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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.
MSC:
65L99Numerical methods for ODE
26A33Fractional derivatives and integrals (real functions)
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
References:
[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)
[4]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[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)
[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)
[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)
[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