ZOU Li, WANG Zhen, ZONG Zhi, WANG Xi-jun, ZHANG Shuo. Analytical and Numerical Investigation of the Variable Coefficient Burgers Equation Under Cauchy Condition With the Exponential Homotopy Method[J]. Applied Mathematics and Mechanics, 2014, 35(7): 777-789. doi: 10.3879/j.issn.1000-0887.2014.07.007
Citation: ZOU Li, WANG Zhen, ZONG Zhi, WANG Xi-jun, ZHANG Shuo. Analytical and Numerical Investigation of the Variable Coefficient Burgers Equation Under Cauchy Condition With the Exponential Homotopy Method[J]. Applied Mathematics and Mechanics, 2014, 35(7): 777-789. doi: 10.3879/j.issn.1000-0887.2014.07.007

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

doi: 10.3879/j.issn.1000-0887.2014.07.007
Funds:  The National Natural Science Foundation of China(51379033; 51221961;51239002;51309040);The National Basic Research Program of China (973 Program)(2013CB036101)
  • Received Date: 2014-01-10
  • Rev Recd Date: 2014-05-15
  • Publish Date: 2014-07-15
  • 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.
  • loading
  • [1]
    Sauchder P L.Nonlinear Diffusive Waves[M]. New York: Cambridge University Press, 1987.
    [2]
    Scott J F. The long time asymptotics of solutions to the generalized Burgers equation[J].Proceedings of the Royal Society of London, Series A ,1981,373(1755): 443-456.
    [3]
    Crighton D G, Scott J F. Asymptotic solution of model equations in nonlinear acoustic[J].Phil Trans R Soc Lond, Series A,1979,292(1389): 101-134.
    [4]
    ZHANG Hui. Global existence and asymptotic behavior of the solution of a generalized Burger’s equation with viscocity[J].Computers and Mathematics With Applications,2001,41(5/6): 589-596.
    [5]
    黄磊, 孙建安, 豆福全, 段文山, 刘兴霞. (3+1)维非线性Burgers系统的新的分离变量解及其局域激发结构与分形结构[J].物理学报, 2007,56(2): 611-619.(HUANG Lei, SUN Jian-an, DOU Fu-quan, DUAN Wen-shan, LIU Xing-xia. New variable separation solutions, localized structures and fractals in the (3+1)-dimensional nonlinear Burgers system[J].Acta Physica Sinica,2007,56(2): 611-619.(in Chinese))
    [6]
    石玉仁, 吕克璞, 段文山, 杨红娟. 变系数Burgers方程的精确解[J]. 兰州大学学报(自然科学版), 2005,41(4): 107-111.(SHI Yu-ren, Lü Ke-pu, DUAN Wen-shan, YANG Hong-juan. Exact solutions to Burgers equation with variable coefficients[J].Journal of Lanzhou University(Natural Sciences),2005,41(4): 107-111.(in Chinese))
    [7]
    史秀珍, 斯仁道尔吉. 变系数Burgers方程与KdV-Burgers方程的试探函数解[J].内蒙古大学学报(自然科学版), 2012,43(1): 23-26.(SHI Xiu-zhen, Sirendaoerji. Trial function solutions of the variable coefficients Burgers equation and the KdV-Burgers equation[J].Journal of Inner Mongolia University(Natural Sciences),2012,43(1): 23-26.(in Chinese))
    [8]
    石玉仁, 汪映海, 杨红娟, 吕克璞, 段文山. 广义变系数Burgers方程的精确解[J]. 华东师范大学学报(自然科学版), 2006,2006(5): 27-33.(SHI Yu-ren, WANG Ying-hai, YANG Hong-juan, L Ke-pu, DUAN Wen-shan. Exact solution of generalized Burgers’ equation with variable coefficients[J].Journal of East China Normal University(Natural Sciences),2006,2006(5): 27-33.(in Chinese))
    [9]
    鲜大权, 戴正德. 应用指数函数法求解变系数耦合Burgers系统[J]. 应用数学学报, 2010,33(3): 559-565.(XIAN Da-quan, DAI Zheng-de. Application of exp-function method to coupled Burgers equation with variable coefficients[J].Acta Mathematicae Applicatae Sinica,2010,33 (3): 559-565.(in Chinese))
    [10]
    Vaganan B M, Jeyalakshmi T. Generalized Burgers equations transformable to the Burgers equation[J].Studies in Applied Mathematics, 2011,127(3): 221-220.
