Solving Duffing equation using an improved semi-analytical method

Abstract

Since achieving the precise analytical solution of most nonlinear problems is not possible, numerical methods are meant to be used. In fact a very limited number of nonlinear problems have the exact analytical solution. Let us consider some other methods which are known as semi-analytical methods and applied to solve nonlinear problems. In these methods by using a primary approximation and an iterative process, the approximate solution can be provided, so that the iterations converge to the desired solution. Methods which have been employed and support researchers in this field are categorized as "Adomian decomposition method"(ADM), "Homotopy perturbation method"(HPM), "Homotopy analysis method"(HAM), and "He variational iteration method"(VIM). The aim of this paper is to suggest an improvement of reliability of Adomian decomposition method to obtain a solution to Duffing equation. In details, Adomian decomposition method using modified Legendre polynomials has been applied in the present work that allows researchers to provide approximation of the solution with high accuracy. This method is tested for example. Moreover the obtained numerical results will be presented.

Improvement of the homotopy perturbation method to nonlinear problems

“…In Table 1, we compare the AEs achieved using our method with those obtained using the ADM with Legendre and Taylor polynomials (Novin and Dastjerd, 2015) at . Figure 8 plot the AEs of the HPM with shifted Jacobi polynomials, which show that our method (Novin and Dastjerd, 2015) at and for Example 3. is more accurate than HPM with Chebyshev and Taylor polynomials (Behrooz and Ebadi, 2011) shown in Figure 9.…”

“…Figure 8 plot the AEs of the HPM with shifted Jacobi polynomials, which show that our method (Novin and Dastjerd, 2015) at and for Example 3. is more accurate than HPM with Chebyshev and Taylor polynomials (Behrooz and Ebadi, 2011) shown in Figure 9. From Table 1, Figures 8 and 9 listed above, it is shown that the method here is surpassed than ADM with Legendre and Taylor polynomials introduced by Novin and Dastjerd (2015) and the HPM with Chebyshev and Taylor polynomials introduced by Behrooz and Ebadi (2011).…”

“…Behrooz and Ebadi (2011) further developed the HPM using Legendre polynomials. Recently, Novin and Dastjerd (2015) improved the adomian decomposition method to obtain solutions for the Duffing equation. This article applies the HPM to the shifted Jacobi polynomials of the right-side function to solve nonlinear differential equations.…”