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Attractors for fractional differential problems of transition to turbulent flows. (English) Zbl 1440.76038

Summary: The complexity of fluid flows remains an intriguing problem and many scientists are still struggling to gain new and reliable insight into the dynamics of fluids. Transition from laminar to turbulent flows is even more complex and many of its features remain surprising and unexplained.
To describe transition to turbulence we introduce some fractional models and use numerical approximations to reveal the existence of attractor points. Two different cases are studied; the classical situation corresponding to the integer dimension one and the pure fractional case. The observed simulations show, in both cases, the presence of attractors near which iterations converge faster than usual. The behavior observed in the conventional case is in concordance with the well-known results that exist in the literature for relatively low order ordinary differential equations. The results observed in the fractional case are innovative since they reveal, not only the persistence of attractors, but also a possible better description of the transition to turbulent flows due to the variation of the fractional parameter that allows the control of the dynamics.

MSC:

76F06 Transition to turbulence
34A08 Fractional ordinary differential equations
33F05 Numerical approximation and evaluation of special functions
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

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[1] Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 73-85 (2015)
[2] Losada, J.; Nieto, J. J., Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1, 2, 87-92 (2015)
[3] Doungmo Goufo, E. F., Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos, 26, 8, 084305 (2016) · Zbl 1378.34011
[4] Baggett, J. S.; Driscoll, T. A.; Trefethen, L. N., A mostly linear model of transition to turbulence, Phys. Fluids, 7, 4, 833-838 (1995) · Zbl 1039.76509
[5] Pausch, M.; Eckhardt, B., Direct and noisy transitions in a model shear flow, Theor. Appl. Mech. Lett., 5, 3, 111-116 (2015)
[6] Singler, J. R., Sensitivity Analysis of Partial Differential Equations with Applications to Fluid Flow (2005), Virginia Polytechnic Institute and State University, (Ph.D. thesis)
[7] Tian, Y.; Zhang, F.; Zheng, P., Global dynamics for a model of a class of continuous-time dynamical systems, Math. Methods Appl. Sci., 38, 18, 5132-5138 (2015) · Zbl 1339.37021
[8] Singler, J. R., Global attractor for a low order ODE model problem for transition to turbulence, Math. Methods Appl. Sci., 40, 8, 2896-2906 (2017) · Zbl 1376.37062
[9] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (2005), Princeton University Press · Zbl 1085.15009
[10] Wang, Z.; Huang, X.; Zhou, J., A numerical method for delayed fractional-order differential equations: based on GL definition, Appl. Math., 7, 2L, 525-529 (2013)
[11] Wang, Z., A numerical method for delayed fractional-order differential equations, J. Appl. Math., 2013 (2013)
[12] Yang, X.-J.; Tenreiro Machado, J.; Baleanu, D.; Cattani, C., On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, 26, 8, 084312 (2016) · Zbl 1378.35329
[13] Yang, X.-J.; Gao, F.; Srivastava, H. M., Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Math. Appl., 73, 2, 203-210 (2017) · Zbl 1386.35460
[14] Caputo, M., Linear models of dissipation whose Q is almost frequency independentii, Geophys. J. Int., 13, 5, 529-539 (1967)
[15] Doungmo Goufo, E. F., A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal., 18, 3, 554-564 (2015) · Zbl 1316.26004
[16] Doungmo Goufo, E. F., Stability and convergence analysis of a variable order replicator-mutator process in a moving medium, J. Theoret. Biol., 403, 178-187 (2016) · Zbl 1343.92340
[17] Hanert, E., On the numerical solution of space-time fractional diffusion models, Comput. & Fluids, 46, 1, 33-39 (2011) · Zbl 1305.65212
[18] Khan, Y.; Sayevand, K.; Fardi, M.; Ghasemi, M., A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations, Appl. Math. Comput., 249, 229-236 (2014) · Zbl 1338.65285
[19] Doungmo Goufo, E. F., Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Math. Model. Anal., 21, 2, 188-198 (2016) · Zbl 1499.35643
[20] Yang, X.-J.; Machado, J. T., A new fractional operator of variable order: Application in the description of anomalous diffusion, Physica A, 481, 276-283 (2017) · Zbl 1495.35204
[21] Crank, J.; Nicolson, P., A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, (Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 43 (1947), Cambridge Univ Press), 50-67, 01 · Zbl 0029.05901
[22] Doungmo Goufo, E. F., Speeding up chaos and limit cycles in evolutionary language and learning processes, Math. Methods Appl. Sci., 40, 8, 3055-3065 (2017) · Zbl 1417.91426
[23] Chen, C.-M.; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227, 2, 886-897 (2007) · Zbl 1165.65053
[24] Chen, H.; Wang, H., Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation, J. Comput. Appl. Math., 296, 480-498 (2016) · Zbl 1342.65168
[25] Lin, R.; Liu, F.; Anh, V.; Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comput., 212, 2, 435-445 (2009) · Zbl 1171.65101
[26] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 1, 65-77 (2004) · Zbl 1126.76346
[27] Tadjeran, C.; Meerschaert, M. M.; Scheffler, H.-P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213, 1, 205-213 (2006) · Zbl 1089.65089
[28] Liu, Y.; Fang, Z.; Li, H.; He, S., A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput., 243, 703-717 (2014) · Zbl 1336.65166
[29] Yuste, S. B.; Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42, 5, 1862-1874 (2005) · Zbl 1119.65379
[30] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47, 3, 1760-1781 (2009) · Zbl 1204.26013
[31] Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Jara, B. M.V., Matrix approach to discrete fractional calculus II: Partial fractional differential equations, J. Comput. Phys., 228, 8, 3137-3153 (2009) · Zbl 1160.65308
[32] Doungmo Goufo, E. F.; Atangana, A., Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, Eur. Phys. J. Plus, 131, 8, 269 (2016)
[33] Young, W. H., On classes of summable functions and their Fourier series, Proc. R. Soc. Lond. Ser. A, Containing Papers of a Mathematical and Physical Character, 87, 594, 225-229 (1912) · JFM 43.0334.09
[34] Henstock, R., Lectures on the Theory of Integration, Vol. 1 (1988), World Scientific · Zbl 0668.28001
[35] Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math., 292-296 (1919) · JFM 47.0399.02
[36] Bellman, R., The stability of solutions of linear differential equations, Duke Math. J., 10, 4, 643-647 (1943) · Zbl 0061.18502
[37] Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36, 1, 31-52 (2004) · Zbl 1055.65098
[38] Li, C.; Tao, C., On the fractional Adams method, Comput. Math. Appl., 58, 8, 1573-1588 (2009) · Zbl 1189.65142
[39] Sternberg, S., Dynamical Systems (2010), Courier Corporation
[40] Cordero, A.; Feng, L.; Magreñán, A.; Torregrosa, J. R., A new fourth-order family for solving nonlinear problems and its dynamics, J. Math. Chem., 53, 3, 893-910 (2015) · Zbl 1318.65028
[41] Ladyzhenskaya, O. A., On the determination of minimal global attractors for the Navier-Stokes and other partial differential equations, Russian Math. Surveys, 42, 6, 27 (1987) · Zbl 0687.35072
[42] Raugel, G., Global attractors in partial differential equations, (Handbook of Dynamical Systems, Vol. 2 (2002)), 885-982 · Zbl 1005.35001
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