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Fractional Calculus and Applied Analysis

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Fractional adsorption diffusion

1Mathematics Department, German University in Cairo, New Cairo City, Egypt

3Department of Mathematical Physics, University of Ulm, Albert-Einstein-Allee 11, D-89069, Ulm, Germany

2School of Computing, University of Utah, 3414 Merrill Engineering Bldg., Salt Lake City, UT, 84112, USA

© 2013 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Fractional Calculus and Applied Analysis. Volume 16, Issue 3, Pages 737–764, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: 10.2478/s13540-013-0046-3, June 2013

Publication History

Published Online:


The aim of this article is to generalize the diffusion based adsorption model to a fractional diffusion and fractional adsorption model. The models are formulated as nonlinear fractional boundary value problems equivalent to a singular Hammerstein integral equation. The novelty is that not only the diffusion component of the model is fractionalized but also the adsorption part. The singular Hammerstein integral equation is solved by Sinc approximations. Specific numerical schemes are presented. Based on these solutions we are able to identify different regimes of adsorption diffusion processes controlled by fractional derivatives verified by experimental data. These regimes allow to classify experiments if examined with respect to their scaling behavior.

MSC: Primary 65-XX, 45-XX, 97-XX; Secondary 65D15, 45E10, 44A35, 97N50

Keywords: Sinc method; fractional calculus; approximation

  • [1] A. Apicella, L. Nicolais, G. Astarita, and E. Drioli, Hygrothermal history dependence of moisture sorption kinetics in epoxy resins. Poly. Eng. Sci. 21 (1981), 18–22. http://dx.doi.org/10.1002/pen.760210104 [CrossRef]

  • [2] G. Baumann and F. Stenger, Fractional calculus and Sinc methods. Fract. Calc. Appl. Anal. 14, No 4 (2011), 568–622; DOI: 10.2478/s13540-011-0035-3; http://link.springer.com/journal/13540/. [CrossRef]

  • [3] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2004). http://dx.doi.org/10.1017/CBO9780511543234 [CrossRef]

  • [4] M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91 (1971), 134–147. http://dx.doi.org/10.1007/BF00879562 [CrossRef]

  • [5] J. Crank, The Mathematics of Diffusion. Oxford Univ. Press, Oxford (2009).

  • [6] P. Delahay and I. Trachtenberg, Adsorption kinetics and electrode processes. J. Amer. Chem. Soc. 79 (1957), 2355–2362. http://dx.doi.org/10.1021/ja01567a004 [CrossRef]

  • [7] P. Delahay and Ch.T. Fike, Adsorption kinetics with diffusion controlthe plane and the expanding sphere. J. Amer. Chem. Soc. 80 (1958), 2628–2630. http://dx.doi.org/10.1021/ja01544a007 [CrossRef]

  • [8] D.D. Do, Adsorption Analysis. Imperial College Press, London (1998).

  • [9] Z. Földes-Papp and G. Baumann, Fluorescence molecule counting for single-molecule studies in crowded environment of living cells without and with broken ergodicity. Cur. Pharm. Biot. 12 (2011), 824–833. http://dx.doi.org/10.2174/138920111795470949 [CrossRef]

  • [10] M. Friedrich, A. Seidel, and D. Gelbin, Measuring adsorption rates from an aqueous solution, AIChE J. 31 (1985), 324–327. http://dx.doi.org/10.1002/aic.690310222 [CrossRef]

  • [11] I.P. Gavrilyuk, P.F. Zhuk, and L.N. Bondarenko, Some inverse problems of internal-diffusion kinetics of adsorption. J. Math. Sci. 66 (1993), 2387–2390. http://dx.doi.org/10.1007/BF01364969 [CrossRef]

  • [12] M. Giona and M. Giustiniani, Adsorption kinetics on fractal surfaces. J. Phys. Chem. 100 (1996), 16690–16699. http://dx.doi.org/10.1021/jp961518l [CrossRef]

  • [13] J.D. Goddard, History effects in transient diffusion through heterogeneous media. Ind. Chem. Res. 31 (1992), 713–721. http://dx.doi.org/10.1021/ie00003a010 [CrossRef]

  • [14] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).

