Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.     

The application of homotopy analysis method to nonlinear equations arising in heat transfer

Here, the homotopy analysis method (HAM), which is a powerful and easy-to-use analytic tool for nonlinear problems and dose not need small parameters in the equations, is compared with the perturbation and numerical and homotopy perturbation method (HPM) in the heat transfer filed. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series.     

Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this Letter; one is He's variational iteration method (VIM) and the other is the homotopy–perturbation method (HPM). Nonlinear convective–radiative cooling equations are used as examples to illustrate the simple solution procedures. These methods are useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. Comparison of the results obtained by both methods with exact solutions reveals that both methods are tremendously effective.     

Exp-function method for N-soliton solutions of nonlinear evolution equations in mathematical physics

In this Letter, the Exp-function method is generalized to construct N-soliton solutions of a KdV equation with variable coefficients. As a result, 1-soliton, 2-soliton and 3-soliton solutions are obtained, from which the uniform formula of N-soliton solutions is derived. It is shown that the Exp-function method may provide us with a straightforward and effective mathematical tool for generating N-soliton solutions of nonlinear evolution equations in mathematical physics.     

New exact approach to models of diffusion in one-dimensional random systems and optimal network kinetics in spin glass models

It is shown that in one-dimensional stochastic models with gaussian random energy levels along a quantum reaction coordinate the dominant, rate-determining time-scale does not follow the conventional Arrhenius law, but rather has a much stronger temperature dependence, of the form τ ～ exp[(B/k_BT)²], where B is proportional to the width of the energy distribution. The new activation law can be ascribed to the large number of energy barriers of varying heights which exist in the random structure, as distinct from the conventional case of a single barrier, leading to the Arrhenius form τ ～ T^p × exp(A/k_BT). In systems with random structure and configuration space which are not strictly one-dimensional it is discussed if the thermal energy bias of detailed balance may lead to a kinetics that is essentially restricted to an energetically optimal network at low temperatures, thus leading to an essentially one-dimensional diffusion. Several recent studies of spin glass models appear to support the relevance of this principle, and include the observation of the new activation law in Monte Carlo experiments.     

关于数学物理方法课程的若干问答(续2)

26如果偏微分方程具有一定的对称性(不变性),它的解是否一定具有同样的对称性(不变性)?不一定!例如,二维拉普拉斯方程具有平移不变性、反射不变性以及旋转不变性,但是它的解x或y就不具有这些不变性.同样,x2-y2也是二维拉普拉斯方程的解,但是它也不具有平移不变性和旋转     

面向21世纪专业教学、物理学史与教学科研相结合教学法

物理学史和教学科研在专业教学中具有不可或缺的重要作用，以此为指导思想，介绍了一种行之有效的新型教学法。     

On convergence,asymptotic behaviour and computational algorithm of generalized exponential integral functions arising in a radiative transfer

In this paper, we investigate uniform convergence of the generalized exponential integral functions (GEI) arising in the study of radiative transfer in a two-dimensional planar-medium and anisotropic scattering in a multidimensional medium. Using the uniform convergence, we study the properties and asymptotic behaviour of the GEI functions. We also give an efficient algorithm for the computations of the values of the GEI functions.     

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.     

Critical layer singularities and complex eigenvalues in some differential equations of mathematical physics

Singular differential equations are a common feature of many problems in mathematical physics. It is often the case that systems with a similar mathematical structure can arise in many different contexts. In this article, mathematically related problems are drawn from areas as diverse as hydrodynamics (with applications to oceanography and meteorology), magnetohydrodynamics and plasma physics (with applications to astrophysics and geophysics, especially solar physics, ionospheric and magnetospheric physics; also nuclear fusion devices), acoustics, electromagnetics, quantum mechanics and nuclear physics. One major unifying feature common to the problems discussed here is the existence of complex eigenvalues, often associated with so-called “classical self-adjoint” equations. No real contradiction is involved here, but the resulting wave functions are often referred to variously as “radioactive states”, “damped resonances”, “leaky waves”, “non-modal solutions” , “singular modes”, “virtual modes”, or “improper eigenfunctions”. In the hydrodynamics of shear flows, such modes are associated with the existence of “critical layers” at which a singularity occurs in the governing (ordinary) differential equation. Similar, but usually more general singular layers are known to occur in equations arising in many of the above-mentioned contexts, and it is the purpose of this review to identify the nature of these singular layers and complex eigenvalues, and the relationships that exist between the different context in which they are found, and in particular to emphasize the occurrence of and interpretation of complex eigenvalues in quantum mechanics. Thus the “exponential catastrophe” is a clearly identified and recurring theme throughout this article by virtue of the similarities that exist between the classical and quantum system discussed here. The examples quoted from quantum mechanics are simple in form, and found in many standard texts, but the virtue of including them here is twofold: the results are easy to understand and relate to the more complicated “classical” systems, and they provide a valuable didactic and pedagogic tool for those readers whose background in quantum mechanics is limited. It is also hoped that this article will be of interest to readers who wish to become more acquainted with some aspects of hydrodynamics and magnetohydrodynamics.     

