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Numerical algorithm for the time fractional Fokker-Planck equation. (English) Zbl 1388.35095
Summary: Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error \(O(k^{\min\{1+2\alpha ,2\}})+O(h^{2})\), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for \(\alpha =1.0\) with the ones of directly discretizing classical Fokker-Planck equation, some numerical results for time fractional Fokker-Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for \(\alpha =0.8\) the convergent order in space is confirmed and the numerical results with different time step sizes are shown.

35K57 Reaction-diffusion equations
35R11 Fractional partial differential equations
35Q84 Fokker-Planck equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
82B35 Irreversible thermodynamics, including Onsager-Machlup theory
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
Full Text: DOI
[1] Agrawal, Om P.; Tenreiro Machado, J.A.; Sabatier, Jocelyn, Introduction, Nonlinear dynam., 38, 1-2, (2004)
[2] Barkai, E.; Metzler, R.; Klafter, J., From continuous time random walks to the fractional fokker – planck equation, Phys. rev. E, 61, 132-138, (2000)
[3] Barkai, E., Fractional fokker – planck equation, solution, and application, Phys. rev. E, 63, 046118, (2001)
[4] Bouchaud, J.; Georges, A., Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. rep., 195, 127-293, (1990)
[5] Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (2000), World Scientific Singapore · Zbl 0987.26005
[6] Deng, W.H., Short memory principle and a predictor – corrector approach for fractional differential equations, J. comput. appl. math., 206, 174-188, (2007) · Zbl 1121.65128
[7] Deng, W.H.; Lü, J.H., Design of multi-directional multi-scroll chaotic attractors based on fractional differential systems via switching control, Chaos, 16, 043120, (2006) · Zbl 1146.37316
[8] Deng, W.H., Generating 3-D scroll grid attractors of fractional differential systems via stair function, Int. J. bifurcation chaos appl. sci. eng., 17, 1-19, (2007)
[9] Deng, W.H.; Li, C.P.; Lü, J.H., Stability analysis of linear fractional differential system with multiple time-delays, Nonlinear dynam., 48, 409-416, (2007) · Zbl 1185.34115
[10] Deng, W.H., Generalized synchronization in fractional order systems, Phys. rev. E, 75, 056201, (2007)
[11] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[12] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam., 29, 3-22, (2002) · Zbl 1009.65049
[13] Diethelm, K.; Ford, N.J.; Freed, A.D., Detailed error analysis for a fractional Adams method, Numer. algorithms, 36, 31-52, (2004) · Zbl 1055.65098
[14] Ford, N.J.; Simpson, A.C., The numerical solution of fractional differential equations: speed versus accuracy, Numer. algorithms, 26, 336-346, (2001) · Zbl 0976.65062
[15] Goychuk, I.; Heinsalu, E.; Patriarca, M.; Schmid, G.; Hänggi, P., Current and universal scaling in anomalous transport, Phys. rev. E, 73, 020101, (2006)
[16] Heaviside, O., Electromagnetic theory, (1971), Chelsea New York · JFM 25.1774.02
[17] Heinsalu, E.; Patriarca, M.; Goychuk, I.; Schmid, G.; Hänggi, P., Fractional fokker – planck dynamics: numerical algorithm and simulations, Phys. rev. E, 73, 046133, (2006)
[18] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with riemann – liouville fractional derivatives, Rheol. acta, 37, 1-7, (2005)
[19] Ichise, M.; Nagayanagi, Y.; Kojima, T., An analog simulation of noninteger order transfer functions for analysis of electrode processes, J. electroanal. chem., 33, 253-265, (1971)
[20] Jumarie, G., A fokker – planck equation of fractional order with respect to time, J. math. phys., 33, 3536-3542, (1992) · Zbl 0761.60071
[21] Kenneth, S.M.; Bertram, R., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley-Interscience Publication US · Zbl 0789.26002
[22] Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity, J. appl. mech., 229-307, (1984) · Zbl 0544.73052
[23] Kusnezov, D.; Bulgac, A.; Dang, G.D., Quantum levy processes and fractional kinetics, Phys. rev. lett., 82, 1136-1139, (1999)
[24] Lavoie, J.L.; Osler, T.J.; Tremblay, R., Fractional derivatives and special functions, SIAM rev., 18, 240-268, (1976) · Zbl 0324.44002
[25] Lenzi, E.K.; Mendes, R.S.; Fa, K.S.; Malacarne, L.C., Anomalous diffusion: fractional fokker – planck equation and its solutions, J. math. phys., 44, 2179-2185, (2003) · Zbl 1062.82043
[26] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional fokker – planck equation, J. comput. appl. math., 166, 209-219, (2004) · Zbl 1036.82019
[27] Lubich, C., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. comp., 45, 463-469, (1985) · Zbl 0584.65090
[28] Lubich, C., Discretized fractional calculus, SIAM J. math. anal., 17, 704-719, (1986) · Zbl 0624.65015
[29] Mandelbrot, B., Some noises with 1/f spectrum, a bridge between direct current and white noise, IEEE trans. inform. theory, 13, 289-298, (1967) · Zbl 0148.40507
[30] Meerschaert, M.M.; Scheffler, H.-P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. comput. phys., 211, 249-261, (2006) · Zbl 1085.65080
[31] Metzler, R.; Barkai, E.; Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: A fractional fokker – planck equation approach, Phys. rev. lett., 82, 3563, (1999)
[32] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep., 339, 1-77, (2000), see also references therein · Zbl 0984.82032
[33] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[34] Samko, S.; Kilbas, A.; Marichev, O., Fractional integrals and derivatives: theory and applications, (1993), Gordon & Breach London · Zbl 0818.26003
[35] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004
[36] Solomon, T.H.; Weeks, E.R.; Swinney, H.L., Observations of anomalous diffusion and levy flights in a 2-dimensional rotating flow, Phys. rev. lett., 71, 3975-3979, (1993)
[37] Sugimoto, N., Burgers equation with a fractional derivative: hereditary effects on nonlinear acoustic waves, J. fluid mech., 225, 631-653, (1991) · Zbl 0721.76011
[38] Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport, Phys. rep., 371, 461-580, (2002) · Zbl 0999.82053
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