扰动mKdV耦合系统的孤子解

采用了一个简单而有效的技巧,研究了一类扰动mKdV耦合系统.首先利用同伦映射方法求解一个相应的复值函数微分方程孤子的近似解.然后得到了原扰动mKdV耦合系统孤子的近似解. 关键词： 孤子 扰动mKdV方程 同伦映射     

Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution

Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the approximate output of the resonant gyroscope.The method of small parameter perturbation is used to analyze the approximate solution of the Mathieu equation.The theoretical analysis and the numerical simulations show that the approximate solution of the Mathieu equation is close to the dynamic output characteristics of the resonant gyroscope.The experimental analysis shows that the theoretical curve and the experimental data processing results coincide perfectly,which means that the approximate solution of the Mathieu equation can present the dynamic output characteristic of the resonant gyroscope.The theoretical approach and the experimental results of the Mathieu equation approximate solution are obtained,which provides a reference for the robust design of the resonant gyroscope.     

Diffusion in a bistable potential: A comparative study of different methods of solution

The problem of diffusion in a bistable potential is studied by considering the associated nonlinear Langevin equation and its equivalent Fokker-Planck equation. Two numerically exact methods of solution, namely, the Monte Carlo solution of the nonlinear Langevin equation and the solution of the Fokker-Planck equation via the finite difference technique, are considered. The latter method has the advantage that it directly gives the evolution of the probability distribution function. Approximate analyses of the fluctuations using the system size expansion, the Gaussian decoupling procedure, and the scaling approach are also carried out. These investigations are performed on a representative problem for two specific cases: (1) evolution from intrinsically unstable states and (2) evolution from extensive regime. The fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods. The study brings out the advantages and limitations of each of the methods considered.     

Infinite series symmetry reduction solutions to the modified KdV--Burgers equation

From the point of view of approximate symmetry, the modified Korteweg--de Vries--Burgers (mKdV--Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV--Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV--Burgers equation satisfies the Painlevé II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

一类非线性方程类孤波的近似解法

采用了一个简单而有效的技巧,研究了一类扰动发展方程.首先引入求解一个相应典型方程的类孤波近似解,然后利用泛函映射方法得到了原扰动发展方程的近似解,指出了近似解级数的收敛性,并用解析方法,讨论了近似解的精度.     

一个广义扰动mKdV耦合系统2极孤子的近似解

采用了一个简单而有效的技巧,研究了一类扰动mKdV耦合系统.首先利用变分迭代方法求解一个相应的复值函数微分方程2-极孤子的近似解.然后得到了原扰动mKdV耦合系统2-极孤子的近似解.     

Exact numerical solution for the diffused channel waveguide

We present an exact numerical solution based on a vector integral equation to investigate the diffused channel waveguide. Various examples are given and compared with other numerical and approximate methods. An approximate numerical solution is also given using the effective index method and the integral equation solution for the inhomogeneous slab.     

一类扰动发展方程近似解

采用了一个简单而有效的技巧, 研究了一类扰动发展方程. 首先引入求解一个相应典型方程的行波孤波解. 然后利用渐近方法得到了原扰动发展方程的近似解. 利用泛函分析的不动点定理, 指出了近似解级数的收敛性, 并讨论了近似解的精度.     

A General Solution of the Fokker-Planck Equation

The Fokker-Planck equation is useful to describe stochastic processes. Depending on the force acting in the system, the solution of this equation becomes complicated and approximate or numerical solutions are needed. The relation with the Schrödinger equation allows building a method to obtain solutions of the Fokker-Planck equation. However, this approach has been limited to the study of confined potentials, restricting its applicability. In this work, we suggest a general treatment for non-confining potentials through the use of series of functions based on the solution of the Schrödinger equation, with part of discrete spectrum and part of continuum spectrum. Two examples, the Rosen-Morse potential and a limited harmonic potential, are analyzed using the suggested approach.     

