SOLVING OF THE FRACTIONAL NON-LINEAR AND LINEAR SCHRÖDINGER EQUATIONS BY HOMOTOPY PERTURBATION METHOD

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1 SOLVING OF THE FRACTIONAL NON-LINEAR AND LINEAR SCHRÖDINGER EQUATIONS BY HOMOTOPY PERTURBATION METHOD DUMITRU BALEANU, ALIREZA K. GOLMANKHANEH,3, ALI K. GOLMANKHANEH 3 Deparme of Mahemaics ad Compuer Sciece, Çakaya Uiversiy, Akara, Turkey ad Isiue of Space Scieces, P.O. BOX, MG-3, RO-0775, Mãgurele Buchares, Romaia, dumiru@cakaya.edu.r Deparme of Physics, Uiversiy of Pue, Pue, 4007, Idia alireza@physics.uipue.ere.i 3 Deparme of Physics, Islamic Azad Uiversiy-Uromia Brach, PO Box 969, Uromia, Ira ali.khalili.gol@gmail.com Received December 8, 008 I his paper, he homoopy perurbaio mehod is applied o obai approximae aalyical soluios of he fracioal o-liear Schrödiger equaios. The soluios are obaied i he form of rapidly coverge ifiie series wih easily compuable erms. We illusraed he abiliy of he mehod for solvig fracioal o liear equaio by some examples.. INTRODUCTION The heory of fracioal calculus goes back o Leibiz, Liouville, Riema, Gruwald ad Leikov has bee foud may applicaio i sciece ad egieerig [ 7]. Fidig accurae ad efficie mehods for solvig fracioal o-liear differeial equaios (FNLDEs) has bee a acive research uderakig. Exac soluios of mos of he FNLDEs ca o be foud easily, hus aalyical ad umerical mehods mus be used. The Adomia Decomposiio Mehod (ADM) was show o be applicable o liear ad oliear fracioal differeial equaios [8]. However, aoher aalyic echique for oliear problems is called he homoopy perurbaio mehod, firs proposed by He [9 4]. I refs [5, 6] homoopy perurbaio mehod is used o solve liear ad oliear fracioal differeial equaios. The soluio of he Schrödiger equaio over a ifiie iegraio ierval by perurbaio mehods was give i [7]. Recely, i [8] a aalyical approximaio o he soluio of Schrödiger equaios has bee obaied. Very recely, aalyical ad approximae soluios for differe oliear Schrödiger equaio of fracioal (NLSF) order ivolvig Capuo derivaives has bee obaied by Adomia Decomposiio Mehod [9]. I he prese, we obai he aalyical Rom. Jour. Phys., Vol. 54, Nos. 9 0, P , Buchares, 009

2 84 D. Baleau, A. K. Golmakhaeh, Ali K. Golmakhaeh ad approximae soluios for fracioal oliear Schrödiger equaio usig homoopy perurbaio mehod. The pla of he paper is as follows: Secio is dedicaed o he oio of fracioal iegral ad fracioal derivaives. Secio 3 coais a brief summary of homoopy perurbaio mehod. Secio 4 deals wih usig homoopy perurbaio mehod for solvig fracioal oliear Schrödiger equaio. Fially, Secio 5 provides some examples for illusraig of he abiliy of mehod.. FRACTIONAL CALCULUS I his secio we give he defiiio of he Riema-Liouville, Capuo derivaives ad fracioal iegral wih properies. Defiiio.. Le f( x ) ad > 0 he [3, 4] f() I x f( x) d x 0 ( ) ( x ) x () 0 is called as he lef sided Riema-Liouville fracioal iegral of order. The properies of he operaor Ix are as followig [3]: x x x () I I f( x) I f( x) x x x x () I I f( x) I I f( x) (3) ( ) x ( ) x I x Defiiio.. Le ux ( ) ad, he parial Capuo fracioal derivaives is defied as: 0 D u( x) ( ) u( x) d ( ) () Noe ha m k ( 0) k ux ( ) u( x ) k k k0 I D u x (3) 3. HOMOTOPY PERTURBATION METHOD The homoopy perurbaio mehod (HPM) iroduced by He [9 4]. The combiaio of he perurbaio mehod ad he homoopy mehods. O he

