Variational Iteration Method for Solving Volterra and Fredholm Integral Equations of the Second Kind

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1 Ge. Mth. Notes, Vol. 2, No. 1, Jury 211, pp ISSN ; Copyright ICSRS Publictio, Avilble free olie t Vritiol Itertio Method for Solvig Volterr d Fredholm Itegrl Equtios of the Secod Kid Mehdi Gholmi Porshokouhi, Behzd Ghbri Deprtmet of Mthemtics, Fculty of sciece, Islmic Azd Uiversity, Tkest Brch, Ir Emil: m_gholmi_p@yhoo.com, b.ghbry@yhoo.com Mjid Rshidi Deprtmet of Agriculturl Mchiery, Fculty of Agriculture, Islmic Azd Uiversity, Tkest Brch, Ir Emil: mjidrshidi81@yhoo.com (Received , Accepted ) Abstrct I this pper, vritiol itertio method (VIM) is used to give the pproimte solutio of Volterr d Fredholm itegrl equtios of the secod kid. The method costructs coverget sequece of fuctios, which pproimtes the ect solutio with few itertios. To illustrte the bility d relibility of the method, some emples re give, revelig its effectiveess d simplicity. Keywords: Vritiol itertio method; Volterr d Fredholm itegrl equtios 2 MSC No: 47G2 1 Itroductio Let u ( ) is ukow fuctios, f ( ) is give kow fuctio, d (, ) kow itegrl kerel. k t

2 Mehdi Gholmi Porshokouhi et l 144 The Volterr itegrl equtio of the secod kid is itegrl equtio of the form ( ) ( ) ( ) ( ) = + ( 1 ) u f k, t u t dt, Ad the Fredholm itegrl equtio of the secod kid is itegrl equtio of the form b ( ) ( ) ( ) ( ) u f k, t u t dt, = + ( 2 ) The vritiol itertio method is ew method for solvig lier d olier problems d ws itroduced by Chiese mthemtici, He [1-3]. I [4] He modified the geerl Lgrge multiplier method [5] d costructed itertive sequece of fuctios which coverges to the ect solutio. I most lier problem the Lgrge multiplier, the pproimte solutio turs ito the ect solutio d is vilble with just oe itertio. To illustrte the method, cosider the followig geerl fuctiol equtio ( ) + ( ) = ( ), ( 3 ) Lu N g Where L is lier opertor, N is o-lier opertor d g ( t ) is kow lyticl fuctio. Accordig to the vritiol itertio method, we c costruct the followig correctio fuctiol { } ( ) ( ) λ ( ξ ) ( ξ ) % ( ξ ) ( ξ ) ξ, ( 4 ) u + = u + Lu + Nu g d 1 Where λ is geerl Lgrge multiplier which c be idetified optimlly vi vritiol theory, u is iitil pproimtio with possible ukows, d u% is cosidered s restricted vritio, i.e., δ % =. 2 Solutio of the Volterr d Fredholm Itegrl Equtio of the Secod Kid Cosider the Volterr d Fredholm itegrl equtio of the secod kid 2 : give i Eqs. ( 1) d ( ) For Eqs. ( 1) d ( ) u 2 first we tke the prtil derivtive with respect to. For the Volterr itegrl equtio of the secod kid we hve

3 145 Vritiol itertio method for solvig d u f k t u t dt d d for the Fredholm itegrl equtio of the secod kid we hve Cosider d ( ) = ( ) + (, ) ( ), ( 5 ) b ( ) ( ) ( ) ( ) u f k, t u t dt, = + ( 6 ) k (, t ) u ( t ) dt, d b d (, ) ( ) k t u t dt, s restricted vritio; we use the vritiol itertio method i directio. The we hve the followig itertio sequece: d ξ u+ 1 ( ) = u ( ) + λ ( ξ ) u ( ) ( ) (, ) ( ), ξ f ξ k ξ t u dξ Tkig the with respect to the idepedet vrible u d oticig tht δ =, we get u ( ) ( 7 ) ( 8 ) + 1 ξ = δu = δu + λδu λ δu dξ =. The we pply the followig sttiory coditios: ( ) λ ( ξ ) 1+ λ ξ =, = ξ = ξ = The geerl Lgrge multiplier, therefore, c be redily idetified: λ = 1, ( 9 ) Ad s result, we obti the followig itertio formul d ξ u+ 1 ( ) = u ( ) u ( ) ( ) (, ) ( ), ξ f ξ k ξ t u dξ 3 Numericl Emples ( 1 ) I this sectio, we pplied the method preseted i this pper to two emples to show the efficiecy of the pproch.

4 Mehdi Gholmi Porshokouhi et l 146 Emple1. Cosider the lier Volterr itegrl equtio ( ) = cos si + 2 si( ) ( ) ( 11 ) u t u t dt The lyticl solutio of the bove problem is give by, ( ) = ep( ). ( 12 ) u I the view of the vritiol itertio method, we costruct correctio fuctiol i the followig form: d ξ u+ 1 ( ) = u ( ) u ( ) si cos 2 { si( ) ( )}, ξ + ξ + ξ ξ t u dξ ( 13 ) Strtig with the iitil pproimtio y cos si 13 successive pproimtios ui ( ) s will be chieved. The plot of ect solutio Eq.( 11 ), the 5th order of pproimte solutio obtied usig the VIM d bsolute error betwee the ect d umericl solutios of this emple re show i Fig. 1. = i Eq. ( ) Fig. 1. The plots of pproimte solutio, ect solutio d bsolute error for Emple 1. Emple 2. Cosider the lier Fredholm itegrl equtio 7 1 u t u t dt ( ) = + ( ) ( 14 ) The lyticl solutio of the bove problem is give by, u ( ) =. ( 15 )

5 147 Vritiol itertio method for solvig I the view of the vritiol itertio method, we costruct correctio fuctiol i the followig form: u+ 1 ( ) = u ( ) u ( ) ( ), ξ t u 8 2 ( 16 ) 7 Strtig with the iitil pproimtio y = i Eq. ( 16 ) successive 8 pproimtios ui ( ) s will be chieved. The plot of ect solutio Eq.( 14 ), the 5th order of pproimte solutio obtied usig the VIM d bsolute error betwee the ect d umericl solutios of this emple re show i Fig. 2. Fig. 2. The plots of pproimte solutio, ect solutio d bsolute error for Emple 2. 4 Coclusio I this pper the vritiol itertio method is used to solve the Volterr d Fredholm itegrl equtios. The results showed tht the covergece d ccurcy of the vritiol itertio method for umericlly lyzed the Volterr d Fredholm itegrl equtios were i good greemet with the lyticl solutios. The computtios ssocited with the emples i this pper were performed usig mple 13. Refereces [1] J.H. He, A ew pproch to lier prtil differetil equtios, Commu. Nolier Sci. Numer. Simul. 2 (4) (1997)

6 Mehdi Gholmi Porshokouhi et l 148 [2] J.H. He, Approimte solutio of olier differetil equtio with covolutio product olierities, Comput. Methods Appl. Mech. Egrg., 167(1998), [3] J.H. He, Some pplictios of olier frctiol differetil equtio d their pproimtios, Bull. Sci. Techol., 15(12)(1999), [4] J.H. He, Vritiol itertio method kid of olier lyticl techique: Some emples, It. J. No-lier Mech., 34(1999), [5] M. Iokuti, H. Sekie d T. Mur, Geerl use of the Lgrge multiplier i olier mthemticl physics, i:s.nemt-nsser (Ed.), Vritiol Method i the Mechics of solids, Pergmo Press, New York, (1978),