Efficient solution of a vibration equation involving fractional derivatives

Attila Pálfalvi

Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980's, and they grow more and more popular nowadays. However, their efficient numerical calculation is nontrivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.

Mots clés: Vibration; Fractional derivative; Fractional differential equation; Adomian decomposition

Attila Pálfalvi. Efficient solution of a vibration equation involving fractional derivatives. International Journal of Non-Linear Mechanics, 2010, 45 (2), pp.169. ⟨10.1016/j.ijnonlinmec.2009.10.006⟩. ⟨hal-00607494⟩