Detailed error analysis for a fractional Adams method with graded meshes

We consider a fractional Adams method for solving the nonlinear fractional differential equation 0 C D t α y ( t ) = f ( t , y ( t ) ) , α > 0 \,C₀Dαₜ y(t) = f(t, y(t)), \, α >0 , equipped with the initial conditions y ( k ) ( 0 ) = y 0 ( k ) , k = 0 , 1 , … , ⌈ α ⌉ − 1 y⁽ᵏ⁾ (0) = y₀⁽ᵏ⁾, k=0, 1, , α -1 . Here, α may be an arbitrary positive number and ⌈α⌉ denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 0 C D t α y ∈ C 2 [ 0 , T ] \,C₀Dαₜ y ∈ C²[0, T] , Diethelm et al. (Numer. Algor. 36, 31–52, 2004) introduced a fractional Adams method with the uniform meshes t n = T(n/N),n = 0,1,2,…,N and proved that this method has the optimal convergence order uniformly in t n , that is O(N −2) if α > 1 and O(N −1−α ) if α ≤ 1. They also showed that if 0 C D t α y ( t ) ∉ C 2 [ 0 , T ] \,C₀Dαₜ y(t) ∉ C²[0, T] , the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C m [0,T] for some m ∈ ℕ m ∈ N and 0 < α < m, the Caputo fractional derivative 0 C D t α y ( t ) \,C₀Dαₜ y(t) takes the form “ 0 C D t α y ( t ) = c t ⌈ α ⌉ − α + smoother terms \,C₀Dαₜ y(t) = c tα -α + smoother terms ” (Diethelm et al. Numer. Algor. 36, 31–52, 2004), which implies that 0 C D t α y \,C₀Dαₜ y behaves as t ⌈α⌉−α which is not in C 2[0,T]. By using the graded meshes t n = T(n/N) r ,n = 0,1,2,…,N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t n even if 0 C D t α y \,C₀Dαₜ y behaves as t σ ,0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.