Conditional Lie–Bäcklund symmetries and solutions of inhomogeneous nonlinear diffusion equations

The second-order conditional Lie–Bäcklund symmetries of nonlinear diffusion equations with variable coefficients are studied. A number of examples are considered and some exact solutions are constructed via the compatibility of conditional Lie–Bäcklund symmetries and the governing equations. These solutions possess the extended forms of the separation of variables, including the extensions of the instantaneous source solutions of the porous medium equations.