Group Classification Invariant Solutions of Burgers' Equation

  • Mohammed Adam Abdualah Khatir
  • Mohammed Ali Basher
  • Blegiss Abdulaziz Abdulrahman Ebyed
Keywords: Infinitesimal generator, Symmetries of Burgers’ equation, optimal system, group classification of algebra


The aims of the present paper is to solve the problem of the group classification of the general Burgers’ equation u_t=f(x,u) u_x^2+g(x,u)u_xx, where f and g are arbitrary smooth functions of the variables x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification: We followed the analysis mathematical method using the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.


Bluman G.W. . S. Kumei. Symmetries and Differential Equations, Springer-Verlage, World Publishing Corp., 1989.

Cantwell B.J. . Introduction to Symmetry Analysis. Cambrige University Press, 2002.

Gandarias M.L. , M. Torrisi. A. Valenti. Symmetry classification and optimal systems of a non-linear wave equation. Lnt. J. Nonlinear Mech 39 (2004)389398.

Gardner C.S. , J. M. Greene. M. . kruskal. R. M. Miura, Method for solving the Kortewegde Vries equation, Phys. Rev. Lett.19 (1967) 10951097.

Hirota R. . J. Satsuma, A variety of nonlinear network equations generated from the Ba ̈cklund transformation for the Tota lattice, Sappl. Prog. Theor. Phys . 59(1976) 64100.

Ibragimov N.H. . M. Tottisi. And A. Valenti. Preliminary group classification of equations u_xx=f(x,u_x ) u_xx+g(x,u_x ).J. Math. Phys, 32. No .11:2988.2995,1991

Ibragimov N.H., M. Tottisi. And A. Valenti. Differential invariants of nonlinear equations u_tt=f(x,u_x ) u_xx+g(x,u_x ), Communications in Nonlinear Science and and Numerical Simulation 9 (2004) 6980.

Lie S. ; Arech. For Math. 6,328 (1881)

Li Y.S. Solution and integrable system, in: Advanced Series in Nonlinear Science, Shanghai Scientific and Technological Education Publishing House. Shang Hai. 1999 (in Chinese) .

Liu H. , Jibm Li and Quanxin Zhangb. Lie symmetry analysis and exact explicit solutions for general Burgers' equation. Journal of computational and Applied Mathematics (2008). 2008.06.009.

Maluleke G.H. , D. P. Masom, Optimal system and group invariant solutions for the nonlinear wave equation, Communications in Nonlinear Science and Numerical Simulation 9 (2004) 93101.

Nadjalikhah M., . Symmetries of Burgers' equation. Adv, appl. Clifford alg. DOI 10.1007s0006-003-0000, 2008.

Nadjafikhah M., . Classification of similarity solutions for inviscid Burgers' equation. Accepted for Adv. Appl. Cliford alg., DOI 10.1007s00006-003-0000.

Olver P.J. , Applications of Lie group to Differential Equations, in. Graduate text Maths, vol. 107, Springer, New York, 1986.

Olver P.J. , Equivalence Invariants and Symmetry, Cambridge University Press, Cambridge. (1995).

Ovsiannikov L.V. . Group Analysis of Differential Equations. Academic Press.New York, 1982.

Popovych R.O. , N.M.Ivanova, New results on group classification of nonlinear diffusion-convection equations. J. Phys. A: Math. Gen, 37 (2004), 7547-7565.

Song L. , and Hongging zhang, Preliminary group classification for the nonlinear wave equation u_tt=f(x,u) u_xx+g(x,u), nonlinear Analysis, (2008). doi. 2008.07.008.

Svirshchevskii S.R. , Group classification and invariant solutions of nonlinear polyharmonic equations, Diffcr. Equ.Diff. Uravn, 29 (1993), 1538-1547.