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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 $A\ne 0$, $B,C$ 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 {\it R. Gorenflo} and {\it F. Mainardi} Fractional calculus: Integral and differential equations of fractional order in {\it A. Carpinteri} and {\it 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.

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)