Random-order fractional differential equation models. (English) Zbl 1203.94056

Summary: This paper proposes a new concept of random-order fractional differential equation model, in which a noise term is included in the fractional order. We investigate both a random-order anomalous relaxation model and a random-order time fractional anomalous diffusion model to demonstrate the advantages and the distinguishing features of the proposed models. From numerical simulation results, it is observed that the scale parameter and the frequency of the noise play a crucial role in the evolution behaviors of these systems. In addition, some potential applications of the new models are presented.


94A12 Signal theory (characterization, reconstruction, filtering, etc.)
60H30 Applications of stochastic analysis (to PDEs, etc.)
34A08 Fractional ordinary differential equations


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[1] West, B. J.; Grigolini, P.; Metzler, R.; Nonnenmacher, T. F.: Fractional diffusion and Lévy stable processes, Physical review E 55, No. 1, 99-106 (1997)
[2] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports 339, 1-77 (2000) · Zbl 0984.82032
[3] Chen, W.: A speculative study of 2/3-order fractional Laplacian modeling of turbulence: some thoughts and conjectures, Chaos 16, 023126 (2006) · Zbl 1146.37312
[4] Weron, K.; Kotulski, M.: On the Cole–Cole relaxation function and related Mittag-lettter distribution, Physica A 232, 180-188 (1996)
[5] Paradisi, P.; Cesari, R.; Mainardi, F.; Tampieri, F.: The fractional fick’s law for non-local transport processes, Physica A 293, 130-142 (2001) · Zbl 0978.82080
[6] Baeumer, B.; Kurita, S.; Meerschaert, M. M.: Inhomogeneous fractional diffusion equations, Fractional calculus and applied analysis 8, No. 4, 371-386 (2005) · Zbl 1202.86005
[7] Caputo, M.: Diffusion with space memory modelled with distributed order space fractional differential equations, Annals of geophysics 46, No. 2, 223-234 (2003)
[8] Sokolov, I. M.; Chechkin, A. V.; Klafter, J.: Distributed-order fractional kinetics, Acta physica polonica B 35, No. 4, 1323-1341 (2004)
[9] Lorenzo, C. F.; Hartley, T. T.: Variable order and distributed order fractional operators, Nonlinear dynamics 29, 57-98 (2002) · Zbl 1018.93007
[10] Samko, S. G.: Fractional integration and differentiation of variable order, Analysis Mathematica 21, 213-236 (1995) · Zbl 0838.26006
[11] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.: Fractional diffusion in inhomogeneous media, Journal of physics A: mathematical and general 38, L679-L684 (2005) · Zbl 1082.76097
[12] Sun, H.; Chen, W.; Chen, Y. Q.: Variable-order fractional differential operators in anomalous diffusion modeling, Physica A 388, 4586-4592 (2009)
[13] Miranda, L. Couto; Riera, R.: Truncated Lévy walks and an emerging market economic index, Physica A 297, 509-520 (2001) · Zbl 0969.91506
[14] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Physical review E 66, 046129 (2002)
[15] Lutz, E.: Fractional transport equations for Lévy stable processes, Physical review letters 86, No. 11, 2208-2211 (2001)
[16] Go, J. -Y.; Pyun, S. -I.: A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes, Journal of solid state electrochemistry 11, 323-334 (2007)
[17] Podlubny, I.: Fractional differential equation, (1999) · Zbl 0924.34008
[18] Coimbra, C. F. M.: Mechanics with variable-order differential operators, Annals of physics 12, No. 11–12, 692-703 (2003) · Zbl 1103.26301
[19] Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear dynamics 29, 3-22 (2002) · Zbl 1009.65049
[20] Deng, W. H.: Short memory principle and a predictor-corrector approach for fractional differential equations, Journal of computational and applied mathematics 206, 174-188 (2007) · Zbl 1121.65128
[21] \langlehttp://www.mathworks.com/matlabcentral/fileexchange/26407 angle.
[22] Chambers, J. M.; Mallows, C. L.; Stuck, B. W.: A method for simulating stable random variables, Journal of the American statistical association 71, No. 354, 340-344 (1976) · Zbl 0341.65003
[23] Fulger, D.; Scalas, E.; Germano, G.: Efficient Monte Carlo simulation of uncoupled continuous-time random walks and approximate solution of the fractional diffusion equation, Physical review E 77, 021122 (2008)
[24] A.J. Turski, B. Atamaniuk, E. Turska, Application of fractional derivative operators to anomalous diffusion and propagation problems, arXiv:math-ph/0701068v2, 2007.
[25] Kobelev, Ya.L.; Kobelev, L. Ya.; Klimontovich, Yu.L.: Anomalous diffusion with time- and coordinate-dependent memory, Doklady physics 48, No. 6, 264-268 (2003) · Zbl 1073.35522
[26] Zhang, H.; Liu, F.; Anh, V.: Numerical approximation of Lévy–Feller diffusion equation and its probability interpretation, Journal of computational and applied mathematics 206, 1098-1115 (2007) · Zbl 1125.26014
[27] Chen, W.; Sun, H. G.; Zhang, X. D.; Korosak, D.: Anomalous diffusion modeling by fractal and fractional derivatives, Computers and mathematics with applications 59, No. 5, 1754-1758 (2010) · Zbl 1189.35355
[28] Hauff, T.; Jenko, F.; Eule, S.: Intermediate non-Gaussian transport in plasma core turbulence, Physics of plasmas 14, No. 10, 102316 (2007)
[29] Santamaria, F.; Wils, S.; De Schutter, E.; Augustine, G. J.: Anomalous diffusion in purkinje cell dendrites caused by spines, Neuron 52, 635-648 (2006)
[30] La Porta, A.; Voth, G. A.; Crawford, A. M.; Alexander, J.; Bodenschatz, E.: Fluid particle accelerations in fully developed turbulence, Nature 409, 1017-1019 (2001) · Zbl 1152.76315
[31] Sugiyama, J.; Mukai, K.; Ikedo, Y.; Nozaki, H.; Mansson, M.; Watanabe, I.: Li diffusion in lixcoo2 probed by muon-spin spectroscopy, Physical review letters 103, 147601 (2009)
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