zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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):

$A{y}^{\text{'}\text{'}}\left(t\right)+B{D}_{*}^{3/2}y\left(t\right)+Cy\left(t\right)=f\left(t\right),y\left(0\right)={y}_{0},{y}^{\text{'}}\left(0\right)={y}_{0}^{\text{'}}$

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)