The authors study the variable-order fractional advection-diffusion equation with a nonlinear source term
and initial boundary conditions
where ; represents the average fluid velocity, is the source term which satisfies the Lipschitz condition
and and is a variable-order fractional derivative defined by
where , . The explicit and implicit Euler methods of approximations for (1) are used and the stability and convergence of the methods are well discussed. Furthermore, other numerical methods such as fractional methods of lines, matrix transfer technique and extrapolation method are also presented and well discussed. Finally, numerical examples were given to demonstrate the effectiveness of the theoretical analysis used.