World Applied Sciences Journal 6 (8): 1139-1146, 2009 ISSN 1818-4952 © IDOSI Publications, 2009
Modified Variation of Parameters Method for Second-order Integro-differential Equations and Coupled Systems
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 second-order integro-differential equations and coupled systems of integro-differential-equations. The proposed modification is made by the elegant coupling of He’s polynomials and the Ma’s variation of parameters method. The proposed MVPM is applied without any discretization, transformation or restrictive assumptions and is free from round off errors, calculation of the so-called Adomian’s polynomials and the identification of Lagrange multipliers. The numerical results are very encouraging.
Key words : Variation of parameters method · he’s polynomials · integro differential-equations · blasius
problem · error estimates
INTRODUCTION paper is the implementation of Modified Variation of Parameters Method (MVPM) for solving for solving The integro-differential equations are of great second-order integro-differential equations and coupled significance and are the governing equations of system of integro-differential equations.. The Modified diversified nonlinear physical problems related to Variation of Parameter Method (MVPM) is applied physics, astrophysics, magnetic dynamics, water without any discretization, transformation or restrictive surface, gravity waves, ion acoustic waves in plasma, assumptions. The suggested method is free from round electromagnetic radiation reactions, engineering and off errors and calculation of the so-called Adomian’s applied sciences [1-21]. Several techniques including polynomials. The numerical results are very decomposition, homotopy perturbation, polynomial and encouraging. non polynomial spline, Sink Galerkin, perturbation, homotopy analysis, finite difference and modified VARIATION OF PARAMETERS variational iteration have been employed to solve such METHOD (VPM) problems [1-21] and the reference therein. Most of these used schemes are coupled with the inbuilt Consider the following second-order partial deficiencies like calculation of the so-called Adomian’s differential equation polynomials, identification of Lagrange multiplier and non compatibility with the physical nature of the ytt=f(t,x,y,z,y,y,y,xyzyxx,yyy,y)zz (1) problems. These facts motivated to suggest alternate techniques such as the Variation of Parameters Method where t such that (-¥ Corresponding Author: Syed Tauseef Mohyud-Din, HITEC University Taxila Cantt., Pakistan 1139 World Appl. Sci. J., 6 (8): 1139-1146, 2009 ¶B ¥ = f u=piuu=+pu+++p23upu (5) ¶t å i0132 L and hence i0= t A(x,y,z,t) =-D(x,y,z) òsfds if p®1, then (5) corresponds to (3) and becomes the 0 t approximate solution of the form, B(x,y,z,t) =-C(x,y,z) ò fds 0 ¥ therefore, f==limuui (6) p1® å i0= y(x,y,z,t) =+y( x,y,z,0) tyt (x,y,z,0) t It is well known that series (6) is convergent for +-tsf(s,x,y,z,y,yy,y,y,y)ds ò( ) xyzxxyyzz most of the cases and also the rate of convergence is 0 dependent on L (u); [3-7]. We assume that (6) has a which can be solved iteratively as [8-10, 15-17] unique solution. The comparisons of like powers of p give solutions of various orders. In sum, according to k1+ [3], He’s HPM considers the nonlinear term N(u) as: y(x,y,z,t) =+y(x,y,z,0) tyt ( x,y,z,0) t +-(ts)f(s,x,y,z,ykkkkkk,y,y,y,y,y)ds ¥ ò xyzxxyyzz N(u)=pi2HH=+pH++pH... 0 å i01 2 i0= k= 0,1,2, . L HOMOTOPY PERTURBATION METHOD where Hn’s are the so-called He’s polynomials [3], (HPM) AND HE’S POLYNOMIALS which can be calculated by using the formula n n To explain the He’s homotopy perturbation 1 ¶ æöi H(u,,u)==N(pu),n0,1,2, n0……nin