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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.

35K55Nonlinear parabolic equations
58J72Correspondences and other transformation methods (PDE on manifolds)
58J70Invariance and symmetry properties
35C05Solutions of PDE in closed form
Full Text: DOI
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