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Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations. (English) Zbl 1024.35042
Summary: We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form \(f(x)u_t=[g(x)u^nu_x]_x\). 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 \(f(x)=x^p\) and \(g(x)=x^q\) 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 \(u_t=[u^{-2}u_x]_x\) and an equivalent equation. Finally, two examples of new exact solutions are given.

MSC:
35K55 Nonlinear parabolic equations
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
58J70 Invariance and symmetry properties for PDEs on manifolds
35C05 Solutions to PDEs in closed form
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