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Anomalous Diffusion: Models, Their Analysis, and Interpretation

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Advances in Applied Analysis

Part of the book series: Trends in Mathematics ((TM))

Abstract

In this chapter, modeling of anomalous diffusion processes in terms of differential equations of an arbitrary (not necessarily integer)ord er is discussed. We start with micro-modeling and first deduce a probabilistic interpretation of normal and anomalous diffusion from basic random walk models. The fractional differential equations are then derived asymptotically in the Fourier-Laplace domain from random walk models and generalized master equations, in the same way as the standard diffusion equation is obtained from a Brownian motion model. The obtained equations and their generalizations are analyzed both with the help of the Laplace-Fourier transforms (the Cauchy problems)and the spectral method (initial-boundary-value problems). In particular, the maximum principle, well known for elliptic and parabolic type PDEs, is extended to initial-boundary-value problems for the generalized diffusion equation of fractional order.

Mathematics Subject Classification (2010). 26A33, 33E12, 35A05, 35B30, 35B45, 35B50, 35K99, 45K05, 60J60, 60J65.

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References

  • [Bazhlekova(1998)] Bazhlekova, E.G. (1998). Duhamel-type representation of the solu-tions of nonlocal boundary value problems for the fractional diffusion-wave equation.Proc. 2nd Int. Workshop “‘TMSF’1996”’. Bulgarian Academy of Sciences, Sofia, pp.32-40. 117

    Google Scholar 

  • [Beghin and Orsingher(2009)] Beghin, L. and E. Orsingher (2009). Fractional Poissonprocesses and related planar random motions. Electronic Journal of Probability 14,paper no. 61. 117

    Google Scholar 

  • [Chechkin et al.(2003)] Chechkin A.V., R. Gorenflo, I.M. Sokolov, and V.Yu. Gonchar(2003). Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal.Vol. 6, pp. 259-279. 117, 141

    Google Scholar 

  • [Chechkin et al.(2005)] Chechkin, A.V., R. Gorenflo, and I.M. Sokolov (2005). Fractionaldiffusion in inhomogeneous media. J. Phys. A, Math. Gen. 38, 679-684. 116

    Google Scholar 

  • [Daftardar-Gejji and Bhalekar(2008)] Daftardar-Gejji, V. and S. Bhalekar (2008).Boundary value problems for multi-term fractional differential equations. J. Math.Anal. Appl. 345, pp. 754-765. 117, 141

    Google Scholar 

  • [Dubbeldam et al.(2007)] Dubbeldam, J.L.A., A. Milchev, V.G. Rostiashvili, and T.A.Vilgis (2007). Polymer translocation through a nanopore: A showcase of anomalousdiffusion. Physical Review E 76, 010801 (R). 116

    Google Scholar 

  • [Eidelman and Kochubei(2004)] Eidelman, S.D and A.N. Kochubei (2004). Cauchy prob-lem for fractional diffusion equations. J. Differ. Equations 199, pp. 211-255. 117

    Google Scholar 

  • [Erdélyi et al. (1953)] Erdélyi, A., W. Magnus, F. Oberhettinger, and F.G. Tricomi(1953). Higher Transcendental Functions, Vol. 3. New York, McGraw-Hill. 126

    Google Scholar 

  • [Freed et al.(2002)] Freed, A., K. Diethelm, and Yu. Luchko (2002). Fractional-order vis-coelasticity (FOV): Constitutive development using the fractional calculus. NASA’sGlenn Research Center, Ohio. 116

    Google Scholar 

  • [Germano et al.(2009)] Germano, G., M. Politi, E. Scalas, and R.L. Schilling (2009).Stochastic calculus for uncoupled continuous-time random walks. Physical ReviewE 79, 066102. 120, 121

    Google Scholar 

  • [Gorenflo and Mainardi(1998)] Gorenflo, R. and F. Mainardi (1998). Random walk mod-els for space-fractional diffusion processes. Frac. Calc. and Appl. Anal. 1, pp. 167-191. 116

