Fractional order state equations for the control of viscoelasticallydamped structures

Solution of the Fundamental Linear Fractional Order Differential Equation

“…Letting x =-at q , this becomes (Bagley and Calico, 1991), which is similar to, but not quite the same as Equation (8). The Laplace transform of this Mittag-Leffler function can also be obtained via term by term transform of series (10b), that is…”

“…Bagley and Calico (1991) obtain a solution in terms of Mittag-Leffler functions. Miller and Ross (1993) obtain a solution in terms of the fractional derivative of the exponential function.…”

Fractional high order methods for the nonlinear fractional ordinary differential equation

“…This topic has received a great 1 Email address :3425linran@sohu.com 2 Corresponding author. Email address :fwliu@xmu.edu.cn,f.liu@qut.edu.au deal of attention in the last decade [7,15,[22][23][24], especially in the fields of viscoelastic materials [1,10,11,27], electrochemical processes [8], dielectric polarization [28], colored noise [29], anomalous diffusion, signal processing [21], control theory [24], advection and dispersion of solutes in natural porous or fractured media [2,3] and chaos [20]. Djrbasjan et al [6] considered the Cauchy problem with multi-term fractional derivatives, and proved the Cauchy problem has a unique solution.…”

Chaos synchronization in fractional differential systems

“…But, in recent decades, this has changed. It was found that fractional calculus is useful, even powerful, for modelling viscoelasticity [4], electromagnetic waves [5], boundary layer effects in ducts [6], quantum evolution of complex systems [7], distributed-order dynamical systems [8] and others. That is, the fractional differential systems are more suitable to describe physical phenomena that have memory and genetic characteristics.…”