Analytic solutions and conservation laws of nonlinear fractional Burgers-type equation with time-dependent diffusion coefficient by Lie group method

In this paper, Lie symmetry analysis method is applied to a nonlinear fractional Burgers-type equation with time-dependent diffusion coefficient, which contains the mixed derivative of the Riemann-Liouville time-fractional derivative and the first order x-derivative. We obtained all the Lie symmetries admitted by the equation and used them to reduce the governing fractional partial differential equation to some fractional ordinary differential equations with Erdélyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.