Symmetry solutions for reaction–diffusion equations with spatially dependent diffusivity

Nonclassical and classical symmetry techniques are employed to analyse a reaction–diffusion equation with a cubic source term. Here, the diffusivity (diffusion term) is assumed to be an arbitrary function of the spatial variable. Classification using Lie point and nonclassical symmetries is performed. It turns out that the diffusivity needs to be given as a quadratic function of the spatial variable for the given governing equation to admit nonclassical symmetries. Both nonclassical and classical symmetries are used to construct some group-invariant (exact) solutions. The results are applied to models arising in population dynamics.