Time-fractional inhomogeneous nonlinear diffusion equation: Symmetries, conservation laws, invariant subspaces, and exact solutions

In this paper, a class of time-fractional inhomogeneous nonlinear diffusion equation (tFINDE) with Riemann–Liouville fractional derivative is studied. All point symmetries admitted by this equation are derived. The optimal system of one-dimensional subalgebras is classified to perform the symmetry reductions. It is shown that the tFINDE can be reduced to fractional ordinary differential equations (FODEs), including Erdélyi–Kober fractional derivatives. As the results, some explicit group-invariant solutions are obtained. Through nonlinear self-adjointness, all conservation laws admitted by tFINDE arising from these point symmetry groups are listed. The method of invariant subspace is also applied to reduce the tFINDE to a two-dimensional dynamical system (DS). The admitted point symmetries of DS are used to derive the exact solutions of DS, which determine the exact solutions of the original tFINDE.