%0 Journal Article
%T Haar Wavelet Operational Matrix Method for Fractional Oscillation Equations
%A Umer Saeed
%A Mujeeb ur Rehman
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/174819
%X We utilized the Haar wavelet operational matrix method for fractional order nonlinear oscillation equations and find the solutions of fractional order force-free and forced Duffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervals. The results are compared with the results obtained by the other technique and with exact solution. 1. Introduction Haar wavelet is the lowest member of Daubechies family of wavelets and is convenient for computer implementations due to availability of explicit expression for the Haar scaling and wavelet functions [1]. Operational approach is pioneered by Chen and Hsiao [2] for uniform grids. The basic idea of Haar wavelet technique is to convert differential equations into a system of algebraic equations of finite variables. The Haar wavelet technique for solving linear homogeneous/inhomogeneous, constant, and variable coefficients has been discussed in [3]. The fractional order forced Duffing-Van der Pol oscillator is given by the following second order differential equation [4]: where is the Caputo derivative; represents the periodic driving function of time with period , where is the angular frequency of the driving force; is the forcing strength; and is the damping parameter of the system. Duffing-Van der Pol oscillator equation can be expressed in three physical situations: (1)single-well , ;(2)double-well , ;(3)double-hump , . The quasilinearization approach was introduced by Bellman and Kalaba [5, 6] as a generalization of the Newton-Raphson method [7] to solve the individual or systems of nonlinear ordinary and partial differential equations. The quasilinearization approach is suitable to general nonlinear ordinary or partial differential equations of any order. The Haar wavelets with quasilinearization technique [8¨C10] are applied for the approximate solution of integer order nonlinear differential equations. In [11], we extend the Haar wavelet - quasilinearization technique for fractional nonlinear differential equations. The aim of the present work is to investigate the solution of the higher order fractional Duffing equation, fractional order force-free and forced Duffing-Van der pol (DVP) oscillator using Haar wavelet-quasilinearization technique. We have discussed the three special situations of DVP oscillator equation such as single-well, double-well, and double- hump. 2. Preliminaries In this section, we review basic definitions of fractional differentiation and fractional integration [12].(1)Riemann-Liouville fractional integral operator of order is as follows: the
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