×

Front propagation in anomalous diffusive media governed by time-fractional diffusion. (English) Zbl 1349.35404

Summary: In this paper, a multi-dimensional model is proposed to study the propagation of random fronts in media in which anomalous diffusion takes place. The front position is obtained as the weighted mean of fronts calculated by means of the level set method, using as weight-function the probability density function which characterizes the anomalous diffusion process. Since anomalous diffusion is assumed to be governed by a time-fractional diffusion equation, its fundamental solution is the required probability density function. It is shown that this fundamental solution can be expressed in the multi-dimensional case in terms of the well-known \(\mathcal{M}\)-Wright/Mainardi function, as in the one-dimensional case. Making use of this representation for the practical purpose of numerical evaluation, the propagation of random fronts in two-dimensional subdiffusive media is discussed and investigated.

MSC:

35R11 Fractional partial differential equations
35K57 Reaction-diffusion equations
60G22 Fractional processes, including fractional Brownian motion
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Barkai, E.; Garini, Y.; Metzler, R., Strange kinetics of single molecules in living cells, Phys. Today, 65, 8, 29-35 (2012)
[2] Höfling, F.; Franosch, T., Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys., 76, 046602 (2013)
[3] Metzler, R.; Jeon, J.-H.; Cherstvy, A. G.; Barkai, E., Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys., 16, 24128-24164 (2014)
[4] del Castillo-Negrete, D., Non-diffusive, non-local transport in fluids and plasmas, Nonlinear Process. Geophys., 17, 795-807 (2010)
[5] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in fractional dynamics descriptions of anomalous dynamical processes, J. Phys. A, Math. Theor., 37, 31, R161-R208 (2004) · Zbl 1075.82018
[6] Blumen, A.; Gurtovenko, A. A.; Jespersen, S., Anomalous diffusion and relaxation in macromolecular systems, J. Non-Cryst. Solids, 305, 71-80 (2002)
[7] Pagnini, G., Short note on the emergence of fractional kinetics, Physica A, 409, 29-34 (2014) · Zbl 1395.82216
[8] Mentrelli, A.; Pagnini, G., Random front propagation in fractional diffusive systems, Commun. Appl. Ind. Math. · Zbl 1338.60110
[9] Sethian, J. A.; Smereka, P., Level set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35, 341-372 (2003) · Zbl 1041.76057
[10] Pagnini, G.; Bonomi, E., Lagrangian formulation of turbulent premixed combustion, Phys. Rev. Lett., 107, 044503 (2011)
[11] Pagnini, G.; Massidda, L., Modelling turbulence effects in wildland fire propagation by the randomized level-set method (July 2012), Tech. Rep 2012/PM12a, CRS4, revised version August 2014
[12] Pagnini, G.; Massidda, L., The randomized level-set method to model turbulence effects in wildland fire propagation, (Spano, D.; Bacciu, V.; Salis, M.; Sirca, C., Modelling Fire Behaviour and Risk. Proceedings of the International Conference on Fire Behaviour and Risk. Modelling Fire Behaviour and Risk. Proceedings of the International Conference on Fire Behaviour and Risk, ICFBR 2011, Alghero, Italy, October 4-6, 2011 (2012)), 126-131
[13] Pagnini, G.; Mentrelli, A., Modelling wildland fire propagation by tracking random fronts, Nat. Hazards Earth Syst. Sci., 14, 2249-2263 (2014)
[14] Pagnini, G., A model of wildland fire propagation including random effects by turbulence and fire spotting, (Proceedings of XXIII Congreso de Ecuaciones Diferenciales y Aplicaciones XIII Congreso de Matemática Aplicada. Proceedings of XXIII Congreso de Ecuaciones Diferenciales y Aplicaciones XIII Congreso de Matemática Aplicada, Castelló, Spain, September 9-13, 2013 (2013)), 395-403
[15] Pagnini, G., Fire spotting effects in wildland fire propagation, (Casas, F.; Martínez, V., Advances in Differential Equations and Applications. Advances in Differential Equations and Applications, SEMA SIMAI Springer Series, vol. 4 (2014), Springer International Publishing: Springer International Publishing Switzerland), 203-216, (eBook: 978-3-319-06953-1) · Zbl 1328.35340
