On a numerical scheme for solving differential equations of fractional order

In this work, on the basis of a modified expansion formula obtained in Atanackovic and Stankovic [Atanackovic, T.M., Stankovic, B., 2004. An Expansion formula for fractional derivatives and its applications. Fractional Calculus and Applied Analysis 7(3), 365–378], we propose a numerical procedure for solving differential equations with fractional derivative by transforming the original system into a system of ordinary differential equations of the first order. Our method is different from the widely used method of Yuan and Agarwal [Yuan, L., Agrawal, O. P., 2002. A numerical scheme for dynamic systems containing fractional derivatives. Journal of Vibration and Acoustics 124, 321–324] and overcomes difficulties in satisfying the initial conditions that where noted by Schmidt and Gaul [Schmidt, A., Gaul, L., 2006. On a critique of a numerical scheme for calculation of fractionally damped dynamical systems. Mechanics Research Communications 33, 99–107]. We tested our procedure on several examples. The results show good agreement with the results obtained by other methods.