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 \(A\neq 0\), \(B,C\) 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.


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
26A33 Fractional derivatives and integrals


Zbl 0917.73004