Pomraning-Eddington approximation for solving the Spencer-Lewis equation

By means of Pomraning-Eddington approximation and maximum entropy method the Spencer-Lewis equation in an infinite medium has been solved explicitly. The behaviour of the approximate solution for the total electron density are shown graphically, and compared with that obtained by using the flux-limited approach. The results reported in this article provide further evidence of the usefulness of both Pomraning-Eddington and flux-limited. Knowing the electron density distribution allows us to calculate directly some physical quantity, such as the reflection function and energy deposition profile.     

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

<正>A class of disturbed evolution equation is considered using a simple and valid technique.We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation.Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method.We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

The factorization method in a rough-surface scattering problem: I

In the present paper the wave scattering problem on rough surface is considered for the Helmholtz equation with the Dirichlet boundary condition. An approximate solution is derived with using a factorization approach to the original Helmholtz equation. As a result, the system of two equations of parabolic type appears. The first system equation has an exact analytical solution whereas for the second one, an approximate solution, is considered in terms of perturbation series. It is shown that the obtained approximate solution is the modified classical small perturbation series with respect to small Rayleigh parameter. In Appendix A it is demonstrated that, when the derived perturbation series is converged, it is possible to summarize it and to represent the exact solution of original boundary problem in an analytical symbolical form.     

Model of a nanoparticle in the theory of inhomogeneous medium

Basic thermodynamic characteristics (density distribution profile near the surface of a nanoparticle, adsorption, and interfacial surface tension) of the structureless nanoparticle-vapor/liquid equilibrium system are calculated using a unified approach. A joint solution of the basic equation in the Van der Waals theory of an inhomogeneous medium for the density distribution profile in the spherical system of coordinates and the Gibbs equation for the interfacial tension and absorption is obtained. The features of nucleation of nanoparticles are considered. The results generalize some familiar formulas and provide a more adequate interpretation of experimental results.     

Investigation of resonant modes in thin microcavities by using electromagnetic theory

A biharmonic differential equation for 3D thin microcavities with uniform thickness is investigated by use of electromagnetic theory, whose exact solution is determined to govern the electromagnetic field distribution inside the thin microcavities. The resonant field patterns of a thin microdisk and thin rectangular microcavity are obtained accordingly. The governing equation can be verified by comparing the results of the thin microdisk presented with the approximate ones in the literature. The fourth-order partial differential equation and its exact solution should be useful in possible applications of the thin microcavities for optical resonators in laser optics and optical devices.     

一类扰动非线性发展方程的类孤子同伦近似解析解

采用一个简单而有效的技巧,研究了一类扰动非线性发展方程.首先,引入一个相应典型方程的孤子近似解.然后,利用同伦映射方法得到了原扰动非线性发展方程的近似解. 关键词： 孤子 扰动发展方程 同伦映射     

自行车系统的计算机模拟

自行车系统是一个具有非完整约束的复杂系统,可以根据最小作用量原理通过欧拉-拉格朗日方程求解,利用差分代替微分及一系列修正截断误差的方法计算近似值.利用MATLAB,只要给定任意质点系的拉格朗日函数以及约束系数,即可求出系统的近似解,并通过这个方法为分析自行车的运动与稳定性提供一种新的思路.     

An efficient technique for solving the space-time fractional reaction-diffusion equation in porous media

In this paper, we obtained the approximate numerical solution of space-time fractional-order reaction-diffusion equation using an efficient technique homotopy perturbation technique using Laplace transform method with fractional-order derivatives in Caputo sense. The solution obtained is very useful and significant to analyze the many physical phenomenons. The present technique demonstrates the coupling of the homotopy perturbation technique and Laplace transform using He’s polynomials for finding the numerical solution of various non-linear fractional complex models. The salient features of the present work are the graphical presentations of the approximate solution of the considered porous media equation for different particular cases and reflecting the presence of reaction terms presented in the equation on the physical behavior of the solute profile for various particular cases.