A new modification of the HPM for the Duffing equation with high nonlinearity

In this work we introduce a new modification of the homotopy perturbation method for solving nonlinear ordinary differential equations. The technique is based on the blending of the Chebyshev pseudo-spectral methods and the Homotopy Perturbation Method (HPM). The method is tested by solving the strongly nonlinear Duffing equation for undamped oscillators. Comparison is made between the proposed technique, the standard HPM, an earlier modification of the HPM and the numerical solutions to demonstrate the high accuracy, applicability and validity of the present approach.


Introduction
Finding accurate and efficient methods for solving nonlinear differential equations in bounded or unbounded domains is an ongoing challenge in engineering and science. For most nonlinear equations with no closed form solutions, recourse is usual made to numerical methods such as the iterative shooting method, the Runge-Kutta schemes and the Keller-box method. These numerical solutions often give very little insight into the structure of the solutions or the effects of the various parameters embedded in the governing equations. Non-numerical approaches include the classical power-series method and its variants such as the homotopy perturbation method for systems of nonlinear differential equations with small or large embedded parameters. For equations with neither small nor large parameters, non-perturbation techniques such as the Adomian decomposition method and the homotopy analysis method (Adomian [1], Liao [14]). He [9,10,11,12] proposed the homotopy perturbation method and he successfully used this method to solve many types of linear and nonlinear differential equations. This method provides a convenient way to obtain analytic or approximate solutions for a wide variety of problems arising in different scientific fields by continuously deforming the difficult problem into a set of simple linear problems that are easy to solve. Modifications of the HPM to improve its accuracy and convergence rate have been reported by, among others, Beléndez et al. [2,3,4], Odibat [17], Odibat and Moman [18] and more recently, by Ganji et al. [7]. The HPM has been used, for example, to solve the Lighthill equation [9] and the Duffing equation [11]. In Beléndez et al. [3] the HPM was modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation and so. Ganji et al. [7] reported some differences between their implementation and the standard HPM in the choice of the linear operator with all the other processes identical. The nonlinear Duffing harmonic oscillation differential equation (see Mickens [15,16]) has been solved using the homotopy perturbation method by Beléndez et al. [5], the variational iteration method (He [13] and Ramos [20]), the Adomian decomposition method (Ghosh et al. [8]), the artificial parameter-decomposition (Ramos [19]) and He's parameter-expanding method (Xu [22]). A comparison between the ADM and exact solution was given in [8] where it was shown that the ADM is only valid for a small region of the domain.
The purpose of the present paper to introduce a new alternative and improved of the HPM called Spectral Homotopy Perturbation Method (SHPM) in order to address some of the perceived limitations of the HPM uses the Chebyshev pseudospectral method. We use the method to find approximate solutions of the equation of motion of Duffing's undamped oscillator. We show that this hybrid method gives rapid convergence and good accuracy. This study proposes a standard way of choosing the linear operators and initial approximations for the SHPM. The obtained results suggest that this newly improvement technique introduces a powerful for solving non-linear differential equations. The new modification demonstrates an accurate solution compared with the numerical solution.

Homotopy perturbation method (HPM)
For the convenience of the reader, we first present a brief review of the standard HPM. This is then followed by a description of the algorithm of the SHPM solving nonlinear ordinary differential equations. To illustrate the basic ideas of the HPM, we consider the following nonlinear differential equation with the boundary conditions , = 0, u B u r n (2) where A is a general operator, B is a boundary operator, () fr is a known analytic function and is the boundary of the domain .
The operator A can, in generally, be divided into two parts L and part N so that equation (1) can be written where L is a simple part which is easy to handle and N contains the remaining parts of A. By the homotopy technique [23,24] , we construct a homotopy ( , ) : where [0,1] p is an embedding parameter, 0 u is an initial approximation of equation (1), wich satisfies the boundary conditions. Obviously, from equation (4) The changing process of p from zero to unity is equivalent to the deformation of ( , ) v r p from 0 () ur to () ur . In topology, this is called deformation and 0 ( ) We can assume that the solution of equation (4) can be written as a power series in p , i.e. ..
The series u is convergent for most cases. However, the convergence rate depends on the nonlinear operator of differential equation (3). The following opinions are suggested and proved by He [9,10] 1. The second derivative of () Nu with respect to u must be small because the parameter p may be relatively large, i.e 1 p .

