Symmetries, conservation laws and exact solutions of the time-fractional diffusivity equation via Riemann-Liouville and Caputo derivatives

In this paper, we deal with the diffusivity equation, by using the Lie group analysis method. The infinitesimal generators of this equation are investigated. The concept of non-linear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on diffusivity equation with the Riemann-Liouville and Caputo time-fractional derivatives. It is shown that this equation is non-linearly self-adjoint and therefore desired conservation laws can be obtained using appropriate formal Lagrangian. Lie group method provides an efficient tool to solve time-fractional diffusivity equation. For this, the similarity reductions are performed with the similarity variables obtained from symmetry operators. As a result, the reduced fractional ordinary differential equations (FODEs) are deduced, and some group invariant solutions in the explicit form are obtained as well. We apply the invariant subspace method for constructing more particular solutions of the desired equation. It allows one to reduce an fractional partial differential equation to a system of non-linear FODEs.