Research Article Application of Hybrid Functions for Solving Duffing-Harmonic Oscillator Mohammad Heydari, 1 Ghasem Brid Loghmani, 1 Seyed Mohammad. Hosseini, 2 and Seyed Mehdi Karbassi 3 1 Department of Mathematics, Yazd University, Yazd, Iran 2 Department of Mathematics, Islamic Azad University, Shahrekord Branch, Shahrekord, Iran 3 Department of Mathematics, Islamic Azad University, Yazd Branch, Yazd, Iran Correspondence should be addressed to Mohammad Heydari; m.heydari85@gmail.com Received 9 April 2014; Revised 26 July 2014; Accepted 28 July 2014; Published 14 August 2014 Academic Editor: Athanassios G. Bratsos Copyright © 2014 Mohammad Heydari et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A numerical method for fnding the solution of Dufng-harmonic oscillator is proposed. Te approach is based on hybrid functions approximation. Te properties of hybrid functions that consist of block-pulse and Chebyshev cardinal functions are discussed. Te 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. Te method is easy to implement and computationally very attractive. Te results are compared with the exact solution and results from several recently published methods, and the comparisons showed proper accuracy of this method. 1. Introduction Most phenomena in our world are essentially nonlinear and are described by nonlinear ordinary diferential equations. Nonlinear oscillation in mechanics, physics, and applied mathematics has been a topic of intensive research for many years. Difculty of solving the nonlinear problems or getting an analytic solution leads one to use numerical methods. Several methods have been used to fnd approximate solu- tions to these nonlinear problems. Some of these well-known methods are harmonic balance method [1], multiple scales method [2], Krylov-Bogoliubov-Mitropolsky method [3, 4], modifed Lindstedt-Poincare method [5], linearized pertur- bation method [6], energy balance method [7], iteration perturbation method [8], bookkeeping parameter pertur- bation method [9], amplitude frequency formulation [10], maximum approach [11], Mickens iteration procedure [12], rational harmonic balance method [13], Adomian decom- position method [14], variational iteration method [15], modifed variational iteration method [16, 17], homotopy perturbation method [18], modifed diferential transform method [19], and modifed homotopy perturbation method [20]. Recently, hybrid functions have been applied extensively for solving diferential equations or systems and proved to be a useful mathematical tool. Te pioneering work in the solu- tion of linear systems with inequality constraints via hybrid of block-pulse functions and Legendre polynomials was led in [21] that frst 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 block- pulse and Chebyshev functions. Razzaghi and Marzban [23] applied the hybrid of block-pulse and Chebyshev functions to fnd approximate solution of systems with delays in state and control. Solution of time-varying delay systems is approxi- mated 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 integrodiferential equation system. Razzaghi and Marzban Hindawi Publishing Corporation Journal of Difference Equations Volume 2014, Article ID 210754, 9 pages http://dx.doi.org/10.1155/2014/210754