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Fractional Diffusion Equations and Anomalous Diffusion
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    Hartmann, P. Reyes, J. C. Kostadinova, E. G. Matthews, L. S. Hyde, T. W. Masheyeva, R. U. Dzhumagulova, K. N. Ramazanov, T. S. Ott, T. Kählert, H. Bonitz, M. Korolov, I. and Donkó, Z. 2019. Self-diffusion in two-dimensional quasimagnetized rotating dusty plasmas. Physical Review E, Vol. 99, Issue. 1,

    Fuziki, M. E. K. Lenzi, M. K. Ribeiro, M. A. Novatski, A. and Lenzi, E. K. 2018. Diffusion Process and Reaction on a Surface. Advances in Mathematical Physics, Vol. 2018, Issue. , p. 1.

    Burazin, K. and Mitrovic, D. 2018. Apriori estimates for fractional diffusion equation. Optimization Letters,


Book description

Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids. Providing a contemporary treatment of this process, this book examines the recent literature on anomalous diffusion and covers a rich class of problems in which surface effects are important, offering detailed mathematical tools of usual and fractional calculus for a wide audience of scientists and graduate students in physics, mathematics, chemistry and engineering. Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equations, this self-contained text presents the possibility of using fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research.

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[1] E., Butkov, Mathematical physics (Addison-Wesley, Reading, Mass., 1973).
[2] A. A., Kilbas, H. M., Srivastava, and J. J., Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, 2006).
[3] M., Abramowitz and I. A., Stegun, Handbook of mathematical functions (National Bureau of Standards, Washington, D.C., 1972).
[4] V. K., Tuan and M., Saigo, Convolution of Hankel transform and its application to an integral involving Bessel functions of first kind, Internat. J Math. Math. Sci. 18, 545–550 (1995). See also L., Britvina, Generalized convolutions for the Hankel transform and related integral operators, Math. Nachr. 280, 962–970 (2007).
[5] R., Piessens, The Hankel transform, in The transforms and applications handbook, 2nd edition. Edited by Alexander D., Poularikas (CRC Press, Boca Raton, 2000).
[6] G., Fikioris, Mellin transform method for integral evaluation-introduction and applications to electromagnetics (Morgan & Claypool, San Rafael, 2007).
[7] J., Bertrand, P., Bertrand, and J. P., Orvalez, The Mellin transform, in The transforms and applications handbook, 2nd edition. Edited by Alexander D., Poularikas (CRC Press, Boca Raton, 2000).
[8] H., Hochstadt, The functions of mathematical physics (Wiley-Interscience, New York, 1971).
[9] H., Cramér, Mathematical methods of statistics (Princeton University Press, Princeton, 1946).
[10] G. N., Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1944).
[11] L., Schwartz, Mathematical for the physical sciences (Addison-Wesley, Reading, Mass., 1966).
[12] P. M., Morse and H., Feshbach, Methods of theoretical physics (McGraw-Hill, New York, 1953), Vol. I.
[13] I. S., Gradshtein and I. M., Ryzhik, Table of integrals, series, and products, 7th edition (Elsevier Inc, Academic Press Inc., Amsterdam, 2007).
[14] H. W., Wyld, Mathematical methods for physics (Perseus Books, Reading, Mass., 1976).
[15] F., Bowman, Introduction to Bessel functions (Courier Corporation, Chelmsford, Mass., 2012).
[16] M. G., Mittag-Leffler, Sur la nouvelle fonction Eα(x), Comptes Rendus Acad. Sci. Paris 137, 554–558 (1903).
[17] M. G., Mittag-Leffler, Sopra la funzione Eα(x), Rend. Acc. Lincei 13, 3–5 (1904).
[18] M. G., Mittag-Leffler, Sur la representation analytique d'une branche uniforme d'une fonction monogène, Acta Math. 29, 101–181 (1905).
[19] A., Wiman, Über den fundamentalsatz in der teorie der funktion Eα(x), Acta Math. 29, 191–201 (1905).
[20] A., Wiman, Über die nulstellen der funktion Eα(x), Acta Math. 29, 217–234 (1905).
[21] R. K., Saxena, A. M., Mathai, and H. J., Haubold, On fractional kinetic equations, Astrophys. Space Sci. 282, 281–287 (2002).
[22] A., Erdélyi, Higher transcendental functions (McGraw-Hill, New York, 1953).
[23] R. P., Agarwal, A propos d'une note de M. Pierre Humbert, Comptes Rendus Acad. Sci. Paris 236, 2031–2032 (1953).
[24] P., Humbert, Quelques résultats relatifs à la fonction de Mittag-Leffler, Comptes Rendus Acad. Sci. Paris 236, 1467–1468 (1953).
[25] P., Humbert and R. P., Agarwal, Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations, Bull. Sci. Math. 77(2), 180–185 (1953).
[26] M. M., Dzherbashyan, Integral transform representations of functions in the complex domain (Nauka, Moscow, 1966).
[27] T. R., Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19, 7–15 (1971).
[28] A. A., Kilbas, M., Saigo, and R. K., Saxena, Solution of Volterra integro-differential equations with generalized Mittag-Leffler functions in the kernels, J. Int. Equat. Appl. 14, 377–396 (2002).
[29] H. J., Haubold, A. M., Mathai, and R. K., Saxena, Mittag-Leffler functions and their applications, J. Appl. Math. 2011, 298628 (2011).
[30] A. M., Mathai, R. K., Saxena, and H. J., Haubold, The H-function: Theory and applications (Springer, New York, 2009).
[31] E. M., Wright, On the coefficients of power series having essential singularities, J. London Math. Soc. 8, 71–80 (1933).
[32] E. M., Wright, The asymptotic expansion of the generalized Bessel function, Proc. London Math. Soc. 38, 257–270 (1934).
[33] E. M., Wright, The generalized Bessel function of order greater than one, Quart. J Math. Oxford Ser. 11, 36–48 (1940).
[34] R., Gorenflo, Y., Luchko, and F., Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal. 2, 383–414 (1999).
[35] G., Pagnini and E., Scalas, Historical notes on the M-Wright/Mainardi function, Comm. Appl. Ind. Math. 6/1, e-495 (2015).
[36] F., Mainardi, Fractional calculus and waves in linear viscoelesticity: An introduction to mathematical models (Imperial College Press, London, 2010).
[37] R., Gorenflo, Y., Luchko, and F., Mainardi, Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math. 118, 175–191 (2000).
[38] C., Fox, The G and H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98, 395–429 (1961).
[39] B. L. J., Braaksma, Asymptotic expansions and analytical continuations for a class of Barnes-integrals, Compositio Math. 15, 239–341 (1962–1963).
[40] F., Mainardi, G., Pagnini, and R. K., Saxena, Fox H functions in fractional diffusion, J. Comp. Appl. Math. 178, 321–331 (2005).
[41] R., Metzler and J., Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1–77 (2000).
[42] R., Metzler and T. F., Nonnenmacher, Space and time fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation, Chem. Phys. 284, 67–90 (2002).
[43] A. A., Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals. I. Transformation and reduction formulae, J. Phys. A 20, 4109–4117 (1987).
[44] A. A., Inayat-Hussain, New properties of hypergeometric series derivable from Feynman integrals. II. A generalisation of the H-function, J Phys. A 20, 4119–4128 (1987).
[45] W. R., Schneider and W., Wyss, Fractional diffusion and wave equations, J Math. Phys. 30, 134–144 (1989).
[46] I., Podlubny, Fractional differential equations: An introdution to fractional derivatives, fractional differential equations, to methods of their solutions and some of their applications (Academic Press, San Diego, 1998).
[47] G. H., Pertz and C. J., Gerhardt, editors, Leibnizens gesalmmelte Werke, Leibnizens mathematische Schriften, Erste Abtheilung, Band II, pp. 301–302. Dritte Folge Mathematik (Erster Band). (A. Asher & Comp., Berlin, 1849).
[48] L., Euler, De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt. The Euler archive at E019.html. A translation by Stacy Langton, titled On transcendental progressions that is, those whose general terms cannot be given algebraically, is available at∼langton/eg.pdf.
[49] J. L., Lagrange, Sur une nouvelle espèce de calcul relatif a la diffentiation et a l'intégration des quantités vairables, in Ouvres de Lagrange, Tome Troisième, edited by J.-A. Serret (Guthier-Villars, Paris, 1869), p. 448.
[50] P. S., Laplace, Théorie analytique des probabilités, 2nd edition (Courcier, Paris, 1814), 85–86.
