Application of Laplace Adomian Padé approximant to solve exponential stretching sheet problem in fluid mechanics

Transcription

1 Joural o Novel ppled Scece valable ole at wwwacorg JNS Joural---S/- ISSN - JNS pplcato o aplace doma Padé approxmat to olve expoetal tretchg heet problem lud mechac Hahoe ad Soltaalzadeh Departmet o Mathematc Zaheda rach Ilamc zad Uverty Zaheda Ira Correpodg author: Soltaalzadeh STRCT: The purpoe o th tudy to apply aplace doma Decompoto Method DM or obtag the aalytcal ad umercal oluto o a olear deretal equato that decrbe a magetohydrodyamc MHD low uder the tretchg heet problem y ug th method the mlarty oluto o the problem are obtaed For obtag computatoal oluto we combed the obtaed ere oluto by the DM wth the Padé approxmato The method eay to apply ad gve hgh accuracy reult From the table ad gure ececy o the preeted techque how eyword: aplace traormato; doma Decompoto Method; Pade approxmato;naver Stoke equato; Sem-te terval; MHD low PCS umber: Hq; Mv; ; Cb INTRODUCTION Nolear pheomea that appear may are cetc eld uch a old tate phyc plama phyc lud mechac populato model ad chemcal ketc ca be deed by olear deretal equato Oe o the mot mportat kd o thee equato the olear deretal equato that characterze boudary layer ubouded doma Frtly Sakad Sakad olved the problem o orced covecto alog a othermal cotatly movg plate whch t a clacal problem o lud mechac Magetohydrodyamc MHD coderg the teracto coductg lud wth electromagetc problm The low o electrcally coductg lud wth the magetc eld oe o the mot applcable ecto varou area o egeerg ad techology The vcou low du to the tretchg boudary mportat extruo proce whe heet materal pulled out o a orce wth rag velocty Thereore ce the umercal/aalytcal o lud low acro a th lqud lm very mportat may brache o cece ad techology the may author pad much atteto to coder the behavor o th problem umercally ad aalytcally I vetgato o boudary layer problem by applyg a good varable traormato we covert the ytem o Naver-Stoke equato to a olear ordary boudary value problem wth a em-te terval I oyd the te doma replaced by [ ] ad the em-te travel wth [] by electg a ucetly large Guo Guo coverted the problem o em-te doma to a model o boudary doma Recetly theory o the boudary layer ha bee ucceully ued ad vetgated to the MHD Falker-Ska low o vcou lud Falker ; Soudalgekar Very recetly Robert etal aalyzed the extece ad uquee reult o the MHD Falker-Ska low Robert et al bbabady etal bbabady et al ; bbabady et al vetgated th problem umercally by ug Hakel-Padé method ad Homotopy aaly method repectvely zal Noor tuded the Falker Ska problem or low pat a tretchg urace wth ucto or blowg Very recetly the MHD Falker-Ska boudary layer low over a permeable wall the preece o a travere magetc eld author have bee examed ad approxmate reult or the mlarty

2 J Nov ppl Sc S: - oluto have bee obtaed by ug the deretal traorm method DTM coupled wth the Padé approxmato Xao-hog The boudary-layer problem ote ca be expreed the orm o a olear two pot boudary value problem wth pecc codto at the two boudare o the doma D [ whch ca ot be ote olved aalytcally ad exactly a cloed orm The doma Decompoto Method ha bee appled to a wde cla o problem phyc bology ad chemcal reacto The method provde the oluto a rapd coverget ere wth computable term doma doma et al The by applyg th method the umercal oluto o ome equato ca be obtaed I th reearch we wll combe the doma Decompoto Method wth the laplace traormato to obta the mlarty oluto o a mportat olear deretal equato Th method propoed hur ; Syam The a combato o the aplace doma Decompoto Method wth the Padé approxmato ha bee preeted by akera aker Th paper ha the ollowg tructer: covertg the model o ytem o olear PDE to the olear ordary deretal equato preeted ecto I ecto we appled the DM to the obtaed ordary deretal equato I ecto combg the DM wth the Padé approxmat how Fally the umercal reult reported Mathematcal ormulato ad dcuo I th ecto uppoe that we coder law o a compreble vcou lud over expoetal tretchg heet at y Suppoe that u be the velocty compoet the x y drecto repectvely I act t the kematc vcoty whch the rato o dyamc vcoty to the dety o the lud e The baed o the above aumpto the correpodg pheomeo ca be troduced a ollow u v x y u u u u v x y y ubect to the ollowg boudary codto x u U e at y u a y where U ad are the reerece velocty a cotat repectvely Now we dee the ollowg mlarty traormato U x e y l x u U e U x e y ug the ad we have wth boudary codto lm aed o our bet kowledge the umercal oluto o ad ha bee dcued [] I th paper we hall propoed oe aother umercal cheme baed o aplace doma decompoto method or aalytcal oluto ad Padé aproxmat or umercal oluto o the preece o boudary codto pplcato o the aplace doma Decompoto Method

