New Parabolic Inequalities and Applications to Thermal Explosion and Diffusion Models

Transcription

1 ew arabolic Ieqaliies Alicaios o Thermal Elosio Diffsio Models AHiwarekar Absrac I his aer we obai ew comariso riciles for arabolic bodary vale roblems i wiho assmig he sal moooic codiio wih resec o Sarig wih he basic heory i secio we develoed ew comariso riciles i secio Alicaios of he heory develoed i his aer is sed o discss he moooiciy of he solio of a hermal elosio roblem We geeralized he resls for arabolic sysems i secio 4 discss is alicaios o diffsio model Ide Terms arabolic bodary vale roblems Comariso riciles Differeial Ieqaliies arabolic Elliic I ITRODUCTIO Eesive lierare o arabolic bodary vale roblems Waler [4] ao [] roer Weiberger [] Taylor [] has esablished heir domia role i he field of alied idsrial Mahemaics Sch bodary vale roblems lay a imora role i he followig fields: combsio heory Arioli Gazzola [] Kasre [7] Thermal bodary layer Dhaigde Kasre [5] chemical eclear egieerig Ladde Lakshmikaham [8] olaio dyamics Al Foso casro Maya Shivaji [] Bodary layer heory Waler [4] Mli comoe diffsio Mc abb [] Aleer L Lee eal [] deecio of roeis C Amaorel eal [4] lbricaio sli model K Ai Hadi [6] bioelecroaalyical sysem M ida F Ivaaskas I Igajev G Vali V Razmas [9] I is ieresig o discss he moooiciy roery of a solio for a give bodary vale roblem wih resec o ay oe of he arameer whe all oher arameers remais fied Sch a sdy is imora for real world roblems govered by arabolic bodary vale roblems Maimm riciles comariso heorems for liear o-liear arabolic eqaios wih resec o are kow roer Weiberger [] Waler [4] However a variey of o-liear oeraors where moooiciy codiio is o saisfied occr arally i may hysical alicaios Therefore we eed he resls wiho moooiciy codiio wih resec o obai more geeral resls Mascri received Feb 4; revised March 4 A Hiwarekar is workig i Vidya raishhas College of Egieerig Baramai (Uiversiy of e) Idia Mob: ( hiwarekarail@gmailcom) Le D be a oe coeced sbse of R wih bodary D D be he closre of D The ois of D are deoed by Le G be he oological rodc of a oe domai D of he -sace R a ierval T Sose G T D D R Le G be he bodary of G The ois of G are deoed by For a real valed fcio which is coiosly differeiable deoes g ra d The bodary disjoi sbses is a mari r where r G ca be divided io hree airwise G G where G G G / U G D G G / U G T D G G G G T D U is a lower half eighborhood of Ths we ge G G G G G G G R G G Frher deails are coaied i Waler [4] II COMARISO RICILE We cosider o liear differeial oeraor of a ariclar ye give by F q r () ISB: ISS: (ri); ISS: (Olie) WCE 4

2 where F is a real valed fcio is he ime variable q D is he sace variable : D R wih C D C D Defiiio : The oeraor give by () is said o be arabolic a a oi if F is moooe decreasig i r weakly moooe icreasig i q we say ha F Defiiio : A comariso ricile wih sric ieqaliy for a oeraor is said o hold if for wo fcios v v i G v o R o R imlies v i G Defiiio : A comariso ricile allowig eqaliy sig (weak comariso ricile) for oeraor is said o hold if for wo fcios v v i G v i G v o R o R imlies Sch comariso riciles are available i roer Weiberger [] for liear arabolic oeraor wiho assmig a moooic codiio wih resec for o-liear arabolic oeraors Waler W [4] assmig moooic codiios Here we obai a weak comariso ricile for o-liear oeraor wiho assmig a moooic codiio wih resec which is coaied i Theorem We reqire he followig v Z F if Theorem : If F v o R o R F v v v v F w w w w i G he v i G [4] Theorem : If F v Z F F is coiosly differeiable wih resec o v q for all ois F F G is boded below > v q v o R o R F v v v v F w w w w i G he v i G roof :- Le v v e > So ha v v e v v v v The F v v v v F v v v v F v v v v F v v v v F v v v v F ve v e v v F v v v v F v v v v Alyig he mea vale heorem we ge F v v v v F v v v v F * * F * * e v v v v v v v v v q * * where v v v v is a oi o he segme joiig v e v e v v v v v v o By assmio F is boded below v large ha F F > q so v ca be chose so ha F v v v v F v v v v > i G Frher v v o R o R sig Theorem we ge v i G Takig he limi as we ge he reqired resl We ge he followig Theorem as a secial case of Theorem wih f Theorem : For wo fcios v Z f v i G v o R o R imlies v i G rovided f is coiosly f differeiable wih resec o v is boded above v for all ois G ow he e secio is a alicaios of he above resls o a hermal elosio roblem ISB: ISS: (ri); ISS: (Olie) WCE 4

