An Efficient Numerical Scheme for a Time-Fractional Burgers Equation with Caputo-Prabhakar Derivative

This paper presents a numerical method to solve a time-fractional Burgers equation, achieving order of convergence $(2-α)$ in time, here $α$ represents the order of the time derivative. The fractional derivative is modeled by Caputo-Prabhakar (CP) formulation, which incorporates a kernel defined by the three-parameter Mittag-Leffler function. Finite difference methods are employed for the discretization of the derivatives. To handle the non-linear term, the Newton iteration method is used. The proposed numerical scheme is proven to be stable and convergent in the $L_{\infty}$ norm. The validity of the theory is supported by two numerical examples.