VARIATIONAL ITERATION METHOD (VIM) FOR SOLVING PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS

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1 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: VARIATIONAL ITERATION ETHOD VI FOR SOLVING PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS 1, AINA KASSI HUSSAIN, FADHEL SUBHI FADHEL, 1 ZAINOR RIDZUAN YAHYA, 1 NURSALASAWATI RUSLI 1 Uiversii lysi Perlis, Isiue of Egieerig hemics, 6 Aru,Perlis, lysi Uiversiy of Al-ussiriyh, Deprme of eril Egieerig, 15 Bghdd, Irq Al-Nhri Uiversiy, College of Sciece, Deprme of hemics d Compuer Applicios,17 Al-Jdriy, Bghdd, Irq E-mil: 1, mi1975_ks@yhoo.com, dr_fdhel67@yhoo.com, 1 zioryhy@uimp.edu.my, 1 urslswi@uimp.edu.my, ABSTRACT I his pper, wo objecives will be chieved, he firs oe is o se d prove he eisece of uique soluio of olier pril iegro-differeil equios by usig Bch fied poi heorem. The secod objecive is o pply He s vriiol ierio mehod for solvig olier pril iegro-differeil equios. This mehod is very powerful mehod for solvig lrge mou of problems. I provides sequece of iered soluios which is coverge o he ec soluio of he problem. Also, i his work he derivio of he ierio formul usig He s mehod hve bee preseed d he prove he coverge of he obied sequece of iered pproime soluios o he ec soluio of he pril iegrodiffereil equio. Filly, illusrive emples were preseed o show he efficie of he ew mehod d he proposed echique ws progrmmed usig hcde 15.. Keywords: Vriiol Ierio ehod, Nolier Pril iegro-differeil Equio, Bch Fied Poi Theorem, Corcio ppig Priciple. 1. INTRODUCTION hemicl modelig is he r of rslig rel life problems io rcble mhemicl formulios, for emple ordiry d pril differeil equios, iegrl d iegro-differeil equios d ohers[1,]. I rece yers, here hs bee growig ieres i he iegro-differeil equios, i priculr olier pril iegro-differeil equios. Sice here re my mhemicl formulios of physicl pheome, such s olier fuciol lysis d heir pplicios i he heory of egieerig, physics, mechics, chemicl kieics, sroomy, ecoomics, biology, poeil heory d elecro sisics coi pril iegro-differeil equios[]. The problem of eisece of uique soluio of differeil equio hve bee cosidered by my uhors, such s, omi [4], omi d Hdid[5], Rbh d S. omi[6], Hu e l. [7]; Shym e l. [8];Krhikey d Trujillo [9]; ATri [1]. Pril iegro-differeil equios usully difficul o be solved lyiclly, herefore, umericl d pproime mehods re required o solve such equios, d here re my such mehods hve bee proposed previously, such s he mehod of successive pproimios dhe s ierio mehod[11,1]. The vriio ierio mehod VI hs esblished o be oe of he useful echiques i solvig my ypes of lier d olier differeil equios for fidig boh lyicl d pproime soluios[1].this echique ws developed by Chiese mhemici He. This mehod successfully pplied o my siuios, for emple, He s proposed he VI o solve Dely differeil equios [14], lier d olier differeil equios[15],seepge flow equio wih frciol derivives i porous medi [16], uoomous ordiry differeil sysems[17], followig by ommi d Abusd used VI o solve Helmholz equio[18],wzwz used VI for solvig lier d olier sysem of pril differeil equios[19],bih e l. used VI o 67

