Detailed error analysis for a fractional Adams method with graded meshes |
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Authors: | Yanzhi Liu Jason Roberts Yubin Yan |
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Institution: | 1.Department of Mathematics,University of Chester,Ince,UK;2.Department of Mathematics,Lvliang University,Lüliang,People’s Republic of China |
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Abstract: | We consider a fractional Adams method for solving the nonlinear fractional differential equation \(\,^{C}_{0}D^{\alpha }_{t} y(t) = f(t, y(t)), \, \alpha >0\), equipped with the initial conditions \(y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots , \lceil \alpha \rceil -1\). 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 \(\,^{C}_{0}D^{\alpha }_{t} y \in C^{2}0, T]\), 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 \(\,^{C}_{0}D^{\alpha }_{t} y(t) \notin C^{2}0, T]\), 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 \(m \in \mathbb {N}\) and 0 < α < m, the Caputo fractional derivative \(\,^{C}_{0}D^{\alpha }_{t} y(t) \) takes the form “\(\,^{C}_{0}D^{\alpha }_{t} y(t) = c t^{\lceil \alpha \rceil -\alpha } + \text {smoother terms}\)” (Diethelm et al. Numer. Algor. 36, 31–52, 2004), which implies that \(\,^{C}_{0}D^{\alpha }_{t} y \) behaves as t ?α??α which is not in C 20,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 \(\,^{C}_{0}D^{\alpha }_{t} y\) behaves as t σ ,0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results. |
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