Efficient solution of a vibration equation involving fractional derivatives

Abstract

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.

Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

“…Among them, Padovan and Sawicki [15] discussed the long time behavior of Duffing oscillator endowed with fractional derivative damping using perturbation method and examined the influence of fractional order on the frequency amplitude response behavior. Palfalvi [16] provided a computationally efficient solution method for the fractionally damped vibration equation using the Adomian decomposition method and Taylor series.…”

Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives

“…Cao et al [20] simulated the fractional-order Duffing equation and investigated the effects of the fractional-order parameters on system dynamics using phase curves, bifurcation diagram and Poincaré map. Palfalvi [21] presented an improved Adomian decomposition method to solve fractional-order differential equation with sine excitation. Sheu et al [22] solved the fractional-order damped Duffing equations by transforming them into a set of fractionalorder integral equations.…”

Cauchy problems for some classes of linear fractional differential equations

“…We refer to [4] and [20] for other methods applied to this equation. Our considerations in previous subsections show that we can take α 3 = 2, …”