Spatiotemporal chaos in diffusive systems with the Riesz fractional order operator. (English) Zbl 07851779
Summary: A reliable and efficient numerical technique for the approximation of Riesz fractional partial differential equations with chaotic and spatiotemporal chaos properties is developed in this work. In such a system, the standard anomalous diffusion terms are modeled using the Riesz fractional derivative which can be extended from one to high dimensional cases. The methodology adopts finite difference schemes, as well as the novel Fourier spectral methods for the approximation of the Riesz fractional derivatives in space. The resulting system of ordinary differential equations is advanced in time with the exponential time-differencing Runge-Kutta method. The performance and applicability of these methods were tested with some practical and real-life problems which are of current and recurring interests arising from computational physics, engineering and other areas of applied sciences. Simulation results are given for different instances of fractional parameter \(\alpha\) in the intervals \((0, 1)\) and \((1, 2)\) which correspond to subdiffusion and superdiffusion scenarios, respectively.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 35K57 | Reaction-diffusion equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 93C10 | Nonlinear systems in control theory |
Keywords:
Fourier spectral method; exponential time-differencing method; Riesz fractional derivative; numerical experiments; chaotic oscillationsReferences:
| [1] | Podlubny, I., Fractional Differential Equations, 1999, Academic Press: Academic Press New York · Zbl 0924.34008 |
| [2] | Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, 1993, Gordon and Breach: Gordon and Breach Amsterdam · Zbl 0818.26003 |
| [3] | Oldham, K. B.; Spanier, J., The Fractional Calculus, 1974, Academic Press: Academic Press New York · Zbl 0292.26011 |
| [4] | Owolabi, K. M.; Atangana, A., Numerical Methods for Fractional Differentiation, 2019, Springer: Springer Singapore · Zbl 1429.65005 |
| [5] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, 2006, Elsevier: Elsevier Amsterdam · Zbl 1092.45003 |
| [6] | Ortigueira, M. D., Fractional Calculus for Scientists and Engineers, 2011, Springer: Springer New York · Zbl 1251.26005 |
| [7] | Das, S., Functional Fractional Calculus, 2011, Springer-Verlag · Zbl 1225.26007 |
| [8] | Guo, B.; Pu, X.; Huang, F., Fractional Partial Differential Equations and their Numerical Solutions, 2011, World Scientific: World Scientific Singapore |
| [9] | Haghighi, A. R.; Dadv, A.; Ghejlo, H. H., Solution of the fractional diffusion equation with the Riesz fractional derivative using McCormack method, Commun. Adv. Comput. Sci. Appl., 2014, 1-11, 2014 |
| [10] | Miller, K. S.; Ross, B., An Introduction To the Fractional Calculus and Fractional Differential Equations, 1993, Wiley: Wiley New York · Zbl 0789.26002 |
| [11] | Zaslavsky, G. M., Hamiltonian Chaos and Fractional Dynamics, 2005, Oxford University Press · Zbl 1083.37002 |
| [12] | Caputo, M., Linear models of dissipation whose \(\mathcal{Q}\) is almost frequency independent: Part II, Geophys. J. Int., 13, 529-539, 1967, Reprinted in: Fractional Calculus and Applied Analysis, 11 (2008) 3-14 · Zbl 1210.65130 |
| [13] | Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1, 73-85, 2015 |
| [14] | Atangana, A.; Baleanu, D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20, 763-769, 2016 |
| [15] | Uçar, S.; Uçar, E.; Ozdemir, N.; Hammouch, Z., Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos Solitons Fractals, 118, 300-306, 2019 · Zbl 1442.92074 |
