solutions for nonlinear differential equations based on the Rach-Adomian-Meyers
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.
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
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
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.
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