[51] S. F., Lacroix, Traité du calcul différentiel et du calcul intégral (Courcier, Paris, 1819), Tome III, pp. 408–409.
[52] J. B. J., Fourier, Théorie analytique de la chaleur (Didot, Paris, 1822), pp. 561–562.
[53] M., Weilbeer, Efficient numerical methods for fractional differential equations and their analytical background (Ph.D. thesis, Technischen Universitat Braunschweig, 2005).
[54] N. H., Abel, Solution de quelques problèmes à l'aide d'integrales définies in Oeuvres Complètes (Grondahl & Son, Christiania, 1881), pp. 15–18.
[55] S. G., Samko, A. A., Kilbas, and O. I., Marichev, Fractional integrals and derivatives: Theory and applications (Gordon and Breach, Yverdon, 1993).
[56] B., Ross, The development of fractional calculus 1695–1900, Historia Mathematica 4, 75–89 (1977).
[57] J., Liouville, Mémoire sur quelches questions de géométrie et de mécanique, et sur un noveau genre de calcul pour résoudre ces questions, J. l' École Roy. Polytéchn., 13, 1–69 (1832).
[58] H. T., Davis, The application of fractional operators to functional equations, Amer. J Math. 49, 123–142 (1927).
[59] J., Liouville, Mémoire sur le théorème des fonctions complémentaires, J. reine angew. Math., 11, 1–19 (1834).
[60] G. F. B., Riemann, Versuch einer Auffassung der integration und differentiation, in Gesammelte mathematische Werke und wissenschaftlicher Nachlass–Bernhard Riemann, edited by H., Weber (Teubner, Leipzig, 1876), pp. 331–344.
[61] N. Y., Sonin, On differentiation with arbitrary index, Moscow Matem. Sbornik 6, 1–38 (1869).
[62] H., Laurent, Sur le calcul des dérivées à indicies quelconques, Nouv. Ann. Math. 3, 240–252 (1884).
[63] A. V., Letnikov, An explanation of the concepts of the theory of differentiation of arbitrary index (in Russian), Moscow Matem. Sbornik 6, 413–445 (1872).
[64] A. K., Grunwald, Über “begrenzte” derivation und deren Anwendung, Z. angew. Math. Phys. 12, 441–480 (1867)
[65] A. V., Letnikov, Theory of differentiation with an arbitrary index (in Russian), Moscow Matem. Sbornik 3, 1–66 (1868).
[66] M., Caputo, Linear models of dissipation whose Q is almost frequency independent–II, Geophys. J. Royal Astr. Soc. 13, 529–539 (1967).
[67] B., West, M., Bologna, and P., Grigolini, Physics of fractal operators (Springer, New York, 2003).
[68] T., Graham, A short account of experimental researches on the diffusion of gas through each other, and their separation by mechanical means, Quart. J. Sci. 2, 74 (1829).
[69] T., Graham, LX. On the molecular mobility of gases, Phil. Mag. 26/177, 409–434 (1863).
[70] J. C., Maxwell, On the dynamical theory of gases (1866), in The scientific papers of James Clerk Maxwell, edited by W. D., Niven (Dover, New York, 1965), p. 59.
[71] A., Fick, Über diffusion, Poggendorff's Annalen der Physik und Chemie 94, 59–86 (1855). English version: On liquid diffusion, Phil. Mag. 10, 30–39 (1855). A more recent English version may be found in J., Memb. Sci. 100, 33–38 (1995).
[72] J. B. J., Fourier, Analytical theory of heat (Great Books of the Western World, vol. 45) (Encyclopedia Britannica, Chicago, 1952), p. 232.
[73] W. C., Roberts-Austen, Bakerian lecture – On the diffusion of metals, Philos. T. R. Soc. London 187, 383–425 (1896).
[74] J., Philibert, One and a half century of diffusion: Fick, Einstein, before and beyond, Diffusion Fundamentals 2, 1.1–1.10 (2005).
[75] Lucretius: The nature of things (Penguin Classics), A. E., Stallings, trans. (Penguin Books, London, 2007).
[76] A., Einstein, Über die von der molekularkinetischen theorie der warme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen, Ann. Phys. 17, 549–560 (1905). See also the English translation, Investigations on the theory of the Brownian movement, edited with notes by R. Fürth (Dover Publications, 1956).
[77] J. B., Perrin, Discontinuous structure of matter, Nobel Lectures, Physics 1922–1941 (Elsevier, Amsterdam, 1965).
[78] M., Smoluchowski, Sur le chemin moyen parcouru par les molécules d'un gaz et son rapport avec la théorie de la diffusion (On the mean path of molecules of gas and its relationship to the theory of diffusion), Bull. Int. Acad. Sci. Cracovie, 202–213 (1906); Essai d'une théorie cinétique du mouvement Brownien et des milieux troubles (Outline of the kinetic theory of Brownian motion of suspensions), ibid., pp. 577–602 (1906); Phys. Z. 17, 557 (1916); ibid., 585 (1916).
[79] A., Fuliński, On Marian Smoluchowski's life and contribution to physics, Acta Phys. Pol. B 29, 1523–1537 (1998).
[80] H., Risken, The Fokker–Planck equation (Springer-Verlag, Berlin, 1996).
[81] P., Langevin, Sur la théorie du mouvement brownien, C. R. Acad. Sci. (Paris) 146, 530–533 (1908).
[82] D. S., Lemons and A., Gythiel, Paul Langevin's 1908 paper “On the theory of Brownian motion” [Sur la théorie du mouvement brownien, C. R. Acad. Sci. (Paris) 146, 530–533 (1908)], Am. J Phys. 65, 1079–1081 (1997).
[83] L., Borland, Microscopic dynamics of the nonlinear Fokker–Planck equation: A phenomenological model, Phys. Rev. E 57, 6634–6642 (1998).
[84] L., Borland, F., Pennini, A. R., Plastino, and A., Plastino, The nonlinear Fokker– Planck equation with state-dependent diffusion – a nonextensive maximum entropy approach, Eur. Phys. J. B 12, 285–297 (1999).
[85] T. D., Frank, Nonlinear Fokker–Planck equations: Fundamentals and applications, (Springer-Verlag, Heidelberg, 2010).
[86] J., Masoliver, K., Lindenberg, and B. J., West, First-passage times for non-Markovian process, Phys. Rev. A 33, 2177–2184 (1986).
[87] K., Pearson, The problem of the random walk, Nature 72, 294 (1905).
[88] Lord Rayleigh, The problem of the random walk, Nature 72, 318 (1905).
[89] K., Pearson, The problem of the random walk, Nature 72, 294 (1905).
[90] L. E., Reichl, A modern course in statistical physics (Wiley, New York, 1998), p. 198.
[91] L. R., Richardson, Atmospheric diffusion shown on a distance neighbour graph, Proc. R. Soc. Lond. A 110, 709–737 (1926).
[92] E. W., Montroll and H., Scher, Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries, J. Stat. Phys. 9, 101–135 (1973).
[93] H., Scher and E. W., Montroll, Anomalous transit-dispersion in amorphous solids, Phys. Rev. B 12, 2455–2477 (1975).
[94] M. F., Schlesinger, Asymptotic solutions of continuous-time random walks, J. Stat. Phys. 10, 421–434 (1974).
[95] B., O'Shaughnessy and I., Procaccia, Analytical solutions for diffusion on fractal objects, Phys. Rev. Lett. 54, 455–458 (1985).
[96] T., Tomé and M. J., Oliveira, Stochastic dynamics and irreversibility (Springer, Heidelberg, 2015).
[97] A. T., Silva, E. K., Lenzi, L. R., Evangelista, M. K., Lenzi, H. V., Ribeiro, and A. A., Tateishi, Exact propagator for a Fokker–Planck equation, first passage time distribution, and anomalous diffusion, J Math. Phys. 52, 083301 (2011).
[98] J., Klafter, I. M., Sokolov, First steps in random walks: From tools to applications (Oxford University Press, Oxford, 2011).
[99] R., Metzler, E., Barkai, J., Klafter, Deriving fractional Fokker–Planck equations from a generalised master equation, Europhys. Lett. 46, 431–436 (1999).
[100] M. A., Zahran, On the derivation of fractional diffusion equation with an absorbent term and a linear external force, Appl. Math. Mod. 33, 3088–3092 (2009).
[101] A. V., Chechkin, R., Gorenflo, and I. M., Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66, 046129 (2002).
[102] E. K., Lenzi, R. S., Mendes, and C., Tsallis, Crossover in diffusion equation: Anomalous and normal behaviors, Phys. Rev. E 67, 031104 (2003).
