Numerical solution of the Bagley-Torvik equation.
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 \textit{R. Gorenflo} and \textit{F. Mainardi} Fractional calculus: Integral and differential equations of fractional order in \textit{A. Carpinteri} and \textit{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.
predictor-corrector methods of Adams-Bashforth-Moulton type
scientific article; zbMATH DE number 1886386