Experimental Implications of Bochner-Levy-Riesz Diffusion

References

[1] R. Bagley and P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheology 27 (1983), 201-210.Search in Google Scholar

[2] S. Bochner, Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley (1955).10.1525/9780520345294Search in Google Scholar

[3] J. Cushman and M. Moroni, Statistical mechanics with threedimensional particle tracking velocimetry in the study of anomalous dispersion, I: Theory. Phys. Fluids 13 (2001), 75-80.Search in Google Scholar

[4] D. del-Castillo-Negrete, Fractional diffusion models of anomalous transport. In: R. Klages, G. Radons, and I. Sokolov (Eds.), Anomalous Transport: Foundations and Applications. Wiley-VCH, Weinheim (2008), 163-212.10.1002/9783527622979.ch6Search in Google Scholar

[5] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes. DeGruyter, Berlin, 2nd Ed. (2011).10.1515/9783110218091Search in Google Scholar

[6] R. Haag, Local Quantum Physics. Springer Verlag, Berlin (1992).10.1007/978-3-642-97306-2Search in Google Scholar

[7] R. Hilfer, Classification theory for anequilibrium phase transitions. Phys. Rev. E 48 (1993), 2466-2475.Search in Google Scholar

[8] R. Hilfer, Foundations of fractional dynamics. Fractals 3 (1995), 549-556.Search in Google Scholar

[9] R. Hilfer, On fractional diffusion and its relation with continuous time random walks. In: A. P. R. Kutner and K. Sznajd-Weron (Eds.), Anomalous Diffusion: From Basis to Applications. Springer, Berlin (1999), 77-82.10.1007/BFb0106834Search in Google Scholar

[10] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapore (2000).10.1142/3779Search in Google Scholar

[11] R. Hilfer, Fractional time evolution. In: R. Hilfer (Ed.), Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000), 87-130.10.1142/9789812817747_0002Search in Google Scholar

[12] R. Hilfer, Threefold introduction to fractional derivatives. In: R. Klages, G. Radons, and I. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008), 17-74; http://www.icp.uni-stuttgart.de/∼hilfer/publikationen/html/ZZ-2008-ATFaA-17/ZZ-2008-ATFaA-17.html.Search in Google Scholar

[13] R. Hilfer and L. Anton, Fractional master equations and fractal time random walks. Phys. Rev. E, Rapid Commun. 51 (1995), R848-R851.10.1103/PhysRevE.51.R848Search in Google Scholar PubMed

[14] J. Klafter, S.C. Lim, R. Metzler (Eds.), Fractional Dynamics. Recent Advances. World Scientific, Singapore (2011).10.1142/8087Search in Google Scholar

[15] R. Klages G. Radons, and I. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008).Search in Google Scholar

[16] N. Landkof, Foundations of Modern Potential Theory. Springer, Berlin (1972).10.1007/978-3-642-65183-0Search in Google Scholar

[17] N. Laskin, Principles of fractional quantum mechanics. In: R. Klages, G. Radons, and I. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim (2008), 393-427; DOI: 10.1142/9789814340595 0017.10.1142/8087Search in Google Scholar

[18] P. Levy, Theorie de l’addition des variables aleatoires. Gauthier-Villars, Paris (1937).Search in Google Scholar

[19] J. Liouville, Mémoire sur quelques questions de geometrie et de mecanique, et sur un nouveau genre de calcul pour resoudre ces questions. Journal de l’Ecole Polytechnique XIII (1832), 1-69.Search in Google Scholar

[20] E. Montroll and G.Weiss, Random walks on lattices, II. J. Math. Phys. 6 (1965), 167-181.10.1063/1.1704269Search in Google Scholar

[21] R. Nigmatullin. The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133 (1986), 425-430.10.1002/pssb.2221330150Search in Google Scholar

[22] M. Riesz, Integrales de Riemann-Liouville et potentiels. Acta Sci. Math. (Szeged) 9 (1938), 1-42.Search in Google Scholar

[23] M. Riesz, L’integrale de Riemann-Liouville et le probleme de Cauchy. Acta Mathematica 81 (1949), 1-222.Search in Google Scholar

[24] Y. Rossikhin and M. Shitikova. Application of fractional calculus for analysis of nonlinear damped vibrations of suspension bridges. J. Eng. Mech. 124 (1998), 1029-1036. 10.1061/(ASCE)0733-9399(1998)124:9(1029)Search in Google Scholar

[25] I. Schäfer and K. Krüger. Modelling of coils using fractional derivatives. J. of Magnetism and Magnetic Materials 307 (2006), 91-98.10.1016/j.jmmm.2006.03.046Search in Google Scholar

[26] W. Schneider and W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134-144.Search in Google Scholar

[27] R. Schumer, D. Benson, M. Meerschaert, and S. Wheatcraft, Eulerian derivation of the fractional advection-dispersion equation. J. Contaminant Hydrol. 48 (2001), 69-86.Search in Google Scholar

[28] V. Uchaikin, Fractional Derivatives for Physicists and Engineers, I. Springer, Berlin (2012).10.1007/978-3-642-33911-0Search in Google Scholar

[29] V. Uchaikin, Fractional Derivatives for Physicists and Engineers, II. Springer, Berlin (2013). 10.1007/978-3-642-33911-0Search in Google Scholar