Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator

A numerical method for finding the solution of Duffing-harmonic oscillator is proposed.The approach is based on hybrid functions approximation.The properties of hybrid functions that consist of block-pulse and Chebyshev cardinal functions are discussed.The associated operational matrices of integration and product are then utilized to reduce the solution of a strongly nonlinear oscillator to the solution of a system of algebraic equations.Themethod is easy to implement and computationally very attractive.The results are compared with the exact solution and results from several recently published methods, and the comparisons showed proper accuracy of this method.

Recently, hybrid functions have been applied extensively for solving differential equations or systems and proved to be a useful mathematical tool. The pioneering work in the solution of linear systems with inequality constraints via hybrid of block-pulse functions and Legendre polynomials was led in [21] that first derived an operational matrix for the integrals of the hybrid function vector. Razzaghi and Marzban in [22] the variational problems are solved using hybrid of blockpulse and Chebyshev functions. Razzaghi and Marzban [23] applied the hybrid of block-pulse and Chebyshev functions to find approximate solution of systems with delays in state and control. Solution of time-varying delay systems is approximated using hybrid of block-pulse functions and Legendre polynomials in [24]. Maleknejad and Tavassoli Kajani in [25] introduced a Galerkin method based on hybrid Legendre and block-pulse functions on interval [0, 1) to solve the linear integrodifferential equation system. Razzaghi and Marzban in [26], a direct method for solving multidelay systems using hybrid of block-pulse functions and Taylor series is presented. Marzban et al. [27] implemented hybrid of blockpulse functions and Lagrange-interpolating polynomials to find approximate solution of Volterra's population model. The Lane-Emden type equations are solved in [28] using hybrid functions of block-pulse and Lagrange-interpolating polynomials. The hybrid of block-pulse functions and Taylor series is employed in [29] to solve the linear quadratic optimal control with delay systems. Application of hybrid of blockpulse functions and Lagrange polynomials for solving the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations is investigated in [30].
In this study, we consider the following nonlinear Duffing-harmonic oscillation [31][32][33][34][35]: where is an example of conservative nonlinear oscillatory systems having a rational form for the restoring force. Note that, for small values of , (1) is that of a Duffing-type nonlinear oscillator; that is, while for large values of the equation approximates that of a linear harmonic oscillator; that is, Hence, (1) is called the Duffing-harmonic oscillator [31]. The system will oscillate between symmetric bounds [− , ], and the frequency and corresponding periodic solution of the nonlinear oscillator are dependent on the amplitude [33].
In this paper, we introduce an alternative numerical method to solve Duffing-harmonic oscillator. The method consists of reducing this equation to a set of algebraic equations by first expanding the candidate function as a hybrid function with unknown coefficients. These hybrid functions, which consist of block-pulse functions plus Chebyshev cardinal functions, are first introduced. The operational matrices of integration and product are given. These matrices are then used to evaluate the coefficients of the hybrid function for the solution of strongly nonlinear oscillators.
The outline of this paper is as follows. In Section 2, the basic properties of hybrid block-pulse functions and Chebyshev cardinal functions required for subsequent development are described. In Section 3, we apply the proposed numerical method to the Duffing-harmonic oscillation. Results and comparisons with existing methods in the literature are presented in Section 4 and finally conclusions are drawn in Section 5.

Properties of Hybrid Functions
Marzban et al. in [27,29,30] used the hybrid of block-pulse functions and Lagrange-interpolating polynomials based on zeros of the Legendre polynomials. But no explicit formulas are known for the zeros of the Legendre polynomials. In this study we used Chebyshev cardinal functions which are special cases of Lagrange-interpolating polynomials based on zeros of the Chebyshev polynomials of the first kind to overcome this problem. In this paper, we present the properties of hybrid functions which consist of block-pulse functions plus Chebyshev cardinal functions similar to [27,29,30]. The hybrid functions are first introduced, and the operational matrices of integration and product are then derived.
So, from (26) and (28) The proof of this lemma is presented in [39].
where ≥ 1 is a positive integer.
By expanding (0) = and ( ) = 1 in terms of hybrid functions we get where U = ( 2 + A ) and is the operational matrix of integration given in (13). Using Lemma 4 the functions 2 ( ) and 3 ( ) can be expanded as Therefore, by using (39) and (42)-(45), the right side of (38) can be approximated as wherẽcan be calculated in a similar way to matrix̃in (24). Since the above equation is satisfied for every ∈ [0, ), we can get This is a system of algebraic equations with equations and unknowns, which can be solved by Newton's iteration method to obtain the unknown vector .

Results and Discussions
In this section, we illustrate the accuracy of the hybrid functions method (HFM) by comparing the approximate solutions previously obtained with the exact angular frequency ex . All the results obtained here are computed using the Intel Pentium 5, 2.2 GHz processor and using Maple 17 with 64-digit precision. The exact angular frequency, ex , of the Duffingharmonic oscillator was found by Lim and Wu in [32] as By using alternative form (38) and applying the harmonic balance method (HBM) [13], Mickens [31] obtained the first approximate angular frequency HBM = ( Ozis and Yildirim [35] obtained the angular frequency using the energy balance method (EBM) in the following form: Ganji et al. in [34] obtained the same approximation as that in (52). Using a single-term approximate solution ( ) = cos( ) to (36) and the Ritz procedure [40], Tiwari et al. [41] obtained an approximate angular frequency as follows: The computed results for the HFM frequency HFM with exact frequency ex [32], HBM frequency HBM [31], EBM frequency EBM [35], and Tiwari's frequency Tiw [41] are listed in Tables 1 and 2. Table 2 shows that the maximum percentage error between HFM and exact frequency ex is 0.118%. Comparison of the exact frequency ex obtained by (50) with HBM , EBM , Tiw , and HFM is shown in Figure 1 for 0 ≤ ≤ 10, 3.0 ≤ ≤ 3.25, 5.0 ≤ ≤ 5.25, and 8.0 ≤ ≤ 8.25.

Conclusion
In this paper, we presented a numerical scheme based on hybrid block-pulse functions and Chebyshev cardinal functions for solving Duffing-harmonic oscillator. This algorithm reduces the solution of Duffing-harmonic oscillator differential equation to the solution of a system of algebraic equations in matrix form. The merit of this method is that the system of equations obtained for the solution does not need to consider collocation points; this means that the system of equations is obtained directly. A comparative study between HBM [31], EBM [35], Tiwari's method [41], and the proposed method was discussed in Section 4. The obtained results showed that the HFM is accurate, capable, and effective technique for the solution of the Duffing-harmonic oscillator. Further research can concentrate on other strongly nonlinear oscillators and more complicated cases.