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Application of Fractional Differential Equations in Modelling the Subdiffusion–Reaction Process

Published online by Cambridge University Press:  24 April 2013

T. Kosztołowicz  and
K. D. Lewandowska
T. Kosztołowicz*
Affiliation:
Institute of Physics, Jan Kochanowski University, ul. Świętokrzyska 15, 25-406 Kielce, Poland
K. D. Lewandowska
Affiliation:
Department of Radiological Informatics and Statistics, Medical University of Gdańsk, ul. Tuwima 15, 80-210 Gdańsk, Poland
*
⋆ Corresponding author. E-mail: tadeusz.kosztolowicz@ujk.edu.pl
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Abstract

We focus on a subdiffusion–reaction system in which substances are separated at the initial moment. This system is described by nonlinear differential subdiffusion–reaction equations with a fractional time derivative. These equations are very difficult to solve but there exist methods which allow us to solve them approximately. We discuss how useful such methods are, in particular, the quasistatic approximation method and the perturbation method in analytical solving subdiffusion–reaction equations.

Keywords

nonlinear fractional differential equationsnumerical calculationsanomalous diffusion
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 44 - 54
Copyright
© EDP Sciences, 2013

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References

Metzler, R., Klafter, J.. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339 (2000), 1–77. CrossRefGoogle Scholar
Metzler, R., Klafter, J.. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys., A 37 (2004), R161–R208. CrossRefGoogle Scholar
Kosztołowicz, T., Dworecki, K., Mrówczyński, S.. How to measure subdiffusion parameters. Phys. Rev. Lett., 94 (2005), 170602-1–170602-4. CrossRefGoogle ScholarPubMed
Kosztołowicz, T., Dworecki, K., Mrówczyński, S.. Measuring subdiffusion parameters. Phys. Rev., E 71 (2005), 041105-1–041105-11. CrossRefGoogle ScholarPubMed
Seki, K., Wojcik, M., Tachiya, M.. Recombination kinetics in subdiffusive media. J. Chem. Phys., 119 (2003), 7525–7533. CrossRefGoogle Scholar
Yuste, S.B., Acedo, L., Lindenberg, K.. Reaction front in an A + B → C reaction–subdiffusion process. Phys. Rev., E 69 (2004), 036126-1–036126-10. CrossRefGoogle Scholar
Kosztołowicz, T., Lewandowska, K.D.. Time evolution of the reaction fornt in a subdiffusive system. Phys. Rev., E 78 (2008), 066103-1–066103-11. CrossRefGoogle Scholar
D. ben-Avraham, S. Havlin. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, 2000.
V. Méndez, S. Fedotov, W. Horsthemke. Reaction–Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities. Springer-Verlag, Berlin, 2010.
Koza, Z.. The long–time behavior of initially separated A + B → 0 reaction–diffusion systems with arbitrary diffusion constants. J. Stat. Phys., 85 (1996), 179–191. CrossRefGoogle Scholar
Koza, Z.. The long–time behaviour of initially separated A + B(static) → 0 reaction–diffusion systems. Physica, A 240 (1997), 622–634. CrossRefGoogle Scholar
Bazant, M.Z., Stone, H.A.. Asymptotics of reaction–diffusion fronts with one static and one diffusing reactant. Physica, D 147 (2000), 95–121. CrossRefGoogle Scholar
Gálfi, L., Rácz, Z.. Properties of the reaction front in an A + B → C type reaction–diffusion process. Phys. Rev., A 38 (1988), 3151–3154. CrossRefGoogle Scholar
Cornell, S., Koza, S., Droz, M.. Dynamic Multiscaling of the Reaction–Diffusion Front for mA + nB → 0. Phys. Rev., E 52 (1995), 3500–3505. CrossRefGoogle ScholarPubMed
Lewandowska, K.D., Kosztołowicz, T.. Application of diffusion-reaction equations to model carious lesion progress. Physica, A 391 (2012), 2608–2616. CrossRefGoogle Scholar
Taitelbaum, H., Koo, Y-E.L., Havlin, S., Kopelman, R., Weiss, G.H.. Exotic Behavior of the Reaction Front in the A + B → C Reaction-Diffusion System. Phys. Rev., A 46 (1992), 2151–2154. CrossRefGoogle ScholarPubMed
Taitelbaum, H., Yen, A., Kopelman, R., Havlin, S., Weiss, G.H.. Effects of bias on the kinetics of A + B → C with initially separated reactants. Phys. Rev., E 54 (1996), 5942–5947. CrossRefGoogle Scholar
Jiang, Z., Ebner, C.. Simulation study of reaction fronts. Phys. Rev., A 42 (1990), 7483–7486. CrossRefGoogle Scholar
Lewandowska, K.D., Kosztołowicz, T.. Numerical study of subdiffusion equation. Acta Phys. Pol., B 38 (2007), 1847–1854. Google Scholar
I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.
J. Crank. The mathematics of diffusion. Clarendon Press, Oxford, 1975.
Kosztołowicz, T.. From solutions of diffusive equation to the solution of subdiffusive one. J. Phys., A 37 (2004), 10779–10789. CrossRefGoogle Scholar
Kosztołowicz, T., Dworecki, K., Lewandowska, K.D.. Subdiffusion in a system with thin membranes. Phys. Rev., E 86 (2012), 021123-1–021123-7. CrossRefGoogle Scholar
Lewandowska, K.D., Kosztołowicz, T., Piwnik, M.. The perturbation method to solve subdiffudion–reaction equations. Acta Phys. Pol., B 43 (2012), 1065–1071. CrossRefGoogle Scholar
K.D. Lewandowska, T. Kosztołowicz. Time evolution of the reaction front in a subdiffusive system. In: Noise and fluctuations, 2007, edited by M. Tacano, Y. Yamamoto, M. Nakao. American Institute of Physics, Melville, 2007.