| [16] | Sadeghi, S.; Jafari, H.; Nemati, S., Operational matrix for Atangana-Baleanu derivative based on Genocchi polynomials for solving FDEs, Chaos Solitons Fractals, 135, Article 109736 pp., 2020 · Zbl 1491.34021 |
| [17] | Ashyralyev, A., A note on fractional derivatives and fractional powers of operators, J. Math. Anal. Appl., 357, 232-236, 2009 · Zbl 1175.26004 |
| [18] | Khan, N. A.; Khan, N. U.; Ara, A.; Jamil, M., Approximate analytical solutions of fractional reaction-diffusion equations, J. King Saud Univ. Sci., 24, 111-118, 2012 |
| [19] | Daftardar-Gejji, V.; Bhalekar, S., Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl., 345, 754-765, 2008 · Zbl 1151.26004 |
| [20] | Li Ding, X.; Lin-Jiang, Y., Analytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions, Nonlinear Anal. RWA, 14, 1026-1033, 2013 · Zbl 1260.35241 |
| [21] | Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293, 104-114, 2015 · Zbl 1349.65263 |
| [22] | Danane, J.; Allali, K.; Hammouch, Z., Mathematical analysis of a fractional differential model of HBV infection with antibody immune response, Chaos Solitons Fractals, 136, Article 109787 pp., 2020 · Zbl 1489.92145 |
| [23] | Singh, A.; Das, S.; Ong, S. H.; Jafari, H., Numerical solution of nonlinear reaction-advection-diffusion equation, ASME. J. Comput. Nonlinear Dynam., 14, 4, Article 041003 pp., 2019, 2019 |
| [24] | Pandey, P. K.; Kumar, Sachin; Jafari, Hossein; Das, Subir, An operational matrix for solving time-fractional order cahn-hilliard equation, Thermal Sci., 23, 2045-2052, 2019 |
| [25] | Das, S., Analytical solution of a fractional diffusion equation by variational iteration method, Comput. Math. Appl., 57, 3, 483-487, 2009 · Zbl 1165.35398 |
| [26] | Das, S.; Vishal, K.; Gupta, P. K., Solution of the nonlinear fractional diffusion equation with absorbent term and external force, Appl. Math. Model., 35, 8, 3970-3979, 2011 · Zbl 1221.35437 |
| [27] | Jafari, H., A new approach for solving nonlinear volterra integro-differential equations with Mittag-Leffler kernel, Proc. Inst. Math. Mech., 46, 1, 144-158, 2020 · Zbl 1455.65106 |
| [28] | Pindza, E.; Owolabi, K. M., Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numerical Simul., 40, 112-128, 2016 · Zbl 1524.65676 |
| [29] | Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56, 80-90, 2006 · Zbl 1086.65087 |
| [30] | Strikwerda, J. C., Partial Difference Schemes and Partial Differential Equations, 2004, SIAM: SIAM Philadelphia · Zbl 1071.65118 |
| [31] | Thomas, J. W., Numerical Partial Differential Equations Numerical Partial Differential Equations - Finite Difference Methods, 1995, Springer-Verlag: Springer-Verlag New York · Zbl 0831.65087 |
| [32] | Ascher, U. M.; Ruth, S. J.; Wetton, B. T.R., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 797-823, 1995 · Zbl 0841.65081 |
| [33] | Ascher, U. M.; Ruth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 151-167, 1997 · Zbl 0896.65061 |
| [34] | Li, D.; Zhang, C.; Wang, W.; Zhang, Y., Implicit-explicit predictor-correctorschemes for nonlinear parabolic differential equations, Appl. Math. Model., 35, 2711-2722, 2011 · Zbl 1219.65098 |
| [35] | Owolabi, K. M., Robust IMEX schemes for solving two-dimensional reaction-diffusion models, Int. J. Nonlinear Sci. Numerical Simul., 16, 271-284, 2015 · Zbl 1401.65100 |
| [36] | Ruuth, S., Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34, 148-176, 1995 · Zbl 0835.92006 |
| [37] | Burrage, K.; Hale, N.; Kay, D., An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34, A2145-A2172, 2012 · Zbl 1253.65146 |
| [38] | Roop, J., Computational aspects of FEM approximations of fractional advection dispersion equations on bounded domains on \(R^2\), J. Comput. Appl. Math., 193, 243-268, 2005 · Zbl 1092.65122 |
