Group classification for one type of space-time fractional quasilinear parabolic equation

In this paper, we present the Lie symmetry analysis for the space-time fractional quasilinear parabolic equation. A group classification is then carried out by investigating the coefficient functions p(t, x, u), q(t, x, u) and s(t, x, u), which includes space-time fractional reaction-diffusion equation, space-time fractional Poisson equation, space-time fractional Black-Scholes equation and space-time fractional cubic Schrödinger equation. We obtain all the Lie symmetries admitted by these equations and use the group generators to reduce these fractional partial differential equations with Riemann-Liouville fractional derivative 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.