[103] C. W., Gardiner, Handbook of stochastic methods for physics, chemistry, and the natural sciences. Springer Series in Synergetics, H., Haken (editor), (Springer- Verlag, Heidelberg, 2004).
[104] F., Mota-Furtado and P. F., O'Mahony, Exact propagator for generalised Ornstein– Uhlenbeck processes, Phys. Rev. E 75, 041102 (2007).
[105] F., Mota-Furtado and P. F., O'Mahony, Eigenfunctions and matrix elements for a class of eigenvalue problems with staggered ladder spectra, Phys. Rev. A 74, 044102 (2006).
[106] V., Zaburdaev, S., Denisov, and J., Klafter, Lévy walks, Rev. Mod. Phys. 87, 483–530 (2015).
[107] T., Srokowski, Fractional Fokker–Planck equation for Lévy flights in nonhomogeneous environments, Phys. Rev. E 79, 040104(R) (2009).
[108] T., Srokowski, Anomalous diffusion in nonhomogeneous media: Time-subordinated Langevin equation approach, Phys. Rev. E 89, 030102(R) (2014)
[109] T., Srokowski, Lévy flights in nonhomogeneous media: Distributed-order fractional equation approach, Phys. Rev. E 78, 031135 (2008).
[110] T., Srokowski, Lévy flights and nonhomogenous memory effects: Relaxation to a stationary state, Phys. Rev. E 92, 012125 (2015).
[111] L. C., Malacarne, R. S., Mendes, E. K., Lenzi, and M. K., Lenzi, General solution of the diffusion equation with a nonlocal diffusive term and a linear force term, Phys. Rev. E 74, 042101 (2006)
[112] P. E., Strizhak, Macrokinetics of chemical processes on porous catalysts having regard to anomalous diffusion, Theor. Exp. Chem. 40, 203–208 (2004).
[113] D., Avnir, The fractal approach to heterogeneous chemistry (Wiley-Interscience, New York, 1990).
[114] R. I., Masel, Principles of adsorption and reaction on solid surfaces (Wiley- Interscience, New York, 1996)
[115] G., Fibich, I., Gannot, A., Hammer, and S., Schochet, Chemical kinetics on surfaces: A singular limit of a reaction–diffusion system, SIAM J Math. Anal. 38, 1371–1388 (2006).
[116] E. K., Lenzi, M. A., Fdos Santos, D. S., Vieira, R. S., Zola, and H. V., Ribeiro, Solutions for a sorption process governed by a fractional diffusion equation, Physica A 443, 32–41 (2016).
[117] S. A., Bradford, Y., Wang, H., Kim, S., Torkzaban, and J., Simunek, Modeling microorganism transport and survival in the subsurface, J. Environ. Qual. 43, 421–440 (2014).
[118] J., Crank, The mathematics of diffusion (Oxford Science Publications, England, 1975).
[119] X., Li and M., Xu, A model for reversible reaction in a sub-diffusive regime, J Math. Phys. 50, 102708 (2009).
[120] B. I., Henry, T. A. M., Langlands, and S. L., Wearne, Anomalous diffusion with linear reaction dynamics: From continuous-time random walks to fractional reaction-diffusion equations, Phys. Rev. E 74, 031116 (2006).
[121] I. L., Novak, P., Kraikivski, and B. M., Slepchenko, Diffusion in cytoplasm: Effects of excluded volume due to internal membranes and cytoskeletal structures, Biophys. J. 97, 758–767 (2009).
[122] E. K., Lenzi, M. A. F., dos Santos, M. K., Lenzi, R. Menechini NetoSolutions for a mass transfer process governed by fractional diffusion equations with reaction terms, Commun. Nonlinear Sci. Numer. Simul. 48, 307–317 (2017).
[123] I. M., Sokolov, A.V., Chechkin, and J., Klafter, Fractional diffusion equation for a power-law-truncated Lévy process, Physica A 336, 245–251 (2004).
[124] T. A. M., Langlands, B. I., Henry, and S. L., Wearne, Anomalous subdiffusion with multispecies linear reaction dynamics, Phys. Rev. E 77, 021111 (2008).
[125] V., Gafiychuk and B., Datsko, Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems, Comp. Math. Appl. 59, 1101–1107 (2010).
[126] D., del-Castillo-Negrete, B. A., Carreras, and V. E., Lynch, Front dynamics in reaction-diffusion systems with Lévy flights: A fractional diffusion approach, Phys. Rev. Lett. 91, 018302 (2003).
[127] V. A., Volpert, Y., Nec, and A. A., Nepomnyashchy, Fronts in anomalous diffusion– reaction systems, Philos. T. Roy. Soc. A 371, 20120179 (2015).
[128] D., Campos, S., Fedotov, and V., Mendez, Anomalous reaction-transport processes: The dynamics beyond the law of mass action, Phys. Rev. E 77, 061130 (2008).
[129] A. A., Nepomnyashchy, Mathematical modelling of subdiffusion–reaction systems, Math. Model. Nat. Phenom. 11, 26–36 (2016).
[130] I. M., Sokolov, M. G. W., Schmidt, and F., Sagués, Reaction–subdiffusion equations, Phys. Rev. 73, 031102 (2006).
[131] A., Yadav and W., Horsthemke, Kinetic equations for reaction–subdiffusion systems: Derivation and stability analysis, Phys. Rev. E 74, 066118 (2006).
[132] M. G. W., Schmidt, F., Sagués, and I. M., Sokolov, Mesoscopic description of reactions for anomalous diffusion: A case study, J Phys. Cond. Matt. 19, 065118 (2007).
[133] K., Seki, M., Wojcik, and M., Tachiya, Fractional reaction–diffusion equation, J. Chem. Phys. 119, 2165–2170 (2003).
[134] K., Seki, M.Wojcik, and M. Tachiya, Recombination kinetics in subdiffusive media, J. Chem. Phys. 119, 7525–7533 (2003).
[135] V., Mendez, D., Campos, and S., Fedotov, Front propagation in reaction-dispersal models with finite jump speed, Phys. Rev. E 70, 036121 (2004).
[136] E. K., Lenzi, H. V., Ribeiro, J., Martins, M. K., Lenzi, G. G., Lenzi, and S., Specchia, Non-Markovian diffusion equation and diffusion in a porous catalyst, Chem. Eng. J. 172, 1083–1087 (2011).
[137] A. V., Weigel, B., Simon, M. M., Tamkun, and D., Krapf, Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking, Procl. Natl. Acad. Sci. U.S.A. 108, 6439–6443 (2011).
[138] M., Giona and M., Giustiniani, Adsorption kinetics on fractal surfaces, J Phys. Chem. 100, 16690–16699 (1996).
[139] E. K., Lenzi, M. K., Lenzi, R. S., Zola, H. V., Ribeiro, F. C., Zola, L. R., Evangelista, and G., Gonçalves, Reaction on a solid surface supplied by an anomalous mass transfer source, Physica A 410, 399–406 (2014).
[140] P. C., Bressloff, Stochastic processes in cell biology (Springer, Heidelberg, 2014).
[141] E. K., Lenzi, P. R. G., Fernandes, T., Petrucci, H., Mukai, and H. V., Ribeiro, Anomalous-diffusion approach applied to the electrical response of water, Phys. Rev. E 84, 041128 (2011).
[142] F., Ciuchi, A., Mazzulla, N., Scaramuzza, E. K., Lenzi, and L. R., Evangelista, Fractional diffusion equation and the electrical impedance: Experimental evidence in liquid-crystalline cells, J Phys. Chem. C 116, 8773–8777 (2012).
[143] R., Rossato, M. K., Lenzi, L. R., Evangelista, and E. K., Lenzi, Fractional diffusion equation in a confined region: Surface effects and exact solutions, Phys. Rev. E 76, 032102 (2007).
[144] X. L., Wun and A., Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath, Phys. Rev. Lett. 84, 3017–3020 (2000).
[145] G., Grégoire, H., Chaté, and Y., Tu, Comment on “Particle diffusion in a quasi-twodimensional bacterial bath,” Phys. Rev. Lett. 86, 556–556 (2001).
[146] X. L., Wun and A., Libchaber, Wu and Libchaber reply, Phys. Rev. Lett. 86, 557–557 (2001).
[147] A., Caspi, R., Granek, and M., Elbaum, Enhanced diffusion in active intracellular transport, Phys. Rev. Lett. 85, 5655–5658 (2000).
[148] A., Caspi, R., Granek, and M., Elbaum, Diffusion and directed motion in cellular transport, Phys. Rev. E 66, 011916 (2002).