ç÷å method, we consider a general equation of the type, n!p¶ èøi0= p0= L (u) = 0 (2) NUMERICAL APPLICATIONS where L is any integral or differential operator. We define a convex homotopy H (u, p) by In this section, we apply the Modified Variation of Parameters Method (MVPM) for solving second-order H(u,p)=(1-+p)F(u)pL(u) (3) integro-differential equations and coupled system of integro-differential equations. Numerical results are where F (u) is a functional operator with known very encouraging. solutions n0, which can be obtained easily. It is clear that, for Example 4.1: Consider the following second-order system of nonlinear integro-differential equations H(u,p)0= (4) x we have 1112 u(x¢¢¢)=1-x3-( v) ++(u22(t)v(t))dt H(u,0)= F(u) 322ò 0 1 x ¢¢ 222 H(u,1)= L(u) v(x)=-1+x-xu(x)+-ò (u(t)v(t))dt 4 0 This shows that H (u, p) continuously traces an with initial conditions implicitly defined curve from a starting point H(n0, 0) to a solution function H (f, 1). The embedding u(0)=1,u(0¢¢)=2,v(0)=-=1,v(0)0 parameter monotonically increases from zero to unit as the trivial problem F (u) = 0 is continuously deforms the original problem L (u) = 0. The embedding The exact solutions for this problem is parameter pÎ(0,1] can be considered as an expanding parameter [3-7, 11-21]. The homotopy perturbation u(x)x=+exx,v(x)xe=- method uses the homotopy parameter p as an expanding parameter [3-7] to obtain Applying Variation of Parameters Method (VPM) 1140 World Appl. Sci. J., 6 (8): 1139-1146, 2009 Table 1: Error estimates *Errors ------x U V -1.0 2.16E-03 8.74E-04 -0.8 4.80E-04 1.85E-04 -0.6 6.87E-05 2.49E-05 -0.4 4.35E-06 1.46E-06 -0.2 3.71E-08 1.15E-08 0.0 0.00000 0.00000 0.2 4.50E-08 1.10E-08 0.4 6.27E-06 1.42E-06 0.6 1.17E-04 2.38E-05 0.8 9.69E-04 1.73E-04 1.0 5.06E-03 7.89E-04 *Error = Exact solution-series solution xx ì æöæö1112 u(x)=u(x)+x-s1-x3-v¢ ++u22(t)v(t)ds ï n+1nòò( )ç÷ç÷( n) ( nn) ï 00èøèø322 í xxæö1 ïv(x)=v(x)+-xs-1+x2-xu(x)+-u22(t)v(t)ds ï n+ 1nòò( )ç÷( n) ( nn) î 00èø2 Applying the Modified Variation of Parameters Method (MVPM) xx ì æöæö1112 u+pu+=1+2xpx+-s1-x3-v¢¢+pv++u2+pu2++++v22pvds ï 01Lòò( )ç÷ç÷( 01L) (( 01LL) ( 01)) ï 00èøèø322 í xxæö1 ïv+pv+=-1px+-s-1+x2-xu+pu++u2+pu2+-++v22pvds ï 01Lòò( )ç÷( ( 01L) ) (( 01LL) ( 01)) î 00èø2 Comparing co-efficient of like powers of p, following approximations are obtained (0) ì:u0 (x)1=+2x p:í î v(x0 )1=- ì 11234511 ïu1 (x)x=+xxx++ (1) ï 261260 p:í 1111 ï v(x)x=--2xxx345-+ îï 1 262460 ì 1114561717895311017 11 u(x)xx=--+x+x+xx+-+xx ï 2 241207205040672120960103680228096 ï (2)ï 111213 p:í ++xx M. ï 19008006177600 ï 115611791317104711112 ï v2 (x)=-xx-x+x+x++xx î 120720100802419201036800114048001267200 The series solution is given by ì 1111234511145617718953 u(x)1=+2xx++xxx++-xx-+x+x++xx ï 261260241207205040672120960 ï ï 1101711111213 í -+xxxx+++L ï 10368022809619008006177600 ï 11234115115611791317 104711112 ï v(x)1=---xxx-+xx-x-xx++x+xx++L î 262460120720100802419201036800 114048001267200 1141 World Appl. Sci. J., 6 (8): 1139-1146, 2009 Fig. 1: Fig. 2: Example 4.2: Consider the following second-order system of nonlinear integro-differential equations x u(x¢¢)x=+2x32+2v(x( ¢¢))22-+ò ((v(t)) u(t)w(t)v(t¢¢ ))dt 0 1 x v(x¢¢)=3x2 -xu(x)+-ò (txv(t)u(t¢¢¢¢)w(t))dt 4 0 x 4 22 w(x¢¢)=2x-3+(u¢¢(x)) -2u2(x)+ò xv( 23(t)++(u(t¢)) tw¢¢(t))dt 3 0 with initial conditions u(0)=1,u(0¢¢)=0,v(0)====0,v(0)1,w(0)0,w(0)0 The exact solutions for this problem is u(x)===x,v(x22)x,w(x)3x Applying Variation of Parameters Method (VPM) ì xxæö u(x)=u(x)+x-sx+2x32+2v¢¢(x)22-+v(t)u(t)w¢¢ (t)v(t)ds ï n+1nòò( )ç÷( ( n) ) ( n) nn ï 00èø ï ï xxæö1 v(x)=v(x)+xs--3x2 xu(x)+-txv¢(t)u¢¢¢(t)w(t))ds í n+ 1nòò( )ç÷( n) (( nnn) ) ï 