一类四阶偏微分方程的对称约化、精确解和守恒律

doi:10.3969/j.issn.1000-5641.2017.06.005

华东师范大学学报(自然科学版) ›› 2017, Vol. 2017 ›› Issue (6): 50-62.doi: 10.3969/j.issn.1000-5641.2017.06.005

一类四阶偏微分方程的对称约化、精确解和守恒律

张丽香, 刘汉泽, 辛祥鹏

聊城大学数学科学学院, 山东聊城 252059

收稿日期:2016-12-14 出版日期:2017-11-25 发布日期:2017-11-25
通讯作者: 刘汉泽,男,教授,研究方向为微分方程理论及应用.E-mail:hnz_liu@aliyun.com E-mail:hnz_liu@aliyun.com
作者简介:张丽香,女,硕士研究生,研究方向为微分方程理论及应用.E-mail:244630623@qq.com.
基金资助:
国家自然科学基金（11171041，11505090）

Symmetry reductions, exact solutions and conservation laws of a class of forth-order partial differential equations

ZHANG Li-xiang, LIU Han-ze, XIN Xiang-peng

School of Mathematical Sciences, Liaocheng University, Liaocheng Shandong 252059, China

Received:2016-12-14 Online:2017-11-25 Published:2017-11-25

摘要/Abstract

摘要： 利用李群分析研究了一类变系数四阶偏微分方程，求出方程的李点对称，把偏微分方程约化为常微分方程，然后结合（G'/G）展开法及椭圆函数展开法，对约化后的常微分方程求其精确解，从而得到原方程的精确解.进一步，给出这类变系数偏微分方程的守恒律.

关键词: 变系数方程, 李群分析, 精确解, 守恒律

Abstract: The partial differential equation with constant coefficients can merely approximately reflect the law of motion of substances. Relatively the partial differential equation with variable coefficients can reflect the complex movement of substances more accurately. Therefore, it is more important to study the partial differential equations with variable coefficients. This paper investigates a class of variable coefficient partial differential equations. By using Lie symmetry analysis, the symmetries of the equations are obtained, Then the partial differential equations are reduced to ordinary differential equations. Moreover, we combine with (G'/G) expansion method and elliptic function expansion, so exact solutions to the original equation are obtained. Furthermore, the conservation laws of this kind of variable coefficient differential equations are given.

Key words: variable coefficient equation, Lie symmetry analysis, exact solution, conservation law

中图分类号:

O175.2

张丽香, 刘汉泽, 辛祥鹏. 一类四阶偏微分方程的对称约化、精确解和守恒律[J]. 华东师范大学学报(自然科学版), 2017, 2017(6): 50-62.

ZHANG Li-xiang, LIU Han-ze, XIN Xiang-peng. Symmetry reductions, exact solutions and conservation laws of a class of forth-order partial differential equations[J]. Journal of East China Normal University(Natural Sc, 2017, 2017(6): 50-62.

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一类四阶偏微分方程的对称约化、精确解和守恒律

Symmetry reductions, exact solutions and conservation laws of a class of forth-order partial differential equations

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