On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions

Under study in this paper is a time fractional generalized nonlinear diffusion equation which can be better to express diffusion phenomena than diffusion equation of integer order. Firstly, we apply the symmetry analysis method to find the symmetry of this considered equation. Then some conservation laws can also be constructed through the above obtained symmetry with the help of the Noether's theorem. Next, we reduce this equation into an ordinary differential equation of fractional order in the symmetry with Erdelyi-Kober fractional differential operator under one-dimensional subalgebras optimal system framework. Finally, some exact solutions contain the invariant solutions have found for this given equation. The results give us a new interpretation of this type diffusion process.