Numerical Schemes for Fractional Ordinary Differential Equations

Fractional calculus, which has almost the same history as classic calculus, did not attract enough attention for a long time. However, in recent decades, fractional calculus and fractional differential equations become more and more popular because of its powerful potential applications. A large number of new differential equations (models) that involve fractional calculus are developed. These models have been applied successfully, e.g., in mechanics (theory of viscoelasticity), biology (modelling of polymers and proteins), chemistry (modelling the anomalous diffusion behavior of Brownian particles), electrical engineering (electromagnetic waves) etc (Bouchaud & Georges, 1990; Hilfer, 2000; Kirchner et al., 2000; Metzler & Klafter, 2000; Zaslavsky, 2002; Mainrdi, 2008). Meanwhile, some rich fractional dynamical behavior which reflect the inherent nature of realistic physical systems are observed. In short, fractional calculus and fractional differential equations have played more andmore important role in almost all the scientific fields. One of the most important fractional models is the following initial value problem