method, we consider a general equation of the type,. L(u) = 0. (2) ... which can be calculated by using the formula n n i n. 0 n i n i 0 p 0. 1 ..... Wronskian solutions.
World Applied Sciences Journal 6 (10): 1372-1376, 2009 ISSN 1818-4952 © IDOSI Publications, 2009
Modified Variation of Parameters Method for Differential Equations 1
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Syed Tauseef Mohyud-Din, Muhammad Aslam Noor and Khalida Inayat Noor 1
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HITEC University, Taxila Cantt, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan
Abstract: In this paper, we apply the Modified Variation of Parameters Method (MVPM) for solving nonlinear differential equations which are associated with oscillators. The proposed modification is made by the elegant coupling of traditional Variation of Parameters Method (VPM) and He’s polynomials. The suggested algorithm is more efficient and easier to handle as compare to decomposition method. Numerical results show the efficiency of the proposed algorithm. Key words: Variation of parameters method He’s polynomials nonlinear oscillator •
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INTRODUCTION The nonlinear oscillators appear in various physical phenomena related to physics, applied and engineering sciences [1-6] and the references therein. Several techniques including variational iteration, homotopy perturbation and expansion of parameters have been applied for solving such problems [1-6]. He [2-9] developed the homotopy perturbation method for solving various physical problems. This reliable technique has been applied to a wide range of diversified physical problems [1-23] and the references therein. Recently, Ghorbani et al. [10, 11] introduced He’s polynomials by splitting the nonlinear term into a series of polynomials. Moreover, it was proved [10, 11] that He’s polynomials are compatible with Adomian’s polynomials but are easier to calculate and are more user friendly. The He’s polynomials are calculated from He’s Homotopy Perturbation Method (HPM) which was developed and formulated by He by merging the standard homotopy and perturbation [1-30] and the references therein. Recently, Mohyud-Din, Noor and Noor [12-18, 23] applied He’s polynomials for solving various nonlinear problems. The basic motivation of this paper is the elegant coupling of traditional Variation of Parameters Method (VPM) [24, 25, 14] and He’s polynomials for solving nonlinear differential equations associated with oscillators. This reliable combination which is called the Modified Variation of Parameters Method (MVPM) [13, 23] has been recently developed by Mohyud-Din, Noor and Noor and is very useful for solving such problems and makes the solution procedure simpler and faster. Moreover, Mohyud-Din, Noor and Noor’s modified version [13, 23] is easier to implement and is independent of
the inbuilt deficiencies of various existing techniques. Numerical results show the complete reliability of the proposed technique. VARIATION OF PARAMETERS METHOD (VPM) Consider the following differential equation:
second-order
partial (1)
ytt = f(t,x,y,z,y x, y y, y z, y xx , y yy ,y zz )
where t such that (-∞
