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Analytical and numerical solutions of multi-term nonlinear fractional orders differential equations. (English) Zbl 1192.65092
Summary: We are concerned with a multi-term nonlinear fractional differential equation. Two methods are used to solve this type of equations. The first is an analytical method: the Adomian decomposition method. A convergence analysis of this method is discussed. This analysis is used to estimate the maximal absolute truncated error of the Adomian series solution. The second method is the proposed numerical method. A comparison between the results of the two methods is given. One of the important applications of these equations is the Bagley-Torvik equation.
MSC:
65L05Initial value problems for ODE (numerical methods)
34A08Fractional differential equations
References:
[1]Abbaoui, K.; Cherruault, Y.: Convergence of Adomian’s method applied to differential equations, Comput. math. Appl. 28, 103-109 (1994) · Zbl 0809.65073 · doi:10.1016/0898-1221(94)00144-8
[2]Adomian, G.: Stochastic system, (1983)
[3]Adomian, G.: Nonlinear stochastic operator equations, (1986)
[4]Adomian, G.: Nonlinear stochastic systems: theory and applications to physics, (1989)
[5]Adomian, G.; Rach, R.; Mayer, R.: Modified decomposition, J. appl. Math. comput. 23, 17-23 (1992) · Zbl 0756.35013 · doi:10.1016/0898-1221(92)90076-T
[6]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1995)
[7]Carpinteri, A.; Mainardi, F.: Fractals and fractional calculus in continuum mechanics, (1997) · Zbl 0917.73004
[8]Cherruault, Y.; Adomian, G.; Abbaoui, K.; Rach, R.: Further remarks on convergence of decomposition method, Int. J. Bio-med. Comput. 38, 89-93 (1995)
[9]Daftarder-Gejji, V.; Babakhani, A.: Analysis of a system of fractional differential equations, J. math. Anal. appl. 293, 511-522 (2004) · Zbl 1058.34002 · doi:10.1016/j.jmaa.2004.01.013
[10]Diethelm, K.; Ford, N. J.: Numerical solution of the bagley – torvik equation, Bit 42, 490-507 (2002) · Zbl 1035.65067
[11]Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solution, Appl. math. Comput. 154, 621-640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2
[12]Diethelm, K.; Luchko, Y.: Numerical solution of linear multi-term initial value problems of fractional order, J. comput. Anal. appl., 243-263 (2004) · Zbl 1083.65064
[13]Diethelm, K.: Multi-term fractional differential equations multi-order fractional differential systems and their numerical solution, J. europ. Syst. autom. 42, 665-676 (2008)
[14]El-Sayed, A. M. A.; El-Mesiry, E. M.; El-Saka, H. A. A.: Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. appl. Math. 23, 33-54 (2004) · Zbl 1213.34025 · doi:10.1590/S0101-82052004000100002 · doi:http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1807-03022004000100002&lng=en&nrm=iso&tlng=en
[15]El-Kalla, I. L.: Error analysis of Adomian series solution to a class of nonlinear differential equations, Appl. math. E-notes 7, 214-221 (2007) · Zbl 1157.65422 · doi:emis:journals/AMEN/2007/2007.htm
[16]El-Mesiry, E. A. M.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. math. Comput. 160, 683-699 (2005) · Zbl 1062.65073 · doi:10.1016/j.amc.2003.11.026
[17]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[18]Li, C. P.; Wang, Y. H.: Numerical algorithm based on Adomian decomposition for fractional differential equations, Comput. math. Appl. 57, No. 10, 1672-1681 (2009) · Zbl 1186.65110 · doi:10.1016/j.camwa.2009.03.079
[19]Li, C. P.; Tao, C. X.: On the fractional Adams method, Comput. math. Appl. 58, No. 8, 1573-1588 (2009)
[20]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[21]Podlubny, I.; El-Sayed, A. M. A.: On two definitions of fractional calculus, (1996)
[22]Podlubny, I.: Fractional differential equations, (1999)
[23]Shawaghfeh, N. T.: Analytical approximate solution for nonlinear fractional differential equations, J. appl. Math. comput. 131, 517-529 (2002) · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9