Approximating Solutions for Ginzburg Landau Equation by HPM and ADM

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1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol. 5, No. Issue (December 1), pp (Previously, Vol. 5, Issue 1, pp ) Applicaios ad Applied Mahemaics: A Ieraioal Joural (AAM) Approximaig Soluios for Gizburg Ladau Equaio by HPM ad ADM J. Biazar, M. Parovi, Z. Ayai Deparme of Mahemaics Faculy of Sciece Uiversiy of Guila P.O. Box P.C Rash, Ira jafar.biazar@gmail.com Received: Jauary 1, 1; Acceped: December 1, 1 Absrac I his paper, a aalyical approximaio o he soluio of Gizburg-Ladauis discussed. A Homoopy perurbaio mehod iroduced by He is employed o derive he aalyic approximaio soluio ad resuls compared wih hose of he Adomia decomposiio mehod. Two examples are preseed o show he capabiliy of he mehods. The resuls reveal ha he mehods are almos equally effecive ad promisig. Keywords: Gizburg-Ladau equaio; Homoopy perurbaio mehod; Adomia decomposiio mehod. MSC (1) No.: 35c1, 65M7 1. Iroducio I his paper we cosider he Gizburg-Ladau equaio i he form iu u u u bu iau, (1) xx 575

2 576 J. Biazar e al. where u is a complex value fucio. Origially discovered by Gizburg ad Ladau (1965), for a phase rasiio i supercoduciviy, he equaio has bee exeded o various fields such as chemical reacios, fluid mechaics ad paer formaio, o meio jus a few. For a deailed iformaio o his equaio, see, [Bechouche ad Jugel ()] ad refereces herei. I he case ab, he equaio reduces o he famous o-liear Schrödiger equaio, [Biazar ad Ghazvii (7)]. The homoopy perurbaio mehod was iroduced by He [(1999), (4), (6)]. I his mehod he soluio assumed o be he summaio of a ifiie series which coverges o he soluio. Usig a echique of opology, a homoopy is cosruced wih a embeddig parameer p [,1] cosidered a small parameer. Cosiderable research has bee coduced o he applicaio of his mehod o a class of liear ad o-liear equaios, [Abbasbady (7), Sadighi ad Gaji (8)]. This mehod has also bee used o solve hyperbolic differeial equaios [Biazar ad Ghazvii (8)], ad oher equaios, [Biazar ad Ghazvii (9, 8)]. Here we exed he mehod o solve he Gizburg-ladau equaio.. Basic Idea of Homoopy Perurbaio Mehod To illusrae he basic ideas of he mehod, we cosider he followig o-liear differeial equaio A( u) f ( r), r, () wih he followig boudary codiios u B u,, r, Where A is a geeral fucioal operaor, B is a boudary operaor, f ( r ) is a kow aalyical fucio, ad is he boudary of he domai. The operaor A ca be decomposed io wo operaors, L ad N, where L is a liear, ad N is a o-liear operaor. Hece, Equaio () ca be rewrie as follows (3) Lu ( ) N( u) f( r). (4) We cosruc a homoopy U ( r, p) : [,1], which saisfies H ( U, p) (1 p)[ L( U) L( u )] p[ A( U) f ( r)], p[,1], r, (5) or

3 AAM: Ier. J., Vol. 5, Issue (December 1) [Previously, Vol. 5, Issue 1, pp ] 577 H( U, p) L( U) L( u ) pl( u ) p[ N( U) f( r)], (6) where p [,1], is a embeddig parameer, u is a iiial approximaio for he soluio of Equaio (), which saisfies he boudary codiios. Obviously, from Equaios (5) ad (6) we will have HU (,) LU ( ) Lu ( ), HU (,1) AU ( ) f( r). (7) The chagig process of p from zero o uiy is jus ha of U ( r, p ) from u ( r ) o u( r ). I opology, his is called homoopy. Accordig o he HPM, we ca firs use he embeddig parameer p as a small parameer, ad assume ha he soluio of Equaio (5) ad (6) ca be wrie as a power series i p U U pu p U (8) 1. Seig p 1, resuls i he approximae soluio of Equaio () u lim U U U U. (9) p 1 1 I is worh meioig ha if k, u, k s are all zero a exac soluio is is coverge uder some esablished crieria [He (199)] for mos oher cases. u U. The series (9) k k 3. Mehods of Soluio 3.1. The Homoopy perurbaio Mehod Applied o Gizburg-Ladau equaio Cosider he followig Gizburg-ladau equaio wih he followig iiial codiio (, ) ux (, ) ux i u u bu iau, x u( x,) u ( x), x, (1) where i ad are wo real cosas. To solve Equaio (1) by homoopy perurbaio mehod, we cosruc he followig homoopy

