In this paper we consider a class of nonlinear dynamical systems with fractional damping. The problem is transformed into a system of fractional-order differential equations. A computationally effective fractional predictor-corrector method is used to simulate and examine the effects and solution behavior of the nonlinear dynamical systems with fractional damping for extensible and inextensible pendulum. The corresponding error analysis is derived. Finally, some numerical examples are given. This method and technique can be applied to solve other fractional-order ordinary differential equations.
BeyerH. and KempfleS., Definition of physically consistent damping laws with fractional derivatives. ZAMM75, (1995), 623–635.
2.
DiethelmK. and FordN.J., Numerical solution of linear and non-linear fractional differential equations involving fractional derivatives of several orders, Numerical Analysis Report No. 379, (2001).
3.
DiethelmK. and FordN.J., Numerical Solution of the Bagley-Torvik Equation, BIT42, (2002), 490–507.
4.
DiethelmK.FordN.J. and FreedA.D., Detailed Error Analysis for a Fractional Adams Method [M], Numerical Algorithms, Kluwer Academic Publishers, (2004) 306–311.
LiuF.AnhV. and TurnerI., Numerical solution of the space fractional Fokker-Planck equation. Journal of Computational and Applied Mathematics, 166, (2004), 209–219.
7.
MaxeyM.R.RileyJ.J., Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids, 26, (1983), 3712–3713.
8.
MetzlerR. and KlafterJ.. The random walk's guide to anomalous diffusion: A fractional dynamics approach[J]. Physics Reports, 339, (2000), 1–77.
9.
PodlubnyI., Fractional Differential Equations [M], New York:Academic Press (1999).
10.
RossikhinY.A. and ShitikovaM.V., A new method for solving dynamic problems of fractional derivative viscelasticity, Int. J. Engng. Sci., 39, (2000), 149–176.
11.
RossikhinY.A. and ShitikovaM.V., Analysis of dynamic behavior of viscoelastic rods whose rheological models contain fractional derivatives of two different orders, ZAMM6, (2001), 363–378.
12.
SeredyńkaM. and HanygaA., Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum, Acta Mechanica, 176, (2005), 169–183.
13.
WangZ.CaoG. and ZhuX., Application of Fractional Calculaus in sysem Modeling. J. Shanghai JiaoTong University, 38, (2004), 802–805.
14.
YangC.LiuF., A Fractional Predictor-Corrector Method of the Fractional Relaxation-Oscillation Equation, J of Xiamen University, 44, (2005), 761–765.
15.
YangC. and LiuF., A computationally effective predictor-corrector method for simulating fractional-order dynamical control system, ANZIAM J., 47, (2006), 168–184.
16.
YusteS.B. and LindenbergK.. Subdiffusion-limited reactions[J]. Chemical Physics, 284, (2002), 169–180.
17.
YusteS.B.AcedoL. and LindenbergK., Reaction front in an A + B → C reactionsubdiffusion process, Phys. Rev., E 69, (2004), 036126.
