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Modified Variation of Parameters Method for Differential Equations 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 (-∞<t<∞) is time and f is linear or non series of polynomials. Moreover, it was proved [10, 11] linear function of y, yx,yy,yz,yxx,yyy,yzz. The that He’s polynomials are compatible with Adomian’s homogeneous solution of (1) is polynomials but are easier to calculate and are more user friendly. The He’s polynomials are calculated from y( t,x,y,z) = A + Bt He’s Homotopy Perturbation Method (HPM) which was developed and formulated by He by merging the where A and B are functions of x, y, z and t. Using standard homotopy and perturbation [1-30] and the Variation of parameters method we have following references therein. Recently, Mohyud-Din, Noor and system of equations Noor [12-18, 23] applied He’s polynomials for solving various nonlinear problems. The basic motivation of ∂∂AB +=0 this paper is the elegant coupling of traditional ∂∂tt Variation of Parameters Method (VPM) [24, 25, 14] and He’s polynomials for solving nonlinear differential ∂B equations associated with oscillators. This reliable = f ∂t combination which is called the Modified Variation of Parameters Method (MVPM) [13, 23] has been recently and hence developed by Mohyud-Din, Noor and Noor and is very tt useful for solving such problems and makes the A (x,y,z,t )(=−=−D x,y,z )∫∫sfds,B (x,y,z,t )(C x,y,z )fds solution procedure simpler and faster. Moreover, 00 Mohyud-Din, Noor and Noor’s modified version [13, 23] is easier to implement and is independent of therefore, 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 − s01 + 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.
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