World Applied Sciences Journal 6 (10): 1372-1376, 2009 ISSN 1818-4952 © IDOSI Publications, 2009

Modified Variation of Parameters Method for Differential Equations

1Syed Tauseef Mohyud-Din, 2Muhammad Aslam Noor and 2Khalida Inayat Noor

1HITEC University, Taxila Cantt, Pakistan 2Department 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

INTRODUCTION the inbuilt deficiencies of various existing techniques. Numerical results show the complete reliability of the The nonlinear oscillators appear in various proposed technique. physical phenomena related to physics, applied and engineering sciences [1-6] and the references therein. VARIATION OF Several techniques including variational iteration, PARAMETERS METHOD (VPM) homotopy perturbation and expansion of parameters have been applied for solving such problems [1-6]. Consider the following second-order partial He [2-9] developed the homotopy perturbation differential equation: method for solving various physical problems. This reliable technique has been applied to a wide range of ytt = f(t,x,y,z,y,y,y,yxyzxx ,yyy ,y)zz (1) 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 where t such that (-∞

Corresponding Author: Dr. Syed Tauseef Mohyud-Din, HITEC University, Taxila Cantt, Pakistan 1372 World Appl. Sci. J., 6 (10): 1372-1376, 2009

∞ y( x,y,z,t) = y( x,y,z,0) + tyt ( x,y,z,0) f= limu = ui (6) t p1→ ∑ i0= +− ∫()t s f(s,x,y,z,yx , y y y z , y xx ,y y y ,y z z )ds 0 It is well known that series (6) is convergent for which can be solved iteratively as [13, 14, 23-25] most of the cases and also the rate of convergence is dependent on L (u); [2-9]. We assume that (6) has a unique solution. The comparisons of like powers of p yk1+ ( x,y,z,t) = y( x,y,z,0) + ty( x,y,z,0) t give solutions of various orders. In sum, according to t +− kkkkkk [10, 11], He’s HPM considers the nonlinear term N(u) ∫()t s f(s,x,y,z,yx ,y y ,y z ,y xx ,y yy ,y zz )ds 0 as: k= 0,1,2, ∞ N(u)= pHi2 =++ H pH pH +... , ∑ i0 1 2 Homotopy Perturbation Method (HPM) i0= and He’s Polynomials To explain the He’s homotopy perturbation where Hn’s are the so-called He’s polynomials [10, 11], method, we consider a general equation of the type, which can be calculated by using the formula

n n 1 ∂ i L(u) = 0 (2) Hn0 (u , ,u n )=  N( p ui ) n= 0,1,2, . ∂ n ∑ n! p i0= p0= where L is any integral or differential operator. We define a convex homotopy H (u, p) by NUMERICAL APPLICATIONS

H(u,p)=−+ (1 p)F(u) pL(u) (3) In this section, we apply the MVPM for solving nonlinear oscillator differential equations. Numerical where F(u) is a functional operator with known results are very encouraging. solutions v0, which can be obtained easily. It is clear that, for Example 1: Consider the following Van Der Pol oscillator problem H (u, p) = 0 (4) 2 d u du23 du 2 + ++u u = 2cost − cost we have dt dt dt

H(u,0)= F(u), H(u,1) = L(u) with initial conditions

This shows that H (u, p) continuously traces an u( 0)= 0, u′( 0)= 1 implicitly defined curve from a starting point H (v 0, 0) to a solution function H (f, 1). The embedding The exact solution of the above problem is given by parameter monotonically increases from zero to unit as the trivial problem F (u) = 0, is continuously deforms u (t) = sin t the original problem L(u) = 0. The embedding parameter p∈(0,1] can be considered as an expanding Applying the Variation of Parameters Method parameter [1-23]. The homotopy perturbation method (VPM) uses the homotopy parameter p as an expanding  parameter [2-9] to obtain dun ( s) t + usn () ds u tt(ts)=−− ∞ n1+ () ∫ i 23 du() s u= p u =+++ u pu p u p u + (5) 0n23 ∑ i0 1 2 3 +un () s −−() 2coss cos s i0= ds

