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.

BEYOND PERTURBATION:THE BASIC CONCEPTS OF THE HOMOTOPY ANALYSIS METHOD AND ITS APPLICATIONS

A new and rather general analytic method for strongly nonlinear problems, namely the homotopy analysis method (HAM), is reviewed. Different from perturbation techniques, the homotopy analysis method is totally independent of small physical parameters, and thus is suitable for most nonlinear problems. Besides, different from all other analytic techniques, it provides us a simple way to ensure the convergence of solution series, so that one can always get accurate enough analytic approximations. Furthermore, different from all other analytic methods, it provides us a great freedom to choose base functions of solution series, so that a nonlinear problem may be approximated more effectively. The homotopy analysis method provides us a completely new way and a different approach to solve nonlinear problems, especially those without small physical parameters. In this review paper, the basic concepts of the homotopy analysis method and its applications in nonlinear mechanics, physics, chemistry, biology, finance, engineering, computational mathematics and so on are discussed, together with its difference and relationship to perturbation techniques, Lyapunov artificial small parameter method, $\delta$-expansion method, Adomian decomposition method, and the so-called homotopy perturbation method.