*(English)*Zbl 1035.65067

This paper is concerned with the numerical solution of initial value problems for a linear differential equation of fractional order (the so-called Bagley-Torvik equation):

where $A\ne 0$, $B,C$ are real constants and $f$ a given real function. Here ${D}_{*}^{q}$ denotes the fractional differential operator of order $q$ in the sense of Canuto [the authors use the definition given by *R. Gorenflo* and *F. Mainardi* Fractional calculus: Integral and differential equations of fractional order in *A. Carpinteri* and *F. Mainerdi* (ed.), Fractal and Fractional Calculus in Continuum Mechanics: pp. 223–276 (1997; Zbl 0917.73004), Chapter 5].

In the paper under consideration the second order equation is written as an equivalent system of four fractional differential equations of order 1/2 arid then linear multistep methods that approximate the fractional order derivatives and are consistent and stable are proposed. In particular, predictor-corrector methods of Adams-Bashforth-Moulton type are given and some convergence results are established.

##### MSC:

65L05 | Initial value problems for ODE (numerical methods) |

65L06 | Multistep, Runge-Kutta, and extrapolation methods |

65L20 | Stability and convergence of numerical methods for ODE |

34A30 | Linear ODE and systems, general |

26A33 | Fractional derivatives and integrals (real functions) |