Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

Abstract

The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIλDμ controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science are also described by differential equations of fractional order. The solution of the differential equation containing a fractional derivative is much involved. Instead of an application of the existing methods, an attempt has been made in the present analysis to obtain the solution of an equation in a dynamic system whose damping behavior is described by a fractional derivative of order 1/2 by the relatively new Adomian decomposition method. The results obtained by this method are then graphically represented and compared with those available in the work of Suarez and Shokooh [Suarez, L. E., and Shokooh, A., 1997, “An Eigenvector Expansion Method for the Solution of Motion Containing Fraction Derivatives,” ASME J. Appl. Mech., 64 , pp. 629–635]. A good agreement of the results is observed.