Lie symmetry analysis and exact explicit solutions for general Burgers’ equation

Abstract

a b s t r a c tIn this paper, the Lie symmetry analysis is performed for the general Burgers' equation. The exact solutions and similarity reductions generated from the symmetry transformations are provided. Furthermore, the all exact explicit solutions and similarity reductions based on the Lie group method are obtained, some new method and techniques are employed simultaneously. Such exact explicit solutions and similarity reductions are important in both applications and the theory of nonlinear science.

Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

“…Furthermore, we can show that the convergence of the power series solution (4.52) to Eq. (4.46) (see e.g., [11,12]). Thus, the power series solution (4.52) is the exact analytic solution to this equation.…”

“…In the past few decades, a wealth of methods have been developed to find these exact solutions of the nonlinear PDEs though it is rather difficult. Some of the most important methods are the inverse scattering method [1,2], Darboux and Bäcklund transformations [3,4], Hirota's bilinear method [4][5][6], Lie symmetry analysis [7][8][9][10][11][12][13][14][15][16][17], CK transformation method [18,19], and so on.…”

“…12) where c 1 , c 2 and α = 0 are arbitrary constants.On the other hand, substituting (4.10) into Eq. (1.4), we reduce the NER equation (1.4) to the following ODE bf f 2 + af − α = 0,(4.13) …”

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

“…Among them, Burgers' equation may be the more intriguing one which serves as a prototype model for turbulent flows [3]. Many works in the literature are devoted to investigation of Lie symmetries and the exact solutions of Burgers' equation(s) and its generalizations [4][5][6][7][8][9][10][11][12][13][14]. Beyond its importance in mathematical physics, it appears also as a geodesic equation on diffeomorphism group of a circle with right invariant 2 metric (see [15] and references therein).…”

Lie Symmetries, Conservation Laws and Exact Solutions for Two Rod Equations

“…scattering transformation (IST) [1], Darboux and Bäcklund transformations [2], Hirota's bilinear method [2][3][4], Lie symmetry analysis [5][6][7][8], CK method [9,10], and so on.…”