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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (3): 298-308.doi: 10.3969/j.issn.1005-3085.2016.03.008

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一类变系数偏微分方程的精确解的求法及其计算机机械化实现(英)

李拔萃   

  1. 中共抚顺市委党校公共管理教研室，辽宁 抚顺  113006
  • 收稿日期:2014-11-17 接受日期:2015-05-04 出版日期:2016-06-15 发布日期:2016-08-15

New Exact Solutions to a Category of Variable-coefficient PDEs and Computerized Mechanization

LI Ba-cui   

  1. Department of Public Management, Party School of CPC Fushun Municipal Committee, Fushun, Liaoning 113006
  • Received:2014-11-17 Accepted:2015-05-04 Online:2016-06-15 Published:2016-08-15

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摘要： 变系数偏微分方程出现在许多物理模型中，在非线性科学领域中有着重要的应用．为了求解某类变系数偏微分方程，本文利用椭圆方程，借助于符号计算软件，构造了辅助椭圆方程方法．新算法的基本思想：只要某个变系数偏微分方程经过合理的变换能变换成椭圆方程的形式，那么该方程的求解问题就会迎刃而解．以变系数Kadomtsev-Petviashvili方程为例，不但说明了该算法的有效性，而且得到了该方程许多新的解，包括暗孤波解、钟形孤波解和雅可比椭圆函数解．这些解可以很好地描述非线性物理现象．

关键词: 变系数偏微分方程, 孤波解, 雅可比椭圆函数解, 符号计算软件, 计算机机械化

Abstract:

The variable-coefficient partial differential equations are not only used in many physical models, but also fundamentally applied in the field of nonlinear science. In order to solve certain variable-coefficient partial differential equations, the auxiliary elliptic-like equation method is introduced in this article by means of the symbolic computation software. The basic idea of the new algorithm is that if certain variable-coefficient partial differential equation can be converted into the form of elliptic equation, then its solutions are readily obtained. By taking the Kadomtsev-Petviashvili equation for an example, not only the effectiveness of the algorithm is demonstrated, but many new solutions are worked out, including dark solitary wave, bell profile solitary wave solutions and Jacobian elliptic function solutions, which may be useful for depicting  nonlinear physical phenomena.

Key words: variable-coefficient partial differential equation, solitary wave solution, Jacobian elliptic function solution, symbolic computation software, computerized mechanization

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