  In this paper, based on the developed nonlinear fourth-order operator and
method of order reduction, a novel fourth-order compact difference scheme is
constructed for the mixed-type time-fractional Burgers' equation, from which
$L_1$-discretization formula is employed to deal with the terms of fractional
derivative, and the nonlinear convection term is discretized by nonlinear
compact difference operator. Then a fully discrete compact difference scheme
can be established by approximating spatial second-order derivative with
classic compact difference formula. The convergence and stability are
rigorously proved in the $L^{\infty}$-norm by the energy argument and
mathematical induction. Finally, several numerical experiments are provided to
verify the theoretical analysis.
