RBF‐FD Method for Solving the Generalized Time‐Fractional Burgers Equation With Initial Weak Singularity

Abstract

ABSTRACT This paper develops an efficient numerical algorithm for solving the generalized time‐fractional Burgers equation subject to three classes of boundary conditions, whose solutions possess weak singularities in the vicinity of the initial instant . Traditional numerical approaches employ uniform temporal grids, which inevitably results in deteriorated numerical accuracy. To overcome this defect, the nonuniform ‐ formula is utilized for the discretization of the Caputo fractional derivative. For the spatial component, the radial basis function finite difference scheme is adopted. The established algorithm attains a global convergence accuracy of , with representing the mesh grading parameter, and standing for the order of the Caputo fractional derivative. The stability of the presented scheme is strictly proven through energy analysis, and its convergence property is also investigated systematically. A series of numerical tests and comparative analyses with existing benchmark methods verify the remarkable performance of the proposed algorithm. Moreover, the computational efficiency is demonstrated by the CPU times.