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

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.