zbMATH — the first resource for mathematics

Statistical analysis for stochastic systems including fractional derivatives. (English) Zbl 1183.70062
Summary: An analytical scheme to determine the statistical behavior of a stochastic system including two terms of fractional derivative with real, arbitrary, fractional orders is proposed. In this approach, Green’s functions obtained are based on a Laplace transform approach and the weighted generalized Mittag-Leffler function. The responses of the system can be subsequently described as a Duhamel integral-type close-form expression. These expressions are applied to obtain the statistical behavior of a dynamical system excited by stationary stochastic processes. The numerical simulation based on the modified Euler method and Monte Carlo approach is developed. Three examples of single-degree-of-freedom system with fractional derivative damping under Gaussian white noise excitation are presented to illustrate application of the proposed method.

70L05 Random vibrations in mechanics of particles and systems
26A33 Fractional derivatives and integrals
Full Text: DOI
[1] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[2] West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)
[3] Lim, S.C.: Fractional derivative quantum fields at positive temperature. Physica A 363, 269–281 (2006) · doi:10.1016/j.physa.2005.08.005
[4] Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983) · Zbl 0515.76012 · doi:10.1122/1.549724
[5] Bagley, R.L., Torvik, P.J.: Fractional calculus: a different approach to the analysis of viscoelastically damped structures. AIAA J. 21, 741–748 (1983) · Zbl 0514.73048 · doi:10.2514/3.8142
[6] Bagley, R.L., Torvik, P.J.: Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA J. 23, 918–925 (1985) · Zbl 0562.73071 · doi:10.2514/3.9007
[7] Koeller, R.C.: Application of fractional calculus to the theory of viscoelasticity. ASME J. Appl. Mech. 51, 299–307 (1984) · Zbl 0544.73052 · doi:10.1115/1.3167616
[8] Metzeler, R., Nonnenmacher, T.F.: Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Int. J. Plast. 19, 941–959 (2003) · Zbl 1090.74673 · doi:10.1016/S0749-6419(02)00087-6
[9] Agrawal, O.P.: Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dyn. 38, 191–206 (2004) · Zbl 1142.74302 · doi:10.1007/s11071-004-3755-7
[10] Deng, R., Davies, P., Bajaj, A.K.: A case study on the use of fractional derivatives: the low-frequency viscoelastic uni-directional behavior of polyurethane foam. Nonlinear Dyn. 38, 247–265 (2004) · Zbl 1142.74313 · doi:10.1007/s11071-004-3759-3
[11] Depollier, C., Fellah, Z.E.A., Fellah, M.: Propagation of transient acoustic waves in layered porous media: fractional equations for the scattering operators. Nonlinear Dyn. 38, 181–190 (2004) · Zbl 1099.74035 · doi:10.1007/s11071-004-3754-8
[12] Mainardi, F., Pagnini, G., Gorenflo, R.: Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187, 295–305 (2007) · Zbl 1122.26004 · doi:10.1016/j.amc.2006.08.126
[13] Meerschaert, M.M., Benson, D.A., Scheffler, H.P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, 1–4 (2002) · Zbl 1244.60080 · doi:10.1103/PhysRevE.65.041103
[14] Laskin, N.: Fractional market dynamics. Physica A 287, 482–492 (2000) · doi:10.1016/S0378-4371(00)00387-3
[15] Jumarie, G.: Fractionalization of the complex-valued Brownian motion of order n using Riemann–Liouville derivative: Applications to mathematical finance and stochastic mechanics. Chaos Solitons Fractals 28, 1285–1305 (2006) · Zbl 1099.60025 · doi:10.1016/j.chaos.2005.08.083
[16] Jumarie, G.: Path integral for the probability of the trajectories generated by fractional dynamics subject to Gaussian white noise. Appl. Math. Lett. 20, 846–852 (2007) · Zbl 1142.82013 · doi:10.1016/j.aml.2006.08.015
[17] Mainardi, F., Pironi, P.L.: The fractional Langevin equation: Brownian motion revisited. Extr. Math. 10, 140–154 (1996)
[18] Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[19] Agrawal, O.P.: Response of a diffusion-wave system subjected to deterministic and stochastic fields. Z. Angew. Math. Mech. 483, 265–274 (2003) · Zbl 1045.35116 · doi:10.1002/zamm.200310033
[20] Spanos, P.D., Zeldin, B.A.: Random vibration of systems with frequency dependent parameters or fractional derivatives. J. Eng. Mech. 123, 290–292 (1997) · doi:10.1061/(ASCE)0733-9399(1997)123:3(290)
[21] Rüdinger, F.: Tuned mass damper with fractional derivative damping. Eng. Struct. 28, 1774–1779 (2006) · doi:10.1016/j.engstruct.2006.01.006
[22] Agrawal, O.P.: Analytical solution for stochastic response of a fractionally damped beam. J. Vib. Acoust. 126, 561–566 (2004) · doi:10.1115/1.1805003
[23] Agrawal, O.P.: Stochastic analysis of dynamic systems containing fractional derivatives. J. Sound Vib. 247, 927–938 (2001) · doi:10.1006/jsvi.2001.3682
[24] Drozdov, A.D.: Fractional oscillator driven by a Gaussian noise. Physica A 376, 237–245 (2007) · doi:10.1016/j.physa.2006.10.060
[25] Ford, N.J., Simpson, A.C.: The approximate solution of fractional differential equations of order greater than 1. Numerical Analysis Report 386, Manchester Center for Numerical Computational Mathematics (2001) · Zbl 0976.65062
[26] Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1995)
[27] Seybold, H., Hilfer, R.: Numerical algorithm for calculating the generalized Mittag–Leffler function. SIAM J. Numer. Anal. 47, 69–88 (2008) · Zbl 1190.65033 · doi:10.1137/070700280
[28] Mainardi, F., Mura, A., Gorenflo, R., Stojanovic, M.: The two forms of fractional relaxation of distributed order. J. Vib. Control 13, 1249–1268 (2007) · Zbl 1165.26302 · doi:10.1177/1077546307077468
[29] Koh, C.G., Kelly, J.M.: Application of fractional derivatives to seismic analysis of base-isolated models. Earthq. Eng. Struct. Dyn. 19, 229–241 (1990) · doi:10.1002/eqe.4290190207
[30] Yuan, L.X., Agrawal, O.P.: A numerical scheme for dynamics systems containing fractional derivatives. In: Proceedings of 1998 ASME Design Engineering Technical Conferences, Atlanta, GA, Sept. 1998, pp. 13–16
[31] Shokooh, A., Suarez, L.: A comparison of numerical methods applied to a fractional model of damping materials. J. Vib. Control 5, 331–354 (1999) · doi:10.1177/107754639900500301
[32] Zhu, Z.Y., Li, G.G., Cheng, C.J.: A numerical method for fractional integral with application. Appl. Math. Mech. 24, 373–384 (2003) · Zbl 1142.74390 · doi:10.1007/BF02439616
[33] Huang, Z.L., Jin, X.L.: Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. J. Sound. Vib. 319, 1121–1135 (2009) · doi:10.1016/j.jsv.2008.06.026
[34] Mainardi, F., Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, 283–299 (2000) · Zbl 0970.45005 · doi:10.1016/S0377-0427(00)00294-6
[35] Suarez, L.E., Shokooh, A.: An eigenvector expansion method for the solution of motion containing fractional derivatives. J. Appl. Mech. 64, 629–635 (1997) · Zbl 0905.73022 · doi:10.1115/1.2788939
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.