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