On the fractional Adams method

Abstract

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 ∗ α y ( t ) = f ( t , y ( t ) ) , y ( k ) ( 0 ) = y 0 ( k ) , k = 0 , 1 , … , n − 1 , where α > 0 , n = ⌈ α ⌉ is the first integer not less than α , and D ∗ α y ( t ) is the α 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 < α , D ∗ α y ( t ) ∈ C 2 [ 0 , T ] , (b) α > 1 , y ∈ C 1 + ⌈ α ⌉ [ 0 , T ] , (c) 0 < α < 1 , y ∈ C 2 [ 0 , T ] , (d) α > 1 , f ∈ C 3 ( G ) , there are still some unsolved problems—(i) the error estimates for α ∈ ( 0 , 1 ) , f ∈ C 3 ( G ) , (ii) the error estimates for α ∈ ( 0 , 1 ) , f ∈ 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.