PLK方法与符号运算  

Poincare-Lighthill-Kuo Method and Symbolic Computation

作　　者：戴世强[1] 

机构地区：[1]上海市应用数学和力学研究所,上海大学上海200072

出　　处：《应用数学和力学》2001年第3期221-227,共7页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目 !(19972 0 35)

摘　　要：阐述将PLK方法与符号运算相结合的途径和有效性·　首先简述PLK方法的思路和发展简史 :其次 ,概述运行符号运算时经常遇到的“中间表达式爆炸”困难 ,为克服这一困难 ,作者提出一种半逆序算法 :通过以符号形式“冻结”中间表达式中冗长的部分 ,到最后阶段再予“解冻” ;并且通过综述作者在一系列非线性波动和非线性振动方面的工作 ,讨论PLK_符号运算方法的具体应用 ,其中 ,Duffing方程的摄动解的计算机延伸表明 ,用PLK方法导得的渐近级数解的收敛半径为 1,从而大大拓广了解的适用范围 ;分层流体中内孤立波和超弹性杆中孤立波对撞的研究表明 ,用所提出的方法可以进行手工计算难以进行的复杂运算 ,借此可得出高阶演化方程和高阶渐近解 ,正确地解释实验结果 ;并说明采用半逆序算法后 ,可在微机上实现繁复的符号运算·　最后得出结论 :借助于符号运算 ,可大大增强PLK方法的生命力 ,至少对保守系统的振动和波动问题的求解 。This paper elucidates the effectiveness of combining the Poincare_Lighthill_Kuo method(PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi_inverse algorithm was proposed by the author, with which the lengthy parts of intermediate expressions are first frozen in the form of symbols till the final stage of seeking perturbation solutions. To discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer_extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head_on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher_order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro_commputers. Finally, it is concluded that with the aid of symbolic computation, the vitality of the PLK method is greatly strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.

关 键 词：PLK方法 摄动方法 符号运算 中间表达式爆炸 边界层 非线性KLEIN-GORDON方程 

分 类 号：O357.4[理学—流体力学] O175.29[理学—力学]