A Fourth-Order Difference Scheme for Solving the Generalized Nonlinear Time-Fractional Burgers-Type Equation

In the present study, we propose a new nonlinear finite difference method for solving the generalized nonlinear time-fractional Burgers-type equation, which can achieve fourth-order accuracy in the spatial direction. The L1 formula is employed to discretize the time-fractional derivative on a graded mesh. Spatial discretization is accomplished by introducing a nonlinear fourth-order difference operator and a linear compact difference operator, and ultimately a nonlinear difference scheme with a temporal accuracy of order 2−α and a spatial accuracy of the fourth order is deduced. For the proposed difference scheme, the existence and boundedness of its solution have been theoretically verified; meanwhile, combined with the cut-off function method, the uniqueness and convergence of the solution to this scheme are further proved. The optimal convergence result is attained under the L2 norm. Eventually, two numerical examples are provided, both of which match the theoretical analysis well.