    [11]
    QU Chang-zheng, WANG Ai-qin. The complete integrability of variable-coefficient Burgers equations[J].Communications in Theoretical Physics,1996,26(3): 369-372.
    [12]
    Liao S J.Beyond Pertubation: Introduction to Homotopy Analysis Method[M]. London: Chapman & Hall/CRC, 2004.
    [13]
    姜丙利, 柳银萍. 带预测参数的同伦分析方法及其在两个非线性系统中的应用[J]. 华东师范大学学报(自然科学版), 2013,2013(3): 131-139, 148.(JIANG Bing-li, LIU Yin-ping. Predictor homotopy analusis method and its application to two nonlinear systems[J].Journal of East China Normal University(Natural Sciences),2013,2013(3): 131-139, 148.(in Chinese))
    [14]
    宋辉, 李芬, 徐献芝. 电池系统建模中Butler-Volmer方程的同伦分析求解[J]. 应用数学和力学, 2013,34(4): 373-382.(SONG Hui, LI Fen, XU Xian-zhi. Analytical solution of Butler-Volmer equation in battery system modeling[J].Applied Mathematics and Mechanics,2013,34(4): 373-382.(in Chinese))
    [15]
    S·侯斯纳因, A·梅姆德, A·阿里. 二阶流体在旋转坐标系中的三维管道流动[J]. 应用数学和力学, 2012,33(3): 280-291.(Hussnain S, Mehmood A, Ali A. Three dimensional channel flow of second grade fluid in a rotating frame[J].Applied Mathematics and Mechanics,2012,33(3): 280-291.(in Chinese))
    [16]
    韩祥临, 欧阳成, 宋涛, 戴孙圣. 交通拥堵迁移问题的同伦分析法[J]. 物理学报, 2013,62(17): 170203.(HAN Xiang-lin, OUYANG Cheng, SONG Tao, DAI Sun-sheng. The homotopy analusis method for a class of jamming transition problem in traffic flow[J].Acta Physica Sinica,2013,62(17): 170203.(in Chinese))
    [17]
    王玉兰, 朝鲁. 利用再生核解一类变系数偏微分方程[J]. 应用数学和力学, 2008,29(1): 118-126.(WANG Yu-lan, CHAO Lu. Partial differential equation with variable coefficients[J].Applied Mathematics and Mechanics,2008,29(1): 118-126.(in Chinese))
    [18]
    朱倩, 商学利, 陈文振. 六组点堆中子动力学方程组的同伦分析解[J]. 物理学报, 2012,61(7): 070201.(ZHU Qian, SHANG Xue-li, CHEN Wen-zhen. Homotopy analysis solution of point reactor kinetics equations with six-group delayed neutrons[J].Acta Physica Sinica,2012,61 (7): 070201.(in Chinese))
    [19]
    钟敏玲, 刘秀湘. 脉冲时滞Hassell-Varley-Holling功能性反应捕食者-食饵系统周期解存在的充要条件[J]. 应用数学学报, 2012,35(2): 297-308.(ZHONG Min-ling, LIU Xiu-xiang. Necessary and sufficient conditions for the existence of periodic solutions in an impulsive predator-prey system with Hassell-Varley-Holling response[J].Acta Mathematicae Applicatae Sinica,2012,35(2): 297-308.(in Chinese))
    [20]
    司新辉, 郑连存, 张欣欣, 司新毅. 微极性流体在上下正交移动的渗透平行圆盘间的流动[J]. 应用数学和力学, 2012,33(8): 907-918.(SI Xin-hui, ZHENG Lian-cun, ZHANG Xin-xin, SI Xin-yi. Flow of a micropolar fluid between two orthogonally moving porous disks[J].Applied Mathematics and Mechanics,2012,33(8): 907-918.(in Chinese))
    [21]
    李永强, 张晨辉, 刘玲, 段俐, 康琦. 微重力下圆管毛细流动解析近似解研究[J]. 物理学报, 2013,62(4): 044701.(LI Yong-qiang, ZHANG Chen-hui, LIU Ling, DUAN Li, KANG Qi. The analytical approximate solutions of capillary flow in circular tubes under microgravity[J].Acta Physica Sinica,2013,62(4): 044701.(in Chinese))
