On the fractional Adams method. (English) Zbl 1189.65142

Li, Changpin; Tao, Chunxing

Comput. Math. Appl. 58, No. 8, 1573-1588 (2009).

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 for ordinary differential equations
26A33	Fractional derivatives and integrals
34A08	Fractional ordinary differential equations and fractional differential inclusions
44A15	Special integral transforms (Legendre, Hilbert, etc.)
45J05	Integro-ordinary differential equations

Keywords:

Adams-Bashforth-Moulton method; Caputo fractional derivative

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