Symmetry group approach to the solution of generalized Burgers equation: Ut + UUx = \lambda Uxx

Abstract

Symmetry of a system of differential equations is a transformation that maps any solution to another solution of the system. In Lie’s framework such transformations are groups that depend on continuous parameters and consist of point transformations (point symmetries), acting on the system’s space of independent and dependent variables, or, more generally, contact transformations (contact symmetries), acting on independent and dependent variables as well as on all first derivatives of the dependent variables. Lie groups, and hence their infinitesimal generators, can be naturally prolonged to act on the space of independent variables, dependent variables, and derivatives of the dependent variables. We present a Lie symmetry approach in solving Burgers Equation:Ut + UU x = λUxx which 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 λ, has the meaning of viscosity. So far in both analytic and numerical approaches the solutions have only been established for 0 ≤ λ ≤ 1. In this paper, we give a global solution to the Burgers equation with no restriction on λ i.e. for λ ∈ (−∞, ∞).