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

T. Kosztołowicz¹ ; K. D. Lewandowska²

¹ Institute of Physics, Jan Kochanowski University, ul. Świętokrzyska 15, 25-406 Kielce, Poland
² Department of Radiological Informatics and Statistics, Medical University of Gdańsk, ul. Tuwima 15, 80-210 Gdańsk, Poland

Mathematical modelling of natural phenomena, Tome 8 (2013) no. 2, pp. 44-54.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20138204

Bibliographie

[1] R. Metzler, J. Klafter Phys. Rep. 2000 1 77

[2] R. Metzler, J. Klafter J. Phys., A 2004 R161 R208

[3] T. Kosztołowicz, K. Dworecki, S. Mrówczyński Phys. Rev. Lett. 2005

[4] T. Kosztołowicz, K. Dworecki, S. Mrówczyński Phys. Rev., E 2005

[5] K. Seki, M. Wojcik, M. Tachiya J. Chem. Phys. 2003 7525 7533

[6] S.B. Yuste, L. Acedo, K. Lindenberg Reaction front in an A + B → C reaction–subdiffusion process Phys. Rev., E 2004

[7] T. Kosztołowicz, K.D. Lewandowska Phys. Rev., E 2008

[8] D. ben-Avraham, S. Havlin. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, 2000.

[9] V. Méndez, S. Fedotov, W. Horsthemke. Reaction–Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities. Springer-Verlag, Berlin, 2010.

[10] Z. Koza The long–time behavior of initially separated A + B → 0 reaction–diffusion systems with arbitrary diffusion constants J. Stat. Phys. 1996 179 191

[11] Z. Koza The long–time behaviour of initially separated A + B(static) → 0 reaction–diffusion systems Physica, A 1997 622 634

[12] M.Z. Bazant, H.A. Stone Physica, D 2000 95 121

[13] L. Gálfi, Z. Rácz Properties of the reaction front in an A + B → C type reaction–diffusion process Phys. Rev., A 1988 3151 3154

[14] S. Cornell, S. Koza, M. Droz Dynamic Multiscaling of the Reaction–Diffusion Front for mA + nB → 0 Phys. Rev., E 1995 3500 3505

[15] K.D. Lewandowska, T. Kosztołowicz Physica, A 2012 2608 2616

[16] H. Taitelbaum, Y-E.L. Koo, S. Havlin, R. Kopelman, G.H. Weiss Exotic Behavior of the Reaction Front in the A + B → C Reaction-Diffusion System Phys. Rev., A 1992 2151 2154

[17] H. Taitelbaum, A. Yen, R. Kopelman, S. Havlin, G.H. Weiss Effects of bias on the kinetics of A + B → C with initially separated reactants Phys. Rev., E 1996 5942 5947

[18] Z. Jiang, C. Ebner Phys. Rev., A 1990 7483 7486

[19] K.D. Lewandowska, T. Kosztołowicz Acta Phys. Pol., B 2007 1847 1854

[20] I. Podlubny. Fractional differential equations. Academic Press, San Diego, 1999.

[21] J. Crank. The mathematics of diffusion. Clarendon Press, Oxford, 1975.

[22] T. Kosztołowicz J. Phys., A 2004 10779 10789

[23] T. Kosztołowicz, K. Dworecki, K.D. Lewandowska Phys. Rev., E 2012

[24] K.D. Lewandowska, T. Kosztołowicz, M. Piwnik Acta Phys. Pol., B 2012 1065 1071

[25] 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.