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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
Transport: Foundations and Applications, Wiley-VCH, Weinheim, Germany, pp. 93-127. 117
[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
[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.
[Hilfer(2000)] Hilfer, R. (ed.) (2000). Applications of Fractional Calculus in Physics.World Scientific, Singapore. 116
[Jacob(2005)] Jacob, N. (2005). Pseudo-Differential Operators and Markov Processes.Vol. 3: Markov Processes and Applications. Imperial College Press, London. 117
[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
[Kochubei(1989)] Kochubei, A.N. (1989). A Cauchy problem for evolution equations offractional order. Differential Equations 25, pp. 967-974. 117
[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
[Luchko(1999)] Luchko, Yu. (1999). Operational method in fractional calculus. Fract.Calc. Appl. Anal. 2, pp. 463-489. 140
[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
[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
[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
[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
[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
[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
[Mainardi(1996)] Mainardi, F. (1996). Fractional relaxation-oscillation and fractionaldiffusion-wave phenomena. Chaos, Solitons and Fractals 7, pp. 1461-1477. 116
[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
[Mainardi and Pagnini(2002)] Mainardi, F. and G. Pagnini (2002). Space-time fractionaldiffusion: exact solutions and probabilistic interpretation. In Monaco, R., M.P.
Bianchi, and S. Rionero (eds.): Proceedings WASCOM 2001, 11th Conference onWaves and Stability in Continuous Media, World Scientific, Singapore. 132
[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
[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
[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
[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
[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
[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
[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
[Podlubny(1999)] Podlubny, I. (1999). Fractional Differential Equations. Academic Press,San Diego. 116, 117, 122, 133, 141
[Pskhu(2005)] Pskhu, A.V. (2005). Partial differential equations of fractional order.Nauka, Moscow (in Russian).
[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
[Schneider and Wyss(1989)] Schneider, W.R. and W. Wyss (1989). Fractional diffusionand wave equations. J. Math. Phys. 30, pp. 134-144. 117
[Vladimirov(1971)] Vladimirov, V.S. (1971). Equations of Mathematical Physics. Nauka,Moscow (in Russian). 137, 139, 141
[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
[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
[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
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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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