Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. (English) Zbl 1189.35151
Summary: We investigate possible scenarios of pattern formations in reaction-diffusion systems with time fractional derivatives. Linear stability analysis is performed for different values of derivative orders. Results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations. Most attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer-van der Pol kinetics and to the Brusselator model.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 45K05 | Integro-partial differential equations |
Keywords:
reaction-diffusion system; fractional differential equations; inhomogeneous oscillations; dissipative structuresReferences:
| [1] | Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 461-580 (2002) · Zbl 0999.82053 |
| [2] | Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032 |
| [3] | Agrawal, O. P.; Tenreiro Machado, J. A.; Sabatier, J., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (2007), Elsevier · Zbl 1116.00014 |
| [4] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier · Zbl 1092.45003 |
| [5] | Uchaikin, V. V., Fractional Derivative Method (2008), Artishok, (in Russian) · Zbl 1151.82430 |
| [6] | Henry, B. I.; Langlands, T. A.M.; Wearne, S. L., Turing pattern formation in fractional activator-inhibitor systems, Phys. Rev. E, 72, 026101 (2005) |
| [7] | Henry, B. I.; Langlands, T. A.M.; Wearne, S. L., Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, 74, 031116 (2006) |
| [8] | Gafiychuk, V.; Datsko, B., Pattern formation in a fractional reaction-diffusion system, Physica A, 365, 300-306 (2006) |
| [9] | Gafiychuk, V.; Datsko, B., Stability analysis and oscillatory structures in time-fractional reaction-diffusion systems, Phys. Rev. E, 75, R, 055201 (2007) |
| [10] | Gafiychuk, V.; Datsko, B., Inhomogeneous oscillatory structures in fractional reaction-diffusion systems, Phys. Lett. A, 372, 619-622 (2008) · Zbl 1217.35103 |
| [11] | Gafiychuk, V.; Datsko, B.; Meleshko, V., Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math., 372, 215-225 (2008) · Zbl 1152.45008 |
| [12] | Golovin, A. A.; Matkovsky, B. J.; Volpert, V. A., Turing pattern formation in the brusselator model with superdiffusion, SIAM J. Appl. Math., 60, 251-272 (2008) · Zbl 1170.35053 |
| [13] | Eule, S.; Friedrich, R.; Jenko, F.; Sokolov, I. M., Continuous-time random walks with internal dynamics and subdiffusive reaction-diffusion equations, Phys. Rev. E, 78, R, 060102 (2008) |
| [14] | Nec, Y.; Nepomnyashchy, A. A.; Golovin, A. A., Oscillatory instability in super-diffusive reaction-diffusion systems: Fractional amplitude and phase diffusion equations, EPL, 82, 58003 (2008) |
| [15] | Nicolis, G.; Prigogine, I., Self-organization in Non-equilibrium Systems (1977), Wiley: Wiley New York · Zbl 0363.93005 |
| [16] | Cross, M. C.; Hohenberg, P. C., Pattern formation outside of equilibrium, Rev. Modern Phys., 65, 851-1112 (1993) · Zbl 1371.37001 |
| [17] | Kerner, B. S.; Osipov, V. V., Autosolitons (1994), Kluwer: Kluwer Dordrecht · Zbl 0779.73003 |
| [18] | Podlubny, I., Fractional Differential Equations (1999), Academic Press · Zbl 0918.34010 |
| [19] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Newark, N.J. · Zbl 0818.26003 |
| [20] | Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal. TMA, 71, 3249-3256 (2009) · Zbl 1177.34084 |
| [21] | Ahmad, B.; Nieto Wearne, J. J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Boundary Value Problems, 2009, 708576 (2009) · Zbl 1167.45003 |
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