2 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: geerl Ricci equio [],Hmid pplied VI o solve wve equios [1], Abbsbdy d Shivi pplied Vriiol Ierio ehod for solvig sysem of olier Volerr iegro- Differeil equios [], Kuruly d Secer pplied Vriiol ierio ehod o solve olier frciol order Iegro-differeil equios []. I his pper, our im is o se d o prove he eisece of uique soluio of pril iegro-differeil equio d he use he vriio ierio mehod o solve such pril iegro-differeil equios, s well s, o prove he covergece of he iered sequece of pproime soluios o he ec soluio of he problem whe i is ssumed o be eis by he sisfcio of he codiios of he eisece of uique soluio of such equios. The form of he cosidered pril iegrodiffereil equio is give by: u, = g, + k y,, u y, dy, [,b], [,T] By usig he followig iiil codiio: u, = u Wherek is he kerel fucio, g is give fucio du is he ukow rel fucio o be evlued.. BASIC CONCEPTS AND DEFINITIONS I order o proceed, some fudmel coceps reled o his work re give i his secio. Defiiios 1,[4]: Le T:X X be mppig o ormed spcex,..a poi X such s T = is clled fied poi of T. Defiiio,[4]: A mppig T o ormed spce X,. is clled corcive if here is o-egive rel umber c, such h c < 1,d for ech 1, X, implies h: T1T1 c 1 The e heorem is well kow i lysis, which is of gre imporce for he eisece of uique soluio of equios 1. 1 Theorem 1, Bch Fied Poi Theorem,[5] LeX,. be complee ormed spce d le T : X X be corcio mppig, he T hs ecly oe fied poi. Defiiios,[6,7]: Le X,. be ormed spce, fucio f,;y 1,y,,y defied o he se: Ω={,;u 1,u,,u m :, b, <u i <, for ech i=1,,,m} is sid o sisfy Lipchiz codiio o Ωwih respec o he vribles u 1,u,,u m if cos L> eiss wih he propery h:, ;,,,, ;, z,, z f u1 u um f z1 m m L yi zi i = 1 Ω. for ll,;u 1,u,,u m d,;z 1,z,,z m i Remrk 1: The spce C [, b] [, will be cosidered i his work s he Bch spce for ll coiuous rel vlued fucios u defied o [,b] [,T] wih coiuous -h order pril derivive wih respec o.. THE EXISTENCE OF A UNIQUE SOLUTION FOR ONE-DIENSIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS I his secio, he seme d he proof of he eisece d uiqueess soluio for equio 1 by usig Bch fied poi heorem d corcio mppig priciple. Theorem : Cosider he pril iegro-differeil equio 1 wih he iiil codiio equio over he regio: {, :, } Q = b T d suppose h k sisfies Lipschiz codiio wih respec o u d cos d T b < 1, he equio1 hs uique soluio. Proof: By iegrig boh sides of equio 1 68

3 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: wih respec o, we ge: u, = u +, g ξ dξ + k y, ξ, u y, ξ dyd ξ Sice, i is kow h he se of ll coiuous fucio defied o he regio Q is complee ormed spce wih u, u, = sup u, u, 1 1 b T Rewrie equio i operor forms s Nu=u N. = u + g, d k y,,. dyd ξ ξ + ξ ξ 5 Ne, o show h N is corcive mppig d for his purpose, ke u1, u C [,b] [, Nu, Nu, = 1 sup u + b T g, ξ dξ+ k y, ξ, u y, ξ 1 u g, ξ dξ k y, ξ, u y, ξ + = sup k y, ξ, u y, ξ k y,, u y, dyd 1 ξ ξ ξ 6 b T sup k y, ξ, u y, ξ k y,, u y, dyd 1 ξ ξ ξ 7 b T 4 6 d sice T b < 1, he N is corcio mppig d herefore N hs uique fied poi, which mes h equio 1 hs uique soluio. 