A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations

Abstract

In this paper we propose an efficient and easy-to-implement numerical method for an $α$-th order Ordinary Differential Equation (ODE) when $α∈ (0, 1)$, based on a one-point quadrature rule. The quadrature point in each sub-interval of a given partition with mesh size $h$ is chosen judiciously so that the degree of accuracy of the quadrature rule is 2 in the presence of the singular integral kernel. The resulting time-stepping method can be regarded as the counterpart for fractional ODEs of the well-known mid-point method for 1st-order ODEs. We show that the global error in a numerical solution generated by this method is of the order $\mathcal{O}(h^{2})$, independently of $α$. Numerical results are presented to demonstrate that the computed rates of convergence match the theoretical one very well and that our method is much more accurate than a well-known one-step method when $α$ is small.