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

TL;DR

This paper explores Lie symmetry methods for solving boundary-value problems of generalized Burgers equations, demonstrating a more general approach and applying it to a specific nonlinear acoustics model.

Contribution

It performs a group classification of generalized Burgers equations with time-dependent viscosity and applies the direct Lie symmetry method to solve related boundary-value problems.

Findings

The direct Lie symmetry approach is more general than previous methods.

A classification of symmetries for generalized Burgers equations was achieved.

Explicit solutions to boundary-value problems in nonlinear acoustics were obtained.

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.