4. FORULATION OF THE VARIATIONAL ITERATION ETHOD FOR NONLINEAR PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS I his secio before derivio he vriio ierio formul for pril iegro-differeil equio will be mde, he mi specs of he VI will be give. The i Aspecs of he VI As meioed bove, he VI which ws suggesed by He i 1998 iesively sudied by severl scieiss d egieers which is fvorbly pplied o my kids of lier d olier problems. The mehod hs bee show o solve lrge clss of lier d olier problems effecively, esily d ccurely. Geerlly, oe or wo ierios led o high ccure soluios. This mehod which is modificio of he well-kow geerl Lgrge muliplier mehod io ierio mehod clled correcio fuciol. Geerlly spekig, he soluio procedure of he VI is very operive, srigh forwrd d coveie [8]. To illusre he bsic ide of he VI, cosider he followig geerl o-lier equio give i operor form: Lu + Nu = g, [, b] 11 wherel is lier operor, N is olier operor d g is y give fucio which is clled he o-homogeeous. Now, rewrie equio 1 i s follows: Lu + Nu g = 1 sup u y, ξ u y, ξ 1 b T sup u1, y u, y b T = supu, u, 1 b T T b supu, u, 1 b T d leu be he h pproime soluio of eq. 14, he i follows h: L u + N u g 1 d herefore he correcio fuciol for eq.15, is give by: u + 1 = u + λ s{ Lu s + Nu % s gs } ds 14 69

4 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: , Whereλ is he geerl Lgrge muliplier, which c be ideified opimlly vi he vriiol heory, d u % is cosidered s resriced vriio which mes δ% u =, [1]. Geerlly spekig, i is obvious h he mi seps of He s vriiol ierio mehod require he deermiio of he Lgrgi muliplierλ firs sep h will be ideified opimlly.afer deermied he Lgrgi muliplier, he successive pproimiosu + 1, of he soluiou will be redily obied upo usig y selecive fuciou. Cosequely, he soluio ucoverge o he ec soluio u u = lim u I he e heorem we will derive he geerl formul for solvig eq.1 usig VI which is bsed o he geerl form eq.16 fer evluig he Lgrge muliplier reled wih he pril iegro-differeil equio 1. Theorem : Cosider heolier pril iegrodiffereil equio 1 wih iiil equio. The he reled vriiol ierio formul is give by: u+ 1, = u, u, ξ g, ξ k y, ξ, u y, ξdy dξ For ll =,1, Proof: 15 The correcio fuciol 16 reled o he pril iegro-differeil equio 1 is give by: u+ 1, = u, + u λ, ξ g, ξ k y, ξ, u y, ξ dy d ξ % 16 Whereλ is he geerl Lgrge muliplier, which mus be evlued opimlly, he subscrip deoes he h pproimio d u% is cosidered s resriced vriio. Tkig he firs vriio δ wih respec ouo he boh sides of equio 18 d seigδu =, yields o: δu+ 1, = δu, + u δ λ, ξ g, ξ k y, ξ, u % y, ξ dy d ξ 17 whereδ u% = d cosequely equio 19will be reduced o 1,, u δ + = δu + δ λ ξ, ξ dξ 18 hece, upo usig he mehod of iegrio by prs of equio will give : δu+ 1, = δu, + λ ξ δu, ξ ξ = δu, ξ λ ξ dξ 19 = 1 + λ ξ δu, ξ u, δ ξ λ ξ= ξ dξ s resul, he followig ecessry codiio is obied for rbirryδ u : λ ξ = 1 wih iiil codiio: 1+ λ ξ = ξ = solvig he ls ordiry differeil equio will yields he geerl Lgrge muliplier o be defied s follows: λ ξ = 1 Hece, subsiuig λ ξ = 1io he correcio fuciol eq. 18 will resuls he followig ierio formul: 1,, u u u, g, k y,, u y, + = ξ ξ ξ % ξ dy dξ 7