[149] V., Latora, A., Rapisarda, and S., Ruffo, Superdiffusion and out-of-equilibrium chaotic dynamics with many degrees of freedoms, Phys. Rev. Lett. 83, 2104–2107 (1999).
[150] V., Latora, A., Rapisarda, and C., Tsallis, Non-Gaussian equilibrium in a long-range Hamiltonian system, Phys. Rev. E 64, 056134 (2001).
[151] R. S., Zola, E. K., Lenzi, L. R., Evangelista, and G., Barbero, Memory effect in the adsorption phenomena of neutral particles, Phys. Rev. E 75, 042601 (2007).
[152] G., Barbero and L. R., Evangelista, Adsorption phenomena and anchoring energy in nematic liquid crystals (Taylor & Francis, London, 2006).
[153] O. M. P., Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dyn. 29, 145–155 (2002).
[154] I. V., Kotov, M. V., Khazimullin, and A. P., Krekhov, Flexoelectric instability in nematic liquid crystals between coaxial cylinders, Mol. Cryst. Liq. Cryst. 366, 885–892 (2001).
[155] H., Tsuru, Stability analysis of nematics between two concentric cylinders, J Phys. Soc. Jpn. 59, 1600–1616 (1990).
[156] D., R. M. Williams and A., Halperin, Nematic liquid crystal in a tube: The Fréedericksz transition, Phys. Rev. E 48, R2366–R2369 (1993).
[157] D. R. M., Williams, Nematic liquid crystals between antagonistic cylinders: Spirals with bend-splay director undulations, Phys. Rev. E 50, 1686–1687 (1994).
[158] I., Jánossy, A. D., Lloyd, and B. S., Wherrett, Anomalous optical Freedericksz transition in an absorbing liquid crystal, Mol. Cryst. Liq. Cryst. 179, 1–12 (1990).
[159] I., Jánossy, L., Csillag, and A. D., Lloyd, Temperature dependence of the optical Fréedericksz transition in dyed nematic liquid crystals, Phys. Rev. A 44, 8410–8413 (1991).
[160] I., Jánossy and T., Kósa, Influence of anthraquinone dyes on optical reorientation of nematic liquid crystals, Opt. Lett. 17, 1183–1185 (1992).
[161] I., Jánossy, Molecular interpretation of the absorption-induced optical reorientation of nematic liquid crystals, Phys. Rev. E 49, 2957–2963 (1994).
[162] R., Muenster, M., Jarasch, X., Zhuang, and Y. R., Shen, Dye-induced enhancement of optical nonlinearity in liquids and liquid crystals, Phys. Rev. Lett. 78, 42–45 (1997).
[163] L. S., Lucena, L. R., Silva, L. R., Evangelista, M. K., Lenzi, R., Rossato, and E. K., Lenzi, Solutions for a fractional diffusion equation with spherical symmetry using Green function approach, Chem. Phys. 344, 90–94 (2008).
[164] M. A., Lomholt, I. M., Zaid, and R., Metzler, Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary, Phys. Rev. Lett. 98, 200603 (2007).
[165] S., Condamin, O., Bénichou, and J., Klafter, First-passage time distributions for subdiffusion in confined geometry, Phys. Rev. Lett. 98, 250602 (2007).
[166] M. O., Vlad, Fractional diffusion equation on fractals: Self-similar stationary solutions in a force field derived from a logarithmic potential, Chaos, Solitons & Fractals 4, 191–199 (1994).
[167] Y. Z., Povstenko, Fractional radial diffusion in a cylinder, J. Mol. Liq. 137, 46–50 (2008).
[168] B. N., Narahari Achar and J. W., Hanneken, Fractional radial diffusion in a cylinder, J. Mol. Liq. 114, 147–151 (2004).
[169] T., Bickel, A note on confined diffusion, Physica A 377, 24–32 (2007) 24; Erratum, Physica A 381, 532 (2007).
[170] G. M., Barrow, Physical chemistry (McGraw-Hill, New York, 1961).
[171] E. M., Gomes, R. R. L., Araújo, M. K., Lenzi, F., R. G. B. Silva, and E. K., Lenzi, Parametric analysis of a heavy metal sorption isotherm based on fractional calculus, Math. Probl. Eng. 2013, 642101 (2013).
[172] J. R., Fernández and M. C., Muniz, Numerical analysis of surfactant dynamics at air–water interface using the Henry isotherm, J Math. Chem. 49, 1624–1645 (2011).
[173] E. K., Lenzi, C. A. R., Yednak, and L. R., Evangelista, Non-Markovian diffusion and the adsorption–desorption process, Phys. Rev. E 81, 011116 (2010).
[174] A. W., Adamson and A. P., Gast, Physical chemistry of surfaces, 6th edition (Wiley, New York, 1997).
[175] E. K., Lenzi, H. V., Ribeiro, A. A., Tateishi, R. S., Zola, and L. R., Evangelista, Anomalous diffusion and transport in heterogeneous systems separated by a membrane, Proc. R. Soc. A 472, 20160502 (2016).
[176] V. G., Guimaraes, H. V., Ribeiro, Q., Li, L. R., Evangelista, E. K., Lenzi, and R. S., Zola, Unusual diffusing regimes caused by different adsorbing surfaces, Soft Matter 11, 1658–1666 (2015).
[177] R., Kopelman, Fractal reaction kinetics, Science 241, 1620–1626 (1988).
[178] S., Schnell and T. E., Turner, Reaction kinetics in intracellular environments with macromolecular crowding: Simulations and rate laws, Prog. Biophys. Mol. Biol. 85, 235–260 (2004).
[179] H., Berry, Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation, Biophys. J. 83, 1891–1901 (2002).
[180] R., Grima and S., Schnell, A systematic investigation of the rate laws valid in intracellular environments, Biophys. Chem. 124, 1–10 (2006).
[181] M. T., Klann, A., Lapin, and M., Reuss, Agent-based simulation of reactions in the crowded and structured intracellular environment: Influence of mobility and location of the reactants, BMC Syst. Biol. 5, 71 (2011).
[182] M. N., Berberan-Santos and J. M. G., Martinho, A linear response approach to kinetics with time-dependent rate coefficients, Chem. Phys. 164, 259–269 (1992).
[183] E. O., Voit, H. A., Martens, and S. W. Omholt, 150 years of the mass action law, PLoS Comput. Biol. 11, 1004012 (2015).
[184] V. A., Parsegian, Van der Waals forces: A handbook for biologists, chemists, engineers, and physicists (Cambridge University Press, Cambridge, 2006).
[185] I. M., Zaid, M. A., Lomholt, and R., Metzler, How subdiffusion changes the kinetics of binding to a surface, Biophys. J. 97, 710–721 (2009).
[186] C., Tsallis and D. J., Bukman, Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54, R2197–R2200 (1996).
[187] M., Bologna, C., Tsallis, and P., Grigolini, Anomalous diffusion associated with nonlinear fractional derivative Fokker–Planck-like equation: Exact time-dependent solutions, Phys. Rev. E 62, 2213–2218 (2000).
[188] C., Tsallis, Non-extensive thermostatistics: Brief review and comments, Physica A 221/1–3, 277–290 (1995).
[189] L. C., Malacarne, I. T., Pedron, R. S., Mendes, and E. K., Lenzi, Nonlinear equation for anomalous diffusion: Unified power-law and stretched exponential exact solution, Phys. Rev. E 63, 30101R (2001).
[190] P. C., Assis, Jr, L. R., da Silva, E. K., Lenzi, L. C., Malacarne, and R. S., Mendes, Nonlinear diffusion equation, Tsallis formalism and exact solutions, J Math. Phys. 46, 123303 (2005).
[191] W. F., Ames, Nonlinear partial differential equations in engineering (Academic Press, New York, 1965).
[192] M., Muskat, The flow of homogeneous fluids through porous media (McGraw-Hill, New York, 1937).
[193] P. Y., Polubarinova-Kochina, Theory of ground water movement (Princeton University Press, Princeton, 1962).
[194] H., Spohn, Surface dynamics below the roughening transition, J Phys. France I, 3, 69–81 (1993).
[195] J., Buckmaster, Viscous sheets advancing over dry beds, J. Fluid Mech. 81, 735–756 (1977).
[196] W. P., Johnson, Some applications of the q-exponential formula, Discrete Math. 157, 207–225 (1996).
[197] C., Tsallis, Introduction to nonextensive statistical mechanics: Approaching a complex world (Springer-Verlag, New York, 2009).
[198] R., Metzler, W. G., Glockle, and T. F., Nonnenmacher, Fractional model equation for anomalous diffusion, Physica A 211, 13–24 (1994).
