On the fractional Adams method

The generalized Adams-Bashforth-Moulton method, often simply called the fractional Adams method, is a useful numerical algorithm for solving a fractional ordinary differential equation: D*(alpha)y(t) = f (t, y(t)), y((k)) (0) = y(0)((k)), k = 0, 1, ... , n - 1. where alpha > 0, n = inverted right perpendicular alpha inverted left perpendicular is the first integer not less than alpha, and D*(alpha)y(t) is the alpha th-order fractional derivative of y(t) in the Caputo sense. Although error analyses for this fractional Adams method have been given for (a) 0 < alpha, D*(alpha)y(t) is an element of C-2[0, T], (b) alpha > 1, y is an element of C1+inverted right perpendicular alpha inverted left perpendicular [0, T], (c) 0 < alpha < 1, y is an element of C-2[0, T], (d) alpha > 1, f is an element of C-3(G), there are still some unsolved problems-(i) the error estimates for alpha is an element of (0, 1), f is an element of C-3(G), (ii) the error estimates for alpha is an element of (0, 1), f is an element of C-2(G), (iii) the solution y(t) having some special forms. In this paper, we mainly study the error analyses of the fractional Adams method for the fractional ordinary differential equations for the three cases (i)-(iii). Numerical simulations are also included which are in line with the theoretical analysis. (C) 2009 Elsevier Ltd. All rights reserved.