    [22]
    郑敏毅, 胡辉, 郭源君, 孙光永. 应用优化的同伦分析法求解非线性Jerk方程[J]. 振动与冲击, 2012,31(5): 21-25.(ZHENG Min-yi, HU Hui, GUO Yuan-jun, SUN Guang-yong. Optimal homotopy analysis method applied to solve a nonlinear Jerk equation[J].Journal of Vibration and Shock,2012,31(5): 21-25.(in Chinese))
    [23]
    Fletcher C A J. Burgers equation: a model for all reasons[C]//Noye J ed.Numerical Solutions of Partial Differential Equations. Amsterdam: North-Holland, 1982.
    [24]
    Cole J D. On a quaslinear parabolic equations occurring in aerodynamics[J].Quart Appl Math,1951,9: 225-236.
    [25]
    Hopf E. The partial differential equation[J].Communications on Pure and Applied Mathematics,1950,3(3): 201-230.
    [26]
    Olver P J.Applications of Lie Groups to Differential Equations[M]. New York: Springer-Verlag, 1993.
    [27]
    Bluman G, Anco S.Symmetry and Integration Methods for Differential Equations[M]. New York : Springer, 2002.
    [28]
    Ibragimov N H.A Practical Course in Differential Equations and Mathematical Modelling[M]. Beijing: Higher Education Press, 2009.
    [29]
    Cheviakov A, Bluman G. Multidimensional partial differential equations systems: nonlocal symmetries, nonlocal conservation laws, exact solutions[J].Journal of Mathematics and Physics,2010,51(10): 103522.
    [30]
    Qu C Z. Allowed transformations and symmetry classes of variable coefficient Burgers equations[J].IMA Journal of Applied Mathematics,1995,54(3): 203-225.
    [31]
    Sophocleous C. Transformation properties of a variable-coefficient Burgers equation[J].Chaos, Solitons & Fractals,2004,20(5): 1047-1057.
    [32]
    Pocheketa O A, Popovych R O. Reduction operators and exact solutions of generalized Burgers equations[J].Physics Letters A,2012,376(45): 2847-2850.
    [33]
    Abd-el-Maleka M B, El-Mansi S M A. Group theoretic methods applied to Burgers’ equation[J].Journal of Computational and Applied Mathematics,2000,115(1/2): 1-12.
    [34]
    Kutluay S, Bahadir A R, Ozdes A. Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods[J].Journal of Computational and Applied Mathematics,1999,103(2): 251-261.
    [35]
    Ozis T, Aksan E N, Ozdes A. A finite element approach for solution of Burgers Equation[J].Applied Mathematics and Computation,2003,139(2/3): 417-428.
    [36]
    Kadalbajoo M K, Awasthi A. A numerical method based on Crank-Nicolson scheme for Burgers’equation[J].Applied Mathematics and Computation,2006,182(2): 1430-1442.
    [37]
    Hon Y C, Mao X Z. An efficient numerical scheme for Burgers’equation[J].Applied Mathematics and Computation,1998,95(1): 37-50.
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (1034) PDF downloads(875) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return