3 3 Solvig of he fracioal o-liear ad liear Schrödiger equaios 85 oher had, his echique ca have full advaage of he radiioal perurbaio echiques. I his mehod he soluio is cosidered as he summaio of a ifiie series which usually coverges rapidly o he exac soluios. I his secio, basic ideas of his mehod has bee explaied. Le us cosider he followig geeral o-liear differeial equaio wih boudary codiios Au ( ) f( r) 0 r (4) u 0 B u r where A is a geeral differeial operaor, B a boudary operaor, f( r ) is a kow aalyical fucio ad is he boudary of he domai. The operaor A ca be geerally divided io liear (L) ad o liear (N) pars. Therefore Eq. (4) ca be wrie as follows: (5) Lu ( ) Nu ( ) fr ( ) 0 (6) Usig he homoopy echique, we cosruc a homoopy Ur ( p) [0], which saisfies: HU ( p) ( p)[ LU ( ) Lu ( )] pau [ ( ) f( r)] 0 p[0] r (7) or 0 HU ( p) LU ( ) Lu ( ) plu ( ) pnu [ ( ) f( r)] 0 (8) 0 0 where p[0 ] is a embeddig parameer, u 0 is a iiial approximaio for he soluio of Eq. (4), which saisfies he boudary codiios. Clearly, from Eqs. (7) ad (8) we have HU ( 0) LU ( ) Lu ( ) 0 (9) 0 HU ( ) AU ( ) f( r) 0 (0) While p chages form zero o uiy, Ur ( p) varies from u 0 () r o ur ( ). I opology, his is called homoopy. O accou a HPM, firs we ca use he embeddig parameer p as a small parameer, ad assume ha he soluio of Eqs. (7) ad (8) ca be wrie as a power series i p: 0 U U pu p U () Leig p =, resuls i he approximae soluio of Eq. (4) u limu U U U () p 0 The series () is coverge for he mos cases [0].

4 86 D. Baleau, A. K. Golmakhaeh, Ali K. Golmakhaeh 4 4. FRACTIONAL NON-LINEAR SCHRÖDINGER The oliear Schrödiger equaio is a model of he evoluio of a oe dimesioal packe of surface waves o sufficiely deep waer. NLS equaio describig he evoluio oliear wave i oliear, srogly dispersive, ad hyperbolic sysems [0]. The propagaio of a guided mode i a perfec oliear moo-mode fiber is modeled by oliear Schrödiger equaio [ 3]. I his secio we have solved he fracioal o liear Schrödiger equaio usig HPM. Le us cosider he followig Schrödiger equaio wih he followig iiial codiio [4]: id ( ) ( ) X Vd X d d (3) ( X0) Xd 0 where Vd ( X ) is he rappig poeial ad d is a real cosa. To solve Eq. (3) by homoopy perurbaio mehod, we cosruc he followig homoopy: H( p) (4) ( p)( i D ) pi D Vd( X) d d 0 Suppose he soluio of Eq. (4) o be as followig form 0 p p (5) Subsiuig (5) io (4), ad equaig he coefficies of he erms wih ideical powers of p, p 0 (6) D 0 0 pd i 0 Vd( X) 0 d0 d ( X0) 0 (7) p D i Vd( X) d ii i0 i dikki 0 ( X 0) 0 i0 k0 (8)

5 5 Solvig of he fracioal o-liear ad liear Schrödiger equaios 87 p3 D 3 i Vd( X) d ii i0 i dikki 0 3( X 0) 0 i0 k0 (9) pj D i V X j j ( ) j d j d i0 i jii j jii dikjki 0 j( X 0) 0 i0 k0 where i p j, here are he muliplicaio of wo series ad. For simpliciy we ake (0) 0 0 ( X0) () Havig his assumpio we ge he followig ieraive equaio j i j ( ) jvd( X) jd ijii ( ) 0 i0 j ji dikjki d i0 k0 The approximae soluio of (3) ca be obaied by seig p =, lim p 0 () 5. EXAMPLES We have used HPM for solvig followig example for efficiecy of he mehod i fracioal oliear ad liear equaio.

6 88 D. Baleau, A. K. Golmakhaeh, Ali K. Golmakhaeh EXAMPLE Cosider he followig oe dimesioal Schrödiger equaio wih he followig iiial codiio [5]. id ( ) x 0 x (3) ( x0) eix He s homoopy perurbaio mehod cosiss of he followig scheme H( p) ( p) i D p i D 0 x (4) Sarig wih 0 0 ( x0) e ix, usig () we obai he recurrece relaio j j i i j j ( ) 3 ( ) 0 ik jki d j x (5) i0 k0 The soluio reads ( ) i x e ( ) ix ( x ) e 4 ( ) ix 3 3( x ) i e 8 (3 ) ix Geeral form ca be wrie as Fially, soluio will be as ( i ) ( ) ix x e ( ) ( i ) ( x ) lim ( x ) eix p ( ) 0

7 7 Solvig of he fracioal o-liear ad liear Schrödiger equaios 89 I Fig., we have preseed he graph of soluio for he values = 0., 0.5 ad 0.9. Fig. Graph of he ( x ) correspodig o he values = 0., 0.5 ad 0.9 from lef o righ. 5.. EXAMPLE Cosider he followig parial differeial equaio [4] id ( ) cos x x 0 x ( x0) six We cosruc a homoopy [0] which saisfies (6) H( p) ( p) i D p( i D cos x 0 x I view of Eq. () we have he followig scheme 0 ( x0) si x, (7) j j i i j ( ) j j cos x ikjki d ( ) 0 x (8) i0 k0 We obai firs few j erm: 3 ( x ) i six ( ) 9 ( x ) six 4 () 3 8 3( x ) six 8 (3)