  • [15] T. Kwon, M. Min, H. Lee, and B.J. Kim, Facile preparation of water soluble CuPt nanorods with controlled aspect ratio and study on their catalytic properties in water. J. Mater. Chem. 21 (2011), 11956–11960. http://dx.doi.org/10.1039/c1jm11318f [CrossRef]

  • [16] C. Lubich, Convolution quadrature and discretized operational calculus: I. Numer. Math. 52 (1988), 129–145. http://dx.doi.org/10.1007/BF01398686 [CrossRef]

  • [17] C. Lubich, Convolution quadrature and discretized operational calculus: II. Numer. Math. 52 (1988), 413–425. http://dx.doi.org/10.1007/BF01462237 [CrossRef]

  • [18] J. McNamee, F. Stenger, and E.L. Whitney, Whittaker’s cardinal function in retrospect. Math. Comp. 23 (1971), 141–154.

  • [19] Y. Meroz, I.M. Sokolov, J. Klafter, Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist. Phys. Rev. E 81 (2010), 010101xxx1–010101xxx4. http://dx.doi.org/10.1103/PhysRevE.81.010101 [CrossRef] [Web of Science]

  • [20] R. Metzler, W.G. Glöckle, and Th. Nonnenmacher, Fractional model equation for anomalous diffusion. Physica A 211 (1994), 13–24. http://dx.doi.org/10.1016/0378-4371(94)90064-7 [CrossRef]

  • [21] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion. Phys. Rep. 339 (2000), 1–77. http://dx.doi.org/10.1016/S0370-1573(00)00070-3 [CrossRef]

  • [22] N. Quirke, Adsorption and Transport at the Nanoscale. CRC/Taylor & Francis, Boca Raton, Fla. (2006).

  • [23] O.J. Redlich and D.L. Peterson, A useful adsorption isotherm. J. Phys. Chem. 63 (1959), 1024–1024. http://dx.doi.org/10.1021/j150576a611 [CrossRef]

  • [24] H. Scher and E.W. Montroll, Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12 (1975), 2455–2477. http://dx.doi.org/10.1103/PhysRevB.12.2455 [CrossRef]

  • [25] A. Seri-Levy and D. Avnir, Kinetics of diffusion-limited adsorption on fractal surfaces. J. Phys. Chem. 97 (1993), 10380–10384. http://dx.doi.org/10.1021/j100142a019 [CrossRef]

  • [26] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993). http://dx.doi.org/10.1007/978-1-4612-2706-9 [CrossRef]

  • [27] F. Stenger, Collocating convolutions. Math. Comp. 64 (1995), 211–235. http://dx.doi.org/10.1090/S0025-5718-1995-1270624-7 [CrossRef]

  • [28] F. Stenger, Handbook of Sinc Numerical Methods. CRC Press, Boca Raton (2011).

  • [29] B. Such, Th. Trevethan, Th. Glatzel, Sh. Kawai, L. Zimmerli, E. Meyer, A.L. Shluger, C.H.M. Amijs, and P. Mendoza, Functionalized truxenes: adsorption and diffusion of single molecules on the KBr(001) surface. ACS Nano 4 (2010), 3429–3439. http://dx.doi.org/10.1021/nn100424g [Web of Science] [CrossRef]

  • [30] Y.J. Weitsman, A continuum diffusion model for viscoelastic materials. J. Phys. Chem. 94 (1990), 961–968. http://dx.doi.org/10.1021/j100365a085 [CrossRef]

  • [31] Y.J. Weitsman and Ya-J. Guo, A correlation between fluid-induced damage and anomalous fluid sorption in polymeric composites. Comp. Sci. Tech. 62 (2002), 889–908. http://dx.doi.org/10.1016/S0266-3538(02)00032-5 [CrossRef]

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