[199] A., Caillard, P., Brault, J., Mathias, C., Charles, R. W., Boswell, and T. Sauvage, Deposition and diffusion of platinum nanoparticles in porous carbon assisted by plasma sputtering, Surf. Coat. Technol. 200, 391–394 (2005).
[200] A. G., Cherstvy and R., Metzler, Ergodicity breaking, ageing, and confinement in generalized diffusion processes with position and time dependent diffusivity, J. Stat. Mech. Theory Exper., 2015, P05010 (2015).
[201] E. K., Lenzi, L. R., Silva, M. K., Lenzi, M. A. F., Santos, H. V., Ribeiro, and L. R., Evangelista, Intermittent motion, nonlinear diffusion equation and Tsallis formalism, Entropy 19, 42 (2017).
[202] V., Mendez, D., Campos, and F., Bartumeus, Stochastic foundations in movement ecology (Springer-Verlag, Heidelberg, 2014).
[203] C., Tsallis, S. V. F., Levy, A. M. C., Souza, and R., Maynard, Statistical-mechanical foundation of the ubiquity of Lévy distributions in nature, Phys. Rev. Lett. 75, 3589–3593 (1995).
[204] E. K., Lenzi, N. R., Menechini, A. A., Tateishi, M. K., Lenzi, and H. V., Ribeiro, Fractional diffusion equations coupled by reaction terms, Physica A 458, 9–16 (2016).
[205] W. H., Press, B. P., Flannery, and S. A., Teukolsky, Numerical recipes in fortran: The art of scientific computing (Cambridge University Press, New York, 1992).
[206] N., Shigesada, Spatial distribution of dispersing animals, J Math. Biol. 9, 85–96 (1980).
[207] C., Tsallis, Possible generalisation of Boltzmann–Gibbs statistics, J. Stat. Phys. 52, 479–488 (1988).
[208] E. K., Lenzi, R., S.Mendes, G., Gonçalves,M. K., Lenzi, and L. R. da Silva, Fractional diffusion equation and Green function approach: Exact solutions, Physica A 360, 215–266 (2006).
[209] E. K., Lenzi, R. S., Mendes, J. S., Andrade, Jr, L. R., da Silva, and L. S., Lucena, N-dimensional fractional diffusion equation and Green function approach: Spatially dependent diffusion coefficient and external force, Phys. Rev. E 71, 052101 (2005).
[210] E. K., Lenzi, L. R., Evangelista, G., Barbero, and F., Mantegazza, Anomalous diffusion and the adsorption–desorption process in anisotropic media, Europhys. Lett. 85, 28004 (2009).
[211] A., Pekalski and K., Sznajd-Weron, eds., Anomalous diffusion: From basics to applications, Lecture Notes in Physics (Springer, Berlin, 1999).
[212] E., Hatta, Anomalous diffusion of fatty acid vesicles driven by adhesion gradients, J Phys. Chem. B 112, 8571–8577 (2008).
[213] H., Stark, Physics of colloidal dispersions in nematic liquid crystals, Phys. Rep. 351, 387–474 (2001).
[214] H. E., Roman, A., Bunde, and W., Dieterich, Conductivity of dispersed ionic conductors: A percolation model with two critical points, Phys. Rev. B 34, 3439–3445 (1986).
[215] F., Brochard and P. G., de Gennes, Theory of magnetic suspensions in liquid crystals, J Phys. (Paris) 31, 691–708 (1970).
[216] R., Eidenschink and W. H., de Jeu, Static scattering in filled nematic: New liquid crystal display technique, Electron. Lett. 27, 1195–1196 (1991).
[217] P., Poulin, V. A., Raghunathan, P., Richetti, and D., Roux, On the dispersion of latex particles in a nematic solution. I. Experimental evidence and a simple model, J Phys. II (France) 4, 1557–1569 (1994).
[218] J. C., Loudet, P., Barois, and P., Poulin, Colloidal ordering from phase separation in a liquid-crystalline continuous phase, Nature 407, 611–613 (2000).
[219] A. V., Chechkin, V., Yu. Gonchar, R., Gorenflo, N., Korabel, and I. M., Sokolov, Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights, Phys. Rev. E 78, 021111 (2008).
[220] G., Barbero and L. R., Evangelista, Adsorption phenomenon of neutral particles and a kinetic equation at the interface, Phys. Rev. E 70, 031605 (2004).
[221] E. R., Weeks and D. A., Weitz, Subdiffusion and the cage effect studied near the colloidal glass transition, Chem. Phys. 284, 361–367 (2002).
[222] G., Marty and O., Dauchot, Subdiffusion and cage effect in a sheared granular material, Phys. Rev. Lett. 94, 015701 (2005).
[223] M., F. de Andrade, E. K., Lenzi, L. R., Evangelista, R. S., Mendes, and L. C., Malacarne, Anomalous diffusion and fractional diffusion equation: Anisotropic media and external forces, Phys. Lett. A 347, 160–169 (2005).
[224] L. R., da Silva, A. A., Tateishi, M. K., Lenzi, E. K., Lenzi, and P. C. da Silva, Green's function for a non-Markovian Fokker–Planck equation: Comb-model and anomalous diffusion, Braz. J Phys. 39, 483–487 (2009).
[225] J. F., Douglas, Polymer science applications of path-integration, integral equations, and fractional calculus, in Applications of fractional calculus in physics, edited by R., Hilfer (World Scientific, Singapore, 2008).
[226] G. A., Mendes, E. K., Lenzi, R. S., Mendes, and L. R. da Silva, Anisotropic fractional diffusion equation, Physica A 346, 271–283 (2005).
[227] L. R., da Silva, L. S., Lucena, P. C., da Silva, E. K., Lenzi, and R. S., Mendes, Multidimensional nonlinear diffusion equation: Spatial time dependent diffusion coefficient and external forces, Physica A 357, 103–108 (2005).
[228] Y., Gefen, A., Aharony, and S., Alexander, Anomalous diffusion on percolating clusters, Phys. Rev. Lett. 50, 77–80 (1983).
[229] H. E., Stanley, Cluster shapes at the percolation threshold: An effective cluster dimensionality and its connection with critical-point exponents, J Phys. A 10, L211 (1977).
[230] S. R., White and M., Barma, Field-induced drift and trapping in percolation networks, J Phys. A 17, 2995–3008 (1984).
[231] A., Bunde, S., Havlin, H. E., Stanley, B., Trus, and G. H., Weiss, Diffusion in random structures with a topological bias, Phys. Rev. B 34, 8129–8132 (1986).
[232] V. E., Arkhincheev and É. M., Baskin, Anomalous diffusion and drift in a comb model of percolation clusters, Zh. Eksp. Teor. Fiz 100, 292–300 (1991).
[233] A., Iomin, Fractional-time quantum dynamics, Phys. Rev. E 80, 022103 (2009).
[234] A., Iomin, Fractional-time Schrödinger equation: Fractional dynamics on a comb, Chaos Solitons & Fractals 44, 348–352 (2011).
[235] K. V., Chukbar, Quasidiffusion of a passive scalar, JETP 82, 719–726 (1996).
[236] A., Iomin, Superdiffusion of cancer on a comb structure, J Phys. Conf. Ser. 7, 57–67 (2005).
[237] A., Iomin, Toy model of fractional transport of cancer cells due to self-entrapping, Phys. Rev. E 73, 061918 (2006).
[238] O. A., Dvoretskaya and P. S., Kondratenko, Anomalous transport regimes and asymptotic concentration distributions in the presence of advection and diffusion on a comb structure, Phys. Rev. E 79, 041128 (2009).
[239] E. K., Lenzi, L. R., da Silva, A. A., Tateishi, M. K., Lenzi, and H. V., Ribeiro, Diffusive process on a backbone structure with drift terms, Phys. Rev. E 87, 012121 (2013).
[240] V. E., Arkhincheev, Random walk on hierarchical comb structures, JETP 88, 710–715 (1999).
[241] A. M., Reynolds, On anomalous transport on comb structures, Physica A 334, 39–45 (2004).
[242] V. E., Arkhincheev, Anomalous diffusion and charge relaxation on comb-model: Exact solutions, Physica A 280, 304–314 (2000).
[243] V. E., Arkhincheev, Random walks on the comb model and its generalizations, Chaos 17, 043102 (2007).
[244] S. C., Weber, J. A., Theriot, and A. J., Spakowitz, Subdiffusive motion of a polymer composed of subdiffusive monomers, Phys. Rev. E 82, 011913 (2010).
[245] J.-H., Jeon and R., Metzler, Analysis of short subdiffusive time series: Scatter of the time-averaged mean-squared displacement, J Phys. A 43, 252001 (2010).
