Solving time fractional Keller–Segel type diffusion equations with symmetry analysis, power series method, invariant subspace method and q-homotopy analysis method

Abstract

In this paper, we investigate the solutions and conservation laws of Keller–Segel (KS) type time fractional diffusion equations. Lie symmetries admitted by this fractional system in Riemann–Liouville sense are derived by means of symmetry analysis. Similarity reductions are performed to construct the group invariant solution and the power series solution is deduced with the help of Erdélyi–Kober (EK) differential operator. Based on the above symmetries, conservation laws are discussed by virtue of the generalized Noether theorem. In addition, analytical solution and numerical solution to initial value problems of time fractional Keller–Segel equations in Caputo sense are established utilizing invariant subspace method and q-homotopy analysis method (q-HAM), respectively.