,并得到相应的经典Hannay角。     

An Exact Solution to the Two-Particle Boltzmann Equation System for Maxwell Gases

An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

厄尔尼诺大气物理机理的变分迭代解法

研究了一个厄尔尼诺大气物理机理振荡.利用变分迭代理论，简捷地得到了该模型解的近似展开式.通过与特殊情形下模型的精确解的比较，得到的一次近似解具有完全符合的精确度. 关键词： 非线性 变分迭代 厄尔尼诺现象     

Non-Markovian and Markovian Dynamics of a V-type Three-Level Atom

The reduced dynamics of a V-type three-level atom in a structure reservoir is presented,which has the exact solution in certain special condition.The Markovian and non-Markovian master equations for this composite system are solved and compared with the exact solution.The solving approach can be directly generalized to the solution of a V-type multilevel system dynamics interacting with a reservoir.The results further testify that these two kinds of master equations are exploited in different coupling regime,providing guidance for further application of these variants master equations to solve multilevel system dynamics without the exact solution.     

The effective conductivity of composite materials with cubic arrays of multi-coated spheres

Two premeditated resistor models have been developed and tested for the prediction of the effective thermal conductivity of a periodic array of multi-coated spheres embedded in a homogeneous matrix of unit conductivity. The results have been compared and evaluated with the exact solution, as obtained by extending a method originally devised by Zuzovski and Brenner. The results for the two models were found to yield bounds for the exact solution. For some situations, the model results match well with the exact solution, but in other cases the results for one of the models could deviate from the exact solution. PACS 41.20.Cv; 44.10.+i; 72.80.Tm     

An exact,closed form,solution for the flexural vibration of a thin annular plate having a parabolic thickness variation

An exact, closed form, solution is obtained for the transverse vibrations, with nodal diameters and circles, of a thin annular plate having a parabolic thickness variation. Representative numerical values for the frequency parameter and typical mode shapes are presented for three different combinations of simple boundary conditions. The corresponding exact solution for an aeolotropic annular plate of the same geometry is also presented. Aside from possible design applications, these exact, closed form, data can be used as test cases for assessing the accuracy of various approximate methods of solution. The analysis involves only the powers of the radius and is simpler than that for the constant thickness solution which involves Bessel functions.     

Multi-channel scattering problems: an analytically solvable model

We have proposed a general method for finding an exact analytical solution for the multi-channel scattering problem in the presence of a delta function coupling. Our solution is quite general and is valid for any set of potentials, if the uncoupled diabatic potential has an exact solution. We have also discussed a few examples, where our method can easily be applied.     

Travelling Wave Solution of Korteweg-de Vries-Burger's Equation

An attempt has been made to obtain exact analytical travelling wave solution of Korteweg-de Vries-Burger's (KdVB) equation by the so-called tanh-method. This equation can be derived for dust ion acoustic shocks by using reduction perturbation method. It is found that an exact solution of the KdVB equation is obtained by tanh-method, provided the parameters involved satisfy a constraint relation. However a special exact analytical solution can be obtained where no such restriction is necessary. This solution has the structure of a shock wave. Numerical solution is also obtained for travelling wave with or without the assumption of the constraint relation. We have also found a singular solution in terms of cosech and coth functions.     

Exact and approximate solutions for the electron nonsinglet structure function in QED

Two exact, valid up to infinite perturbative order, numerical solutions of the Lipatov equation for the nonsinglet electron structure function in the QED are presented. One of them is of the Monte Carlo type and another is based on the numerical inversion of the Mellin transform. They agree numerically to a very high precision (better than 0.05%). Within the leading logarithmic approximation, the exact solution is compared with the perturbative second and third order exponentiated solutions. It is shown that the perturbative second order solution inspired by the Yennie-Frautschi-Suura (exclusive) exponentiation is much closer to the exact solution than the other ones. New compact analytical formula for the third order exponentiated solution is given. It is shown to be in perfect numerical agreement with the infinite order solution of the Monte Carlo and Mellin type.     

Gravitation of the Klein-Gordon scalar field

This paper presents an exact solution of the Einstein-Klein-Gordon equations in the static and spherically symmetric case and points out the differences between it and Yilmaz's solution. In addition, the essential difference between the exact solution and the post-Newtonian approximate solution is also shown.     

VIBRATIONS OF NON-UNIFORM CONTINUOUS BEAMS UNDER MOVING LOADS

The dynamic behavior of multi-span non-uniform beams transversed by a moving load at a constant and variable velocity is investigated. The continuous beam is modelled using Bernoulli-Euler beam theory. The solution is obtained by using both the modal analysis method and the direct integration method. The natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross-section along the beam using the exact element method. The mode shapes are expressed as infinite polynomial series. Using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity. Numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study, and the results are compared to previously published results.     

Exterior and interior metrics with quadrupole moment

We present the Ernst potential and the line element of an exact solution of Einstein’s vacuum field equations that contains as arbitrary parameters the total mass, the angular momentum, and the quadrupole moment of a rotating mass distribution. We show that in the limiting case of slowly rotating and slightly deformed configuration, there exists a coordinate transformation that relates the exact solution with the approximate Hartle solution. It is shown that this approximate solution can be smoothly matched with an interior perfect fluid solution with physically reasonable properties. This opens the possibility of considering the quadrupole moment as an additional physical degree of freedom that could be used to search for a realistic exact solution, representing both the interior and exterior gravitational field generated by a self-gravitating axisymmetric distribution of mass of perfect fluid in stationary rotation.     

Perturbative solution of the schwinger model

One interesting question for the exactly solvable Schwinger model is how to infer the exact solution from perturbation theory. We give a systematic procedure of deriving the exact solution from Feynman diagrams of arbitrary order for arbitraryn-point functions. As a byproduct, we derive from perturbation theory exact integral equations that then-point functions have to obey. This work was supported by a research stipendium of the University of Vienna.     

Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry

In this paper we present the exact solution of the Riemann problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave. We provide examples of possible wave patterns. Numerical solution of a first-order dissipative scheme as well as an implementation of our Riemann solver in the second-order upwind method are compared with the proposed exact Riemann problem solution. A practical implementation of the proposed exact Riemann solver in the framework of a second-order upwind TVD method is also illustrated.     

New Exact Solutions for （1 ＋ 1）-Dimensional Dispersion-Less System

Using improved homogeneous balance method, we obtain complex function form new exact solutions for the （1＋1）-dimensional dispersion-less system, and from the exact solutions we derive real function form solution of the field u. Based on this real function form solution, we find some new interesting coherent structures by selecting arbitrary functions appropriately.     

Numerical solution of 3D Green's function for the dynamic system of anisotropic elasticity

In this Letter, we used homotopy perturbation method to obtain numerical solution of the 3D Green's function for the dynamic system of anisotropic elasticity. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results obtained from convolution of Green's function and data of the Cauchy problem are compared with the exact solution for cubic media. The results reveal that the proposed method is very effective and simple.     

BERRY PHASES IN THE QUANTUM STATE OF THE ISOTROPIC HARMONIC OSCILLATOR WITH TIME-DEPENDENT FREQUENCY AND BOUNDARY CONDITIONS

The problem of the isotopic harmonic oscillator of time-dependent frequency confined in a spherical box with time-dependent radius is studied. We show that the exact solution and the Lewis invariant operator can be obtained by performing two consecutive gauge transformations on the time-dependent Schr?dinger equation. On the basis of the exact solution the non-adiabatic Berry phases for the system are calculated.