[246] W. F., Marshall, A., Straight, J. F., Marko, J., Swedlow, A., Dernburg, A., Belmont, A. W., Murray, D. A., Agard, and J. W., Sedat, Interphase chromosomes undergo constrained diffusional motion in living cells, Curr. Biol. 7, 930–939 (1997).
[247] I., Golding and E. C., Cox, Physical nature of bacterial cytoplasm, Phys. Rev. Lett. 96, 098102 (2006).
[248] N., Leijnse, J.-H., Jeon, S., Loft, R., Metzler, and L. B., Oddershede, Diffusion inside living human cells, Eur. Phys. J. – Special Topics 204, 75–84 (2012).
[249] J.-H., Jeon, V., Tejedor, S., Burov, E., Barkai, C., Selhuber-Unkel, K., Berg-Sorensen, L., Oddershede, and R. Metzler, In vivo anomalous diffusion and weak ergodicity breaking of lipid granules, Phys. Rev. Lett. 106, 048103 (2011).
[250] D., ben-Avraham and S., Havlin, Diffusion and reactions in fractals and disordered systems (Cambridge University Press, Cambridge, 2000).
[251] D., Villamaina, A., Sarracino, G., Gradenigo, A., Puglisi, and A., Vulpiani, On anomalous diffusion and the out of equilibrium response function in onedimensional models, J., Stat. Mech. L01002 (2011).
[252] D., Panja, Anomalous polymer dynamics is non-Markovian: Memory effects and the generalized Langevin equation formulation, J., Stat. Mech. P06011 (2010).
[253] D., Panja, Generalized Langevin equation formulation for anomalous polymer dynamics, J., Stat. Mech. L02001 (2010).
[254] D., Panja, Probabilistic phase space trajectory description for anomalous polymer dynamics, J Phys. Cond. Matt. 23, 105103 (2011).
[255] B. N., Narahari Achar, B. T., Yale, and J. W., Hanneken, Time fractional Schrödinger equation revisited, Adv. Math. Phys. 2013, 290216 (2013).
[256] B., Baeumer, M. M., Meerschaert, and M., Naber, Stochastic models for relativistic diffusion, Phys. Rev. E 82, 011132 (2010).
[257] E. K., Lenzi, B. F., de Oliveira, N. G. C., Astrath, L. C., Malacarne, R. S., Mendes, M. L., Baesso, and L. R., Evangelista, Fractional approach, quantum statistics, and non-crystalline solids at very low temperatures, Eur. Phys. J. B 62, 155–158 (2008).
[258] X., Jiang, H., Qi, and M., Xu, Exact solutions of fractional Schrödinger-like equation with a nonlocal term, J Math. Phys. 52, 042105 (2011).
[259] E. K., Lenzi, M. K., Lenzi, R., Rossato, and L. C. M., Filho, Solutions for diffusion equation with a nonlocal term, Acta Sci. Technol. 31, 81–86 (2009).
[260] E. K., Lenzi, B. F., de Oliveira, L. R., da Silva, and L. R., Evangelista, Solutions for a Schrödinger equation with a nonlocal term, J Math. Phys. 49, 032108 (2008).
[261] N., Laskin, Lévy flights over quantum paths, Commun. Nonlinear Sci. Nonlinear Simul. 12, 2–18 (2007).
[262] T. A. M., Langlands, Solution of a modified fractional diffusion equation, Physica A 367, 136–144 (2006).
[263] N., Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268, 298–305 (2000).
[264] N., Laskin, Fractional quantum mechanics, Phys. Rev. E 62, 3135–3145 (2000).
[265] N., Laskin, Fractals and quantum mechanics, Chaos 10, 780–790 (2000).
[266] N., Laskin, Fractional Schrödinger equation, Phys. Rev. E 66, 056108 (2002).
[267] M., Naber, Time fractional Schrödinger equation, J Math. Phys. 45, 3339 (2004).
[268] J., Dong, Green's function for the time-dependent scattering problem in the fractional quantum mechanics, J Math. Phys. 52, 042103 (2011).
[269] H., Ertik, D., Demirhan, H., Sirin, and F., Büyükkiliç, Time fractional development of quantum systems, J Math. Phys. 51, 082102 (2010).
[270] S. S., Bayin, Time fractional Schrödinger equation: Fox's H-functions and the effective potential, J Math. Phys. 54, 012103 (2013).
[271] S. S., Bayin, On the consistency of the solutions of the space fractional Schrödinger equation, J Math. Phys. 53, 042105 (2012).
[272] E., Capelas de Oliveira, F., Silva Costa, and J., Vaz, The fractional Schrödinger equation for delta potentials, J Math. Phys. 51, 123517 (2010).
[273] E. K., Lenzi, H. V., Ribeiro, H., Mukai, and R. S., Mendes, Continuous-time random walk as a guide to fractional Schrödinger equation, J Math. Phys. 51, 092102 (2010).
[274] M., Jeng, S.-L.-Y., Xu, E., Hawkins, and J. M., Schwarz, On the nonlocality of the fractional Schrödinger equation, J Math. Phys. 51, 062102 (2010).
[275] Y., Luchko, Fractional Schrödinger equation for a particle moving in a potential well, J Math. Phys. 54, 012111 (2013).
[276] E. K., Lenzi, H. V., Ribeiro, M. A., F. dos Santos, R., Rossato, and R. S., Mendes, Time dependent solutions for a fractional Schrödinger equation with delta potentials, J Math. Phys. 54, 082107 (2013).
[277] E. C., de Oliveira and Jayme Vaz, Jr, Tunneling in fractional quantum mechanics, J Phys. A Math. Theor. 44, 185303 (2011).
[278] S. I., Muslih, Solutions of a particle with fractional δ-potential in a fractional dimensional space, Int. J. Theor. Phys. 49, 2095–2104 (2010).
[279] M., Kleber, Exact solutions for time-dependent phenomena in quantum mechanics, Phys. Rep. 236, 331–393 (1994).
[280] S. M., Blinder, Green's function and propagator for the one-dimensional δ-function potential, Phys. Rev. A 37, 973–976 (1988).
[281] I., Cacciari and P., Moretti, Propagator for the double delta potential, Phys. Lett. A 359, 396–401 (2006).
[282] J., Martins, H. V., Ribeiro, L. R., Evangelista, L. R., da Silva, and E. K., Lenzi, Fractional Schrödinger equation with noninteger dimensions, App. Math. Comp. 219, 2313–2319 (2012).
[283] G. W., Weiss, Aspects and applications of the random walk (North Holland, Amsterdam, 1994).
[284] E., Barkai, Fractional Fokker–Planck equation, solution, and application, Phys. Rev. E, 63, 046118 (2001).
[285] R., Gorenflo and F., Mainardi, Random walk models for space-fractional diffusion processes, Fractional Calculus Appl. Anal. 1, 167–191 (1998).
[286] F., Mainardi, G., Pagnini, and R., Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, App. Math. Comp. 187, 295–305 (2007).
[287] P., Brockmann and I. M., Sokolov, Lévy flights in external force fields: From models to equations, Chem. Phys. 284, 409–421 (2002).
[288] N., Krepysheva, L., Di Pietro, and M.-C., Néel, Space-fractional advection–diffusion and reflective boundary condition, Phys. Rev. E 73, 021104 (2006).
[289] A. V., Chechkin, R., Metzler, V. Y., Gonchar, J., Klafter, and L. V., Tanatarov, First passage and arrival time densities for Lévy flights and the failure of the method of images, J Phys. A 36, L537 (2003).
[290] X., Guo and M., Xu, Some physical applications of fractional Schrödinger equation, J Math. Phys. 47, 082104 (2006).
[291] J., Dong and M., Xu, Some solutions to the space fractional Schrödinger equation using momentum representation method, J Math. Phys. 48, 072105 (2007).
[292] J., Dong and M., Xu, Applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics, J Math. Phys. 49, 052105 (2008).
[293] T., Sandev, I., Petreska, and E. K., Lenzi, Time-dependent Schrödinger-like equation with nonlocal term, J Math. Phys. 55, 092105 (2016).
[294] A. A., Tateishi, E. K., Lenzi, L. R., da Silva, H. V., Ribeiro, S., Picoli, Jr, and R. S. Mendes, Different diffusive regimes, generalized Langevin and diffusion equations, Phys. Rev. E 85, 011147 (2012).
[295] I. M., Sokolov and J., Klafter, From diffusion to anomalous diffusion: A century after Einstein's Brownian motion, Chaos 15, 026103–026109 (2005).