4 578 J. Biazar e al. or U (1 )( u U U p ) p( i( u u bu iau)), x U u u U p( i( u u bu iau)). x (11) Suppose he soluio of Equaio (1) o be i he followig form U U pu p U (1) 1. Subsiuig (1) io (11), ad equaig he coefficies of he erms wih he ideical powers of, p leads o he followig p : U u, 1 p : U u 1 U i( U U bu iau ), U1( x,), x p : U U 1 i( ( U U U 1U 1U ) bu 1iaU 1), U ( x,), x 3 p : U 3 U i( ( U U U U U U U1U1U1U1U ) bu iau), x U ( x,), 3 U j U j p : i( ( U U U ) bu iau ), U j ( x,), j1 j i 1 j1 i k jki1 j1 j1 x i k (13) where i p j, here are he muliplicaio of wo series U ad U. For he sake of he simpliciy we ake U( x, ) u ( x, ) u ( x). (14) Havig his assumpio we ge he followig ieraive equaio U U i ( ( U U U ) bu iau ) d, j 1,,3,. (15) j 1 ji 1 j 1 j i k jk i 1 j1 j1 x i k The approximae soluio of (1) ca be obaied by seig p 1,

5 AAM: Ier. J., Vol. 5, Issue (December 1) [Previously, Vol. 5, Issue 1, pp ] 579 u lim U U U U. p 1 1 The resuls of he followig examples are compared wih he resuls of he origial ADM, They seem o be i good agreeme, as expeced, [Biazar, Ayai ad Ebrahimi (8)]. I is worh o oig ha i was Wazwaz ha iroduced a reliable modificaio of he ADM, which acceleraes he covergece of he series soluio. 3.. The Adomia Decomposiio Mehod Applied o Gizburg-Ladau Equaio Cosider he followig Gizburg-ladau equaio wih he followig iiial codiio iu u u u bu iau, xx u( x,) u ( x), x, where ad are wo real cosas. Pay aeio o iiial codiios operaor L. Therefore, we have u i( u u u bu iau). (16) xx The iverse operaor of L is L 1. Applyig he iverse operaor (.) d (16), we ge u ux (, ) ux (,) i ( u ubuiaud ), x or 1 L o boh sides of u (, ) (,) ( ). x ux ux i u u bu iaud (17) To solve Equaio (17) by Adomia decomposiio mehod we cosider, as usual i his mehod, he series soluio u iegrad o he righ side is he sum of a series. u. So ha he compoes u ca be deermied recursively. The u u A, (18) where A ( u, u1,, u ) called he Adomia polyomials are compued usig mehods iroduced i [Wazwaz ()]. We have

6 58 J. Biazar e al. u x u u( x,) i ( A b u ia u ) d, (19) which i ur, yields: u ( x, ) u( x,) u u 1( x, ) i ( A ),,1,,. bu iau d x () Resulig i he followig approximaios for he Adomia polyomials, A u u, A u u u u u, A u u u u u u u u u u, A u u u u u u u u u u u u u u u u, A u u u u u u u u u u u u u u u u u uu 1 uuu 1 3uu 4, From () we have u ( x, ) u( x,) u u1( x, ) i ( u u bu iau ) d, x u u ( x, ) i ( ( u u u u u ) bu iau ) d, x u u ( x, ) i ( ( u u u u u u u u u u ) bu iau ) d, x u u ( x, ) i ( ( u u u u u u u u u u u u u u u u ) x bu iau ) d, 3 3

7 AAM: Ier. J., Vol. 5, Issue (December 1) [Previously, Vol. 5, Issue 1, pp ] 581 u u x i u u u u u u u u u u u u u u u u u u u 4 5(, ) ( ( x u u u u u ) bu iau ) d, We ca deermie he compoes u as far as we like o ehace he accuracy of he approximaio. 4. Examples To illusrae he mehods ad o demosrae heir capabiliy, wo examples are preseed. Example 1. Cosider he followig parial differeial equaio u u i u u u iu,, x ix ux (,) e. (1) We cosruc a homoopy [,1] which saisfies U (1 )( u U U p ) p( i( u u u iu)). x From (14), (15) we have he followig scheme ix u u( x,) e, U u i ( ( U U U ) U iu ) d, j 1,,3,... j 1 ji 1 j 1 j i k jk i 1 j1 j1 x i k For he firs few j, we derive 1 ix u1( x, ) e, 1! 1 1 u x e i e! 6 ix ix (, ),

