Time fractional differential equation model with random derivative order

Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 2009, DETC2009

Volumn 4, Issue PART B, 2010, Pages 1301-1306

  Sun, Hongguang ^a,b   Chen, Yangquan ^b   Chen, Wen ^a  

^a HOHAI UNIVERSITY   (China)

^b Utah State University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFUSION PROCESS; FRACTIONAL DIFFERENTIAL EQUATIONS; NEW MODEL; NOISE TERMS; NUMERICAL SIMULATION; POTENTIAL APPLICATIONS; SCALE PARAMETER; TIME DERIVATIVE;

DIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS;

COMPUTER SIMULATION;

EID: 77953699468     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1115/DETC2009-87483     Document Type: Conference Paper

References (26)

1. West, B. J., Grigolini, P., Metzler, R., and Nonnenmacher, T. F., 1997. "Fractional diffusion and Lévy stable processes". Physical Review E, 55(1), pp. 99-106.
2. Metzler, R., and Klafter, J., 2000. "The random walk's guide to anomalous diffusion: a fractional dynamics approach". Physics Reports, 339, pp. 1-77.
3. Chen, W., 2006. "A speculative study of 2/3-order fractional Laplacian modeling of turbulence: some thoughts and conjectures". Chaos, 16, p. 023126.
4. Paradisi, P., Cesari, R., Mainardi, F., and Tampieri, F., 2001. "The fractional Fick's law for non-local transport processes". Physica A, 293, pp. 130-142. (Pubitemid 32875425)
5. Baeumer, B., Kurita, S., and Meerschaert, M. M., 2005. "Inhomogeneous fractional diffusion equations". Fractional Calculus and Applied Analysis, 8(4), pp. 130-142.
6. Caputo, M., 2003. "Diffusion with space memory modelled with distributed order space fractional differential equations". Annals of Geophysics, 46(2), pp. 223-234.
7. Sokolov, I., Chechkin, A., and Klafter, J., 2004. "Distributed-order fractional kinetics". arXiv:cond-mat/0401146vl [cond-mat.stat-mech], .
8. Samko, S. G., 1995. "Fractional integration and differentiation of variable order". Analysis Mathematica, 21, pp. 213-236.
9. Chechkin, A., Gorenflo, R., and Sokolov, I., 2005. "Fractional diffusion in inhomogeneous media". J. Phys. A: Math. Gen., 38, pp. L679-L684.
10. Lorenzo, C. F., and Hartley, T. T., 2002. "Variable order and distributed order fractional operators". Nonlinear Dynamics, 29, pp. 57-98.
11. Miranda, L. C., and Riera, R., 2001. "Truncated Lévy walks and an emerging market economic index". Physica A, 297, pp. 509-520.
12. Chechkin, A., Gorenflo, R., and Sokolov, I. M., 2002. "Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations". Physical Review E, 66, p. 046129.
13. Lutz, E., 2001. "Fractional transport equations for Lévy stable processes". Physical Review Letters, 86(11), pp. 2208-2211.
14. Su-Il Pyun, J.-Y. G., 2007. "A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion- controlled transfer processes". J Solid State Electrochem, 11, pp. 323-334.
15. Weron, K., and Kotulski, M., 1996. "On the Cole-Cole relaxation function and related Mittag-Lettter distribution". Physica A, 232, pp. 180-188.
16. Podlubny, I., 1999. Fractional differential equation. Academic press.
17. Coimbra, C. F. M., 2003. "Mechanics with variable-order differential operators". Ann. Phys. (Leipzig), 12(11-12), pp. 692-703.
18. Mainardi, F., Mura, A., Gorenflo, R., and Stojanovic, M., 2007. "The two forms of fractional relaxation of distributed order". Journal of Vibration and Control, 13(9-10), pp. 1249-1268.
19. Chambers, J. M., Mallows, C. L., and Stuck, B. W., 1976. "A method for simulating stable random variables". Journal of the American Statistical Association, 71(354), pp. 340-344.
20. Fulger, D., Scalas, E., and Germano, G., 2008. "Efficient Monte Carlo simulation of uncoupled continuous-time random walks and approximate solution of the fractional diffusion equation". Physical Review E, 77, p. 021122.
21. Turski, A. J., Atamaniuk, B., and Turska, E., 2007. "Application of fractional derivative operators to anomalous diffusion and propagation problems". arXiv:math-ph/0701068v2.
22. Barkai, E., and Sokolov, I., 2007. "On Hilfer's objection to the fractional time diffusion equation". Physica A, 373, pp. 231-236.
23. Metzler, R., Chechkin, A. V., Gonchar, V. Y., and Klafter, J., 2007. "Some fundamental aspects of Lévy flights". Chaos, Solitons and Fractals, 34, pp. 129-142.
24. Hauff, T., Jenko, F., and Eule, S., 2007. "Intermediate non-Gaussian transport in plasma core turbulence". Physics of Plasmas, 14(10), p. 102316.
25. Porta, A. L., Voth, G., Crawford, A., Alexander, J., and Bodenschatz, E., 2001. "Fluid particle accelerations in fully developed turbulence". Nature, 409, pp. 1017-1019.
26. Santamaria, F., Wils, S., Schutter, E. D., and Augustine, G. J., 2006. "Anomalous diffusion in Purkinje cell dendrites caused by spines". Neuron, 52, pp. 635-648.