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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.

65L06Multistep, Runge-Kutta, and extrapolation methods
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
44A15Special transforms (Legendre, Hilbert, etc.)
45J05Integro-ordinary differential equations
Full Text: DOI
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