[296] W., Feller, An introduction to probability theory and its applications, vol. II (Wiley, New York, 1968).
[297] A. N., Kochubei, General fractional calculus, evolution equations, and renewal processes, Integr. Equ. Oper. Theory 71, 583 (2011).
[298] F., Mainardi, A., Mura, R., Gorenflo, and M., Stojanovic, The two forms of fractional relaxation of distributed order, J. Vib. Control 13, 1249–1268 (2007).
[299] A. V., Chechkin, J., Klafter, and I. M., Sokolov, Fractional Fokker–Planck equation for ultraslow kinetics, Europhys. Lett. 63, 326–332 (2003).
[300] T., Sandev and Z., Tomovski, Langevin equation for a free particle driven by power-law behaviour type of noises, Phys. Lett. A 378, 1–9 (2014).
[301] E. K., Lenzi, N. G. C., Astrath, R., Rossato, and L. R., Evangelista, Nonlocal effects on the thermal behavior of non-crystalline solids, Braz. J Phys. 39, 507–510 (2009).
[302] R. C., Zeller and R. O., Pohl, Thermal conductivity and specific heat of noncrystalline solids, Phys. Rev. B 4, 2029–2041 (1971).
[303] S., Takeno and M., Goda, A theory of phonon-like excitations in non-crystalline solids and liquids, Prog. Theor. Phys. 47, 790–806 (1972).
[304] S., Takeno and M., Goda, Frequency spectrum and low-temperature specific heat of non-crystalline solids, Prog. Theor. Phys. 48, 1468–1473 (1972).
[305] W. H., Tanttila, Anomalous thermal properties of glasses, Phys. Rev. Lett. 39, 554–557 (1977).
[306] W. H., Tanttila, J., Cooper, and D. J., Toms, Demonstration of possible excitations in liquids, Am. J Phys. 45, 395 (1977).
[307] L. P., Kadanoff and G., Baym, Quantum statistical mechanics (Benjamin, New York, 1962).
[308] C. M., Varma, R. C., Dynes, and J. R., Banavar, Thermally created tunnelling states in glasses, J Phys. C 15, L1221 (1982).
[309] A., Nittke, P., Esquinazi, H.-C. Semmelhak, A. L. Burin, and A. Z., Patashinski, Thermodynamic properties of small amorphous and crystalline silica particles at low temperatures, Eur. Phys. J., B 8, 19–30 (1999).
[310] C., Talón, M. A., Ramos, and S., Vieira, Low-temperature specific heat of amorphous, orientational glass, and crystal phases of ethanol, Phys. Rev. B 66, 012201 (2002).
[311] N. G. C., Astrath, A. C., Bento, M. L., Baesso, E. K., Lenzi, and L. R., Evangelista, Semiclassical approximation for the specific heat of non-crystalline solids at intermediate temperatures, Phil. Mag. 87, 291–297 (2007).
[312] J. R., Macdonald and W. B., Johnson, Fundamental of impedance spectroscopy, in Impedance spectroscopy, theory, experiment, and applications, edited by E., Barsoukov and J. R., Macdonald (Wiley, New York, 2005), pp. 1–26.
[313] I. D., Raistrick, J. R., Macdonald, and D. R., Franceschetti, Theory, in Impedance spectroscopy, theory, experiment, and applications, edited by E., Barsoukov and J. R., Macdonald (Wiley, New York, 2005).
[314] E. K., Lenzi, R. S., Zola, H. V., Ribeiro, D. S., Vieira, F., Ciuchi, A., Mazzulla, N., Scaramuzza, and L. R., Evangelista, Ion motion in electrolytic cells: Anomalous diffusion evidences, J Phys. Chem. B 121/13, 2882–2886 (2017).
[315] E. K., Lenzi, R. S., Zola, R., Rossato, H. V., Ribeiro, D. S., Vieira, and L. R., Evangelista, Asymptotic behaviours of the Poisson–Nernst–Planck model, generalizations and best adjust of experimental data, Electrochim. Acta 226, 40–45 (2017).
[316] R. S., Zola, F. C. M., Freire, E. K., Lenzi, L. R., Evangelista, and G., Barbero, Kinetic equation with memory effect for adsorption–desorption phenomena, Chem. Phys. Lett. 438, 144–147 (2007).
[317] J., Bisquert and A., Compte, Theory of the electrochemical impedance of anomalous diffusion, J. Electroanal. Chem. 499, 112–120 (2001).
[318] J., Bisquert, G., Garcia-Belmonte, and A., Pitarch, An explanation of anomalous diffusion patterns observed in electroactive materials by impedance methods, ChemPhysChem 4, 287–292 (2003).
[319] J., Bisquert, Interpretation of a fractional diffusion equation with nonconserved probability density in terms of experimental systems with trapping or recombination, Phys. Rev. E 72, 011109 (2005).
[320] J., Bisquert, Fractional diffusion in the multiple-trapping regime and revision of the equivalence with the continuous-time random walk, Phys. Rev. Lett. 91, 010602 (2003).
[321] E., Warburg, Ueber die polarisationscapacitat des platins, Ann. Phys. 311, 125–135 (1901).
[322] J. N., Israelachvili, Intermolecular and surface forces, 3rd edition (Elsevier– Academic Press, Amsterdam, 2017).
[323] G., Barbero and M., Scalerandi, Similarities and differences among the models proposed for real electrodes in the Poisson–Nernst–Planck theory, J. Chem. Phys. 136, 136, 084705 (2012).
[324] E. K., Lenzi, L. R., Evangelista, and G., Barbero, Fractional diffusion equation and impedance spectroscopy of electrolytic cells, J Phys. Chem. B 113, 11371–11374 (2009).
[325] I. M., Sokolov, A. V., Chechkin, and J., Klafter, Distributed-order fractional kinetics, Acta Phys. Pol. B 35, 1323–1341 (2004).
[326] F., Mainardi and G., Pagnini, The role of the Fox–Wright functions in fractional sub-diffusion of distributed order, J. Comput. Appl. Math. 207, 245–257 (2007).
[327] T., Pajkossy and L., Nyikos, Scaling-law analysis to describe the impedance behavior of fractal electrodes, Phys. Rev. B 42, 709–719 (1990).
[328] H., Sanabria and J. H., Miller, Relaxation processes due to the electrode-electrolyte interface in ionic solutions, Phys. Rev. E 74, 051505 (2006).
[329] T., Kosztolowicz and K. D., Lewandowska, Hyperbolic subdiffusive impedance, J Phys. A Math. Theor. 42, 055004 (2009).
[330] J. R., Macdonald, The impedance of a galvanic cell with two plane-parallel electrodes at a short distance, J. Electroanal. Chem. 32, 317–328 (1971).
[331] A., Alexe-Ionescu and G., Barbero, Role of the diffuse layer of the ionic charge on the impedance spectroscopy of a cell of liquid, Liq. Cryst. 32, 943–949 (2005).
[332] J. R., Macdonald, Utility of continuum diffusion models for analyzing mobile-ion immittance data: Electrode polarization, bulk, and generation–recombination effects, J Phys. Cond. Matt. 22, 495101 (2010).
[333] J. R., Macdonald, L. R., Evangelista, E. K., Lenzi, and G., Barbero, Comparison of impedance spectroscopy expressions and responses of alternate anomalous Poisson–Nernst–Planck diffusion equations for finite-length situations, J Phys. Chem. C 115, 7648–7655 (2011).
[334] L. R., Evangelista, E. K., Lenzi, and G., Barbero, The Kramers–Kronig relations for usual and anomalous Poisson–Nernst–Planck models, J Phys. Cond. Matt. 46, 465104 (2013).
[335] J. R., Macdonald and D. R., Franceschetti, Theory of small-signal ac response of solids and liquids with recombining mobile charge, J. Chem. Phys. 68, 1614–1637 (1978).
[336] P. A., Santoro, J. L., de Paula, E. K., Lenzi, and L. R., Evangelista, Anomalous diffusion governed by a fractional diffusion equation and the electrical response of an electrolytic cell, J. Chem. Phys. 135, 114704 (2011).
[337] D. L., Sidebottom, Colloquium: Understanding ion motion in disordered solids from impedance spectroscopy scaling, Rev. Mod. Phys. 81, 999–1014 (2009).
[338] F., Batalioto, A. R., Duarte, G., Barbero, and A. M. F., Neto, Dielectric dispersion of water in the frequency range from 10 mHz to 30 MHz, J Phys. Chem. B 114, 3467–3471 (2010).
