Error Analysis and Numerical Investigation of an L1-2 Fourth-Order Difference Scheme for Solving the Time-Fractional Burgers Equation

This paper presents a finite difference approach for solving the time-fractional Burgers’ equation, which is a model for nonlinear flow with memory effects. The method leverages the L1-2 formula for the fractional derivative and provides a novel linearization strategy to efficiently transform the system into a stable linear problem. Rigorous analysis establishes the existence, uniqueness, and pointwise-in-time convergence of the numerical solution in the L2 norm. The proposed formulation achieves second-order time accuracy and fourth-order spatial accuracy under smooth initial conditions, with numerically verified temporal convergence rates of O(τ1+α+τ2tnα−2) for solutions with weak singularities. Critically, numerical findings demonstrate that the method is robust and highly efficient, offering high-resolution solutions at a substantially lower computational cost than equivalent graded-mesh formulations.