8 58 J. Biazar e al. 1 3 u x e i e 3! 9 3 ix 3 ix 3(, ), u x e i e e 4! ix 4 ix 4 ix 4(, ), u x e i e e 5! ix 5 ix 5 ix 5(, ), These approximaios are preseed as follows u x e ie e e! i x i x i x 5 i x (, ) ( 1) ( ) 4. Example. Cosider he followig parial differeial equaio u u i u u u iu,, x ix ux (,) e. From () we obai ix u e u u 1( x, ) i ( A ),,1,,. u iu d x For he firs few, we have 1 ix u1( x, ) e, 1! 1 1 u x e i e! 6 ix ix (, ), 1 3 u x e i e 3! 9 3 ix 3 ix 3(, ), u x e i e e 4! ix 4 ix 4 ix 4(, ), u x e ie e 5! ix 5 ix 5 ix 5(, ),

9 AAM: Ier. J., Vol. 5, Issue (December 1) [Previously, Vol. 5, Issue 1, pp ] 583 These approximaios are preseed as follows u x e ie e e! i x i x i x 5 i x (, ) ( 1) ( ) Coclusios I his paper, he homoopy perurbaio mehod is proposed for solvig o-liear Gizburg- Ladau equaios. This mehod reveals ha soluio are exacly he same as hose obaied by Adomia decomposiio mehod, which has o overcome he difficulies i he calculaio of he Adomia, s polyomials, as demosraed i [Biazar, e al (8)]. As was poied ou, here is a reliable modificaio of he ADM, which acceleraes he covergece. Compariso of kow mehods, such as HPM, HAM, VIM, wih MADM [Wazwaz (1999)] is o-goig. The aalyical approximaio o he soluio is clearly more reliable ad cofirms he power ad capabiliy of He, s homoopy perurbaio mehod as a easie procedure for obaiig he soluio of oliear equaios. Compuaios i his work were performed by usig Maple 11. REFERENCES Abbasbady, S. (7). Applicaio of He, s homoopy perurbaio mehod o fucioal iegral Equaios, Chaos, solios ad Fracals, 31, Bechouche, P. ad Jugel, A. (). Iviscid limis of he complex Gizburg Ladau equaio, Commu. Mah. Phys. 14,1-6. Biazar, J. ad Ghazvii, H. (7). Exac soluios for o-liear Schrödiger equaios by He s homoopy perurbaio mehod, Physics Leers, A 366, Biazar, J. ad Ayai, Z. ad Ebrahimi, H. (8). Comparig Homoopy Perurbaio Mehod ad Adomia Decomposiio Mehod, Numerical Aalysis Ad Applied Mahemaics, 148, Biazar, J. ad Ghazvii, H. (8). Homoopy perurbaio mehod for solvig hyperbolic parial differeial equaios, Compuers ad Mahemaics wih Applicaios, 56, Biazar, J. ad Ghazvii, H. (8). Numerical soluio for special o-liear Fredholm iegral equaio by HPM, Applied Mahemaics ad Compuaio. 195, Biazar, J. ad Ghazvii, H. (9). He s homoopy perurbaio mehod for solvig sysem of Volerra iegral equaios of he secod kid, Chaos, Solios ad Fracals, 39, Giibre, J. ad Velo, G. (1996). The Cauchy problem i local spaces for he Complex Gizburg- Ladau Equaio, l. Compacess mehods, Physica D: No-liear Pheomea. 95, Gizburg, V. ad Ladau, L. (1965). O he heory of supercoduciviy. Zh Eksp Fiz 195; : 164; Eglish rasl. I: Ladau LD, Ter Haar D, ediors, Me of physics, vol. I. New York: Pergamo Press; P

10 584 J. Biazar e al. He, J. H. (4). Asympology by homoopy perurbaio mehod, Applied Mahemaics ad Compuaio 156 (3), He, J. H. (4). The homoopy perurbaio mehod for o-liear oscillaors wih discoiuiies, Applied Mahemaics ad Compuaio, 151, He, J. H. (6). New ierpreaio of HPM. I J. Mod. Phys. B,, He, J.H. (1999) Homoopy perurbaio echique, Compuer Mehods i Applied Mechaics ad Egieerig, 178, He, J.H. (5). Homoopy perurbaio mehod for bifurcaio of o-liear problems, Ieraioal Joural of No-liear Sciece Numerical Simulaio, 6, 7-8. Sadighi, A. ad Gaji, D.D. (8). Aalyic reame of liear ad oliear Schrödiger equaios: A sudy wih Homoopy perurbaio ad Adomia Decomposiio mehods, Physics Leers, A 37, Wazwaz, A.M. (1999). A reliable modificaio of Adomia decomposiio mehod, Applied operaors, Applied Mahemaics ad Compuaio, 111(1), Wazwaz, A.M. (). A ew algorihm for calculaio Adomia polyomials for o-liear Mahemaics ad Compuaio, 1,