5 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: ANALYSIS OF CONVERGENCE FOR NONLINEAR PARTIAL INTEGRO- DIFFERENTIAL EQUATION I he e heorem, he covergece of he sequece of iered pproime soluio 17 of he pril iegro-differeil equio 1 o he ec soluio will be proved. Theorem 4 Le u, u C [, b] [,T] be he ec d pproime soluios of equio 1 d 17, respecively. If E, = u, u, d he kerel k sisfies Lipschiz codiio wih cos, he he sequece {u } coverge o he ec soluio u. Proof: The pproime soluio usig he VI is give by: u, = u, + 1 u,, y,,, % ξ g ξ k ξ u y ξ dy d ξ 17 dsice u is he ec soluio of equio 1, hece i sisfies he VI formul, i.e., u, = u, u, g, k y,, u y, dy d ξ ξ ξ ξ ξ 4 subsrce u from u +1 d recll h E, = u, u,, implies o : u+ 1, u, = u, u, u, ξ u, ξ g, ξ + g, ξ k y, ξ, u y, ξ k y, ξ, u y, ξ dy dξ Hece E+ 1, = E, u E, ξ k y, ξ, u y, ξ k y, ξ, u y, ξ dy d ξ = E, E, E, + k y, ξ, u y, ξ k y, ξ, u y, ξ dy dξ = k y, ξ, u y, ξ k y, ξ, u y, ξ dy dξ, 7 where E, = kig he orm o he boh sides eq.9, give E+ 1, = 1 k y, ξ, u y, ξ k y, ξ, u y, ξ 8 ky, ξ,u y, ξ ky, ξ,uy, ξ dy dξ u y, ξ u y, ξ dy dξ 9 Hece E+ 1, E y, ξ dy dξ, for ll =,1,, Now, if =, he : E1 E 1 = E = E While if =1, he: E E1 4 E ξ y 5 = E 6 lso if =, he: 71

6 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: E E 7 y E ξ dy dξ 8 L E = 9 E = 4 E E, b, T ! sice = d kig he supermom vlue of d over [,b] d [,T], respecively, o ge E E T b 4! d i is cler h if d is o lrge i mgiude, he! u, = 1+ Srig wih he iiil pproimio u, = 1, he firs four pproime soluios usig he VI 18 re foud o be: 4 1, 1 u 4 4 9, 1 u = , 1 u , 1 u Compriso bewee he ec d pproime soluiosu 1, u, u d u 4 usig he bsolue error re give i ble 1. TABLE 1: The Absolue Error Of Emple 1 HeceE,i.e, u u s 6. NUERICAL SIULATION AND IILLUSTRATIVE, EXAPLES A B C D I his secio, wo illusrive emples will be cosidered i order o emie he vlidiy d illusrive he covergece of he vriio ierio formul give by eq. 18. Two illusrive emples re cosidered, for lier d olier pril iegrodiffereil equios, i which he ccurcy of he resuls regive by schedulig he bsolue error bewee he ec soluio give here for compriso purpose d ech iered soluio. Emple 1: Cosider he followig lier pril iegrodiffereil equio u, = + y + u dy, where, [,1] [,1] wih he iiil codiio u, = u = 1 The ec soluio is give by Where A: u, u1, Emple :,.5, e e 6.649e 8 6.6e 11.5, e e e 8.75, e 1.78e 4.911e 6 1, e 6.944e 5 B: u, u, C: u, u, D: u, u4, 7

7 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: , A B C D,.5, u, Compriso bewee he ec d pproime soluiosu 1, u, u du 4 usig he bsolue error re give i ble. TABLE : The bsolue error of emple.5, Where A: u, u1,.75, B: u, u, C: u, u, 1, Cosider he followig olier pril iegro-differeil equio: u, 4 = + y uy, dy, 4 Where, [,1] [,1] wih he iiil codiio: u, = u = 1 The ec soluio is give by: u, = Srig wih he iiil pproime soluio u, =, he firs four pproime soluios usig he VI eq. 18 re foud o be: 1, u , u 5 16, u D: u, u4, 7. CONCLUSIONS The vriio ierio mehod VI hs bee show o solve lrge clss of o-lier problems effecively,wih he pproimios which re coverge re rpidly o he ec soluios.i his work, he VI hs bee successfully employed o obi he pproime soluio o lyicl soluio of lier d o- lier pril iegro-differeil equios. For his purpose, we hve show h he VI hs rpid covergece by emples. REFERENCES: [1] B.Bih,. S. Noori d HshimI,Numericl Soluios of he Nolier Iegro-Differeil Equios, Jourl of Ope Problems Comp. h., Vol.1, No.1, 4-41, 8. [] N.H. Sweilm, Fourh order iegrodiffereil equios usig vriiol ierio mehod, Compuers d hemics wih Applicios, Vol.54, , 7. [] J.. Yoo, S. Xic d V. Hrykiv, A series Soluio o Pril Iegro-Differeil Equio Arisig i Viscoelsiciy, IAE N G ieriol Jourl of Applied hemics, 4, 4, 1. [4] S.omi, Locl d globl eisece heorems o frciol iegro-differeil equios, Jourl of Frciol Clculus, Vol. 18, 81-86,. [5] S. omi d S. Hdid,Lypuov sbiliy soluios of frciol iegro-differeil equios,ijs,vol. 47, 5-57, 4. 7

