Fractional dynamics of one-dimensional linear chain

Abstract

We consider a finite one-dimensional linear chain of N identical masses connected by Hookean springs. A classical treatment for this system is considered before the second order time derivative of the equations of motions is changed into Caputo time fractional derivative of arbitrary order α, where 1 < α < 2. Laplace transformation was used to get a solution for the motion of particles in terms of the generalized Mittag-Leffler function. Diffusion equation is also considered for the same system, using a Caputo fractional derivative of order 0 < α < 1. Laplace transformation is also used and the solution is also expressed in the Mittag-Leffler function.