Reconstruction of Variational Iterative Method for Solving Fifth Order Caudrey-Dodd-Gibbon (CDG) Equation

Transcription

1 Shraz Unvery of Technology From he SelecedWor of Habbolla Lafzadeh Reconrcon of Varaonal Ierave Mehod for Solvng Ffh Order Cadrey-Dodd-Gbbon (CDG Eqaon Habbolla Lafzadeh, Shraz Unvery of Technology Avalable a: hp://wor.bepre.com/habb_lafzadeh/9/

2 Inernaonal Jornal of Scence and Engneerng Invegaon vol., e 6, Jly ISSN: -884 Reconrcon of Varaonal Ierave Mehod for Solvng Ffh Order Cadrey-Dodd-Gbbon (CDG Eqaon A. Nar, J. Vahd, M. Jafarnejad Ghom, M. Mghan 4 Deparmen of Cvl Engneerng, Shomal Unvery, Amol, Iran, P. O. Bo 7 Deparmen of Mahemac, Iran Unvery of Scence and Technology, Behhahr, Iran Olomrayaneh Pblher, Babol, Iran 4 Deparmen of Archecral Engneerng, Shomal Unvery, Amol, Iran, P. O. Bo 7 ( al.nar@homal.ac.r, j.vahd@.ac.r Abrac-Engneerng problem and he olon of many mporan phycal problem are cenered on fndng accrae olon o Nonlnear fncon. Varo appromaon mehod have been ed for comple eqaon. One of he newe appromaon mehod Reconrcon of varaonal eraon mehod (RVIM. In h Paper we e RVIM o olon N-olon olon for he ffh order Cadrey-Dodd- Gbbon (CDG Eqaon. Rel compared wh hoe of on how ha he Reconrcon of RVIM very effecve and overcome he edo wor ha radonal mehod reqre and qe accrae o yem of non-lnear paral dfferenal eqaon. Keyword-Reconrcon of Varaonal Ieraon Mehod (RVIM, Cadrey-Dodd-Gbbon (CDG Eqaon, Adoman decompoon mehod (ADM. I. INTRODUCTION In recen year he heory of olary wave ha araced mch concern for reamen of PDE decrbng nonlnear and developmen concep. Paral dfferenal eqaon whch emanae n real-world phycal problem are ofen oo complcaed o be olved eacly, and even f an eac olon obanable, ch a nvere caerng mehod, Ba clnd -], Here mehod, Panleve analy [4], and oher mehod, he reqred calclaon may be praccally oo complcaed, or mgh be dffcl o epond he ocome. Wh he rapd promoon of lnear and nonlnear cence, n he pa everal decade, varo mehod for obanng olon of DE have been preened, ch a Homoopy perrbaon mehod, varaonal eraon mehod, ep-fncon mehod and RVIM and o on. RVIM ha been hown o olve a large cla of nonlnear problem wh appromaon convergng o olon rapdly, effecvely, ealy, and accraely. bede he am of h paper o how ha RVIM rongly and mply capable of olvng a large cla of lnear or nonlnear dfferenal eqaon who he angble rercon of envy o he degree of he nonlnear erm. The mo enble advanage of RVIM are ng Laplace Tranform and choong nal condon mply and ealy n olvng lnear and nonlnear eqaon. In h arcle we e RVIM o olve he N-olon olon for he ffh order Cadrey-Dodd-Gbbon (CDG Eqaon. 8 ( wh (, a adeqaely ofen dfferenable fncon. The CDG eqaon domnae he Panleve propery a demonraed by We n [4]. A efl dy nrodced n [4] ng he Panleve propery and he Ba clnd ranformaon n handlng he CDG eqaon and oher eqaon a well. I wa fond n [4] ha he CDG Eqaon ( ha he Baclnd ranformaon ln where afe he CDG eqaon, and 6 and ; 4 ; The la wo eqaon can be epreed a he La par 6 8 6(6 The objecve of h wor o promoe oher de relaed o he CDG eqaon. The anh mehod [, 6], and he anh-coh ( 8

