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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

ISSN号：1000-0887
期刊名称：应用数学和力学
时间：2014.7
页码：777-789
分类：O242.1[理学—计算数学;理学—数学]
作者机构：[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
相关基金：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）
相关项目：海洋环境灾害作用与结构安全防护

关键词：变系数Burgers方程, 解析解, 指数同伦法, variable coefficient Burgers equation, analytical solution, exponential homotopy

中文摘要：

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．通过有限善分法讲行盲棒数值樟拟．已聆证该方法的准确件和有效件．

英文摘要：

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 an- alytical solution expression in time domain. Analytical solutions to the variable coefficient Bur- gem 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. Di- rect numerical simulation with the finite difference method proves accuracy and effectiveness of the proposed exponential homotopy method.

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