[339] G., Barbero, Influence of adsorption phenomenon on the impedance spectroscopy of a cell of liquid, Phys. Rev. E 71, 062201 (2005).
[340] L. R., Evangelista, E. K., Lenzi, G., Barbero, and J. R., Macdonald, Anomalous diffusion and memory effects on the impedance spectroscopy for finite-length situations, J Phys. Cond. Matt. 23, 485005 (2011).
[341] P. A., Santoro, E. K., Lenzi, L. R., Evangelista, F., Ciuchi, A., Mazzulla, and N., Scaramuzza, Anomalous diffusion effects on the electrical impedance response of liquid-crystalline systems, Mol. Cryst. Liq. Crist. 576, 23–31 (2013).
[342] J. R., Macdonald and L. D., Potter, A flexible procedure for analysing impedance spectroscopy results: Description and illustrations, Solid State Ionics 24, 61–79 (1987).
[343] J. R., Macdonald, Equivalent circuits for the binary electrolyte in the Warburg region, Electroanal. Chem. Interfacial Electrochem. 47, 182–189 (1973).
[344] J. R., Macdonald, Three to six ambiguities in immittance spectroscopy data fitting, J Phys. Cond. Matt. 24, 175004 (2012).
[345] J., Jorcin, M. E., Orazem, N., Pébére, and B., Tribollet, CPE analysis by local electrochemical impedance spectroscopy, Electrochim. Acta 51, 1473–1479 (2006).
[346] P., Córdoba-Torres, T. J., Mesquita, O., Devos, B., Tribollet, V., Roche, and R. P., Nogueira, On the intrinsic coupling between constant-phase element parameters alpha and in electrochemical impedance spectroscopy, Electrochim. Acta 72, 172–178 (2012).
[347] H. S., Liu, Fractal model for ac response of a rough interface, Phys. Rev. Lett. 55, 529–532 (1985).
[348] U., Rammelt and G., Reinhard, On the applicability of a constant phase element (CPE) to the estimation of roughness of solid metal-electrodes, Electrochim. Acta, 35, 1045–1049 (1990).
[349] B., Sapoval, Transport across irregular interfaces: Fractal electrodes, membranes and catalysts, in Fractals and disordered systems, edited by A., Bunde and S., Havlin (Springer-Verlag, Heidelberg, 1996), pp. 233–261.
[350] C., Hitz and A., Lasia, Experimental study and modeling of impedance of the her on porous Ni electrodes, J. Electroanal. Chem. 500, 213–222 (2001).
[351] V. M., Huang, V., Vivier, M. E., Orazem, N., Pébère, and B., Tribollet, The apparent constant-phase-element behaviour of an ideally polarized blocking electrode – A global and local impedance analysis, J. Electrochem. Soc. 152, C81–C88 (2007).
[352] J., Bisquert, Analysis of the kinetics of ion intercalation ion trapping approach to solid-state relaxation processes, Electrochim. Acta 47, 2435–2449 (2002).
[353] J., Bisquert, Beyond the quasistatic approximation: Impedance and capacitance of an exponential distribution of traps, Phys. Rev. B 77, 235203 (2008).
[354] E. K., Lenzi, J. L., de Paula, F. R., G. B. Silva, and L. R., Evangelista, A connection between anomalous Poisson–Nernst–Planck model and equivalent circuits with constant phase elements, J Phys. Chem. C 117, 23685–23690 (2013).
[355] B., Sapoval, J.-N. Chazalviel, and J. Peyriere, Electrical response of fractal and porous interfaces, Phys. Rev. A 38, 5867–5887 (1988).
[356] L., Nyikos and T., Pajkossy, Fractal dimension and fractional power frequencydependent impedance of blocking electrodes, Electrochim. Acta 30, 1533–1540 (1985).
[357] B., Sapoval, Fractal electrodes and constant phase-angle response – Exact examples and counter examples, Solid State Ionics 23, 253–259 (1987).
[358] T., Pajkossy, Electrochemistry at fractal surfaces, J. Electroanal. Chem. 300, 1–11 (1991).
[359] B. Y., Park, R., Zaouk, C., Wang, and M. J., Madou, A case for fractal electrodes in electrochemical applications, J. Electrochem. Soc. 154, P1–P5 (2007).
[360] T. C., Halsey and M., Leibig, The double-layer impedance at a rough-surface – Theoretical results, Ann. Phys. 219, 109–147 (1992).
[361] B., Sapoval, General formulation of Laplacian transfer across irregular surfaces, Phys. Rev. Lett. 73, 3314–3316 (1994).
[362] D. S., Grebenkov, M., Filoche, and B., Sapoval, Mathematical basis for a general theory of Laplacian transport towards irregular interfaces, Phys. Rev. E 73, 021103 (2006).
[363] E. K., Lenzi, P. R. G., Fernandes, T., Petrucci, H., Mukai, H. V., Ribeiro, M. K., Lenzi, and G., Gonçalves, Anomalous diffusion and electrical response of ionic solutions, Int. J. Electrochem. Sci. 8, 2849–2862 (2013).
[364] E. K., Lenzi, M. K., Lenzi, F. R., G. B. Silva, G., Gonçalves, R., Rossato, R. S., Zola, and L. R., Evangelista, A framework to investigate the immittance responses for finite-length-situations: Fractional diffusion equation, reaction term, and boundary conditions, J. Electroanal. Chem. 712, 82–88 (2014).
[365] J., Klafter, A., Blumen, and M. F., Shlesinger, Stochastic pathway to anomalous diffusion, Phys. Rev. A 35, 3081–3085 (1987).
[366] G., Derfel, E. K., Lenzi, C. R., Yednak, and G., Barbero, Electrical impedance of an electrolytic cell in the presence of generation and recombination of ions, J. Chem. Phys. 132, 224901 (2010).
[367] J. L., de Paula, P. A., Santoro, R. S., Zola, E. K., Lenzi, L. R., Evangelista, F., Ciuchi, A., Mazzulla, and N., Scaramuzza, Non-Debye relaxation in the dielectric response of nematic liquid crystals: Surface and memory effects in the adsorption–desorption process of ionic impurities, Phys. Rev. E 86, 051705 (2012).
[368] J. D., Murray, Mathematical biology I: An introduction (Springer-Verlag, Heidelberg, 2002).
[369] J. D., Murray, Mathematical biology II: Spatial models and biomedical applications (Springer-Verlag, Heidelberg, 2003).
[370] G., Barbero and J. R., Macdonald, Transport process of ions in insulating media in the hyperbolic diffusion regime, Phys. Rev. E 81, 051503 (2010).
[371] C., Criado, P., Galan Montenegro, P., Velasquez, and J. R. Ramos Barrado, Diffusion with general boundary conditions in electrochemical systems, J., Electroanal. Chem. 488, 59–63 (2000).
[372] J. R., Ramos Barrado, P., Galan Montenegro, and C., Criado Gambom, A generalized Warburg impedance for a nonvanishing relaxation process, J. Chem. Phys. 105, 2813–2815 (1996).
[373] K. D., Lewandoska and T., Kosztolowicz, Application of generalized Cattaneo equation to model subdiffusion impedance, Acta Phys. Pol. B 39, 1211–1220 (2008).
[374] F., Batalioto, O. G., Martins, A. R., Duarte, and A. M. Figueiredo Neto, The influence of adsorption phenomena on the impedance spectroscopy of an electrolytic cell, Eur. Phys. J., E 34, 10 (2011).
[375] T., Basu, M. M., Goswami, T. R., Middya, and S., Tarafdar, Morphology and ionconductivity of gelatin-LiClO4 films: Fractional diffusion analysis, J Phys. Chem. B 116, 11362–11369 (2012).
[376] D. R., Franceschetti and J. R., Macdonald, Numerical analysis of electrical response: Statics and dynamics of space-charge regions at blocking electrodes, J. Appl. Phys. 50, 291–302 (1979).
[377] H., Chang and G., Jaffé, Polarization in electrolytic solutions. Part I. Theory, J. Chem. Phys. 20, 1071–1077 (1952).
[378] R. J., Friauf, Polarization effects in the ionic conductivity of silver bromide, J. Chem. Phys. 22, 1329–1338 (1954).
[379] J. R., Macdonald, Theory of space-charge polarization and electrode-discharge effects, J. Chem. Phys. 58, 4982–5001(1973); Erratum: Theory of space-charged polarization and electrode-discharge effects, 60, 343 (1974).
[380] J. R., Macdonald, Effects of various boundary conditions on the response of Poisson– Nernst–Planck impedance spectroscopy analysis models and comparison with a continuous time random walk model, J Phys. Chem. A 115, 13370–13380 (2011).