Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations

Abstract

We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form f(x)ut = [g(x)unux]x. We present a complete classification of Lie symmetries and form-preserving point transformations in the case where f(x) = 1 which is equivalent to the original equation. We also introduce certain nonlocal transformations. When f(x) = xp and g(x) = xq we have the most known form of this class of equations. If certain conditions are satisfied, then this latter equation can be transformed into a constant coefficient equation. It is also proved that the only equations from this class of partial differential equations that admit Lie-Bäcklund symmetries is the well-known nonlinear equation ut = [u-2ux]x and an equivalent equation. Finally, two examples of new exact solutions are given. © 2003 Elsevier Science B.V. All rights reserved.