00èø4 ï xx æö4 2 2 ïw(x)=w(x)+( x-s2x) æö-3+--(u¢¢ (x)) 2u22(x)xv(t)++(u¢(t)) tw3 ¢¢ (t)ds ï n+ 1nòòç÷ç÷nnn( nn) î 00èøèø3 Applying the Modified Variation of Parameters Method (MVPM) ì xæöxx 3 ¢¢2¢¢22¢¢¢¢ ïu0++pu1L=1px+-ò( s)ç÷(x++++2x2(v0pv1L) ) +òò(v0+pv1+L) ds-(xsu-)(( 0+pu1+L)(v0+pv1+LL) (w01++pw))ds ï 0èø00 ï xxx ï æö21 22 ív0+pv1+=xpxs+( -)ç÷3x-x(u0+pu1+) +sx(v¢0+pv¢1+) u0+pu1+(u¢0¢+pu¢1¢+) dspxsw+-( )( ¢¢01++pw)ds Lòç÷( L) 4 òò( L( L) LL) ï 0èø00 ï xxx æöæö4 2222 ïw+pw+=-pxs2xuç÷-33+++¢¢pu¢¢-2u+pu+-xv+pv+ds--pxxsu¢++pu¢+tw¢¢++pw¢¢ ds ï 01Lò( )ç÷ç÷( 01L) ( 01L) òò( 01L) ( )(( 01LL) ( 01)) î 0èøèø3 00 Comparing co-efficient of like powers of p, following approximations are obtained 1142 World Appl. Sci. J., 6 (8): 1139-1146, 2009 Table 2: Error estimates *Errors ------x U V W -1.0 2.33E-02 3.48E-02 3.48E-01 -0.8 4.79E-03 1.25E-03 1.25E-01 -0.6 5.77E-04 3.12E-04 3.12E-02 -0.4 2.83E-05 4.17E-05 4.17E-03 -0.2 1.72E-07 1.29E-07 1.29E-04 0.0 0.00000 0.00000 0.00000 0.2 7.20E-08 1.26E-08 1.26E-04 0.4 2.80E-06 4.00E-06 4.00E-03 0.6 7.15E-05 3.07E-05 3.07E-02 0.8 1.64E-03 1.34E-03 1.34E-01 1.0 1.53E-02 4.37E-02 4.37E-01 *Error = Exact solution-series solution Fig. 3: ì:u0 (x)1= (0) ï p:í v0 (x)x= ï îw0 (x)0= ì 251 u(x)=+xx ï 1 10 ï (1)4ï 1 p:í v1 (x)x= ï 4 ï 11 w(x)xxx256 ï 1 =-+ î 1560 ì 11568197 u(x)=x-+xx ï 2 6605040 ï 1117 ï -x9+-xx1112 ï 336742526400 ï (2)ï 1147411811 p:í v(x2 )x=--x-+xx 126303360440 ï Fig. 4: ï 161389227 ïw2 (x)x=-+-xx ï 2016830240 ï 111 ++-xxx101112 îï 144039606600 M. The series solution is given by ì 25511168197 u(x)1=+x++-+xxxx ï 106605040 ï 1117 ï -x9+xx11-+12 ï 336742526400 L ï ï 141417411811 í v(x)=x+x-x-x-++xxL ï 4126303360440 ï 11113227 w(x)=xxx256-+-x6+-xx89 ï 15602016830240 ï ï 111101112 ++-+xxxL îï 144039606600 Fig. 5: 1143 World Appl. Sci. J., 6 (8): 1139-1146, 2009 Table 3: Pade´ approximants and numerical value of a Pade´ approximant a [2/2] 0.5778502691 [3/3] 0.5163977793 [4/4] 0.5227030798 Fig. 7: a = 0.5163977793 Fig. 6: a = 0.5778502691 Example 4.3: Consider the two dimensional nonlinear inhomogeneous initial boundary value problem for the integro differential equation related to the Blasius problem Fig. 8: a = 0.5227030798 1 x y(x¢¢)=a-y(t)y(t)dt¢¢ ,-¥< Applying the Modified Variation of Parameters Method (MVPM) x 22xx y0+py1+py2+=x+p(x-s) ( y0+py1+)(y¢0¢+py12¢¢++py¢¢ ) ds Lòòò00( LL) 0 Proceeding as before, the series solution is given as 1241112561127æö1131843 29 y(x)=+axx-axx-a+ax+ax+ç÷a+axx-a 24824096020160èø161280960967680 æ153ö10æ587254ö11æö11312 +ça-a÷xxx+ça-a÷+ç÷-a+a+ è52960387072øè2128896004257792øèø162201607257792 L and consequently 1144 World Appl. Sci. J., 6 (8): 1139-1146, 2009 13112451126æö1137143 28 y(x¢ )=1+ax-a-axx+ax+axç÷a-axx-a 12481602880èø201602688107520 æ153ö9æ587254ö10æö11311 +10ça-a÷x+11ça-a÷x+12xç÷-a+a+ è552960387072øè2128896004257792øèø16220160725760 L The diagonal Pade´ approximants can be applied to nanomechanics in textile engineering. Int. J. Mod. determine a numerical value for the constant a by Phys., B22 (21): 3487-4578. using the given condition. 5. He, H., 2008. Recent developments of the homotopy perturbation method. Top. Meth. Nonlin. CONCLUSION Anal., 31: 205-209. 