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Numerical solution of the Bagley-Torvik equation. (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):

Ay '' (t)+BD * 3/2 y(t)+Cy(t)=f(t),y(0)=y 0 ,y ' (0)=y 0 '

where A0, 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:
65L05Initial value problems for ODE (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
34A30Linear ODE and systems, general
26A33Fractional derivatives and integrals (real functions)