EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 8 / No 2 (2013) Math. Model. Nat. Phenom., 8 2 (2013) 1-16 References
Free Access
Issue
Math. Model. Nat. Phenom.
Volume 8, Number 2, 2013
Anomalous diffusion
Page(s) 1 - 16
DOI https://doi.org/10.1051/mmnp/20138201
Published online 24 April 2013
  1. W. Arendt, C. J. K. Batty, M. Hieber,F. Neubrander. Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics 96, Birkhäuser Verlag, Basel, 2001. [Google Scholar]
  2. B. Baeumer, M. M. Meerschaert. Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 (2001), 481–500. [MathSciNet] [Google Scholar]
  3. E. Barkai, R. Metzler, J. Klafter. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E 61 (2000), 132. [CrossRef] [MathSciNet] [Google Scholar]
  4. D.A. Benson, M.M. Meerschaert. A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv. Water Resour. 32 (2009), 532–539. [CrossRef] [Google Scholar]
  5. B. Berkowitz, A. Cortis, M. Dentz, H. Scher. Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44 (2006), 1–49. [CrossRef] [Google Scholar]
  6. N. H. Bingham. Limit theorems for occupation times of Markov processes. Z. Wahrsch. verw. Geb. 17 (1971), 1–22. [CrossRef] [Google Scholar]
  7. R. Durrett. Probability: Theory and Examples. Cambridge University Press, New York, 2010. [Google Scholar]
  8. A. Einstein. On the movement of small particles suspended in a stationary liquid demanded by the molecular kinetic theory of heat. Ann. Phys. 17 (1905), 549–560. [CrossRef] [Google Scholar]
  9. W. Feller. An Introduction to Probability Theory and Its Applications. 2nd ed., Wiley, New York, 1971. [Google Scholar]
  10. M. G Hahn, K. Kobayashi, S. Umarov. Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc., 139 (2011), 691–705. [CrossRef] [MathSciNet] [Google Scholar]
  11. N. Jacob. Pseudo differential operators and Markov processes. Vol. I. Imperial College Press, London, 2001. [Google Scholar]
  12. A. N. Kochubei. A Cauchy problem for evolution equations of fractional order. Diff. Eq. 25 (1989), 967–974. [Google Scholar]
  13. A. N. Kochubei. Fractional-order diffusion. Diff. Eq. 26 (1990), 485–492. [Google Scholar]
  14. V. N. Kolokoltsov. Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Th. Prob. Appl. 53 (2009), 594. [CrossRef] [Google Scholar]
  15. R.L. Magin. Fractional Calculus in Bioengineering. Begell House, 2006. [Google Scholar]
  16. M. Magdziarz, A. Weron. Competition between subdiffusion and Lévy flights: Stochastic and numerical approach. Phys. Rev. E 75 (2007), 056702. [NASA ADS] [CrossRef] [Google Scholar]
  17. F. Mainardi. Fractals and fractional calculus in continuum mechanics. Springer Verlag, 1997. [Google Scholar]
  18. F. Mainardi, R. Gorenflo. On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118 (2000), 283–299. [CrossRef] [Google Scholar]
  19. J. Masoliver, M. Montero, J. Perelló, G.H. Weiss, J. Perello. The continuous time random walk formalism in financial markets. J. Econ. Behav. Org. 61 (2006), 577–598. [CrossRef] [Google Scholar]
  20. R. Metzler, J. Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77. [NASA ADS] [CrossRef] [Google Scholar]
  21. R. Metzler, J. Klafter. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37 (2004), R161–R208. [NASA ADS] [CrossRef] [Google Scholar]
  22. M. M. Meerschaert, H.-P. Scheffler. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley, New York, 2001. [Google Scholar]
  23. M. M. Meerschaert, H.-P. Scheffler, P. Kern. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004), 455–466. [Google Scholar]
  24. M. M. Meerschaert, H.-P. Scheffler. Triangular array limits for continuous time random walks. Stoch. Proc. Appl. 118 (2008), 1606–1633. [CrossRef] [Google Scholar]
  25. M. M. Meerschaert, E. Nane, P. Vellaisamy. Fractional Cauchy problems on bounded domains. Ann. Probab. 37 (2009), 979–1007. [CrossRef] [MathSciNet] [Google Scholar]
  26. M. M. Meerschaert, A. Sikorskii. Stochastic Models for Fractional Calculus. De Gruyter, Berlin, 2012. [Google Scholar]
  27. R. Metzler, E. Barkai, J. Klafter. Deriving fractional Fokker-Planck equations from a generalised master equation. Europhys. Lett. 46 (1999), 431–436. [CrossRef] [Google Scholar]
  28. E. Montroll, G. Weiss. Random walks on lattices. II. J. Math Phys. 6 (1965), 167–181. [Google Scholar]
  29. R. R. Nigmatullin. The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B 133 (1986), 425–430. [Google Scholar]
  30. A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. [Google Scholar]
  31. A. Piryatinska, A. I. Saichev, W. Woyczynski. Models of anomalous diffusion: The subdiffusive case. Phys. A 349 (2005), 375–420. [CrossRef] [Google Scholar]
  32. I. Podlubny. Fractional differential equations, Academic press, 1999. [Google Scholar]
  33. R Development Core Team. R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2010. [Google Scholar]
  34. J. Sabatier, O.P. Agrawal, J.A.T. Machado. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, 2007. [Google Scholar]
  35. S. G. Samko, A. A. Kilbas, O. I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London, 1993. [Google Scholar]
  36. E. Scalas. Five years of continuous-time random walks in econophysics. Complex Netw. Econ. Inter. 567 (2006), 3–16. [Google Scholar]
  37. A.I. Saichev, W.A. Woyzczynski. Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms, and Wavelets. Birkhäuser, 1997. [Google Scholar]
  38. A.I. Saichev, G.M. Zaslavsky. Fractional kinetic equations: solutions and applications. Chaos, 7 (1997), 753–764. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  39. H. Scher, M. Lax. Stochastic transport in a disordered solid. I. Theory. Phys. Rev. B 7 (1973), 4491–4502. [CrossRef] [MathSciNet] [Google Scholar]
  40. W. R. Schneider, W. Wyss. Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144. [CrossRef] [Google Scholar]
  41. I. N. Sneddon. Fourier Transforms. Dover, New York, 1995. [Google Scholar]
  42. I. Stakgold, M. J. Holst. Green’s functions and boundary value problems. Wiley, New York, 1998. [Google Scholar]
  43. V. V. Uchaikin, V. M. Zolotarev. Chance and Stability. Stable Distributions and Their Applications. VSP, Utrecht, 1999. [Google Scholar]
  44. G. Zaslavsky. Fractional kinetic equation for Hamiltonian chaos. Phys. D 76 (1994), 110–122. [CrossRef] [MathSciNet] [Google Scholar]
  45. Y. Zhang, D. Benson, M. M. Meerschaert, H. Scheffler. On using random walks to solve the space-fractional advection-dispersion equations. J. Stat. Phys. 123 (2006), 89–110. [CrossRef] [Google Scholar]
  46. V. Zolotarev. One-dimensional Stable Distributions. Translations of Mathematical Monographs 65, American Mathematical Society, Providence, RI, 1986. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.