Lie symmetry solutions of the Generalized burgers equation

Abstract

Burgers equation: u, + UUx = luxx is a nonlinear partial differential equation which arises in model studies of turbulence and shock wave theory. In physical application of shock waves in fluids, coefficient 1 ,has the meaning of viscosity. For light fluids or gases the solution considers the inviscid limit as 1 tends to zero. The solution of Burgers equation can be classified into two categories: Numerical solutions using both finite difference and finite elements approaches; the analytic solutions found by Cole and Hopf In both cases the solutions have been valid for only 0 ~ 1 ~ 1. In this thesis, we have found a global solution to the Burgers equation with no restriction on 1 i.e. 1 E (- 00 , 00). In pursuit of our objective, we have used, the Lie symmetry analysis. The method includes the development of infinitesimal transformations, generators, prolongations, and the invariant transformations of the Burgers equation. We have managed to determine all the Lie groups admitted by the Burgers equation, and used the symmetry transformations to establish all the solutions corresponding to each Lie group admitted by the equation. These solutions, which are appearing in literature for the first time are more explicit and more general than those previously obtained. This is a big contribution to the mathematical knowledge in the application of Burgers equation.