Boundary layer problem for system of first order of ordinary differential equations with linear non-local boundary conditions

Transcription

1 IJS 3 37A3 Secil issue-mheics: Irnin Journl of Science & echnology h://ijssshirzucir Boundry lyer role for syse of firs order of ordinry differenil euions wih liner non-locl oundry condiions M Jhnshhi * A R Srhsi S Ashrfi nd N Aliev Deren of Mheics Azrijn Shhid Mdni Universiy riz Irn Deren of Mheics Bu Se Universiy Bu Azerijn E-il: jhnshhi@zrunivedu Asrc In his er we sudy he oundry lyer roles in which oundry condiions re non-locl Here we ry o find he necessry condiions y he hel of fundenl soluion o he given djoin euion By geing hel fro hese condiions firs he oundry condiion is chnged fro non-locl o locl he in i of his er is o idenify he locion of he oundry lyer In oher words which oin he oundry lyer is fored Keywords: Singulr erurion roles; oundry lyer; fundenl soluion; necessry condiions Inroducion An iorn sujec in lied heics is he heory of singulr erurion roles he heicl odel for his ind of role is usully in he for of eiher ordinry differenil euions ODE or ril differenil euions PDE in which he highes derivive is ulilied y soe owers of s osiive sll reer [-3] he ojec heory of singulr erurion is o solve differenil euion wih soe iniil or oundry condiions wih sll reer hese roles re essenilly he her of oundry vlue nd iniil vlue roles [3-] hrough hese sudies we cn find ou wheher he oundry condiions ecoe locl ye Dirichle nd he soluion of he oundry lyer role is sisfied in oundry condiions hen here is no oundry lyer If he lii soluion when is no sisfied in he oundry condiion hen here will e oundry lyer In oo [4] fer he firs nd second chers he unsolved oundry lyer role is seen which shows h oundry lyer roles wih non-locl oundry condiions hve no een sudied crefully So in his er nd soe oher wors: M Jhnshhi & A R Srhsi [-4] nd N Aliev & S Ashrfi [5] [6] we sudy he oundry lyer roles in which oundry condiions re non-locl Here n e is de o find he necessry condiions wih he hel of he fundenl soluion of he given djoin *orresonding uhor Received: 8 Augus / Acceed: Ferury 3 euion By ing dvnge of hese condiions firs he oundry condiions re chnged for nonlocl o locl nd finlly s efore here will e locl cse nd he reson for he oundry lyer loc will e sudied [] Mheicl seen of role We consider he following oundry lyer role: l [ ] f where is sll reer re he sure rices of n order in which he eleens re rel coninuous funcions nd f nd re colun vecors whose funcions re rel coninuous nd f coefficiens of euion re nown funcions while is unnown vecor funcion D of oundry condiion in he role h is nd is sure rices of n order wih rel consn eleens nd of colun vecor hs n coonens wih rel consn eleens Euion esily shows h when i chnges o n lgeric syse Bsed on his fc i cn e verified wheher soluions of liner lgeric syse exis in oundry condiion or no As fr s we now if

2 IJS 3 37A3 Secil issue-mheics: he lii funcion is sisfied in oundry condiion here is no oundry lyer in ny of he nd oins If he lii funcion is no sisfied in oundry condiion hen oundry lyer exiss 3 he djoin euion of in euion o oin he djoin euion we firs eslish he following le 3 Le If re coninuous funcions hen he ssocied djoin euion of euion will e: ly y [ ] y 3 where nd rices re he rnsoses of Proof: o do his we e o ge he Lgrnge forul of syse [5]onsider he following sclr roduc of rel funcions: y l y y y [ y l d y [ ] y { y y [ ] d d d ] y } d 4 Forl soluion Now we refer o consrucing he forl soluion of he syse y [ ] y 4 wih reer A firs we verify roof of he nex le: 4 Le If in he syse 4 rix eleens of nd re infiniely differenile funcions nd he roos of he chrcerisic euion of his syse de[ E] 5 E is he ideniy rix of order n re disinc nd he rel rs of he re no zero h is: Re s s [ ] n hen he forl soluion of syse will e d y e ; [ ] 6 n where n Here is scler funcions Moreover re he colun vecors of n coonens in which he eleens re funcions h re oined fro hese syses: 7 or According o Lgrnge forul he inside of he ove inegrl er gives he ssocied djoin euion 3 If we use r of he inegrl ehod for he firs senence on he righ hnd side he Lgrnge forul leds o he resen inegrls of he le in forul 3