If p→1, then (5) corresponds to (3) and becomes Applying the Modified Variation of Parameters the approximate solution of the form: Method (MVPM) 1373 World Appl. Sci. J., 6 (10): 1372-1376, 2009

du du tt01++ p  2 du du ++23 +=−−ds ds − − ++01 + ++ − u0 pu 1 p u 2 t p∫∫() t s ds p( t s)( u01 pu  )p 2coss cos s ds ds ds 00+++2 + ()u01 pu p u 2

Table 1: Error estimates Table 2: Error estimates T Exact solution Series solution *Errors t Exact solution Series solution *Errors 0.0 0.0000000000 0.000000000 0.000000 0.0 2.0000000000 2.0000000000 0.000000 0.1 0.0998291400 0.099833416 4.27×10-6 0.1 1.9975000000 1.9950041650 2.49×10-3 0.2 0.1985983230 0.198669330 7.10×10-5 0.2 1.9900000000 1.9800665780 9.93×10-3 0.3 0.2951435550 0.295520206 3.76×10-4 0.3 1.9775000000 1.9553364890 2.21×10-2 0.4 0.3881614730 0.389418342 1.25×10-3 0.4 1.9600000000 1.9210609940 3.89×10-2 0.5 0.4761672760 0.479425538 3.25×10-3 0.5 1.9375000000 1.8775825620 5.99×10-2 0.6 0.5574415350 0.564642473 7.20×10-3 0.6 1.9100000000 1.8253356150 8.46×10-2 0.7 0.6299709850 0.644217687 1.42×10-3 0.7 1.8775000000 1.7648421870 1.12×10-1 0.8 0.6913890112 0.717356090 2.59×10-2 0.8 1.8400000000 1.6967067090 1.43×10-1 0.9 0.7389216471 0.783326909 4.44×10-2 0.9 1.7975000000 1.6216099680 1.75×10-1 1.0 0.7693445486 0.841470988 7.21×10-2 1.0 1.7500000000 1.5403023060 2.09×10-1 *Error =Exact solution-Series solution *Error =Exact solution-Series solution

Comparing the co-efficient of like powers of p Applying the Variation of Parameters Method following approximants are obtained (VPM)

(0) t p:utt0 ( )= du() s = +− −2 −n + un1+ () t u n (t)∫ ( t s ) un () s u n () s 1 0 ds

(1) 11212 3 4 3 4 11 p:uttttcostcost1 ()=−−−+−+ 2! 3! 4! 9 3 9 Applying the Modified Variation of Parameters  Method (MVPM)

The series solution after one iteration is given by ∂∂22uu t 01++p  2 ∂∂ss22 u012+pu +p u + =+2pt∫ () −s ds 02 1121234 3 4 11 −++−++(u pu )( u pu ) utttttcostcost()=−−−+ − ++ 0 1 01 t 2! 3! 4! 9 3 9 du du2 du −p∫ () t − s01 + p + p 2 ++ 1 ds 0 ds ds ds Table 1 exhibits the exact and the series solutions along with the errors obtained by using the MVPM by Comparing the co-efficient of like powers of p, using single iteration only. following approximants are obtained

(0) Example 2: Consider the following nonlinear oscillator p:ut20 ( ) = differential equation

(1) 1112346 1 2 p:ut1 ()=−++− t t t t d u − +2 +du −= 8 24 32 960 2 u u 10 dt dt  with initial conditions The series solution after one iteration is given by

111 1 u( 0) = 2, u′( 0) = 0 ut2tttt()=−++−+234 6 8 24 32 960

The exact solution of the above problem is given by Table 2 exhibits the exact and the series solutions along with the errors obtained by using the MVPM by ut( )= 1 +cost using single iteration only. 1374 World Appl. Sci. J., 6 (10): 1372-1376, 2009

