Modeling subdiffusion using reaction diffusion systems

SIAM Journal on Applied Mathematics

Volumn 70, Issue 1, 2009, Pages 112-132

  Mommer, Mario S ^a   Lebiedz, Dirk ^b  

^a UNIVERSITY OF HEIDELBERG   (Germany)

^b UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Anomalous diffusion; Diffusion; Reaction diffusion equations; Subdiffusion

Indexed keywords

ANOMALOUS DIFFUSION; CELLULAR BIOLOGY; CONTINUOUS-TIME RANDOM WALK MODELS; DIFFUSING SPECIES; FICK'S LAW; LINEAR REACTION; LIVING CELL; MASS CONSERVATION; MICROSCOPIC DESCRIPTION; REACTION DIFFUSION EQUATIONS; REACTION DIFFUSION SYSTEMS; SPATIO-TEMPORAL MODELS; SUBDIFFUSION; SUBDIFFUSIVE BEHAVIOR;

CELL MEMBRANES; CONTINUOUS TIME SYSTEMS; CYTOLOGY;

DIFFUSION;

EID: 67649306579     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/070681648     Document Type: Article

References (26)

1. G. BEYLKIN AND L. MONZÓN, On approximation of functions by exponential sums, Appl. Com- put. Harmon. Anal., 19 (2005), pp. 17-48.
2. J. P. BOUCHAUD AND A. GEORGES, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. , 195 (1990), pp. 127-293.
3. I. D. CHUESHOV, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Electronic Library of Mathematics (ELibM), 2002; http://www.emis.de/monographs/ Chueshov/.
4. L. DEVROYE, Non-Uniform Random Variate Generation, Springer-Verlag, New York, Berlin, 1986.
5. L. DEVROYE, A Course in Density Estimation, Birkhäuser Boston, Cambridge, MA, 1987.
6. A. B. fULTON, How crowded is the cytoplasm?, Cell, 30 (1982), pp. 345-347.
7. M. K. GHOSH, A. ARAPOSTATHIS, AND S. I. mARCUS, Optimal control of switching diffusions with application to flexible manufacturing systems, SIAM J. Control Optim., 31 (1993), pp. 1183-1204.
8. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., North Oxford Academic, London, 1996.
9. R. GORENFLO AND F. MAINARDI, Fractional diffusion processes: Probability distributions and continuous time random walk, in Long Range Dependent Processes: Theory and Applications, G. Rangarajan and M. Ding, eds., Springer-Ver lag, Heidelberg, 2003, pp. 148-166.
10. J. W. hAUS AND K. W. KEHR, Diffusion in regular and disordered lattices, Phys. Rep.,150 (1987), pp. 263-406.
11. V. M. kENKRE, E. W. MONTROLL, AND M. F. SHLESINGER, Generalized master equations for continuous time random walks, J. Statist. Phys., 9 (1973), pp. 45-50.
12. R. Z. KHASMINSKII, C. ZHU, AND G. YIN, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), pp. 1037-1051.
13. K. D. lEWANDOWSKA AND T. KOSZTOLOWICZ, Numerical study of subdiffusion equation,Acta Phys. Polon. B, 39 (2008), pp. 1211-1220.
14. F. MAINARDI, P. PARADISI, AND R. GORENFLO, Probabi lity distributions generated by fractional diffusion equations, in Econophysics: An Emerging Science (Proceedings of the first Econophysics workshop, Budapest 1997), J. Kertész and I. Kondor, eds., published online, 2002; http://newton.phy.bme.hu/~kullmann/Egyetem/konyv.html.
15. R. METZLER AND J. KLAFTER, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1-77.
16. E. W. MONTROLL AND G. H. wEISS, Random walks on lattices. II, J. Math. Phys., 6 (1965), pp. 167-181.
17. A. V. pETERSON AND R. A. kRONMAL, On mixture methods for the computer generation of random variables, Amer. Statist., 36 (1982), pp. 184-191.
18. S. PRAJNA, A. JADBABAIE, AND G. J. PAPPAS, A frame work for worst-case and stochastic safety verification using barrier certificates, IEEE Trans. Automat. Control, 52 (2007), pp. 1415-1428.
19. M.S. rahimi, Applications of Percolation Theory, Taylor & Francis, London, 1994.
20. M. Sahimi, Applications of Percolation Theory, Taylor & Francis, London, 1994.
21. M. J. SAXTON, Chemically limited reactions on a percolation cluster, J. Phys. Chem., 116 (2002), pp. 203-208.
22. M. J. SAXTON, A biological interpretation of transient anomalous diffusion. I. Qualitative model, Biophys. J., 92 (2007), pp. 1178-1191.
23. E. SCALAS, R. GORENFLO, F. MAINARDI, AND M. RABERTO, Revisiting the derivation of the fractional diffusion equation, Fractals, 11 (2003), pp. 281-289.
24. M. F. SCHLESINGER AND B. D. HUGHES, Analogs of renormalization group transformations in random processes, Phys. A, 109 (1981), pp. 597-608.
25. M. WEISS, M. ELSNER, F. KARTBERG, AND T. NLLSSON, Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells, Biophys. J., 87 (2004), pp. 3518-3524.
26. G. YIN AND H. ZHANG, Two-time-scale Markov chains and applications to quasi-birth-death queues, SIAM J. Appl. Math., 65 (2005), pp. 567-586.