Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 5 March 2018
Abstract
Purpose
The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics.
Design/methodology/approach
The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods.
Findings
First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions.
Originality/value
The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.
Keywords
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11661060, 11571008), Natural Science Foundation of Inner Mongolia Autonomous Region of China (2016MS0116) and High Education Science Research Program of Inner Mongolia (NJZY094). The first author would like to express his deep gratitude to Prof Temuer Chaolu who gave valuable suggestions to help finishing the paper.
Citation
Bilige, S. and Han, Y. (2018), "Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 28 No. 3, pp. 518-531. https://doi.org/10.1108/HFF-08-2016-0304
Publisher
:Emerald Publishing Limited
Copyright © 2018, Emerald Publishing Limited