Generalized Burgers equations transformable to the Burgers equation

Abstract

Using the mappings which involve first-order derivatives, the Burgers equation with linear damping and variable viscosityis linearized to several parabolic equations including the heat equation, by applying a method which is a combination of Lie's classical method and Kawamota's method. The independent variables of the linearized equations are nott,xbutz(x,t), τ(t), wherezis the similarity variable. The linearization is possible only when the viscosityΔ(t)depends on the damping parameterαand decays exponentially for larget. And the linearization makes it possible to pose initial and/or boundary value problems for the Burgers equation with linear damping and exponentially decaying viscosity. Bäcklund transformations for the nonplanar Burgers equation with algebraically decaying viscosity are also reported. © 2011 by the Massachusetts Institute of Technology.