3 mehod [7] wll be ed o emphaze power n he deermnaon of ngle-olon olon and oher ravellng wave olon. We plan o e RVIM o olve h eqaon. II. BASIC IDEA OF RVIM To clarfy he bac dea of or propoed mehod n [8], we conder he followng dfferenal eqaon ame a VIM baed on Lagrange mlpler [9]: L(,, N(,, f (,, ( By ppoe ha L(,, L ( (4 where L a lnear operaor, N a nonlnear operaor and f,..., an nhomogeneo erm. ( we can rewre eqaon ( down a correcon fnconal a follow: L ( j j f (,, N(,, (,,, (,, j L ( ( herefore Lj( j (,,, (,, (6 Wh arfcal nal condon beng zero regardng he ndependen varable j. By ang Laplace ranform of boh de of he eqaon (6 n he al way and ng he arfcal nal condon, we oban he rel a follow P(. U(,,,,, H((,,,,,, (7 Where P( a polynomal wh he degree of he hghe dervave n eqaon (7, (he ame a he hghe order of he lnear operaor L. The followng relaon are poble; j [h] H (8-a B( P( (8-b [b( ] B( (8-c Whch ha n eqaon (8-a he fncon H((,,,,,, and (,,,,,, have been abbrevaed a H, h repecvely. Hence, rewre he eqaon (7 a; U,,,,, H((,,,,,,. B( ( (9 Now, by applyng he nvere Laplace Tranform on boh de of eqaon (9 and by ng he (8-a - (8-c, we have; (,, -,,, ( (,,,,,,. b( d Now, we m mpoe he acal nal condon o oban he olon of he eqaon (. Th, we have he followng eraon formlaon: (,,,,, (,,,,, n - (, -,,,,,. b( d ( Where nal olon wh or who nnown parameer. Amng he olon of L, wh nal/bondary condon of he man problem, In cae of no nnown parameer, hold afy nal/ bondary condon. When ome nnown parameer are nvolved n, he nnown parameer can be denfed by nal/bondary condon afer few eraon, h echnology very effecve n dealng wh bondary problem. I worh menonng ha, n fac, he Lagrange mlpler n he He' varaonal eraon mehod b ( a hown n ( []. The nal vale are ally ed for elecng he zeroh appromaon. Wh deermned, hen everal appromaon n, follow mmedaely. Coneqenly, n he eac olon may be obaned by ng (,,,,, lm (,, - III. n APPLYING RVIM n -,,,. ( Before To demonrae he effecvene of he mehod we conder here Eq.( wh gven nal condon. Conder he Cadrey-Dodd-Gbbon (CDG Eqaon ( wh he followng nal condon: ( (, ec h ( A fr rewre eq. ( baed on elecve lnear operaor a ( (,, 8 (4 Now Laplace ranform mplemened wh repec o ndependen varable on boh de of eq. (4 and by Inernaonal Jornal of Scence and Engneerng Invegaon, Volme, Ie 6, Jly 9 ISSN: -884 Paper ID: 6-7

4 ng he new arfcal nal condon (whch all of hem are zero we have (,,, (,, (, (6 And wherea Laplace nvere ranform of / a follow [ ] (7 Therefore by ng he Laplace nvere ranform and convolon heorem conclded ha (,,, d (8 Hence, we arrve he followng erave formla for he appromae olon of bjec o he nal condon (. So, n echange wh applyng recrve algorhm, followng relaon are acheved n ( 8 (9 Now we ar wh an arbrary nal appromaon (, ec h ( ha afe he nal condon and by ng he RVIM eraon formla (9, we have he followng cceve appromaon (co n (, coh ( 8 (, (co co n co n 4 4 co 8 co 49 co n 4 6 co n 4 co co d co Fgre.. The rface on boh colmn repecvely how he olon, (,, for RVIM on he op and ADM on he boom when µ=. Inernaonal Jornal of Scence and Engneerng Invegaon, Volme, Ie 6, Jly 4 ISSN: -884 Paper ID: 6-7

5 [] -de Vre Eqaon for 8, 97, pp [] W. Hereman Aca Applcandae Mahemacae, Phy-c Leer A, Vol. 76, 98, pp [4] 984, pp. -4. [] -an Clae of Nonlnear Evolon and Wave Eqaand Appled Mahe-mac, Vol. 64-6, 4, pp [6] Amercan Jornal of Phyc, Vol. 6, No. 7, 99, pp [7] -Coh Mehod for Solon and Kn -pled Mahemac and Compaon, Vol. 88, 7, pp [8] Nmer. Smlaon ( ( [9] A.M. Mahemac wh Applcaon, ( [] arbrary Lagrangan Jornal for Nmercal Mehod n Engneerng 6; 67(9:7 89. Fgre.. The rface on boh colmn repecvely how he olon, (,, for RVIM on he op and ADM on he boom when µ=. IV. CONCLUSION In h paper, we cceflly apply Reconrcon of Varaonal Ieraon Mehod (RVIM o appromae he olon of ffh order Cadrey-Dodd-Gbbon (CDG Eqaon. Alo, comparon were made beween He varaonal eraon mehod and Adoman decompoon mehod (ADM for Cadrey-Dodd-Gbbon (CDG Eqaon. Moreover, he RVIM redce he ze of calclaon by no reqrng he edo Adoman polynomal, and hence he eraon drec and raghforward. The rel repored here provde frher evdence of he eflne of RVIM for fndng he analyc and nmerc olon for he lnear and nonlnear dffon eqaon and, alo a promng mehod o olve dfferen ype of nonlnear eqaon n mahemacal phyc. REFERENCES [] -brdge Unvery Pre, Cambrdge, 4. Al Nar a reearcher n he Deparmen of Cvl Engneerng, Shomal Unvery, Amol, Iran. He ha ome pblcaon n appled mahemac. H feld of nere are cenfc compng and nmercal analy. H crren reearch manly cover n appromae, analycal and nmercal olon of nonlnear problem arng n he mechanc and Cvl Engneerng. He revewer of ome Inernaonal Jornal. Javad Vahd an aan profeor n he Deparmen of Mahemac, Iran Unvery of Scence and Technology, Behhahr, Iran. He Chef Edor of he Inernaonal Jornal of Nonlnear Analy and Applcaon (JNAA n Inda, Chef Edor of he Inernaonal and ml-naonaly Jornal of Nonlnear Scence and Applcaon (JNSA, member of referee commee of he Inernaonal Jornal of Comper Scence and Mahemac n Poland. Releang more han 4 boo n varo relaed opc, havng more han cenfc arcle, nernaonal magazne and ec. H crren reearch manly cover n appromae, analycal and nmercal olon of nonlnear problem arng n he mechanc/fld mechanc. Inernaonal Jornal of Scence and Engneerng Invegaon, Volme, Ie 6, Jly 4 ISSN: -884 Paper ID: 6-7