Some high order formulae for approximating Caputo fractional derivatives

In order to approximate the Caputo fractional derivative of order alpha, 0 < alpha < 1, we construct here a new class of formulae on the basis of B-spline interpolation. These new formulae called S1, S2 and S3 have 2 - alpha, 3 - alpha and 4 - alpha order of convergence, respectively. The proposed formulae are as simple as the well-known L1 formula and the main advantage of them lies in the fact that their accuracy is fixed in the whole interval of integration while the previous formulae such as L1-2 have lower accuracy at the beginning of the interval. Hence in comparison with the previous formulae, new ones have better accuracy and their computational costs are comparable. We then modify S2 and S3 formulae for approximating the Caputo fractional derivative of order alpha, 1 < alpha < 2. Some numerical examples as well as two applications in solving fractional ordinary and partial differential equations (PDEs) are provided to demonstrate the applicability and accuracy of the new formulae. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.