3 39 IJS 3 37A3 Secil issue-mheics: n n n n nn n n n n n n n n n Proof: In order o verify le we u he forl soluion 6 in syse 4 nd oi exonenil ers fro oh sides I cn e esily seen h relions 7 re he coefficiens of successive owers of sll reer According o he generl for of 7 he firs syse of euions is: [ E] 8 or n n n n nn n In order o hve non rivil soluions he deerinn of syse 8 should e zero I ens h for n re he roos of chrcerisic euion 5 In his cse for fixed he eleens of will e: j jr jr P e j n 9 where j re rirry funcions nd P includes he whole of ssocied co-fcors of r row of rix P E so i is shown s P jr e jr Here we should choose r in such wy h co-fcors of he row do no ecoe zero siulneously I should e h os one row of rix P E cn e zero s resul of hving differen roos Also e jr is del of Kroncer h is: e jr j r j r he colun vecor which consiss of he rirry funcions in 9 is resuled fro wriing insed of h is soluion of syse 8 Becuse he syse 8 is hoogeneous so funcions of j re rirry funcions By fcoring hese rirry funcions finlly we ulily he eleens of one row of rix P E o co -fcors of row r in h rix his one gin is zero sed on liner lger heores [6] So he unnown sclrs funcions of nd lso vecors of firs syse in he syse se 7 cn e oined Now in order o verify he soluion of he second syse in 7 in which we should find he resuled syse fro unnown colun vecor = o e he syse solvle i is necessry h he righ hnd side of he resuled syse fro ecoes orhogonl o he eigenvecor of djoin euions Becuse he righ hnd includes he rirry funcion so we cn ly he orhogonliy condiion ecuse he deerinn of coefficiens of he resuled syse fro is zero By following his rocess he whole soluions of he firs nd second euions of yses 7 will e clculed s he deerined funcions of Now Le us confir his forl soluion s rel nd clssic soluion For his y idenifying he

4 IJS 3 37A3 Secil issue-mheics: nd of unnown funcions in syse 7 nd ccording o forl soluion 6 he nex er h should e considerd is exonenil ers in 6 Bsed on choosing which ws discussed in he ove le when exonenil er is will led o zero Finlly ccording o infiniy series in he for of forl soluion of 6 he syoic exnsions of liner indeenden soluions of 4 cn e oined [4-6] We cn conclude his le: 4 Le On he condiion of le 4 he syoic exnsions of y for n y : leds o d M M y e O ; [ ] n M where O is error er 5 Fundenl soluion Now for clculing he fundenl soluion consider he non-hoogeneous syse reled o syse 4: [ ] g where sure rices of n order in h consis of rel funcions nd is unnown vecor Here g is he nown colun vecor of n order A firs we eslish he following le 5 Le Under he condiions of le 4 fundenl soluion of syse will e: Here is he funcion of Heviside nd is he rix soluion of syse 4 Proof: A firs we oin he generl soluion of non hoogeneous syse hrough vriion of vrile ehod: A 3 Here A is he unnown colun vecor By using 3 in we will hve: [ A ] A A g where is he rix soluion of inhoogeneous syse 4: or A A g g Here we cn find his relion for A A A g d 4 If in 4 we suose A nd u his one in he soluion of he non hoogeneous syse will e: y g d g d g d g d Here we cn see h he fundenl soluion of 4 is in he for of In fc if we u soluion in he lef side of 4 i will e:

5 393 IJS 3 37A3 Secil issue-mheics: ] [ ] [ } ]} [ { ] [ Noe h he second er of he ls relion will e zero ecuse is soluion of reled hoogenous syse 4 So his soluion is he fundenl soluion of he syse 4 h is: [ ] E 5 Where is he Dirc-del funcion nd E is he ideniy rix of n order 6 Fundenl relions nd necessry condiions A firs we verify he following le: 6 Le Under he condiions of le 4 he rirry soluion of syse is sisfied in he following relions: f d 6 In which he second cse gives us he necessry condiions Proof: Nex we ulily he wo sides of syse o rnsose fundenl soluion in he lef hnd nd hen oin he inegrl in ] [ d f d d ] [ or ] [ d f d d he relion 5 nd fro he roeries of he Dirc-del funcion [8] [9] give us relions 6 { [ ]} f d d d We use he second cse of ls relion nd ge he necessry condiion s follows: d f d f According o firs relion we will hve: f d 7

6 IJS 3 37A3 Secil issue-mheics: Noe h we show We should consider h in he second relion he ove will e s idenified s follows f d he inegrl er will e zero fro 7 Loclizion of oundry condiions In his r we conver non-locl condiions y using he ove necessry condiions for locl oundry condiions A firs we rove he following le 7 Le Under he condiions of le 4 we ssue: de[ ] 8 so for oundry vlues of unnown funcion we hve he following relions: f d [ ] [ f d] [ ] [ f d] 9 Proof: If we consider he relion 7 nd oundry condiion s n lgeric syse for he unnowns nd nd solve his syse y rer rule we will hve he relions 9 for oundry vlues of In his cse he nonlocl oundry condiions were convered o locl oundry condiions 8 Soluion of in oundry vlue role In order o give nlyicl seen o soluion of in role - we rove his heore: 8 heore Assue in he syse of in role is sll reer nd re he sure rices of n order in which he eleens re funcions of infiniely differenile f is funcion of colun vecors h in coninuous is unnown vecor of n order nd re sure rices of n order which hs consn eleens nd is consn colun vecor of n order nd non-locl oundry condiions re liner indeenden Moreover he roos of he chrcerisic euion 5 re disinc nd heir rel rs will no e zero Also we ssue condiion 8 will hold hen he soluion of he oundry lyer role - will e: f d [ f d ] [ ] [ f d] [ ] [ f d] where is he rnsose of fundenl soluion Proof: According o he disinc roos in euion 5 nd he for of forl soluion 6 in he syse 4 we cn oin norl for of i in le 5 On he oher hnd syoic exnsion of norl liner indeenden soluion 4 hs een given in le 4 nd in le 5 he fundenl soluion of syse 4 sed on he relion nd lso he sic relion nd necessry condiion y relion 7 is offered Finlly in le 7 we cn find oundry condiion in locl for In his siuion in order o verify he heore i is enough o u he oundry condiions of loclizion 9 in he lef hnd side of 6 Firs cse of relion 6 gives he soluion of role - 9 Liiing ehvior of soluion Now le us sudy he liied siuion of soluion where For his when he lii of syse is f If he deerinn of he ove syse is no zero h is:

7 395 IJS 3 37A3 Secil issue-mheics: de[ ] 3 So f 4 In he syse 5 where he fundenl soluion is usle when he soluion will e: E 5 or such s we will hve 6 [ ] Now y considering 6 he liiing relions 9 nd when will e: f f nd 7 f 8 We consider he relion 8 is he se s relion 4 which is liiing se of role - he forion of oundry lyers As enioned in he inroducion he forion or non forion of oundry lyers is iorn for consrucing roxie soluions [] [3] hey esily see he liiing siuion of soluion of he in role is djusle in liiing siuion of syse hence we hve his heore: heore Under he condiions of heore 8 nd he condiion If we hve his relion de[ ] f f 9 hen here will e no oundry lyer in ny of he oundry oin's Proof: We cn see fro 9 h he liiing siuion of soluion in role - when in re sisfied wih he liiing siuion of oundry condiion Hence in his cse ccording o oundry lyer roles we hve no oundry lyers heore Under he condiions of heore nd he condiion: de[ ] Also he condiion 9 does no hold so I if f he oundry lyer fors jus in II if f oundry lyer will e in oh nd Proof: In fc if we don hve condiion 9 here will e fored oundry lyer Bu if f we cn do he liiing siuion of soluion in firs relion 9 However does no sisfy wih second relion 9 Becuse of his oundry lyer in will e fored Referring o he fc h he lii funcion is no sisfied wih relions 9 finlly in oh nd oins we will hve oundry lyer Acnowledgeen he uhors re greful o ProfM K Mirni for ediing he er nd reviewers for heir suggesions o irove nuscri of he er References [] Prndle L 934 Uer Flussigeis-ewegung ei Kleiner Reiung Verhndlungen III IN Mh Gongress uener Leizig [] O Mlley J R E 974 Inroducion o Singulr Perurion New or Acdeic Press [3] O Mlley J R E 99 Singulr Perurion Mehods for O D E s Sringer Verlg [4] Dooln E P Miller J J & Schilders W H 98 Unifor Nuericl Mehods for Proles wih Iniil nd Boundry Lyers Boole Press Dulin [5] Birhoff G D 98 On he syeoic of he soluion of cerin liner differenil euions conining reer rns Aer Mh Soc 9-3 & [6] ohen D S 973 Mulile soluions of nonliner ril differenil euions Nonliner roles in he hysicl sciences nd iologg eds I Sgold D D Joseh D H Singer Lecure Noes in Mheics 3 Sringer Verlg Berlin 5-77 [7] Keller H B 968uulr cheicl recors wih recycles SIAM-AMS Proc 8 ed cohen [8] Nir M A 968 Liner differenil oerors r II New or Unger [9] Veldiirov V S 97 Euion of Mheicl Physics 3rd edn Nu Moscow

8 IJS 3 37A3 Secil issue-mheics: [] Jhnshhi M Invesigion of oundry lyers in singulr erurion role including 4h order ordinry differenil euions Journl of sciences Islic Reulic of Irn ehrn [] Srhsi A R & Jhnshhi M 9 Asyoic soluion of singulr erurion role for second order liner O D E wih locl oundry condiions Exended Asrcs 4h Annul Irnin Mheics onference [] Srhsi A R & Jhnshhi M Invesigion of oundry lyers in singulr erurion roles wih generl liner non-locl oundry condiions Exended Asrcs o IV ongress of he uric world heicl sociey Bu Azryjn [3] Srhsi A R & Jhnshhi M Asyoic Soluion of Prole of Singulr Perurion of Second-Order Liner wih onsn oefficiens wih Dirichle ondiion Journl of Sciences ehrn Universiy of ri Molle [4] Srhsi A R Jhnshhi M Ashrfi S Srhsi M 3 Invesigion of Boundry Lyers in Soe Singulr Perurion Proles Including Fourh Order Ordinry Differenil Euion World Alied Sciences Journl [5] Ashrfi S Medov & Aliev N Invesigion of Arising of Boundry Lyer in Boundry Vlue Prole for he Fourh Order Ordinry Differenil Euions J Bsic Al Sci Res [6] Ashrfi S & Aliev N Invesigion of Boundry Lyers for Second Order Euions under Locl nd Non- Locl Boundry ondiions J Bsic Al Sci Res