3 J Nov ppl Sc S: - I th ecto we ue the aplace traorm algorthm or thrd order olear tal value problem to have mlarty oluto o the olear ordary deretal equato wth boudary codto Frtly we take the aplace traormato o both de o Eq the preece o the boudary codto The we get Wth the our computatoal proce we wll determe the value o The we dee we ca olve the equato ubect to the ollowg tal value codto where a ukow cotat that t mut to be oud Now by applyg the codto to we obta y ug the aplace decompoto method we wll be able to obta a aalytcal oluto o the orm o the ollowg te ere y preetg a teratve proce we wll d the compoet or I addto we decompoed the olear term ad deed to doma polyomal doma doma etal to the ollowg cae The we have y ug above relato the ew compoet o the doma polyomal o above olear term cab be gve a ollow: ad

4 J Nov ppl Sc S: - I th way by ug the above reult ad doma polyomal to we get The we have where repreet the term arg rom precrbe tal codto the baed o the moded aplace decompoto method [?] we ca decompoe the ucto to two part a the Thereore or obtag the rtly we compare both de o the equato ad the ue rom the vere aplace traorm Fally by applyg the ollowg teratve proce we ca obta the value o or y ug vere aplace traorm the equato we ca obta the tal term Now we ca compute the value o by ug the kow value o y cotug th proce we ca d the ucceve term The we have

5 J Nov ppl Sc S: - y obtag the compoet or the approxmate aalytc oluto o the ukow ucto ca be oud rom equato The the approxmate aalytc oluto or the te terato tep

6 J Nov ppl Sc S: - From Eq t evdet that the obtaed aalytc oluto through DM are power ere the depedet varable The accordg to boudary codto thee oluto have ot the correct behavor at ty ad o thee oluto ca ot be drectly appled So to reolve th problem we combe the ere oluto obtaed by the DM wth the Padé approxmat The DM-Padé pproxmato Combg the obtaed ere oluto by the DM the prevou ecto wth the Padé approxmato the ma part o th ecto To th ed we apply th proce or obtag ome hgh accuracy computatoal reult or problem wth boudary codto The we traorm the power ere obtaed by the aplace doma Decompoto Method to a ratoal ucto a ollow [ S/ N] S a N b We kow that N S the the lmt at ty the boudary codto ha a correct behavor So the ratoal ucto ha S N coecet that we ca elect them I [ S / N] exactly a Padé approxmato SN the [ S/ N] O The we ca obta the coecet b a S b S S N a b ad by the ollowg relato where ak b k k > N From ad we ca obta the value o a S b N ad We kow that the ucto t bouded e or all t > we have t < M ad the lmt t be ext the lm F where F t the laplace traorm o the ucto t For dg the ukow parameter we ca utlze the above pot or laplace traorm to t or by ug Pade equece Plot o the approxmate oluto o ad whch obtaed by the DM-Pade how Fgure ad Fgure The accuracy o propoed method ca be udertad rom thee plot Fgure Plot o DM-Padé approxmate oluto o

7 J Nov ppl Sc S: - Fgure CONCUSION I th artcle oe o the thrd order olear autoomou equato ubect to a boudary codto whch deed at ty codered ew em-aalytcal method aplace doma Decompoto coupled wth Pade approxmat ucceully apled or olvg th equato The obtaed computatoal reult by ug our method are preeted a table It evdece that th method gve hgh accuracy reult very ew tterato ad ca be appled to other mlar problem CNOWEDGEMENT Th work upported by Grat--d rom the Ilamc zad Uverty Zaheda rach The author thak very much rom ther upport REFERENCES Sakad C oudary layer behavor o cotou old urace: The boudary layer o a cotou lat urace IChE J : - oyd JP Chebyhev ad Fourer Spectral Method ecod edto Dover New York Guo Y Jacob approxmato certa Hlbert pace ad ther applcato to gular deretal equato J Math al ppl : - Falker VM Ska SW Some approxmate oluto o the boudary layer equato Phlo Mag ;:- Soudalgekar VM Takhar HS ad Sgh M Velocty ad temperature eld MHD Falker-Ska low Joural o the Phycal Socety o Japa - Robert VG ad Varavelu Extece ad uquee reult or a olear deretal equato arg MHD Falker- Ska low Commucato Nolear Scece ad Numercal Smulato - bbabady S hayat T Soluto o the MHD Falker Ska low by Hakel Padé methodphyc etter - bbabady S hayat T Soluto o the MHD Falker-Ska low by homotopy aaly method Commucato Nolear Scece ad Numercal Smulato - - Noor Falker Ska equato or low pat a tretchg urace wth ucto or blowg: alytcal oluto ppled Mathematc ad Computato - Xao-hog SU ad a-cu ZHENG pproxmate oluto to MHD Falker-Ska low over permeable wall ppl Math Mech -Egl Ed doma G Solvg roter problem o phyc: the decompoto method Dordrecht: luwer cademc Publher; doma G Rach R Moded doma polyomal Math ComputModel - hur S aplace decompoto algorthm appled to cla o olear deretal equato J Math ppl Syam MI Hamda ecet method or olvg ratu equato ppl Math Comput : aker G Eetal o Padé pproxmat cademc Pre odo