8 Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No JATIT & LLS. All righs reserved. ISSN: E-ISSN: [6] W. I. Rbh d S. omi,o he Eisece d Uiqueess of Soluios of Clss Frciol Differeil Equios, J. h. Al. Appl., Vol.4, 1-1, 7. [7] H. Lyig, R.Yog d R. Skhivel, Eisece d Uiqueess of ild Soluio for SemilierIegro-differeil Equio of Frciol Order wih Nolocl Iiil Codiios d Delys, Semigroup Form, Vol.79, , 9. [8] A.. Shym, J. Z. Hussei d H. Smir,Eisece d Uiqueess Theorem of Frciol ied Volerr-FredholmIegrodiffereil Equio wih Iegrl Boudry Codiios, Ieriol Jourl of Differeil Equios, Vol.11, 11. [9] K. Krhikey d J. J. Trujillo, Eisece d Uiqueess Resuls for Frciol Iegro- Differeil Equios wih Boudry Vlue Codiios, Commu Nolier NumerSimul, Vol.17,1. [1] A. Tri, "O he Eisece Uiqueess d Soluio of he Nolier Volerr Pril Iegro-Differeil Equios", Ieriol Jourl of Nolier Sciece, Vol.16, No.,15-16, 1. [11] H. Jrd, Al-Syyed O. d Al-Shr S., Numericl Soluio of Lier Iegro- Differeil Equios, Jourl of hemics d Sisics, Vol. 44, 5-54,8. [1] il R. C., d Nigm R,Soluio of Frciol Iegro-Differeil Equios by A Domi Decomposiio ehod, Ieriol Jourl of Appl. h. Ad ech., 4, 87-94, 8. [1] J. Bizr,. G. Porshokouhi d B. Ghbri, Numericl soluio of fuciol iegrl equios by he Vriiol ierio mehod,jourl of Compuiol d Applied hemics,vol. 5, , 11. [14] J.H. He, Vriiol Ierio ehod for Dely differeil equios,commuicios i Nolier Sciece & Numericl Simulio, Vo1., No.4,1997. [15] J.H. He, Vriiol ierio mehod kid of o-lier lyicl echique: Some emples, Ier. J. Nolier ech. 4, , [16] J.H. He, Approime lyicl soluio for seepge flow wih frciol derivives i porous medi, Compu. ehods Appl. ech. Egrg. 167,57 68,1998, [17] J.H.He, Vriiol ierio mehod for uoomous ordiry differeil sysems,applied hemics d Compuio,Vol. 114, 115 1,. [18] A.. Wzwz, The vriiol ierio mehod for solvig lier d olier sysems of PDEs, Compuers d hemics wih Applicios,Vol. 54, 895 9, 7. [19] S. omi, d S. Abusd, Applicio of He s vriiol ierio mehod o Helmholz equio, Chos Solio Frcls. 7, , 6. [] B.Bih,. S. Noori d I.Hshim, Applicio of Vriiol Ierio ehod o Geerl Ricci Equio, Ieriol hemicl Form, No.56,759-77, 7,. [1] A.A. Hemed,Vriiol ierio mehod for solvig wve equio, Compuers d hemics wih Applicios 56, , 8. [] S.Abbsbdy, E.Shivi, Applicio of he Vriiol Ierio ehod for Sysem of Nolier Volerr'sIegro-Differeil Equios, Jourl of hemicl d Compuiol Applicios, Vol.14, No., , 9 [].Kuruly d Secer A., Vriiol Ierio ehod for Solvig Nolier Frciol Iegro-Differeil Equios, Ieriol Jourl of Compuer Sciece d Emergig Techologies, Vol., 18-, 11. [4]. Reed d B. Simo,Fuciol Alysis, Acdemic Press Ic., New York, 198. [5] A. J. Jerri, Iroducio o Iegrl Equios wih Applicios, reel Dekker, Ic, [6] S. K.Berberi, Iroducio o Hilber Spce, Chelse publishig Compy, New York, [7] R. L. Burde. dfires J. D.,Numericl Alysis, Sih Ediio, Thomso Lerig, Ic, 1997 [8] J. S Ghorbid, J. S. Ndjfi, Covergece of He's vriiol ierio mehod for olier oscillors, Nolier Sci. Vol. 1 4, 79-84,1. 74