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Anomalous diffusion modeling by fractal and fractional derivatives. (English) Zbl 1189.35355
Summary: This paper makes an attempt to develop a fractal derivative model of anomalous diffusion. We also derive the fundamental solution of the fractal derivative equation for anomalous diffusion, which characterizes a clear power law. This new model is compared with the corresponding fractional derivative model in terms of computational efficiency, diffusion velocity, and heavy tail property. The merits and distinctions of these two models of anomalous diffusion are then summarized.

35R11Fractional partial differential equations
26A33Fractional derivatives and integrals (real functions)
35A08Fundamental solutions of PDE
Full Text: DOI
[1] 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 · doi:10.1063/1.2208452
[2] Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for fractional advection--dispersion flow equations, Journal of computational and applied mathematics 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[3] Greenenko, A. A.; Chechkin, A. V.; Shul’ga, N. F.: Anomalous diffusion and Lévy flights in channelling, Physics letters A 324, 82-85 (2004) · Zbl 1123.82346 · doi:10.1016/j.physleta.2004.02.053
[4] Schulz, B. M.; Schulz, M.: Numerical investigations of anomalous diffusion effects in glasses, Journal of non-crystalline solids 352, 4884-4887 (2006)
[5] Anh, V. V.; Angulo, J. M.; Ruiz-Medina, M. D.: Diffusion on multifractals, Nonlinear analysis 63, e2043-e2056 (2005) · Zbl 1224.60111
[6] Paradisia, 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 · doi:10.1016/S0378-4371(00)00491-X
[7] 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 · doi:10.1016/j.cam.2006.09.017
[8] Chen, W.: Time--space fabric underlying anomalous diffusion, Chaos, solitons and fractals 28, 923-929 (2006) · Zbl 1098.60078 · doi:10.1016/j.chaos.2005.08.199
[9] W. Chen, Fractional and fractal derivatives modeling of turbulence [J], Arxiv preprint nlin/0511066, 2005
[10] Kanno, R.: Representation of random walk in fractal space-time, Physica A 248, 165-175 (1998)
[11] Podlubny, I.: Fractional differential equation [M], (1999) · Zbl 0924.34008
[12] Gorenflo, R.; Mainardi, F.: Random walk models for space-fractional diffusion processes, Fractional calculus applied analysis 1, 167-191 (1998) · Zbl 0946.60039
[13] Chen, W.; Holm, S.: Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, Journal of the acoustical society of America 115, No. 4, 1424-1430 (2004)
[14] Vlad, M. O.; Metzler, R.; Nonnenmacher, T. F.; Mackey, M. C.: Universality classes for asymptotic behavior of relaxation processes in systems with dynamical disorder: dynamical generalizations of stretched exponential, J. math. Phys. 37, 2279-2306 (1996) · Zbl 0872.60097 · doi:10.1063/1.531509
[15] Balescu, R.: V-Langevin equations, continuous time random walks and fractional diffusion, Chaos, solitons and fractals 34, 62-80 (2007) · Zbl 1142.82356 · doi:10.1016/j.chaos.2007.01.050
[16] Mainardi, F.; Luchko, Y.; Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation, Fractional calculus and applied analysis 4, No. 2, 153-192 (2001) · Zbl 1054.35156
[17] Mainardi, F.; Pagnini, G.: The role of the fox--wright functions in fractional sub-diffusion of distributed order, Journal of computational and applied mathematics 207, 245-257 (2007) · Zbl 1120.35002 · doi:10.1016/j.cam.2006.10.014
[18] Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations, Applied numerical mathematics 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[19] Abdel-Rehim, E. A.; Gorenflo, R.: Simulation of the continuous time random walk of the space-fractional diffusion equations, Journal of computational and applied mathematics 222, No. 2, 274-283 (2008) · Zbl 1153.65007 · doi:10.1016/j.cam.2007.10.052
[20] Gorenflo, R.; Vivoli, A.; Mainardi, F.: Discrete and continuous random walk models for space-time fractional diffusion, Nonlinear dynamics 38, 101-116 (2004) · Zbl 1125.76067 · doi:10.1007/s11071-004-3749-5
[21] Abe, S.; Thurner, S.: Anomalous diffusion in view of Einstein’s 1905 theory of Brownian motion, Physica A 356, 403-407 (2005)
[22] Go, J. -Y.; Pyun, Su-Il: 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)
[23] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K.: Stability and convergence of the difference methods for the space--time fractional advection--diffusion equation, Applied mathematics and computation 191, 12-20 (2007) · Zbl 1193.76093 · doi:10.1016/j.amc.2006.08.162
[24] Lin, Y.; Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of computational physics 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001