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Higher-Order Numeric Solutions for Nonlinear Systems Based on the Modified Decomposition Method

DOI: 10.4236/jamp.2014.21001, PP. 1-7

Keywords: Adomian Polynomials, Modified Decomposition Method, Adomian-Rach Theorem, Nonlinear Differential Equations, Numeric Solution

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Abstract:

Higher-order numeric solutions for nonlinear differential equations based on the Rach-Adomian-Meyers modified decomposition method are designed in this work. The presented one-step numeric algorithm has a high efficiency due to the new, efficient algorithms of the Adomian polynomials, and it enables us to easily generate a higher-order numeric scheme such as a 10th-order scheme, while for the Runge-Kutta method, there is no general procedure to generate higher-order numeric solutions. Finally, the method is demonstrated by using the Duffing equation and the pendulum equation.

References

[1]  G. Adomian, “Nonlinear Stochastic Operator Equations,” Academic, Orlando, 1986.
[2]  G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic, Dordrecht, 1994. http://dx.doi.org/10.1007/978-94-015-8289-6
[3]  G. Adomian and R. Rach, “Inversion of Nonlinear Stochastic Operators,” Journal of Mathematical Analysis and Applications, Vol. 91, 1983, pp. 39-46. http://dx.doi.org/10.1016/0022-247X(83)90090-2
[4]  A. M. Wazwaz, “Partial Differential Equations and Solitary Waves Theory,” Higher Education and Springer, Beijing and Berlin, 2009. http://dx.doi.org/10.1007/978-3-642-00251-9
[5]  R. Rach, “A New Definition of the Adomian Polynomials,” Kybernetes, Vol. 37, 2008, pp. 910-955. http://dx.doi.org/10.1108/03684920810884342
[6]  R. Rach, “A Convenient Computational Form for the Adomian Polynomials,” Journal of Mathematical Analysis and Applications, Vol. 102, 1984, pp. 415-419. http://dx.doi.org/10.1016/0022-247X(84)90181-1
[7]  A. M. Wazwaz, “A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators,” Applied Mathematics and Computation, Vol. 111, 2000, pp. 53-69. http://dx.doi.org/10.1016/S0096-3003(99)00063-6
[8]  J. S. Duan, “Recurrence Triangle for Adomian Polynomials,” Applied Mathematics and Computation, Vol. 216, 2010, pp. 1235-1241. http://dx.doi.org/10.1016/j.amc.2010.02.015
[9]  J. S. Duan, “An Efficient Algorithm for the Multivariable Adomian Polynomials,” Applied Mathematics and Computation, Vol. 217, 2010, pp. 2456-2467. http://dx.doi.org/10.1016/j.amc.2010.07.046
[10]  J. S. Duan, “Convenient Analytic Recurrence Algorithms for the Adomian Polynomials,” Applied Mathematics and Computation, Vol. 217, 2011, pp. 6337-6348. http://dx.doi.org/10.1016/j.amc.2011.01.007
[11]  R. Rach, G. Adomian and R. E. Meyers, “A modified Decomposition,” Computers & Mathematics with Applications, Vol. 23, 1992, pp. 17-23. http://dx.doi.org/10.1016/0898-1221(92)90076-T
[12]  G. Adomian and R. Rach, “Transformation of Series,” Applied Mathematics Letters, Vol. 4, 1991, pp. 69-71. http://dx.doi.org/10.1016/0893-9659(91)90058-4
[13]  G. Adomian and R. Rach, “Nonlinear Transformation of Series—Part II,” Computers & Mathematics with Applications, Vol. 23, 1992, pp. 79-83. http://dx.doi.org/10.1016/0898-1221(92)90058-P
[14]  G. Adomian and R. Rach, “Modified Decomposition Solution of Linear and Nonlinear Boundary-Value Problems,” Foundations of Physics Letters, Vol. 23, 1994, pp. 615-619. http://dx.doi.org/10.1016/0362-546X(94)90240-2
[15]  G. Adomian, R. Rach and N. T. Shawagfeh, “On the Analytic Solution of the Lane-Emden Equation,” Foundations of Physics Letters, Vol. 8, 1995, pp. 161-181. http://dx.doi.org/10.1007/BF02187585
[16]  J. C. Hsu, H. Y. Lai and C. K. Chen, “An Innovative Eigenvalue Problem Solver for Free Vibration of Uniform Timoshenko Beams by Using the Adomian Modified Decomposition Method,” Journal of Sound Vibration, Vol. 325, 2009, pp. 451-470. http://dx.doi.org/10.1016/j.jsv.2009.03.015
[17]  G. Adomian, R. C. Rach and R. E. Meyers, “Numerical Integration, Analytic Continuation, and Decomposition,” Applied Mathematics and Computation, Vol. 88, 1997, pp. 95-116. http://dx.doi.org/10.1016/S0096-3003(96)00052-5

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