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Numerical algorithm based on Adomian decomposition for fractional differential equations. (English) Zbl 1186.65110
Summary: A novel algorithm based on Adomian decomposition for fractional differential equations is proposed. Comparing the present method with the fractional Adams method, we use this derived computational method to find a smaller “efficient dimension” such that the fractional Lorenz equation is chaotic. We also apply this new method to the time-fractional Burgers equation with initial and boundary value conditions. Numerical results and computer graphics show that the constructed numerical is efficient.

65L99Numerical methods for ODE
26A33Fractional derivatives and integrals (real functions)
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
Full Text: DOI
[1] Zaslavsky, G. M.: Chaos, fractional kinetics, and anomalous transport, Phys. rep. 371, 461-580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[2] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J.: Fractional calculus models and numerical methods, (2009)
[3] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[4] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[5] Li, C. P.; Deng, W. H.: Remarks on fractional derivative, Appl. math. Comput. 187, 774-784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163
[6] C.P. Li, X.H. Dao, P. Guo, Fractional derivatives in complex plane. Nonlinear Anal. TMA, 2009, doi:10.1016/j.na.2009.01.021 (in press) · Zbl 1173.26305
[7] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[8] 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 · doi:10.1023/A:1016592219341
[9] 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 · doi:10.1023/B:NUMA.0000027736.85078.be
[10] Adomian, G.: A review of the decomposition method in applied mathematics, J. math. Anal. appl. 135, 501-544 (1988) · Zbl 0671.34053 · doi:10.1016/0022-247X(88)90170-9
[11] Adomian, G.: Solving frontier problems of physics: the decomposition method, (1995) · Zbl 0802.65122
[12] Cherruault, Y.; Adomian, G.: Decomposition methods: A new proof of convergence, Math. comput. Model. 18, No. 12, 103-106 (1993) · Zbl 0805.65057 · doi:10.1016/0895-7177(93)90233-O
[13] Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys. rev. Lett. 91 (2003) · Zbl 1234.49040
[14] Li, C. P.; Peng, G. J.: Chaos in Chen’s system with a fractional order, Chaos solitons fractals 22, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[15] Wang, Y. H.; Li, C. P.: Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?, Phys. lett. A 363, 414-419 (2007)
[16] Deng, W. H.; Li, C. P.: The evolution of chaotic dynamics for fractional unified system, Phys. lett. A 372, 401-407 (2008) · Zbl 1217.37026 · doi:10.1016/j.physleta.2007.07.049
[17] Hu, T. C.; Wang, Y. H.: Numerical detection of the lowest ”efficient dimensions” for chaotic fractional differential systems, Open mathe. J. 1, 11-18 (2008) · Zbl 1185.34006 · doi:10.2174/1874117700801010011
[18] Sun, J. Q.; Qin, M. Z.: A kind of explicit stable method to solve the Burgers equation, Math. numer. Sinica 29, 67-72 (2007) · Zbl 1121.65350
[19] Momani, S.: An explicit and numerical solutions of the fractional KdV equation, Math comput simulation 70, 110-118 (2005) · Zbl 1119.65394 · doi:10.1016/j.matcom.2005.05.001