    Google Scholar 

  • [Gorenflo and Mainardi(2008)] Gorenflo, R. and F. Mainardi (2008). Continuous timerandom walk, Mittag-Leffler waiting time and fractional diffusion: mathematical as-pects. Chap. 4 in R. Klages, G. Radons and I.M. Sokolov (editors): Anomalous

    Google Scholar 

  • Transport: Foundations and Applications, Wiley-VCH, Weinheim, Germany, pp. 93-127. 117

    Google Scholar 

  • [Gorenflo and Mainardi(2009)] Gorenflo, R. and F. Mainardi (2009). Some recent ad-vances in theory and simulation of fractional diffusion processes. Journal of Compu-tational and Applied Mathematics 229, pp. 400-415. 117

    Google Scholar 

  • [Gorenflo et al.(2000)] Gorenflo, R., Yu. Luchko, and S. Umarov (2000). On the Cauchyand multi-point problems for partial pseudo-differential equations of fractional order.Frac. Calc. and Appl. Anal. 3, pp. 249-277.

    Google Scholar 

  • [Hilfer(2000)] Hilfer, R. (ed.) (2000). Applications of Fractional Calculus in Physics.World Scientific, Singapore. 116

    Google Scholar 

  • [Jacob(2005)] Jacob, N. (2005). Pseudo-Differential Operators and Markov Processes.Vol. 3: Markov Processes and Applications. Imperial College Press, London. 117

    Google Scholar 

  • [Kilbas et al.(2006)] Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo (2006). Theory andApplications of Fractional Differential Equation. Elsevier, Amsterdam. 116, 117, 122,123, 133

    Google Scholar 

  • [Kochubei(1989)] Kochubei, A.N. (1989). A Cauchy problem for evolution equations offractional order. Differential Equations 25, pp. 967-974. 117

    Google Scholar 

  • [Luchko and Gorenflo(1999)] Luchko, Yu. and R. Gorenflo (1999). An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica 24, pp. 207-233. 140

    Google Scholar 

  • [Luchko(1999)] Luchko, Yu. (1999). Operational method in fractional calculus. Fract.Calc. Appl. Anal. 2, pp. 463-489. 140

    Google Scholar 

  • [Luchko(2009a)] Luchko, Yu. (2009a) Boundary value problems for the generalized time-fractional diffusion equation of distributed order. Fract. Calc. Appl. Anal. 12, pp.409-422. 117, 141

    Google Scholar 

  • [Luchko(2009b)] Luchko, Yu. (2009b). Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. and Appl. 351, pp. 218-223. 117, 133

    Google Scholar 

  • [Luchko(2010)] Luchko, Yu. (2010). Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comp. and Math. with Appl. 59, pp. 1766-1772. 117, 133

    Google Scholar 

  • [Luchko et al.(2010)] Luchko, Yu., M. Rivero, J.J. Trujillo, and M.P. Velasco (2010).Fractional models, non-locality, and complex systems. Comp. and Math. with Appl.59, pp. 1048-1056. 115

    Google Scholar 

  • [Luchko(2011)] Luchko, Yu. (2011). Initial-boundary-value problems for the generalizedmulti-term time-fractional diffusion equation. J. Math. Anal. and Appl. 374, pp.538-548. 117, 141

    Google Scholar 

  • [Luchko and Punzi(2011)] Luchko, Yu. and A. Punzi (2011). Modeling anomalous heattransport in geothermal reservoirs via fractional diffusion equations. InternationalJournal on Geomathematics, 1, pp. 257-276. 116

    Google Scholar 

  • [Mainardi(1996)] Mainardi, F. (1996). Fractional relaxation-oscillation and fractionaldiffusion-wave phenomena. Chaos, Solitons and Fractals 7, pp. 1461-1477. 116

    Google Scholar 

  • [Mainardi and Tomirotti(1997)] Mainardi, F. and M. Tomirotti (1997). Seismic pulsepropagation with constant and stable probability distributions. Annali di Ge-ofisica 40, pp. 1311-1328. 116

    Google Scholar 

  • [Mainardi and Pagnini(2002)] Mainardi, F. and G. Pagnini (2002). Space-time fractionaldiffusion: exact solutions and probabilistic interpretation. In Monaco, R., M.P.

