指数同伦法对Cauchy条件下变系数Burgers方程的解析与数值分析

邹丽; 王振; 宗智; 王喜军; 张朔

doi:10.3879/j.issn.1000-0887.2014.07.007

指数同伦法对Cauchy条件下变系数Burgers方程的解析与数值分析

doi: 10.3879/j.issn.1000-0887.2014.07.007

邹丽^1,2,
王振³,
宗智^1,2,
王喜军^1,2,
张朔^1,4

¹工业装备结构分析国家重点实验室(大连理工大学),辽宁大连 116085;²大连理工大学船舶工程学院, 辽宁大连 116085;³大连理工大学数学科学学院, 辽宁大连 116085;⁴大连理工大学航空航天学院, 辽宁大连 116085)

基金项目: 国家自然科学基金（51379033; 51221961；51239002；51309040）;国家重点基础研究发展计划（973计划）（2013CB036101）；中央高校基本科研业务费专项资金（DUTBJS01）

详细信息

作者简介:
邹丽（1981—），女，辽宁盘锦人，副教授，博士(通讯作者. Tel：+86-411-84706373; E-mail: lizou@dlut.edu.cn)

中图分类号: O242.1
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出版历程
- 收稿日期: 2014-01-10
- 修回日期: 2014-05-15
- 刊出日期: 2014-07-15

Analytical and Numerical Investigation of the Variable Coefficient Burgers Equation Under Cauchy Condition With the Exponential Homotopy Method

ZOU Li^1,2,
WANG Zhen³,
ZONG Zhi^1,2,
WANG Xi-jun^1,2,
ZHANG Shuo^1,4

¹State Key Laboratory of Structural Analysis for Industrial Equipment(Dalian University of Technology),Dalian, Liaoning 116085, P.R.China;²School of Naval Architecture, Dalian University of Technology, Dalian, Liaoning 116085, P.R.China;³School of Mathematical Sciences, Dalian University of Technology,Dalian, Liaoning 116085, P.R. China;⁴School of Aeronautics and Astronautics, Dalian University of Technology,Dalian, Liaoning 116085, P.R. China

Funds: The National Natural Science Foundation of China（51379033; 51221961;51239002；51309040）;The National Basic Research Program of China (973 Program)（2013CB036101）

摘要

摘要: 使用近似解析法来研究在给定初始条件和边界条件下变系数Burgers方程，引入一种新式同伦来解决微分方程中由变系数带来的问题，这种新同伦比传统方法计算更高效，并能给出时域上的一致解析表达式.分别计算了有限空间域上变系数Burgers方程的解析解，讨论了在有限空间区域上激波的形成，并对所得解析解进行了范数意义下收敛性研究的探索.基于Lie（李）变换群理论，研究了该方程的对称性质，给出了其无穷小生成子，守恒律和群不变解.文中给出的解是从非线性偏微分方程中直接得到的，未经过行波变换.通过“h-curve”准则探讨了近似解的收敛性.通过有限差分法进行直接数值模拟，已验证该方法的准确性和有效性.
- 变系数Burgers方程 /
- 解析解 /
- 指数同伦法
Abstract: The variable coefficient Burgers equation was studied with an approximate analytical method under the given initial and boundary conditions. A new-form homotopy was introduced to overcome the problem brought by the variable coefficient, this new-form homotopy enhanced the computational efficiency in comparison with the traditional forms, and gave a consistent analytical solution expression in time domain. Analytical solutions to the variable coefficient Burgers equation in finite space domain were determined respectively, and shock wave formation in finite space domain was also discussed. Convergence of the presented analytical solution was explored in the sense of norm. Based on the Lie transformtion group theory, symmetry of the variable coefficient Burgers equation was studied with its infinitesimal generators, conservation law and group invariant solution obtained. The presented solution was directly deduced from the nonlinear partial differential equation without travelling wave transformation. Convergence of the approximate analytical solution was discussed with the so-called‘h-curve’criteria. Direct numerical simulation with the finite difference method proves accuracy and effectiveness of the proposed exponential homotopy method.
- variable coefficient Burgers equation /
- analytical solution /
- exponential homotopy

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参考文献(37)

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指数同伦法对Cauchy条件下变系数Burgers方程的解析与数值分析

doi: 10.3879/j.issn.1000-0887.2014.07.007

作者简介:
邹丽（1981—），女，辽宁盘锦人，副教授，博士(通讯作者. Tel：+86-411-84706373; E-mail: lizou@dlut.edu.cn)

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Analytical and Numerical Investigation of the Variable Coefficient Burgers Equation Under Cauchy Condition With the Exponential Homotopy Method

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指数同伦法对Cauchy条件下变系数Burgers方程的解析与数值分析

doi: 10.3879/j.issn.1000-0887.2014.07.007

作者简介: 邹丽（1981—），女，辽宁盘锦人，副教授，博士(通讯作者. Tel：+86-411-84706373; E-mail: lizou@dlut.edu.cn)

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Analytical and Numerical Investigation of the Variable Coefficient Burgers Equation Under Cauchy Condition With the Exponential Homotopy Method

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出版历程

目录

作者简介:
邹丽（1981—），女，辽宁盘锦人，副教授，博士(通讯作者. Tel：+86-411-84706373; E-mail: lizou@dlut.edu.cn)