8 830 D. Baleau, A. K. Golmakhaeh, Ali K. Golmakhaeh 8 Thus, soluio will be as: ( 3 i ) ( x ) lim ( x ) six p ( ) EXAMPLE 3 Cosider liear Schrödiger equaio as followig id ( x) i 0 x wih iiial codiio ( x0) sih( x) Le cosruc he followig homoopy: (9) H( p) ( p) i D ( x ) p i D ( x ) i 0 x (30) Subsiuig from Eq. (5) io Eq. (30), rearragig based o powers of p-erms ad solvig he resuled equaios, we have: 0( x ) sih( x) 4i x ( ) sih( x) ( ) ( 4 i ) ( x ) sih( x) ( ) The soluio of he Eq. (9) whe p will be as follows: ( 4 i ) ( x ) lim ( x ) sih( x) ( ) p 5.4. EXAMPLE 4 Cosider he followig hree dimesioal Schrödiger equaio wih he followig iiial codiio [4]

9 9 Solvig of he fracioal o-liear ad liear Schrödiger equaios 83 id ( x) V( xyz) x y z 0 ( xyz) [0 ] [0 ] [0 ] (3) ( xyz0) si xsi ysi z where V( xyz) si xsi ysi z. We cosruc a homoopy [0] which saisfies H( p) ( p) i D ( x ) p id x y z I virue of Eq. () we ge he recurrece relaio ( ) V xyz ( xyz0) sixsiysiz i j j j j ( ) ( ) 0 x y z j ji V( xyz) ) d We derive he followig resuls j i k jki i0 k0 (3) (33) 5 ( xyz) i sixsiysiz ( ) 5 ( xyz) sixsiysiz 4 () 3 5 3( xyz) i sixsiysiz 8 (3) ( 5 i ) ( xyz) sixsiy ( )

10 83 D. Baleau, A. K. Golmakhaeh, Ali K. Golmakhaeh 0 Soluio of Eq. (3) will be derived by hese erms, so ( 5 i ) ( xyz) lim ( x) si xsi ysi z p ( ) 0 6. CONCLUSION I he prese paper we obai aalyical approximae soluio for fracioal oliear Schrödiger equaio by homoopy perurbaio mehod i shows ha abiliy of he homoopy perurbaio mehod i oliear fracioal equaio. REFERENCES. K. B. Oldham, J. Spaier, The Fracioal Calculus, Academic Press, K. S. Miller, B. Ross, A Iroducio o he Fracioal Iegrals ad Derivaives Theory ad Applicaio, Joh Wiley ad Sos, S. G. Samko, A. A. Kilbas, O. I. Marichev, Fracioal Iegrals ad Derivaives Theory ad Applicaios, Gordo ad Breach, R. Hilfer, Applicaios of Fracioal Calculus i Physics, World Scieific, I. Podluby, Fracioal Differeial Equaios, Academic Press, G. M. Zaslavsky, Hamiloia Chaos ad Fracioal Dyamics, Oxford Uiversiy Press, A. A. Kilbas, H. M. Srivasava, J. J. Trujillo, Theory ad Applicaios of Fracioal Differeial Equaios, Elsevier, V. Dafardar-Gejji, H. Jafari, J. Mah. Aal., 30, 508 (005). 9. J. H. He, Compu. Mahods Appl. Mech. Egrg., 78, 57 (999). 0. J. H. He, I. J. No-Liear Mech., 35, 37 (000).. J. H. He, Appl. Mah. Compu., 56, 57 (004).. J. H. He, Appl. Mah. Compu., 35, 73 (003). 3. J. H. He, Appl. Mah. Compu., 5, 87 (004). 4. J. H. He, Chaos Solios Fracals, 6, 695 (005). 5. D. D. Gaji, M. Rafi, Phys. Le. A, 356, 3 (006). 6. Z. M. Odiba, S. Momai, I. J. Noliear Sci. Nummer. Simulaio, 7, 7 (006). 7. V. Ledoux, L. Gr.Ixaru, M. Rizea, M. Va Daele, G. Vade Berghe, Compu. Phys. Commu. 75, 6 (006); 8. J. Biazar, H. Ghazvii, Phys. Le. A, 366, 79 (007). 9. S. Z. Rida, H. M. El-Sherbiy, A. A. M. Arafa, Phys. Le. A, 37, 553 (008). 0. O. Bag, P. L. Chrisiase, F. K. Rasmusse, Appl. Aal., 57, 3 (995).. A. Hasegawa, F. Tapper, Appl. Phys. Le., 3, 4 (973).. A. Hasegawa, Opical Solios i Fibers, Spriger, Berli, L. F. Molleaur, R. H. Sole, J. P. Gordo, Phys. Rev. Le., 45, 095 (980). 4. H. Wag, Appl. Mah. Compu., 70, 7 (005). 5. S. A. Khuri, Appl. Mah. Compu., 97, 5 (998).