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 + UUx = 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 ∈ (−∞,∞).
