Lie symmetries of generalized Burgers equations: application to boundary-value problems

Abstract

There exist several approaches exploiting Lie symmetries in the reduction of boundary-value problems for partial differential equations modelling real-world phenomena to those problems for ordinary differential equations. Using an example of generalized Burgers equations appearing in nonlinear acoustics we show that that the "direct" procedure of solving boundary-value problems using Lie symmetries firstly described by Bluman is more general and straightforward than the method suggested by Moran and Gaggioli in [J. Eng. Math. 3 (1969), 151-162]. After the group classification of a class of generalized Burgers equations with time-dependent viscosity is performed we solve an associated boundary-value problem using the symmetries obtained.