PLK方法与符号运算

戴世强

PLK方法与符号运算

戴世强

上海市应用数学和力学研究所, 上海大学, 上海 200072

基金项目: 国家自然科学基金资助项目(19972035)

详细信息

作者简介:
戴世强(1941- ),男,浙江定海人,教授,博导,从事流体力学和应用数学研究,已发表论文80余篇.

中图分类号: O175
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- 被引次数: 0
出版历程
- 收稿日期: 2000-01-23
- 修回日期: 2000-11-27
- 刊出日期: 2001-03-15

Poincare-Lighthill-Kuo Method and Symbolic Computation

DAI Shi-qiang

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P R China

摘要

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,它是一个非常有效的工具.
- PLK方法 /
- 摄动方法 /
- 符号运算 /
- 中间表达式爆炸 /
- 半逆序算法
Abstract: This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method(PLK method,for short) and symbolic computation.Firstly,the idea and hist ory 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.
- PLK method /
- perturbation methods /
- symbolic computation /
- intermediate expression swell /
- semi-inverse algorithm

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参考文献(41)

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