DAI Shi-qiang. Poincare-Lighthill-Kuo Method and Symbolic Computation[J]. Applied Mathematics and Mechanics, 2001, 22(3): 221-227.
 Citation: DAI Shi-qiang. Poincare-Lighthill-Kuo Method and Symbolic Computation[J]. Applied Mathematics and Mechanics, 2001, 22(3): 221-227.

# Poincare-Lighthill-Kuo Method and Symbolic Computation

• Rev Recd Date: 2000-11-27
• Publish Date: 2001-03-15
• 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.
•  [1] 戴世强.PLK方法[A].奇异摄动理论及其在力学中的应用[M].(钱伟长主编),北京:科学出版社,1981:33-86. [2] Poincare H.New Methods of Celestial Mechanics[M].NASA TTF-450:English edition,1967. [3] Lighthill M J.A technique for rendering approximate solutions to physical problems uniformly valid[J].Phil Mag,1949,40(5):1179-1120. [4] Kuo Y H.On the flow of an incompressible viscous fluid past a flat plate at moderate Reynolds numbers[J].J Math and Phys,1953,32(1):83-51. [5] Kuo Y H.Viscous flow along a flat plate moving at high supersonic speeds[J].J Aero Sci,1956,23(1):125-136. [6] Tsien H S.The Poincare-Lighthill-Kuo method[J].Advan Appl Math,1956,4(2):281-349. [7] DAI Shi-qiang.On the generalized PLK method and its applications[J].Acta Mech Sinica,1990,6(2):111-118. [8] 戴世强.完全近似法的推广及其应用[J].应用数学和力学,1991,12(3):237-244. [9] 戴世强,Sigalov G F,Diogenov A V.若干强非线性问题的近似解析解[J].中国科学(A辑),1990,33(2):153-162. [10] 戴世强.两个界面孤立波之间的迎撞[J].力学学报,1983,15(6):623-632. [11] 戴世强.一个二流体系统中两对孤立波的相互作用[J].中国科学(A辑),1983,26(11):1007-1017. [12] 戴世强.分层流体中推广的Boussinesq方程和斜相互作用的孤立波,应用数学和力学,1984,5(4):499-509. [13] 戴世强,张社光.分层流体中凸孤立波与凹孤立波的相互作用[J].科学通报,1986,31(1):96-99. [14] 张社光,戴世强.分层流体中不同模式孤立波的迎撞[J].上海工业大学学报,1986,7(4):375-383. [15] 张社光,戴世强.分层流体中相同模式孤立波的迎撞[J].应用数学与计算数学学报,1986,1(1):61-69. [16] 刘宇陆,戴世强.二流体系统中自由面及界面上的二阶椭圆余弦波[J].应用数学和力学,1987,8(6):479-484. [17] 朱勇,戴世强.缓变深度分层流体中的准周期波和准孤立波[J].应用数学和力学,1989,10(3):202-210. [18] 戴世强.关于振荡型的界面孤立波[J].水动力学研究与进展(A辑),1992,7(1):1-6. [19] 朱勇,戴世强.分层流体中gKdV型孤立波的迎撞[J].力学学报,1992,24(1):9-18. [20] 朱勇.一个二流体系统中mKdV型孤立波的迎撞[J].应用数学和力学,1992,13(5):389-399. [21] DAI Shi-qiang,ZHU Yong.Perturbation solution of gKdV equation and interaction of gKdV solitary waves[A].In:S Xiao,X Hu,Eds.Nonlinear Problems in Engineering and Science[C].Beijing,New York:Science Press,1992. [22] 唐苓,戴世强.KdV-Burgers方程的一类渐近解:见:单调激波解[A].黄黔,潘立宙主编:应用数学和力学(钱伟长八十诞辰祝寿文集[M].北京:科学出版社,重庆:重庆出版社,1993,400-404. [23] 臧宏鸣,戴世强.一个非线性振动方程的计算机代数解,上海工业大学学报,1993,14(3):189-197. [24] 臧宏鸣.若干力学问题的计算机代数-摄动研究[D].上海:上海工业大学硕士学位论文,1993. [25] 王明祺,戴世强.Duffing方程摄动解的计算机延伸[J].上海工业大学学报,1994,15(3):384-389. [26] 王明祺,戴世强.一个非线性波动方程的计算机代数-摄动解[J].应用数学和力学,1995,16(5):403-408. [27] ZANG Hong-ming,DAI Shi-qiang.Higher-order solutions for interfacial solitary waves in a two-fluid system[A].In:Editorial Board of Journal of Hydrodynamics,Ed.Proc 1st Int Conf on Hydrodynamics[C].Beijing:China Ocean Press,1994. [28] 戴世强,臧宏鸣.内孤立波的计算机代数研究[J].自然杂志,1995,17(3):177-179. [29] 田梅.Klein-Gordon方程的九阶计算机代数-摄动解[A].见:戴世强,刘曾荣,黄黔主编.现代数学和力学(MMM-Ⅵ)[C].苏州:苏州大学出版社,1995. [30] CHENG You-liang.The evolution equation for second-order internal solitary waves in stratified fluid of great depth[J].J Shanghai Univ,1997,1(2):130-134. [31] CHENG You-liang,DAI Shi-qiang.Higher-order solutions for internal solitary waves via symbolic computation[A].In:H Kim,S H Lee,S J Lee,Eds.Proc 3rd Int Conf on Hydrodynamics[C].Seoul:Ulam Publishers,1998. [32] 程友良.分层流体中孤立波的理论分析和符号运算研究[D].上海:上海大学博士学位论文,1998. [33] DAI Hui-hui,DAI Shi-qiang,HUO Yi.Head-on collision between two solitary waves in a compressible Mooney-Rivlin elastic rod[J].Wave Motion,2000,32(1):93-111. [34] 戴世强.非线性力学问题的计算机代数研究[A].见:戴世强,刘曾荣,黄黔 主编,现代数学和力学(MMM-Ⅵ)[C].苏州:苏州大学出版社,1995. [35] Calmet J,Van Hulzen J A.Computer algebra applications[A].In:B Buchberger,G E Collins,R Loos Eds.Computer Algebra Symbolic and Algebraic Computation[M].Beijing:World Publishing Corporation,1988. [36] Beltzer A I B.Engineering analysis via symbolic computation:a breakthrough[J].Appl Mech Rev,1990,403(6):119-127. [37] Heck A.Introduction to MAPLE[M].New York:Springer-Verlag,1993. [38] Rand H R.Armbruster D.Perturbation Methods,Bifurcation Theory and Computer Algebra[M].New York:Springer-Verlag,1987. [39] 戴世强,臧宏鸣.计算机代数应用中的一个半逆序算法[J].应用数学和力学,1997,18(2):105-111. [40] 戴世强.约化摄动法和非线性波远场分析[J].力学进展,1982,12(2):2-22. [41] 戴世强.两层流体界面上的孤立波[J].应用数学和力学,1982,3(6):721-731.

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