[16] Mainardi, F.; Mura, A.; Pagnini, G., The M-Wright function in time-fractional diffusion processes: a tutorial survey, Int. J. Differ. Equ., 2010, 104505 (2010) · Zbl 1222.60060
[17] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity (2010), Imperial College Press: Imperial College Press London · Zbl 1210.26004
[18] Pagnini, G., The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes, Fract. Calc. Appl. Anal., 16, 2, 436-453 (2013) · Zbl 1312.33061
[19] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[20] Hanyga, A., Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. Lond. A, 458, 933-957 (2002) · Zbl 1153.35347
[21] Mainardi, F.; Pagnini, G.; Gorenflo, R., Mellin transform and subordination laws in fractional diffusion processes, Fract. Calc. Appl. Anal., 6, 441-459 (2003) · Zbl 1083.60032
[22] Osher, S. J.; Sethian, J. A., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12-49 (1988) · Zbl 0659.65132
[23] Peters, N., Turbulent Combustion (2004), Cambridge University Press: Cambridge University Press Cambridge
[24] Mallet, V.; Keyes, D. E.; Fendell, F. E., Modeling wildland fire propagation with level set methods, Comput. Math. Appl., 57, 1089-1101 (2009) · Zbl 1186.65140
[25] Jettestuen, E.; Helland, J. O.; Prodanović, M., A level set method for simulating capillary-controlled displacements at the pore scale with nonzero contact angles, Water Resour. Res., 49, 4645-4661 (2013)
[26] Machacek, M.; Danuser, G., Morphodynamic profiling of protrusion phenotypes, Biophys. J., 90, 1439-1452 (2006)
[27] Guo, W.; Sawin, H. H., Review of profile and roughening simulation in microelectronics plasma etching, J. Phys. D, Appl. Phys., 42, 194014 (2009)
[28] Monaghan, J. J., Smoothed particle hydrodynamics, Rep. Prog. Phys., 68, 1703-1759 (2005)
[29] Zimont, V. L., Gas premixed combustion at high turbulence. Turbulent flame closure combustion model, Exp. Therm. Fluid Sci., 21, 179-186 (2000)
[30] Sabelnikov, V. A.; Lipatnikov, A. N., Towards an extension of TFC model of premixed turbulent combustion, Flow Turbul. Combust., 90, 387-400 (2013)
[31] Ohta, T.; Jasnow, D.; Kawasaki, K., Universal scaling in the motion of random interfaces, Phys. Rev. Lett., 49, 1223-1226 (1982)
[32] Soner, H. M.; Touzi, N., A stochastic representation for the level set equations, Commun. Partial Differ. Equ., 27, 2031-2053 (2002) · Zbl 1036.49010
[33] Juan, O.; Keriven, R.; Postelnicu, G., Stochastic motion and the Level Set method in Computer Vision: stochastic active contours, Int. J. Comput. Vis., 69, 7-25 (2006) · Zbl 1477.68380
[34] Klimontovich, Y. L., Nonlinear Brownian motion, Phys. Usp., 37, 737-767 (1994)
[35] Waterman, M. S.; Whiteman, D. E., Estimation of probability densities by empirical density functions, Int. J. Math. Educ. Sci. Technol., 9, 127-137 (1978) · Zbl 0374.62039
[36] Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 1, 134-144 (1989) · Zbl 0692.45004
[37] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, 1461-1477 (1996) · Zbl 1080.26505
[38] Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9, 6, 23-28 (1996) · Zbl 0879.35036
[39] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4, 2, 153-192 (2001) · Zbl 1054.35156
[40] Mainardi, F.; Pagnini, G., The Wright functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput., 141, 1, 51-62 (2003) · Zbl 1053.35008
[41] 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
[42] Gorenflo, R.; Loutchko, J.; Luchko, Y., Computation of the Mittag-Leffler function \(E_{\alpha, \beta}(z)\) and its derivative, Fract. Calc. Appl. Anal., 5, 491-518 (2002) · Zbl 1027.33016
[43] Jones, E.; Oliphant, T.; Peterson, P., SciPy: open source scientific tools for Python (2001)
[44] Mainardi, F.; Gorenflo, R., On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118, 283-299 (2000) · Zbl 0970.45005
[45] Hunter, J. D., Matplotlib: a 2D graphics environment, Comput. Sci. Eng., 9, 3, 90-95 (2007)
[46] Pérez, F.; Granger, B. E., IPython: a system for interactive scientific computing, Comput. Sci. Eng., 9, 3, 21-29 (2007)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.