六组点堆中子动力学方程组的同伦分析解

鉴于目前六组点堆中子动力学方程仍然无法获得解析解,本文尝试将同伦分析方法应用于六组缓发中子动力方程组的求解,获得了它的级数解析解,并对级数解析解算法的有效性进行了检验.结果表明,该级数解析解算法从计算时间和精度上都能达到工程应用的要求,可适宜于反应堆中子动力学控制的设计分析和仿真计算.     

Application of ultraspherical polynomials to non-linear autonomous systems

This paper deals with the approximate solutions of non-linear autonomous systems by the application of ultraspherical polynomials. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by a generalized averaging technique based on the ultraspherical polynomial expansions. The method is illustrated with examples and the results are compared with the digital and analog computer solutions. There is a close agreement between the analytical and exact results.     

Perturbation method for periodic solutions of nonlinear jerk equations

A Lindstedt-Poincaré type perturbation method with bookkeeping parameters is presented for determining accurate analytical approximate periodic solutions of some third-order (jerk) differential equations with cubic nonlinearities. In the process of the solution, higher-order approximate angular frequencies are obtained by Newton's method. A typical example is given to illustrate the effectiveness and simplicity of the proposed method.     

Piecewise optimal linearization method for nonlinear stochastic differential equations

A method for estimating the dynamical statistical properties of the solutions of nonlinear Langevin-type stochastic differential equations is presented. The non-linear equation is linearized within a small interval of the independent variable and statistical properties are expressed analytically within the interval. The linearization procedure is optimal in the sense of the Chebyshev inequality. Long-term behavior of the solution process is obtained by appropriately matching the approximate solutions at the boundaries between intervals. The method is applied to a model nonlinear equation for which the exact time-dependent moments can be obtained by numerical methods. The calculations demonstrate that the method represents a significant improvement over the method of statistical linearization in time regimes far from equilibrium.Supported in part by the National Science Foundation under Grants CHE77-16307 and PHY76-04761.     

Physiological Transportation of Casson Fluid in a Plumb Duct

This paper is devoted to a study of the peristaltic motion of a Casson fluid of a non-Newtonian fluid accompanied in a horizontai tube.To characterize the non-Newtonian fluid behavior,we have considered the Casson fluid model.Suitable similarity transformations are utilized to transform the governing partial differential momentum into the non-linear ordinary differential equations.Exact analytical solutions of these equations are obtained and are the properties of velocity,pressure and profiles are then studied graphically.     

Analytical solution of transverse oscillation in cyclotron using LP method

We have carried out an approximate analytical solution to precisely consider the influence of magnetic field on the transverse oscillation of particles in a cyclotron.The differential equations of transverse oscillation are solved from the Lindstedt-Poincare method.After careful deduction,accurate first-order analytic solutions are obtained.The analytical solutions are applied to the magnetic field from an isochronous cyclotron with four spiral sectors.The accuracy of these analytical solutions is verified and confirmed from comparison with a numerical method.Finally,we discussed the transverse oscillation at v_0=N/2,using the same analytical solution.     

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.     

An Analysis of the Transition Zone Between the Various Scaling Regimes in the Small-World Model

We analyse the so-called small-world network model (originally devised by Strogatz and Watts), treating it, among other things, as a case study of non-linear coupled difference or differential equations. We derive a system of evolution equations containing more of the previously neglected (possibly relevant) non-linear terms. As an exact solution of this entangled system of equations is out of question we develop a (as we think, promising) method of enclosing the “exact” solutions for the expected quantities by upper and lower bounds, which represent solutions of a slightly simpler system of differential equation. Furthermore we discuss the relation between difference and differential equations and scrutinize the limits of the spreading idea for random graphs. We then show that there exists in fact a “broad” (with respect to scaling exponents) crossover zone, smoothly interpolating between linear and logarithmic scaling of the diameter or average distance. We are able to corroborate earlier findings in certain regions of phase or parameter space (as e.g. the finite size scaling ansatz) but find also deviations for other choices of the parameters. Our analysis is supplemented by a variety of numerical calculations, which, among other things, quantify the effect of various approximations being made. With the help of our analytical results we manage to calculate another important network characteristic, the (fractal) dimension, and provide numerical values for the case of the small-world network.     

再入过载的匹配渐进展开分析与卸载方法研究

探月飞船返回地球时将以第二宇宙速度再入地球大气层,面临极其严苛的气动环境,因此对于再入气动过载的分析具有重要意义.再入运动方程是一组非线性很强的常微分方程,数值方法计算量大,不适用于在线任务.因此,本文采用一种近似解法对气动过载进行分析.首先,基于匹配渐进展开方法将再入纵向运动解在大气外层区域与内层区域分别展开,得到统一形式的闭型近似解,在此基础上分段求解气动过载,并与精确解进行对比分析.其次,利用闭型近似解,通过当前状态反解虚拟初始条件,在此基础上提出初次过载峰值的解析预测方法,并分析了不同条件下预测的相对误差变化规律.最后,基于过载峰值的解析预测对飞船的初次再入过程进行卸载,将飞船在再入过程中耗散的总能量进行重新分配,并通过蒙特卡罗飞行仿真试验验证了卸载方法的有效性.     

