一类弱非线性振动问题的插值摄动解法

用插值摄动法［１］求解一类有阻尼的弱非线性振动问题。算例表明，当小参数很小时，本文结果和多尺度法的一级近似结果十分接近。当小参数不是很小时（即接近于强非线性振动时），本文结果，仍然相当准确，并优于多尺度法的一级近似结果。     

超越摄动:同伦分析方法基本思想及其应用

介绍一种新的、求解强非线性问题解析近似的一般方法------同伦分析方法.该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 因此,适用范围广.此外, 不同于所有其他解析近似方法,同伦分析方法提供了一个简单的途径, 确保所得到的级数解收敛, 从而获得足够精确的解析近似.而且, 不同于所有其他解析近似方法, 同伦分析方法(HAM)提供了选取基函数之自由, 从而可以选择较好的基函数, 更有效地逼近问题的解.同伦分析方法为非线性问题的解析近似求解提供了一个全新的思路, 为非线性问题(特别是不含小参数的强非线性问题)的求解开辟了一个全新的途径.简要描述同伦分析方法的基本思想, 其在非线性力学、物理、化学、生物、金融、工程和计算数学等领域的应用举例, 以及与摄动方法、Lyapunov 人工小参数法、$\delta$展开法、Adomian 分解法、同伦摄动方法之区别和联系.      

同伦摄动理论及其应用

本文应用同伦技术和摄动技术，提出了一种新的摄动理论。这方法有别于传统的摄动理论，它不依赖于小参数，而是应用同伦技术，构造一个含嵌入参数ｐ∈（０，１）的方程，然后把嵌入参数作为“小参数”，因此这种新方法可以克服传统摄动理论的不足，而又可以充分应用各种摄动方法。其结果显示这种方法简单而有效，并且其一阶近似往往具有很高的精度。     

摄动法在研究弹-粘塑性波问题中的应用

采用控制金属材料宏观塑性流动的两个无量纲物理参数作为小参数,将一维弹/粘-塑性问题的解摄动展开,从而,求解非线性波动方程的问题可以转化成求解相应的齐次或非齐次电报方程的问题,用Laplace积分变换或级数展开技术首先得到零次精确解。然后,用Riemann函数方法可获得一次和高次摄动解。与非线性问题的数值解比较,在恒应力或恒速度边界条件下,一次摄动解给出了波动问题的良好近似。这就表明,摄动技术在研究一类广泛的弹/粘-塑波问题中是有效的。     

瞬态波作用下非线性岩土与非圆结构的相互作用

本文讨论了瞬态波作用下非线性岩土与非圆结构相互作用的问题。文中引用了一种多参数的非线性弹性岩土模型,应用复变函数方法求解了由小参数摄动展开得到的各阶渐近线性方程,并利用映射函数使计算得到简化。文中给出了瞬态波水平和垂直入射时直墙拱结构动力响应的位移和应力数值结果。     

非线性岩土中球壳对冲击波的动力响应分析

采用小参数摄动法和Ｌａｐｌａｃｅ变换研究了非线性弹性岩土中球壳对冲击波的动力响应问题。系统的非线性方程由小参数摄动渐近展开后，利用Ｓｔｏｋｅｓ－Ｈｅｌｍｈｏｌｔｚ矢量分解定理把它们简化为一系列的线性波动方程，并由Ｌａｐｌａｃｅ变换和本征函数给出了各线性波动方程的求解。最后，对平面冲击波和球面冲击波给出了球壳动力响应的应力结果。     

摄动理论在求解流固耦合渗流问题中的应用

通过引进小参数ε，对工程中一类强非线性耦合渗流问题采用摄动理论进行解析求解，探讨了数学模型中压力动态分布特征，并通过数值算例分析了压力随饱和度s及时间t变化的情况，可为实验确定压力-饱和度-渗透率之间的关系提供理论依据，同时为求解非线性偏微分数学方程组提供一种解析求解方法．     

摄动权余法及在薄板大挠度问题上的应用

本文结合了摄动法及权余法的优点,提出一种处理非线性问题的新方法——摄动权余法(PWRM)。这种方法吸取摄动法能将非线性问题化成许多级线性问题及权余法的灵活性的优点,避免了每级摄动方程难于精确求解及单纯使用权余法将导致非线性代数方程组的缺点。作为本方法的特例,提出摄动里兹法及摄动伽辽金法,并用它们处理了两个实例。     

应用同伦技术构造摄动参数

本文应用同伦技术和摄动技术,提出了一种新的摄动思想。这种方法有别于传统的摄动理论,它不依赖于小参数,而且是应用同伦技术,构造一个含嵌入参数ｐ∈「０,１」的方程,然后把嵌入参数为“小参数”,因此这种新方法可以克服传统摄动理论的不足,而又充中种摄动方法。     

二自由度耦合非线性随机系统的最大Lyapunov指数和稳定性

利用摄动方法讨论了一类耦合二自由度非线性系统，在小强度白噪声参数激励下系统运动模态的稳定性，获得了系统扩散过程的稳态概率密度的渐近表达式，由此获得了系统运动模态几乎必然稳定的充分必要条件。     

A Modified Perturbation Technique Depending Upon an Artificial Parameter

In this paper, a modified perturbation method is proposed to search for analytical solutions of nonlinear oscillators without possible small parameters. An artificial perturbation equation is carefully constructed by embedding an artificial parameter, which is used as expanding parameter. It reveals that various traditional perturbation techniques can be powerfully applied in this theory. Some examples, such as the Duffing equation and the van der Pol equation, are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weak nonlinear systems, but also for strongly nonlinear systems. In applying the new method, some special techniques have been emphasized for different problems.     

A brief review of the homotopy analysis method

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.     

同伦分析方法：一种不依赖于小参数的非线性分析方法

本文进一步一般化了作者所提出的一种新的非线性分析方法，称为“同伦分析方法”。“同伦分析方法”的最大优点，在于其不依赖小参数，从而可求解更多的非线性问题，甚至包括那些不含小参数的问题。本文给出了几个应用实例，以说明“同伦分析方法”的有效性及巨大的潜力。     

同伦分析方法进展综述

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.      

Asymptotic solution of nonlocal nonlinear reaction-diffusion Robin problems with two parameters

In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed in relation to two small parameters. The asymptotic solutions of the problem are given.     

Accurate Higher-Order Analytical Approximate Solutions to Large-Amplitude Oscillating Systems with a General Non-Rational Restoring Force

A new approximate analytical approach for accurate higher-order nonlinear solutions of oscillations with large amplitude is presented in this paper. The oscillatory system is subjected to a non-rational restoring force. This approach is built upon linearization of the governing dynamic equation associated with the method of harmonic balance. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. This approach also explores large parameter regions beyond the classical perturbation methods which in principle are confined to problems with small parameters. It has significant contribution as there exist many nonlinear problems without small parameters. Through some examples in this paper, we establish the general approximate analytical formulas for the exact period and periodic solution which are valid for small as well as large amplitudes of oscillation.     

Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects

In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton's principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.     

Nonlinear oscillations with parametric excitation solved by homotopy analysis method

An analytical technique, namely the homotopy analysis method （HAM）, is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations.     

A perturbation solution in the nonlinear theory of circular plates

In this paper, we analyzed some problems of nonlinear circular plates by means of perturbation method. The perturbation parameters chosen here are obtained from solving the equations and are not certain mechanical quantities given precedently. This is an extension of W. Z. Chien’s perturbation method, which uses the central deflection as the perturbation parameter.