Gravitational radiation,source behavior,and the method of matched asymptotic expansions

It is conjectured that a suitably modified Bondi-type expansion of the gravitational field in the radiation zone is a rapidly convergent series. It is also conjectured that the source behavior in the inner zone is insensitive to the initial conditions imposed on the gravitational field in solving the initial-value problem in this zone. Consequences of these conjectures for the problem of relating source motion to the Bondi news function are discussed.     

Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method

A new generalized solitary solution of the Jaulent-Miodek equations is obtained using the Exp-function method. By a transformation, the solitary solution can be easily converted into a generalized compacton-like solution. The free parameters in the obtained generalized solutions might imply some meaningful results in physical process.     

三Riccati 方程的新展开法及其在非线性数学物理方程中的应用

通过对tanh函数法和射影Riccati方程法的探讨, 提出一种新的算法来求解非线性发展方程. 并且通过求解高阶 schrodinger 方程和 mKdV 方程来说明该算法.得到了这些方程的新形式解,包括新的孤立波解, 周期解等, 并图示一些新形式解.     

类比推理教学法在非线性物理中的应用

在物理学的方法论中,类比法是一种逻辑思维方法,也是一种重要的创新思维方式.文中以刚性摆的运动为例,应用类比推理法,对约瑟夫森结、滑移电荷密度波等非线性物理中的现象进行了研究,所得结果与实验和数值计算结果相符合.     

Reconstruction of holograms generated by using a combination of an iterative method and an error diffusion method

Two-level quantized phase holograms produced by using a combination of an iterative method and an error diffusion method are presented. By using a previously proposed iterative method, images reconstructed from two-level quantized phase holograms are rapidly improved with small amounts of iterative operations but further improvement of the image quality is difficult even if the iterations are greatly increased. Therefore, in order to exceed the improvement limitation of the image quality, the combination of the iterative method and an error diffusion method was used for the hologram-making process. Experimental results using the combination method showed better images than those obtained by the iterative method alone, when referred to results from synthesized holograms.     

核物理实验中模数转换器选型及其供电方案

分析了核物理实验的特点及其对模数转换器(ADC)的要求,比较了3种ADC芯片的主要技术指标,选取了具有逐次逼近寄存器型架构和16位分辨力的ADS8472作为采集芯片,其微分非线性和积分非线性分别是±0.50 LSB(最低有效位,least significant bit)和±0.65 LSB.根据该芯片在实际电路中的应...     

核物理实验中模数转换器选型及其供电方案

分析了核物理实验的特点及其对模数转换器（ADC）的要求,比较了3种ADC芯片的主要技术指标,选取了具有逐次逼近寄存器型架构和16位分辨力的ADS8472作为采集芯片,其微分非线性和积分非线性分别是±0.50 LSB(最低有效位,least significant bit)和±0.65 LSB。根据该芯片在实际电路中的应用,介绍了多道脉冲幅度采集分析系统架构,并对其工作原理进行了说明。利用线性电源芯片给出了合理的供电方案,经过长时间（72 h）满负荷运行考验,各扇出电压波动均小于1 mV。     

The invariant integral of physical mesomechanics as the foundation of mathematical physics: Some applications to cosmology,electrodynamics, mechanics and geophysics

Phe general invariant integral based on the energy conservation law is introduced into physical mesomechanics, with taking into account the cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter. Phe physical mesomechanics thus becomes a mega-mechanics embracing most of the scales of nature. Some basic laws following from the general invariant integral are indicated, including Coulomb’s law of electricity generalized for moving electric charges, Newton’s law of gravitation generalized for coupled gravitational/cosmic field, the Archimedes’ law of buoyancy generalized for bodies partially submerged in water, and others. Using the invariant integral the temperature track behind moving cracks and dislocations is found out, and the coupling of elastic and thermal energies is set up in fracturing and plastic flow, namely for opening mode cracks and edge dislocations. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding invariant integrals. As applied to the horizontal drilling and hydrofracturing of boreholes in the Earth’ crust, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a diskshape fracture issuing the borehole, are derived in the fluid extraction regime. A theory of fracking in shale gas/oil reservoirs is suggested for three basic regimes of the drill mud permeation into the multiply fractured rock region, with calculating the shape and volume of this region in terms of the geometry parameters and pressures of rock, drill mud and shale gas. Phe method of functional equations in the theory of a complex variable and the boundary layer method are also used to solve these problems.     

实验方法教育的内容体系和实施途径

系统构建了中学物理实验方法教育的内容体系，并指出了实施实验方法教育的具体途径．