| [39] | Bueno-Orovio, A.; Kay, D.; Burrage, K., Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numer. Math., 54, 937-954, 2014 · Zbl 1306.65265 |
| [40] | Owolabi, K. M., High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology, Chaos Solitons Fractals, 34, Article 109723 pp., 2020 · Zbl 1483.35117 |
| [41] | Zhang, H.; Liu, F., The fundamental solutions of the space, space-time Riesz fractional partial differential equations with periodic conditions, Numer. Math. A J. Chin. Univ. Engl. Ser., 16, 181-192, 2007 · Zbl 1174.35328 |
| [42] | Owolabi, K. M.; Atangana, A., Numerical solution of fractional-in-space Schrödinger equation with the Riesz fractional derivative, Eur. Phys. J. Plus, 131, 335, 2016 |
| [43] | Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker-Planck equation, J. Comput. Appl. Math., 166, 1, 209-219, 2004 · Zbl 1036.82019 |
| [44] | Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Gu, Y. T., Finite element method for space-time fractional diffusion equation, Numer. Algorithms, 72, 749-767, 2016 · Zbl 1343.65122 |
| [45] | Lai, J.; Liu, F.; Anh, V.; Liu, Q. A., Space-time finite element method for solving linear Riesz space fractional partial differential equations, Numer. Algorithms, 88, 499-520, 2021 · Zbl 1480.65262 |
| [46] | Sabatier, J.; Agrawal, O. P.; Machado, J. A.T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 2007, Springer: Springer Netherlands · Zbl 1116.00014 |
| [47] | Owolabi, K. M.; Atangana, A., Chaotic behaviour in system of noninteger-order ordinary differential equations, Chaos Solitons Fractals, 115, 362-370, 2018 · Zbl 1416.65180 |
| [48] | Zhang, S.; Zeng, Y.; Li, Z., One to four-wing chaotic attractors coined from a novel 3D fractional-order chaotic system with complex dynamics, Chinese J. Phys., 56, 793-806, 2018 · Zbl 07819466 |
| [49] | Vishal, K.; Agrawal, S. K., On the dynamics, existence of chaos, control and synchronization of a novel complex chaotic system, Chinese J. Phys., 55, 519-532, 2017 · Zbl 1539.34059 |
| [50] | Luo, R.; Su, H.; Zeng, Y., Synchronization of uncertain fractional-order chaotic systems via a novel adaptive controller, Chinese J. Phys., 55, 342-349, 2017 · Zbl 1539.34062 |
| [51] | Sarwar, S.; Iqbal, S., Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction, Chinese J. Phys., 56, 374-384, 2018 · Zbl 07816181 |
| [52] | Owolabi, K. M.; Atangana, A., Numerical simulation of noninteger order system in subdiffussive, diffusive, and superdiffusive scenarios, J. Comput. Nonlinear Dynam., 12, Article 031010-1, 2017 |
| [53] | Owolabi, K. M.; Atangana, A., Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system, Adv. Appl. Math. Mech., 9, 1438-1460, 2017 · Zbl 1488.65272 |
| [54] | Magin, R. L., Fractional Calculus in Bioengineering, 2006, Begell House Publisher. Inc.: Begell House Publisher. Inc. Connecticut |
| [55] | Zaslavsky, G. M., Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371, 461-580, 2002 · Zbl 0999.82053 |
| [56] | Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34, 200-218, 2010 · Zbl 1185.65200 |
| [57] | Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9, 333-349, 2006 · Zbl 1132.35507 |
| [58] | Kassam, A. K.; Trefethen, L. N., Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput., 26, 1214-1233, 2005 · Zbl 1077.65105 |
| [59] | Cox, S. M.; Matthews, P. C., Exponential time differencing for stiff systems, J. Comput. Phys., 176, 430-455, 2002 · Zbl 1005.65069 |
| [60] | Owolabi, K. M.; Patidar, K. C., Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Appl. Math. Comput., 240, 30-50, 2014 · Zbl 1334.65136 |
| [61] | Munthe-Kaas, H., High order runge-kutta methods on manifolds, Appl. Numer. Math., 29, 115-127, 1999 · Zbl 0934.65077 |
