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

ZOU Li; WANG Zhen; ZONG Zhi; WANG Xi-jun; ZHANG Shuo

doi:10.3879/j.issn.1000-0887.2014.07.007

Volume 35 Issue 7

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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

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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

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）

Received Date: 2014-01-10
Rev Recd Date: 2014-05-15
Publish Date: 2014-07-15

Abstract

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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References(37)

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