    Google Scholar 

  • Bianchi, and S. Rionero (eds.): Proceedings WASCOM 2001, 11th Conference onWaves and Stability in Continuous Media, World Scientific, Singapore. 132

    Google Scholar 

  • [Mainardi and Pagnini(2003)] Mainardi, F. and G. Pagnini (2003). The Wright functionas solution of the time-fractional diffusion equation. Applied Mathematics and Com-putation 141, pp. 51-62. 132

    Google Scholar 

  • [Mainardi et al.(2001)] Mainardi, F., Yu. Luchko, and G. Pagnini (2001). The fundamen-tal solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal.4, pp. 153-192. 117, 124, 128, 132

    Google Scholar 

  • [Mainardi et al.(2004)] Mainardi, F., R. Gorenflo, and E. Scalas (2004). A fractional generalization of the Poisson processes. Vietnam Journal of Mathematics 32, pp. 53–64. 117

    Google Scholar 

  • [Meerschaert et al.(2009)] Meerschaert, M.M., E. Nane, and P. Vellaisamy (2009). Frac-tional Cauchy problems in bounded domains. The Annals of Probability 37, pp.979-1007. 117

    Google Scholar 

  • [Meerschaert et al.(2011)] Meerschaert, M.M., E. Nane, and P. Vellaisamy (2011). Thefractional Poisson process and the inverse stable subordinator. Available online at http://arxiv.org/abs/1007.5051. 117

    Google Scholar 

  • [Metzler and Klafter(2000a)] Metzler, R. and J. Klafter (2000a). The random walk’sguide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339,pp. 1-77. 115, 116, 120, 121, 123

    Google Scholar 

  • [Metzler and Klafter(2000b)] Metzler, R. and J. Klafter (2000b). Boundary value prob-lems for fractional diffusion equations. Phys. A 278, pp. 107-125. 117

    Google Scholar 

  • [Podlubny(1999)] Podlubny, I. (1999). Fractional Differential Equations. Academic Press,San Diego. 116, 117, 122, 133, 141

    Google Scholar 

  • [Pskhu(2005)] Pskhu, A.V. (2005). Partial differential equations of fractional order.Nauka, Moscow (in Russian).

    Google Scholar 

  • [Scalas et al.(2004)] Scalas, E., R. Gorenflo, and F. Mainardi. Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. PhysicalReview E 69, pp. 011107/1-8. 117

    Google Scholar 

  • [Schneider and Wyss(1989)] Schneider, W.R. and W. Wyss (1989). Fractional diffusionand wave equations. J. Math. Phys. 30, pp. 134-144. 117

    Google Scholar 

  • [Vladimirov(1971)] Vladimirov, V.S. (1971). Equations of Mathematical Physics. Nauka,Moscow (in Russian). 137, 139, 141

    Google Scholar 

  • [Voroshilov and Kilbas(2006)] Voroshilov, A.A. and A.A. Kilbas (2006). The Cauchyproblem for the diffusion-wave equation with the Caputo partial derivative. Dif-ferential Equations 42, pp. 638-649. 117

    Google Scholar 

  • [Zacher(2008)] Zacher, R. (2008). Boundedness of weak solutions to evolutionary partialintegro-differential equations with discontinuous coefficients. J. Math. Anal. Appl.348, pp. 137-149. 117

    Google Scholar 

  • [Zhang(2006)] Zhang, S. (2006). Existence of solution for a boundary value problem offractional order. Acta Math. Sci., Ser. B 26, pp. 220-228. 117

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics II, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, D-13353, Berlin, Germany

    Yury Luchko

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  1. Yury Luchko
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Correspondence to Yury Luchko .

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Editors and Affiliations

  1. , Department of Economics, Belarusian State University, Nezavisimosti ave 4, Minsk, 220030, Belarus

    Sergei V. Rogosin

  2. , Department of Economics, Belarus State University, Nezavisimosti ave 4, Minsk, 220030, Belarus

    Anna A. Koroleva

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Luchko, Y. (2012). Anomalous Diffusion: Models, Their Analysis, and Interpretation. In: Rogosin, S., Koroleva, A. (eds) Advances in Applied Analysis. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0417-2_3

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