Mathematical Modelling of Subdiffusion-reaction Systems

A. A. Nepomnyashchy¹

¹ Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel

Mathematical modelling of natural phenomena, Tome 11 (2016) no. 1, pp. 26-36.

Résumé

A review of recent developments in the theoretical description and mathematical modelling of subdiffusion-reaction systems is presented. A number of subdiffusion-reaction models are discussed, including models of subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators. Specific features of the instability development and pattern formation are discussed. Some applications of subdiffusion-reaction models to specific natural problems are mentioned.

Détail

DOI : 10.1051/mmnp/201611102

Bibliographie

[1] J.-P. Bouchaud, A. Georges Phys. Rep. 1990 127 293

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

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

[4] R. Metzler, J. Klafter, J. Phys A: Math Gen. 2004 R161 R208

[5] W. Feller. An Introduction to Probability Theory and Its Applications, v. I. John Wiley Sons, New York et al., 1968.

[6] W. Feller. An Introduction to Probability Theory and Its Applications, v. II. John Wiley Sons, New York et al., 1971.

[7] B. B. Mandelbrot, J. W. Van Ness SIAM Rev. 1968 422 437

[8] I. Goychuk, P. Hänngi Phys. Rev. Lett. 2007 200601

[9] I. Goychuk Adv. Chem. Phys. 2012 187 253

[10] A. Taloni, A. Chechkin, J. Klafter Math. Model. Nat. Phenomen. 2013 127 143

[11] Y. Meroz, I. M. Sokolov, J. Klafter Phys. Rev. E 2010 010101(R)

[12] Y. Meroz, I. M. Sokolov, J. Klafter Phys. Rev. Lett. 2011 260601

[13] Y. Meroz, I. M. Sokolov Phys. Rep. 2015 1 29

[14] Y. Meroz, I. M. Sokolov, J. Klafter Phys. Rev. Lett. 2013 090601

[15] D. H. Zanette Phys. Rev. E 1997 1181 1184

[16] D. Del-Castillo-Negrete Phys. Rev. E 2009 031120

[17] B. I. Henry, T. A. M. Langlands, S. L. Wearne Phys. Rev. E 2006 031116

[18] K. Seki, M. Wojcik, M. Tachiya J. Chem. Phys. 2003 2165 2170

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

[20] S. B. Yuste, L. Acedo, K. Lindenberg Phys. Rev. E 2004 036126

[21] I. M. Sokolov, M. G. W. Schmidt, F. Sagués Phys. Rev. E 2006 031102

[22] T. A. M. Langlands, B. I. Henry, S. L. Wearne Phys. Rev. E 2008 021111

[23] M. O. Vlad, J. Ross Phys. Rev. E 2002 061908

[24] A. Yadav, W. Horsthemke Phys. Rev. E 2006 066118

[25] Y. Nec, A. A. Nepomnyashchy J. Phys. A: Math. Theor. 2008 385101

[26] V. Méndez, S. Fedotov, W. Horsthemke. Reaction-Transport Systems. Springer, Berlin, 2010.

[27] M. G. W. Schmidt, F. Sagués, I. M. Sokolov J. Phys.: Condens. Matter 2007 065118

[28] Y. Nec, A. A. Nepomnyashchy J. Phys. A: Math. Theor. 2007 14687 14702

[29] D. Campos, V. Méndez Phys. Rev. E 2009 021133

[30] H. H. Schmidt-Martens, D. Froemberg, I. M. Sokolov, F. Sagués Phys. Rev. E 2009 041135

[31] T. Kosztołowicz, K. D. Lewandowska Math. Model. Nat. Phenom. 2013 44 54

[32] T. Kosztołowicz, K. D. Lewandowska Phys. Rev. E 2008 066103

[33] T. Kosztołowicz, K. D. Lewandowska Acta Phys. Polon. B 2006 1571 1577

[34] Y. Nec, V. A. Volpert, A. A. Nepomnyashchy Discr. Cont. Dyn. Syst. 2010 827 846

[35] T. Kosztołowicz Phys. Rev. E 2014 042151

[36] A. A. Nepomnyashchy, V. A. Volpert J. Phys. A: Math. Theor. 2013 065101

[37] S. K. Hansen, B. Berkowitz Phys. Rev. E 2015 032113

[38] D. Froemberg, I. . Sokolov Phys. Rev. Lett. 2008 108304

[39] K. Seki, A. I. Shushin, M. Wojcik, M. Tachiya J. Phys.: Condens. Matter 2007 065117

[40] S. B. Yuste, E. Abad, K. Lindenberg Phys. Rev. E 2010 061123

[41] S. B. Yuste, E. Abad, K. Lindenberg. A reaction-subdiffusion model of fluorescence recovery after photobleaching (FRAP). J. Stat. Mech. - Theory and Exp., (2014), P11014.

[42] C. N. Angstmann, I. C. Donelly, B. I. Henry Math. Model. Nat. Phenom. 2013 17 27

[43] S. Fedotov, S. Falconer Phys. Rev. E 2012 031132

[44] S. Fedotov, S. Falconer Phys. Rev. E 2013 052139

[45] S. Fedotov, A. O. Ivanov, A. Y. Zubarev Math. Model. Nat. Phenom. 2013 28 43

[46] D. Froemberg, H. Schmidt-Martens, I. M. Sokolov, F. Sagués Phys. Rev. E 2008 011128

[47] S. Fedotov Phys. Rev. E 2010 011117

[48] B. I. Henry, S. L. Wearne SIAM J. Appl. Math. 2002 870 887

[49] D. Matignon Comput. Eng. Syst. Appl. 1996 963 970

[50] Y. Nec, A. A. Nepomnyashchy Math. Model. Nat. Phenom. 2007 77 105

[51] V.V. Gafiychuk, B.Y. Datsko Phys. Rev. E 2007 055201(R)

[52] Y. Nec, A. A. Nepomnyashchy Euro. J. Appl. Math. 2008 329 349

[53] V. Gafiychuk, B. Datsko, V. Meleshko J. Comp. Appl. Math. 2008 215 225

[54] Y. Nec, M. J. Ward Math. Model. Nat. Phenom. 2013 55 87

[55] V. A. Volpert, Y. Nec, A. A. Nepomnyashchy Philos. Trans. R. Soc. Lond., A 2013 20120179

[56] B. I. Henry, T. A. M. Langlands, S. L. Wearne Phys. Rev. E 2005 026101

[57] V. Gafiychuk, B. Datsko Comp. Math. Appl. 2010 1101 1107

[58] V. Gafiychuk, B. Datsko J. Comp. Nonlin. Dyn. 2012 031001

[59] Y. Nec, M. J. Ward Physica D 2012 947 963

[60] Y. Nec, M. J. Ward Math. Model. Nat. Phenom. 2013 206 232

[61] B. I. Henry, T. A. M. Langlands, S. L. Wearne Phys. Rev. Lett. 2008 128103

[62] S. Fedotov, A. Iomin Phys. Rev. Lett. 2007 118101

[63] S. Fedotov, A. Iomin Phys. Rev. E 2008 031911

[64] S. Fedotov, S. Falconer Phys. Rev. E 2014 012107