| [62] | Rössler, O., An equation for continuous chaos, Phys. Lett. A, 57, 397-398, 1976 · Zbl 1371.37062 |
| [63] | Lorenz, E. N., Deterministic non-periodic flow, J. Atmos. Sci., 20, 130-141, 1963 · Zbl 1417.37129 |
| [64] | Chen, G.; Ueta, T., Yet another chaotic attractor, Int. J. Bifurcation Chaos Appl. Sci. Eng., 9, 1465-1466, 1999 · Zbl 0962.37013 |
| [65] | Sprott, J. C., Elegant Chaos Algebraically Simple Chaotic Flows, 2010, World Scientific: World Scientific Singapore · Zbl 1222.37005 |
| [66] | Wu, J.; Wang, L.; Chen, G.; Duan, S., A memristive chaotic system with heart-shaped attractors and its implementation, Chaos Solitons Fractals, 92, 20-29, 2016 · Zbl 1372.37073 |
| [67] | Wu, J.; Wang, C., A new simple chaotic circuit based on memristor, Int. J. Bifurcation Chaos Appl. Sci. Eng., 26, Article 1650145 pp., 2016, 1-11 · Zbl 1347.34078 |
| [68] | Doungmo Goufo, E. F.; Nieto, J. J., Attractors for fractional differential problems of transition to turbulent flows, J. Comput. Appl. Math., 339, 329-342, 2018 · Zbl 1440.76038 |
| [69] | Zidan, M. A.; Radwan, A. G.; Salama, K. N., Controllable V-shape multiscroll butterfly attractor: system and circuit implementation, Int. J. Bifurcation Chaos, 22, Article 1250143 pp., 2012 · Zbl 1270.34112 |
| [70] | Liu, H.; Kadir, A.; Li, Y., Asymmetric color pathological image encryption scheme based on complex hyper chaotic system, Optik, 127, 5812-5819, 2016 |
| [71] | Radwan, A. G.; Moaddy, K.; Salama, K. N.; Momani, S.; Hashim, I., Control and switching synchronization of fractional order chaotic systems using active control technique, J. Adv. Res., 5, 125-132, 2014 |
| [72] | Azar, A. T., A novel chaotic system without equilibrium: dynamics, synchronization, and circuit realization, Complexity, 2017, 1-11, 2017, 7871467 · Zbl 1367.34073 |
| [73] | Ivancevic, V. G.; Tijana, T. I., Complex Nonlinearity: Chaos, Phase Transitions, Topology Change, and Path Integrals, 2008, Springer · Zbl 1152.37002 |
| [74] | Safonov, L. A.; Tomer, E.; Strygin, V. V.; Ashkenazy, Y.; Havlin, S., Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic, Chaos, 12, 1006-1014, 2002 · Zbl 1080.34564 |
| [75] | Wang, Z.; Sun, Y.; van Wyk, B. J.; Qi, G.; van Wyk, M. A., A 3-D four-wing attractor and its analysis, Braz. J. Phys., 39, 547-553, 2009 |
| [76] | Wang, Z.; Qi, G.; Sun, Y.; van Wyk, B. J.; van Wyk, M. A., A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems, Nonlinear Dynam., 60, 443-457, 2010 · Zbl 1189.34100 |
| [77] | Liu, J. Y.; Tsai, S. H.; Wang, C. C.; Chu, C. R., Acoustic wave reflection from a rough seabed with a continuously varying sediment layer overlying an elastic basement, Sound Vib., 275, 739-755, 2004 |
| [78] | Murray, J. D.; Myerscough, M. R., Pigmentation pattern formation on snakes, J. Theoret. Biol., 149, 339-360, 1991 |
| [79] | Kress, R.; Roach, G. F., Transmission problems for the Helmholtz equation, J. Math. Phys., 19, 1433-1437, 1978 · Zbl 0433.35017 |
| [80] | Kleinman, R. E.; Roach, G. F., Boundary integral equations for the three-dimensional Helmholtz equation, SIAM Rev., 16, 214-236, 1974 · Zbl 0253.35023 |
| [81] | Karageorghis, A., The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation, Appl. Math. Lett., 14, 837-842, 2001 · Zbl 0984.65111 |
| [82] | Heikkola, E.; Rossi, T.; Toivanen, J., A parallel fictitious domain method for the three-dimensional Helmholtz equation, SIAM J. Sci. Comput., 24, 1567-1588, 2003 · Zbl 1035.65126 |
| [83] | Samuel, M. S.; Thomas, A., On fractional Helmholtz equations, Fract. Calc. Appl. Anal., 13, 3, 295-308, 2010 · Zbl 1223.26013 |
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