# Optimization of the Adomian Decomposition Method for Solving Differential Equation with Fractional Order

Authors

Department of Mathematics, Shiraz University of Technology, Shiraz, Iran

Abstract

Up to now, Adomian Decomposition Method (ADM) has been widely employed in solving different kinds of differential equations. However, in many cases it is observed that the ADM has a lower precision in comparison with other methods, especially that of Homotopic ones. ADM is a relatively general and powerful method for finding analytical approximate results from different equations. In this paper, we seek to raise Optimal Adomian Decomposition Method (OADM) precision by employing the standard pattern of ADM. The main character of this repetitive method is based on employment of a controlling parameter in convergence, which resemble the parameters used in Homotopy Analysis Method (HAM). This parameter is indicated in such a way to reasonably increase the precision of obtained results. To indicate the optimizing parameter, the Least Squares Method has been used. The presented examples demonstrate that,  how  the above mentioned method has validity, applicability and a high degree of precision in solving differential equations of  fractional order so that it can be generally used in solving differential equations.

#### References

 He, J.H. (1999), Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262.  Hesameddini, E. and Latifizadeh, H. (2009), A new vision of the He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10, 1415-1424.  Hesameddini, E. and Latifizadeh, H. (2009), An optimal choice of initial solutions in the homotopy perturbation method. International اسماعیل حسامالدینی، محسن ریاحی 42 Journal of Nonlinear Sciences
and Numerical Simulation, 10, 1389- 1398.  Liao, S.J. (1992), On the proposed homotopy analysis technique for nonlinear problems and its applications, Ph.D. Dissertation. Shanghai Jiao Tong University, Shanghai, China.  Liao, S.J. (2004), On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147(2), 499-513.  West, B.J. and
Bolognab, M. and Grigolini, P. (2003), Physics of Fractal Operators, Springer, New York.  Miller, K.S. and Ross, B. (1993), An
introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.  Samco, S.G. and Kilbas, A.A. and Marichev, O.I. (1993), Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon.  Einicke, G.A. and White, L.B. and Bitmead, R.R. (2003), The use of fake algebraic Riccati Equations for co-channel demodulation, IEEE Transactions on Signal Processing, 51(9) 129-134.  Adomian, G. (1990), A review of the decomposition method and
some recent results for nonlinear equation, Mathematical and Computer Modelling, 13(7) 17-42.  Adomian, G. (1994), Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, HA.  Hosseini, M.M. (2006), Adomian decomposition method with Chebyshev polynomials, Applied Mathematics and Computation, 175, 1685-1693.  Duan, J.S. and Rach, R. (2011), A new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Applied Mathematics and Computation, 218, 4090-4118.  Kumar, M. and Singh, N. (2010), Modified Adomian Decomposition Method and computer implementation for solving singular boundary value problems arising in
various physical problems, Computers and Chemical Engineering, 34, 1750-1760.  Abbaoui, K. and Cherruaul, Y. (1995), New ideas for proving convergence of decomposition methods, Computers & Mathematics with Applications, 29, 103-8.

### History

• Receive Date: 11 June 2013
• Accept Date: 11 June 2013
• First Publish Date: 11 June 2013