CONCLUSION 9. He, J.H., 2000. A coupling method of homotopy technique and perturbation technique for nonlinear In this paper, we applied Modified Variation of problems. Int. J. Nonlinear Mech., 35 (1): 115-123. Parameters Method (MVPM) for solving nonlinear 10. Ghorbani, A. and J.S. Nadjfi, 2007. He’s homotopy oscillator differential equations. The method is perturbation method for calculating Adomian’s applied in a direct way without using linearization, polynomials. Int. J. Nonlin. Sci. Num. Simul., transformation, discretization or restrictive 8 (2): 229-332. assumptions. The fact that the proposed technique 11. Ghorbani, A., 2007. Beyond Adomian’s solves nonlinear problems without using Adomian’s polynomials: He polynomials, Chaos, Solitons & Fractals, In Press. polynomials is a clear advantage of this algorithm over 12. Mohyud-Din, S.T. and M.A. Noor, 2007. the decomposition method. Homotopy perturbation method for solving fourth-order boundary value problems. Math. ACKNOWLEDGMENT Prob. Engg., pp: 1-15. Article ID 98602, doi:10.11 55/2007/98602. The authors are highly grateful to the referee and 13. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, Professor Ghasem Najafpour for very constructive 2009. Modified variation of parameters method comments. We would like to thank Dr S. M. Junaid for solving partial differential equations. Int. J. Zaidi, Rector CIIT for providing excellent research Mod. Phys. B. environment and facilities. The first author is also 14. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, extremely thankful to Brig (R) Qamar Zaman, Vice 2009. Ma’s variation of parameters method for Chancellor HITEC University for the provision of very nonlinear oscillator differential equations. Int. J. conducive environs for research. Mod. Phys. B. 15. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, REFERENCES 2009. Homotopy perturbation method for unsteady flow of gas through a porous medium. Int. J. Mod. 1. Barari, A., M. Omidvar, A.R. Ghotbi and D.D. Phys. B. Ganji, 2008. Applications of homotopy 16. Noor, M.A. and S.T. Mohyud-Din, 2008. perturbation method and variational iteration Homotopy perturbation method for nonlinear method to nonlinear oscillator differential higher-order boundary value problems. Int. J. equations. Acta Appl. Mathca. DOI Nonlin. Sci. Num. Simul., 9 (4): 395-408. 17. Noor, M.A. and S.T. Mohyud-Din, 2008. Modified 10.1007/s10440-008-9248-9. variational iteration method for Goursat and 2. He, J.H., 2008. An elementary introduction of Laplace problems. World Appl. Sci. J., 4 (4): recently developed asymptotic methods and 487-498. nanomechanics in textile engineering. Int. J. Mod. 18. Noor, M.A., S.T. Mohyud-Din and M. Tahir, 2008. Phys. B 22 (21): 3487-4578. Modified variational iteration methods for 3. He, J.H., Some asymptotic methods for strongly Thomas-Fermi equation. World Appl. Sci. nonlinear equation. Int. J. Mod. Phys., 20 J., 4 (4): 479-498. (20)10: 1144-1199. 19. Saadati, R., B. Rafbari, H. Abibi, S.M. Maezpour 4. He, J.H., 2008. Recent developments of the and S. Shakeri, 2008. A comparison between homotopy perturbation method. Top. Meth. Nonlin. variational iteration method and trapezoidal rule Anal., 31: 205-209. for solving linear integro-differential equations. 5. He, J.H., 2006. Homotopy perturbation method for World Appl. Sci. J., 4 (3): 321-325. solving boundary value problems. Phys. Lett., A 20. Xu, L., 2007. He’s homotopy perturbation method 350: 87-88. for a boundary layer equation in unbounded 6. He, J.H., 2004. Comparison of homotopy domain. Comput. Math. Appl., 54: 1067-10 perturbation method and homotopy analysis 21. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, method. Appl. Math. Comput., 156: 527-539. 2009. Solving second-order singular problems 7. He, J.H., 2005. Homotopy perturbation method for using He’s polynomials. World Appl. Sci. J. 6 (6): bifurcation of nonlinear problems. Int. J. Nonlin. 769-775. Sci. Num. Simul., 6 (2): 207-208. 22. Mohyud-Din, S.T. and M.A. Noor, 2009. 8. He, J.H., 2004. The homotopy perturbation method Travelling wave solutions of seventh-order for nonlinear oscillators with discontinuities. Appl. generalized KdV equations using He’s Math. Comput., 151: 287-292. polynomials. Int. J. Nonlin. Sci. Num. Sim., 10 (1): 1375 World Appl. Sci. J., 6 (10): 1372-1376, 2009

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