A robust numerical method for a fractional differential equation

This paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order α ∈ (0, 1). First the fractional differential equation is transformed into an equivalent Volterra integral equation of the second kind with a weakly singular kernel. Based on the a priori information about the exact solution, an integral discretization scheme on an a priori chosen adapted mesh is proposed. By applying the truncation error estimate techniques and a discrete analogue of Gronwall's inequality, it is proved that the numerical method is first-order convergent in the discrete maximum norm. Numerical results indicate that this method is more accurate and robust than finite difference methods when α is close to 0.