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On the fractional Adams method. (English) Zbl 1189.65142
Summary: 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: $D^{\alpha}_*y(t)=f(t,y(t))$, $y^{(k)}(0)=y_0^{(k)}$, $k=0,1,\dots,n-1$, where $\alpha >0$, $n=\lceil \alpha\rceil$ 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) $0<\alpha$, $D^{\alpha}_*y(t)\in C^2[0,T]$, (b) $\alpha >1$, $C^{1+\lceil\alpha\rceil}[0,T]$, (c) $0<\alpha <1$, $y\in C^2[0,T]$, (d) $\alpha >1, f\in C^{3}(G)$, there are still some unsolved problems- (i) the error estimates for $\alpha\in (0,1)$, $f\in C^{3}(G)$, (ii) the error estimates for $\alpha\in (0,1)$, $f\in C^{2}(G)$, (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.

##### MSC:
 65L06 Multistep, Runge-Kutta, and extrapolation methods 26A33 Fractional derivatives and integrals (real functions) 34A08 Fractional differential equations 44A15 Special transforms (Legendre, Hilbert, etc.) 45J05 Integro-ordinary differential equations
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##### References:
 [1] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011 [2] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002 [3] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008 [4] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003 [5] Diethelm, K.; Ford, N. J.; Freed, A. D.: Detailed error analysis for a fractional Adams method, Numer. algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be [6] Zhou, T. S.; Li, C. P.: Synchronization in fractional-order differential systems, Physica D 212, 111-125 (2005) · Zbl 1094.34034 · doi:10.1016/j.physd.2005.09.012 [7] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194 [8] Li, C. P.; Peng, G. J.: Chaos in Chen’s system with a fractional order, Chaos solitons fractals 22(, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013 [9] Lubich, C.: Runge--Kutta theory for Volterra and Abel integral equations of the second kind, Math. comput. 41, 87-102 (1983) · Zbl 0538.65091 · doi:10.2307/2007768