Detailed Error Analysis for a Fractional Adams Method

We investigate a method for the numerical solution of the nonlinear fractional differential equation D _* y(t)=f(t,y(t)), equipped with initial conditions y ^(k)(0)=y ₀ ^(k), k=0,1,...,–1. Here may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.