6. He, H., 2006. Some asymptotic methods for n this paper, we applied Modified Variation of strongly nonlinear equation. Int. J. Mod. Phys., Parameters Method (MVPM) for solving integro- 20 (10): 1144-1199. differential equations and coupled systems of integro- 7. He, H., 2004. Comparison of homotopy differential equations. The proposed technique is perturbation method and homotopy analysis method. Appl. Math. Comput., 156: 527-539. employed without using linearization, discretization or 8. Ma, X. and Y. You, 2004. Solving the Korteweg- restrictive assumptions. Moreover, the suggested de Vries equation by its bilinear form: Wronskian method is free from round off errors, calculation solutions. Transactions of the American of the so-called Adomian’s polynomials and mathematical society, 357: 1753-1778. identification of Lagrange multiplier. It may be 9. Ma, W.X. and Y. You, 2004. Rational solutions of concluded that the MVPM is very powerful and the Toda lattice equation in Casoratian form. efficient in finding the analytical solutions for a wide Chaos, Solitons and Fractals, 22: 395-406. class of boundary value problems and can be 10. Ma, W.X., C.X. Li and J.S. He, 2008. A second considered as an alternative for solving nonlinear Wronskian formulation of the Boussinesq initial and boundary value problems. equation. Nonlin. Anl. Th. Meth. Appl., 70 (12): 4245-4258. ACKNOWLEDGMENT 11. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, 2009. Travelling wave solutions of seventh-order he authors are highly grateful to Dr S. M. Junaid generalized KdV equations using He’s Zaidi, Rector CIIT for providing excellent research polynomials . Int. J. Nonlin. Sci. Num. Sim., 10 (2): environment and facilities. The first author would like 223-229. to thank to Brig (R) Qamar Zaman, Vice Chancellor 12. Mohyud-Din, S.T. and M.A. Noor, 2009. HITEC University for the provision of very conducive Homotopy perturbation method for solving partial environs for research. differential equations. Zeitschrift für Naturforschung A, 64a. 13. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, REFERENCES 2009. Some relatively new techniques for nonlinear problems . Math. Prob. Eng. 1. Abbasbandy, 2006. The application of homotopy 14. Mohyud-Din, S.T. and M.A. Noor, 2007. analysis method to nonlinear equations arising in Homotopy perturbation method for solving heat transfer. Phys. Lett., A360: 109-113. fourth-order boundary value problems . Math. 2. Abdou, A. and A.A. Soliman, 2005. Variational Prob. Engg., Article ID 98602, doi:10.1155/2007/ iteration method for solving Burger’s and 98602, pp: 1-15. coupled Burger’s equations. J. Comput. Appl. 15. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, Math., 181: 245-251. 2009. Modified variation of parameters method 3. Ghorbani and J.S. Nadjfi, 2007. He’s homotopy for solving partial differential equations. Int. J. perturbation method for calculating Adomian’s Mod. Phys. B. polynomials . Int. J. Nonlin. Sci. Num. Simul., 16. Mohyud-Din, S.T., M.A. Noor, K.I. Noor and A. 8 (2): 229-332. Waheed, 2009. Modified variation of parameters 4. He, H., 2008. An elementary introduction of method for solving nonlinear boundary value recently developed asymptotic methods and problems . Int. J. Mod. Phys. B. 1145 World Appl. Sci. J., 6 (8): 1139-1146, 2009 17. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, 2009. Ma’s variation of parameters method for 20. Noor, M.A. and S.T. Mohyud-Din, 2008. Modified nonlinear oscillator differential equations. Int. J. variational iteration method for Goursat Mod. Phys. B. and Laplace problems . Wd. Appl. Sci. J., 4 (4): 487-498. 18. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, 21. Noor, M.A., S.T. Mohyud-Din and M. Tahir, 2008. 2008. Comparison and coupling of polynomials Modified variational iteration methods for with correction functional for travelling wave Thomas-Fermi equation. Wd. Appl. Sci. J., 4 (4): solutions of seventh order generalized KdV 479-498. equations, Wd. Appl. Sci. J. 19. Mohyud-Din, S.T., M.A. Noor and K.I. Noor, 2009. Solving second-order singular problems using He’s polynomials . Wd. Appl. Sci. J., 6 (6): 769-775. 1146