On the modelling of non-linear guided-wave optics for all-optical signal processing

The application of a reduced variational formalism, to model guided-wave optical signal processing devices is reviewed. The focus is mainly on second-order non-linearities. First, approximate analytical solutions (solitary waves) are derived for propagation in second-order non-linear media. Next, these analytical results are validated through numerical simulation of the governing equations. To show the potential of this formalism by means of a practical example, the analytical results are applied to a soliton-based LiNbO3 non-linear directional coupler exploiting second-order non-linear effects. This revised version was published online in November 2006 with corrections to the Cover Date.     

干斜压大气拉格朗日原始方程组的半解析解法和非线性密度流数值试验

以差分方程代替微分方程给大气原始方程组求解带来了诸多难以解决的问题, 对于(半)拉格朗日模式来说质点轨迹的计算与Helmholtz方程的求解是两大难题. 本文通过对气压变量代换, 并在积分时间步长内将原始方程组线性化, 近似为常微分方程组, 求出方程组的半解析解, 再采用精细积分法求解半解析解. 半解析方法可同时计算风、气压和位移, 无需求解Helmholtz方程, 质点的位移采用积分风的半解析解得到, 相比采用风速外推的计算方法, 半解析方法更科学合理. 非线性密度流试验检验表明: 半解析模式能够清晰地模拟Kelvin-Helmholtz 切变不稳定涡旋的发生和发展过程; 模拟的气压场和风场环流结构与标准解非常相似, 且数值解是收敛的, 同时, 总质量和总能量具有较好的守恒性. 试验初步证明了采用半解析方法求解大气原始方程组是可行的, 为大气数值模式的构建提供了一个新的思路.     

Marangoni flow and mass transfer of power-law non-Newtonian fluids over a disk with suction and injection

We scrutinize the approximate analytical solutions by the optimal homotopy analysis method (OHAM) for the flow and mass transfer within the Marangoni boundary layer of power-law fluids over a disk with suction and injection in the present paper. Concentration distribution on the surface of a disk varies in a power-law form. The non-Newtonian fluid flow is due to the surface concentration gradient without considering gravity and buoyancy. According to the conservation of mass, momentum and concentration, the governing partial differential equations are established, and the appropriate generalized Kármán transformation is found to reduce them to the nonlinear ordinary differential equations. OHAM is used to access the approximate analytical solution. The influences of Marangoni the number, suction/injection parameters and power-law exponent on the flow and mass transfer are examined.     

Non-linear responses of suspended cables to primary resonance excitations

We investigate the non-linear forced responses of shallow suspended cables. We consider the following cases: (1) primary resonance of a single in-plane mode and (2) primary resonance of a single out-of-plane mode. In both cases, we assume that the excited mode is not involved in an autoparametric resonance with any other mode. We analyze the system by following two approaches. In the first, we discretize the equations of motion using the Galerkin procedure and then apply the method of multiple scales to the resulting system of non-linear ordinary-differential equations to obtain approximate solutions (discretization approach). In the second, we apply the method of multiple scales directly to the non-linear integral-partial-differential equations of motion and associated boundary conditions to determine approximate solutions (direct approach). We then compare results obtained with both approaches and discuss the influence of the number of modes retained in the discretization procedure on the predicted solutions.     

Differential transform method for solving linear and non-linear systems of partial differential equations

In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

On the Newtonian limit of general relativity

We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einstein's equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einstein's equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones).Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.     

A Method for Solving Initial-value Problem and Finding Exact Solutions of Nonlinear Partial Differential Equations

Homogeneous balance method for solving nonlinear partial differential equation(s) is extended to solving initial-value problem and getting new solution(s) from a known solution of the equation(s) under consideration. The approximate equations for long water waves are chosen to illustrate the method, infinitely many simple-solitary-wave solutions and infinitely many rational function solutions, especially the closed form of the solution for initial-value problem, are obtained by using the extended homogeneous balance method given here.     

Weak nonlinear propagation of sound in a finite exponential horn.

This article presents an approximate solution for weak nonlinear standing waves in the interior of an exponential acoustic horn. An analytical approach is chosen assuming one-dimensional plane-wave propagation in a lossless fluid within an exponential horn. The model developed for the propagation of finite-amplitude waves includes linear reflections at the throat and at the mouth of the horn, and neglects boundary layer effects. Starting from the one-dimensional continuity and momentum equations and an isentropic pressure-density relation in Eulerian coordinates, a perturbation analysis is used to obtain a hierarchy of wave equations with nonlinear source terms. Green's theorem is used to obtain a formal solution of the inhomogeneous equation which takes into account linear reflections at the ends of the horn, and the solution is applied to the nonlinear horn problem to yield the acoustic pressure for each order, first in the frequency and then in the time domain. In order to validate the model, an experimental setup for measuring fundamental and second harmonic pressures inside the horn has been developed. For an imposed throat fundamental level, good agreement is obtained between predicted and measured levels (fundamental and second harmonic) at the mouth of the horn.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

On non-linear oscillations with slowly varying system parameters

The paper deals with an approximate analysis of non-linear oscillation problems with slowly varying system parameters. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by using the generalized averaging method of Sinha and Srinivasan based on ultraspherical polynomial expansions. The Bogoliubov-Mitropolsky results are given by a particular set of these polynomials. Problems of a single degree of freedom system as well as monofrequency oscillations in systems with multiple degrees of freedom are considered. The approach has been illustrated by an example and the results are compared with the numerical solutions. A close agreement is found.