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 JAMP  Vol.7 No.10 , October 2019
The Adomian Decomposition Method for a Type of Fractional Differential Equations
Abstract: Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.
Cite this paper: Guo, P. (2019) The Adomian Decomposition Method for a Type of Fractional Differential Equations. Journal of Applied Mathematics and Physics, 7, 2459-2466. doi: 10.4236/jamp.2019.710166.
References

[1]   Oldham, K.B. and Spainer, J. (1974) The Fractional Calculus. Academic Press, New York.

[2]   Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wily and Sons Inc., New York.

[3]   Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.

[4]   Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon.

[5]   Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
https://doi.org/10.1142/9789812817747

[6]   Sabatier, J., Argawal, O.P. and Machado, A.T. (2007) Advances in Fractional Calculus. Springer, Dordrecht.
https://doi.org/10.1007/978-1-4020-6042-7

[7]   Bagley, R.L. and Torvik, J. (1983) Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures, AIAA Journal, 21, 741-748.
https://doi.org/10.2514/3.8142

[8]   Momani, S. and Al-Khaled, K. (2005) Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method. Applied Mathematics and Computation, 162, 1351-1365.
https://doi.org/10.1016/j.amc.2004.03.014

[9]   Momani, S. (2006) Non-Perturbative Analytical Solutions of the Space- and Time-Fractional Burgers Equations. Chaos, Solitons and Fractals, 28, 930-937.
https://doi.org/10.1016/j.chaos.2005.09.002

[10]   Chen, Y. and Moore, K. (2002) Discretization Schemes for Fractional-Order Differentiators and Integrators. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49, 363-367.
https://doi.org/10.1109/81.989172

[11]   Song, B., Xu, L. and Lu, X. (2015) Discrete Approximation of Fractional-Order Differentiator Based on Tustin Transform. Science Technology and Engineering, 15, 92-95.

[12]   Varieschi, G. (2018) Applications of Fractional Calculus to Newtonian Mechanics. Journal of Applied Mathematics and Physics, 6, 1247-1257.
https://doi.org/10.4236/jamp.2018.66105

[13]   Tamsir, M. and Srivastava, V.K. (2016) Analytical Study of Time-Fractional Order Klein-Gordon Equation. Alexandria Engineering Journal, 55, 561-567.
https://doi.org/10.1016/j.aej.2016.01.025

[14]   Chen, H., Lü, S. and Chen, W. (2017) A Fully Discrete Spectral Method for the Nonlinear Time Fractional Klein-Gordon Equation. Taiwanese Journal of Mathematics, 21, 231-251.
https://doi.org/10.11650/tjm.21.2017.7357

[15]   Adomian, G. (1988) A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis & Applications, 135, 501-544.
https://doi.org/10.1016/0022-247X(88)90170-9

[16]   Adomian, G. (1994) Solution of Nonlinear Evolution Equations. Mathematical and Computer Modelling, 20, 1-2.
https://doi.org/10.1016/0895-7177(94)90120-1

[17]   Cherruault, Y., Adomian, G., Abbaoui, K. and Rach, R. (1995) Further Remarks on Convergence of Decomposition Method. International Journal of Bio-Medical Computing, 38, 89-93.
https://doi.org/10.1016/0020-7101(94)01042-Y

[18]   Abbaoui, K. and Cherruault, Y. (1995) New Ideas for Proving Convergence of Decomposition Methods. Computers & Mathematics with Applications, 29, 103-108.
https://doi.org/10.1016/0898-1221(95)00022-Q

 
 
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