A lattice Boltzmann method(LBM) for two-dimensional fractional reaction-diffusion equations involving Riemann-Liouville derivatives is proposed. Firstly, fractional integral terms in equations are discretized. Then, with Chapman-Enskog analysis and Taylor-expansion technology two-dimensional fractional reaction-diffusion equations are recovered from the LB model. Equilibrium state of distribution function in each velocity direction is derived. Finally, the LB model is tested with numerical examples. It was found that numerical results agree well with analytical solutions.

A D1Q3 evolution model of lattice Boltzmann method (LBM) is established to numerically solve a class of spatial fractional convection-diffusion equation in Riesz sense. By discretizing the integral term of fractional order operator, the fractional convection-diffusion equation is transformed into a standard one with Riesz derivative. With Taylor expansion, Chapman-Enskog and multi-scales expansion, equilibrium distribution functions of the model are derived in all directions. Furthermore, the macroscopic equation to be solved is recovered correctly. Finally, numerical experiments are carried out to verify the model.