The norm of
1 N L u must be smaller than one so that the series converges.
The coupling of the perturbation method and the homotopy method gives the homotopy perturbation method (HPM), which has eliminated limitations of the traditional perturbation methods.

The Duffing oscillator equation
The nonlinear differential equation for the cubic free undamped Duffing oscillator is given by (Ganji et al. [7]); (10) subject to the initial conditions It is also convenient to make the boundary conditions homogeneous by making use of the transformation 0 ( ) = ( ) ( ), u t f y u t (13) where 0 ( ) = cos( ) u t a t is chosen to satisfy the initial conditions (11) and is a spatial scaling parameter.

Applications and discussions
In order to assess the advantages and the accuracy of our method for solving nonlinear Duffing equation, we have applied it to a variety of initial-value problems arising in nonlinear dynamics.

Example 1:
Consider the governing equations of motion of free undamped Duffing's oscillator as follows O c t 5 , 2 0 1 3 3 ( ) ( ) = 0, u t u t u (14) and then substitute (12) and (13) in (14) gives Tu T T T a t u a t a t t u u u (16) Equation (15) is solved subject to the initial conditions

y a t f y a t f y a t f y t
We thus construct the homotopy: is an embedding parameter and v is an approximate series solution of (15) given by where m is the order of the method. The linear operator L is taken to be a y dy (20) and 0 f is the initial approximation for the solution of (18) and it is obtained from the solution to the nonhomogeneous linear part of (18) which is with the boundary conditions 00 ( 1) = ( 1) = 0 ff (22) Equation (18) can be written as   and D is the Chebyshev spectral differentiation matrix whose entries see [6,21] are given by Finally, the solution of the Duffing oscillator differential equation (14) is obtained by substitute the series i f in (13). This problem has been solved by Ganji et al. [7] using a different modification of the homotopy perturbation method. Their algorithm adds and subtracts a term to the main governing equation. One of the additional terms is then added to the linear operator. The only difference between the scheme used by Ganji et al. [7] and the standard HPM is in the choice of the linear operator with other procedures the same. They found that the first two terms in the solution series are The current method is convergent if 1 ( ) = / norm L N u is smaller than one, this is the same strategy that used in the standard HPM approach, it is observed that from Figure 1(a) the solution of (14) is convergent when =1.4 (the correspond minimum value of () ). If we consider =1.5, =1, = 0.5 A , the optimum value for is 1.35 as shown in Figure 1(b). values of a although not sufficiently to match the current method. To improve the accuracy of the Ganji et al. [7] scheme it is necessary to increase the number of terms in the solution series. This shows that for the same O c t 5 , 2 0 1 3 number of terms, the proposed spectral modification of the HPM gives superior accuracy and convergence to the numerical solution.

Example 2:
We also solved the nonlinear undamped Duffing oscillator equation for fifth nonlinearity which it takes the form: We made a comparison between HPM, SHPM and numerical results of Example 2 in Table 2 to validate the SHPM procedure, it seen it is evident that our results are in excellent agreement with that tnumerical solution.

Conclusion
In this work we have proposed a novel spectral modification of the standard homotopy perturbation method (SHPM) for solving nonlinear ordinary differential equations. The method has been used to solve the strongly nonlinear Duffing oscillator equation. Comparison with the standard HAM, the modified homotopy perturbation method of Ganji et al. [7] and numerical solutions show that the SHPM is highly accurate, efficient and converges rapidly with only a few iterations required to achieve the accuracy of the numerical results. The main difference between the HPM and the SHPM is that the solutions are obtained by solving a system of higher order ordinary differential equations in the HPM while for the SHPM solutions are obtained by solving a system of linear algebraic equations that are easier to solve. The advantages of this new method is that it converges much faster than the standard HPM and other recent modifications of the HPM such as the recent scheme by Ganji et al. [7]. The method can be used as alternative to the traditional Runge-Kutta, finite difference, finite element and Keller-Box methods. O c t 5 , 2 0 1 3