An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. (English) Zbl 1082.65562

Ray, S. Saha; Bera, R. K.

Appl. Math. Comput. 167, No. 1, 561-571 (2005).

Summary: The aim of the present analysis is to apply the Adomian decomposition method to the solution of a nonlinear fractional differential equation. Finally, the solution obtained by the decomposition method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method. A good agreement of the results is observed.

MSC:

65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
26A33	Fractional derivatives and integrals

Keywords:

numerical examples; Adomian decomposition method; Fractional differential equation; Fractional derivative; Fractional integral

References:

[1]	Adomian, G., Nonlinear Stochastic Operator Equations (1986), Academic Press: Academic Press New York, NY · Zbl 0614.35013
[2]	Adomian, G.; Rach, R., On linear and nonlinear integro-differential equations, J. Math. Anal. Appl., 133, 199-201 (1986) · Zbl 0601.45009
[3]	Adomian, G., Nonlinear Stochastic Systems Theory and Applications to Physics (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Netherlands · Zbl 0659.93003
[4]	Adomian, G.; Rach, R., Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Comput. Math. Appl., 10, 12, 9-12 (1990) · Zbl 0702.35058
[5]	Adomian, G., An analytical solution of the stochastic Navier-Stokes system, Found. Phys., 21, 7, 831-843 (1991)
[6]	Adomian, G.; Rach, R., Linear and nonlinear Schrödinger equations, Found. Phys., 21, 983-991 (1991)
[7]	Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[8]	Adomian, G., Solution of physical problems by decomposition, Comput. Math. Appl., 27, 9-10, 145-154 (1994) · Zbl 0803.35020
[9]	Adomian, G., Solutions of nonlinear P.D.E., Appl. Math. Lett., 11, 3, 121-123 (1998) · Zbl 0933.65121
[10]	Abbaoui, K.; Cherruault, Y., The decomposition method applied to the Cauchy problem, Kybernetes, 28, 103-108 (1999) · Zbl 0937.65074
[11]	Kaya, D.; El-Sayed, S. M., On a generalized fifth order KdV equations, Phys. Lett. A, 310, 1, 44-51 (2003) · Zbl 1011.35114
[12]	Kaya, D.; El-Sayed, S. M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals, 17, 5, 869-877 (2003) · Zbl 1030.35139
[13]	Kaya, D., An explicit and numerical solutions of some fifth-order KdV equation by decomposition method, Appl. Math. Comput., 144, 2-3, 353-363 (2003) · Zbl 1024.65096
[14]	Kaya, D., A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput., 149, 3, 833-841 (2004) · Zbl 1038.65101
[15]	Wazwaz, A., A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102, 1, 77-86 (1999) · Zbl 0928.65083
[16]	Podlubny, I., Fractional Differential Equations (1999), Academic Press · Zbl 0918.34010
[17]	Bagley, R. L.; Torvik, P. J., On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51, 294-298 (1984) · Zbl 1203.74022
[18]	Caputo, M., Elasticità e Dissipazione (1969), Zanichelli: Zanichelli Bologna
[19]	Suarez, L. E.; Shokooh, A., An eigenvector expansion method for the solution of motion containing fractional derivatives, ASME J. Appl. Mech., 64, 629-635 (1997) · Zbl 0905.73022
[20]	Ochmann, M.; Makarov, S., Representation of the absorption of nonlinear waves by fractional derivatives, J. Acoust. Soc. Am., 94, 6 (1993)
[22]	Glockle, W. G.; Nonnenmacher, T. F., Fractional integral operators and Foxfunctions in the theory of viscoelasticity, Macromolecules, 24, 6424-6434 (1991)
[26]	Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991)
[27]	Diethelm, K.; Ford, A. D., On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Voß, H.; Werther, J., Scientific Computing in Chemical Engineering II. Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg), 217-224
[29]	George, A. J.; Chakrabarti, A., The Adomian method applied to some Extraordinary differential equations, Appl. Math. Lett., 8, 3, 91-97 (1995) · Zbl 0828.65081
[30]	Arora, H. L.; Abdelwahid, F. I., Solutions of non-integer order differential equations via the Adomian decomposition method, Appl. Math. Lett., 6, 1, 21-23 (1993) · Zbl 0772.34009
[31]	Shawagfeh, N. T., The decompostion method for fractional differential equations, J. Frac. Calc., 16, 27-33 (1999) · Zbl 0956.34004
[32]	Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 517-529 (2002) · Zbl 1029.34003
[35]	Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[36]	Abdelwahid, F., A mathematical model of Adomian polynomials, Appl. Math. Comput., 141, 447-453 (2003) · Zbl 1027.65072
[37]	Cherruault, Y., Convergence of Adomian’s method, Kybernetes, 18, 31-38 (1989) · Zbl 0697.65051
[38]	Abbaoui, K.; Cherruault, Y., Convergence of Adomian’s method applied to differential equations, Comput. Math. Appl., 28, 103-109 (1994) · Zbl 0809.65073
[39]	Abbaoui, K.; Cherruault, Y., New Ideas for proving convergence of decomposition methods, Comput. Math. Appl., 29, 103-108 (1995) · Zbl 0832.47051
[40]	Himoun, N.; Abbaoui, K.; Cherruault, Y., New results of convergence of Adomian’s method, Kybernetes, 28, 4-5, 423-429 (1999) · Zbl 0938.93019