一维非定常完全气体热声流的代数显式解析解

热声学是一门较新的学科,其理论与实验研究尚在蓬勃发展中,已知的代数显式解析解很少.本文对一维非定常完全气体热声流推导出一组非常简明的代数显式解析解.它除了有其理论基础价值外,还可以作为标准解来推动计算热声学的发展.     

考虑旋流的非定常几何一维流动解析解

对文献[1]给出的能反映管内非定常完全气体有旋流流动的一维计算模型,首次给出了其多套代数显式解析解,以便于在理论上了解此流动模型,尤其是有助于计算流体力学工作者验证与发展其计算方法与技巧。本文导出一些解析解时利用了作者以前推导普通(无旋流)非定常一维流所得的解与推导经验,也是一个特色。     

关于完全气体无粘可压流动的解析解

本文导出了一套不定常二、三维完全气体无粘可压流动的代数显式解析解，这些解除在理论上有重要意义外，还可以用来作为检验与发展各种数值解法的标准解．另外，文中还将提出和讨论二维可压等熵流的一个很有意思的疑难．     

变热物性非定常导热方程的一些显式解析解

各种变热物性（热传导系数、密度与比热为变数）的非定常导热方程的解析解在理论上是有意义的；而且它们对计算传热学也很有实用价值；可以作为标准解来校核各种数值计算以及用来启发发展各种计算技巧例如差分格式、网格生成等等。但已知的解析解很少。本文对直角座标下沿几何座标变热物性的非定常几何一元及二元的导热方程导出了一些代数显式解析解，其中有些解包含有任意函数，其实是无限多个解；可作为发展导热学理论及计算导热学之用．     

非定常动态演化伴随优化设计方法

研究了基于定常流动解和伴随方程定常解基础上的传统的伴随方法.在此基础上对定态飞行气动外形的优化设计提出了基于非定常流动控制方程瞬态解和非定常伴随方程瞬态解的新的优化方法,称之为动态演化伴随方法.这种新的优化方法保留了传统伴随方法适用于具有大数量设计变量的气动优化问题,而且比传统的伴随方法可节省大量的计算时间.大量算例计算结果表明,新方法与传统方法具有相同的精度.     

生物导热基本方程的三维非定常代数显式解析解

对目前比较通用的生物导热基本方程Pennes方程,给出了其三维非定常情况下的代数显式解析解。据知,以前很少有发表过这类严格的几何多维解析解。它除了有不可代替的理论价值外,还可以作为标准解来校验日益发达的数值解以及作为基础来研究计算传热学的数值计算方法。另外,文章的特殊求解方法也有发展价值之处。     

一维非定常流体力学数值模拟

非线性的流体偏微分方程中的激波间断解是物理中很有挑战的问题,其中的Riemann问题可采用解析的方法求出理论解,但采用近似解的方法进行流体动力学数值模拟也可以有效地追踪和捕捉激波.本文以一维激波管为例,对非定常流体基本方程组采用Roe通量差分裂格式,并进行数值模拟.通过与理论解对比,发现其符合度较好.     

可压涡卷空间演化的迎风紧致差分数值模拟

从数值算法的耗散和色散特征的时空全离散Fourier分析出发,通过直接求解二维非定常可压Navier-Stokes方程,将发展的5阶迎风紧致差分格式用于无约束可压平面受迫剪切层中基频涡卷空间演化过程的数值模拟,采用被动守恒标量等方法显示了基频涡卷的饱和、一次对并、二次对并等现象,据此探讨了入口来流亚谐扰动引起的初值效应问题,表明可压大尺度涡结构空间演化形态与受迫扰动方式之间存在关联。     

N—S方程隐式分区并行计算

将WuZN和ZouH提出的一套适全三点块三角隐式格式的定常和非定常可压无粘流的覆盖分区并行计算方法推广到N－S方程的计算,在二维定常亚音速、跨音速和非定常情况下得到了较好的绝对并行效率。     

可压剪切层线性稳定性特征直接数值模拟

基于伪随机数生成技术促白噪声扰动,以高精度迎风／对称紧致混合差分算法求解二维／三维非定常可压Navier－Stokes方程,揭示了可压自由剪切层初始剪切过程中扰动的线性演化特征,以及该过程对扰动波数和方向的内在选择性,验证了所用算法的有效性,表明线性理论同数值模拟相结合是可压剪切层研究的合理途径之一。     

On a certain method of solving nonlinear differential equations

A new method of exactly solving nonlinear differential partial equations is developed. This method is applied to an equation of a potential flow of compressible gas. A new solution is obtained in the supersonic region, which may be treated as an analogon of a nonlinear stationary wave.Partially supported by NSF contract INT 73-20002A01.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

A transformed rational function method for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

A direct method, called the transformed rational function method, is used to construct more types of exact solutions of nonlinear partial differential equations by introducing new and more general rational functions. To illustrate the validity and advantages of the introduced general rational functions, the (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama (YTSF) equation is considered and new travelling wave solutions are obtained in a uniform way. Some of the obtained solutions, namely exponential function solutions, hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions and rational solutions, contain an explicit linear function of the independent variables involved in the potential YTSF equation. It is shown that the transformed rational function method provides more powerful mathematical tool for solving nonlinear partial differential equations.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

Matched solutions of nonlinear forced equations and Rossby vortices

An analytical method for solving nonlinear equations with local forcing is proposed. It is shown in an example that a nonlinear forced equation may have many solutions, which generally do not turn to the solution of a linear equation in the limit of the nonlinear term becoming small. The solution of the Korteweg-de Vries (KdV) equation with forcing is applied to the problem of topographic Rossby vortices in shear flow. Solutions of other nonlinear equations with forcing are also obtained.     

A Generalized Method and Exact Solutions in Bose-Einstein Condensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schr(o)dinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function's soliton solutions, but also Jacobi elliptic function solutions and Weierstrass elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient.     

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

In this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.     

Exponential-fraction trial function method to the 5th-order mKdV equation

This paper obtains some solutions of the 5th-order mKdV equation by using the exponential--fraction trial function method, such as solitary wave solutions, shock wave solutions and the hopping wave solutions. It successfully shows that this method may be valid for solving other nonlinear partial differential equations.