3 III MOOTOICITY WITH RESECT TO ARAMETERS AD ALICATIOS TO THERMAL EXLOSIO ROBLEM Comariso riciles osiiviy-egaiviy Theorems are imora ools for obaiig moooiciy of a solio wih resecive arameers I is ieresig o discss moooiciy roery of a solio for a give bodary vale roblem wih resec o ay oe arameer whe all oher arameers remai fied wiho fidig he solio elicily Sch a sdy is very imora for real world roblems Here we cosider he followig A Thermal Elosio roblem We cosider he followig arabolic bodary vale roblem: D e / i G g o G o G The vales of R so rescribed o R G G will be deoed by G o R The roblem () () wih = rereses a hysical roblem of hermal elosio ao [] Here D rereses he hermal diffsiviy he cosa > is called he Arrheis mber > is a cosa is he emerare a ime For he above roblem icrease of o i oe of he arameers D is a idicaio of a qicker blow Therefore he sdy of moooiciy wih resec o a arameer for sch arabolic ieqaliies is imora We sdy he dimesioal aaloge of he hermal elosio roblem; for R R We reqire he followig codiio: m Mi { G } : Le R Sch a codiio is legiimae becase for blow roblems he iiial bodary codiios are eeced o be high ow we have he followig Theorem : The solio of roblem () () is sricly osiive boded below der codiio Theorem : The solio of roblem () () icreases wih der codiio roof: I his roblem he solio of he roblem () () saisfies m i G de o codiio Differeiaig () arially wih resec o assmig U we ge U D U U e e Takig f U U U D U U e he f e which is boded above U by e Ths m LU U f U U U e i G U o R Hece i follows from Theorem ha U i G Ths is a icreasig fcio of Theorem : The solio of roblem () () icreases wih der codiio where m roof: Differeiaig () arially wih resec o assmig V we ge V D V e V e Takig F V V V D V V e F e e which is boded V m above ow V V F V V V e i G V o R Hece i follows from Theorem ha V i G Ths is a icreasig fcio of ow we will rove he moooiciy wih resec o he hird arameer D for which we reqire he followig relimiary lemma g saisfy he followig Lemma : Le fcio codiio g o G : The i G i G Theorem 4: Le all he codiios of Theorem hold I addiio sose codiio is saisfied he he solio of roblem () () decreases wih D ISB: ISS: (ri); ISS: (Olie) WCE 4

4 IV O-LIEAR ARABOLIC SYSTEM AD ALICATIO Le R wih comoes give by R q E for grad Sose r is a i j Mari for i ; j a d We cosider he followig sysem of arabolic oeraors give by F F F F defied by F F 4 F We eed he followig defiiio Defiiio 4: The fcio F if i is moooic icreasig i r sricly icreasig i q Defiiio 4: If each F for he we say ha fcio F We refer o Waler[4] for frher deails of oaios We eed he followig v Z f Comariso ricile 4: If F we have v o R o R : : F F v v v v i G imly v i G rovided F is qasimoooic icreasig i We eed he above resl by obaiig a comariso ricile allowig eqaliy sig which is coaied i he followig v Z f Comariso ricile 4: Le F for The : v o R a d o R : F F v v v v i G ISB: ISS: (ri); ISS: (Olie) Imlies v i G rovided F saisfies a Lischiz s codiio of he followig ye: For here eis egaive fcios c c c c c cl cl l sch ha l c l is boded below F F l c for l l l F is differeiable wih resec o q F q o D f roof: Le v v e herefore v v e v v v v Cosider F v v v v F v v v v F v e v e v v F v v v v Usig he mea vale heorem codiio we ge F v v v v F v v v v cl e e l e cl l F v v v v * * q F v v v v q F Sice q * * l c l (4) is boded below we ca choose sfficiely large so ha righ h side of (4) is greaer ha zero i G By assmios we have F F v v v v i G for v o R a d o R So by comariso ricile 4 v i G Takig he limi as we ge v i G May hysical heomea are govered by arabolic sysems sch as he diffsio of ios oi defecs i meals codcio of hea i heerogeeos media rasor of waer hrogh la isse rom walk model Aleer L Lee James M Hills [] I he e secio we cosider a alicaio of he above resls o he rom walk model WCE 4

5 A Alicaios o Diffsio Model Here we cosider he followig sysem i R which describes he diffsio i resece of hree diffsio ahs i R fo r ; [] D a a a D a a a D a a a i G (4) g g g o R (44) are he coceraios where D D D are diffsiviies are o-egaive cosas The o-egaive cosas a ij rereses he rasiio robabiliies i he diffsio model Aleer L Lee James M Hills [] obaied he codiios der which aleas oe of he hree aais is comoes maimm o R I wha follows we obai he codiios der which all hree comoes aais heir maimm o R Sice ors is a maimm ricile for a bodary vale roblem he resl deeds aar from he coefficies o he bodary vales g g g also Theorem 4: For he sysem (4) (44) we assme he followig codiios: 6 : are coios wih heir derivaives o secod order i G saisfy codiios (4) (4) 6 : g g g o R are coios here a a a m 6 : k Mi a a a a a a m where m M a m m m m M i m m m are R R osiive o acco of (6 ) M Ma m Mi i The i R i i i R aai heir maimm o R roof: Le L D a a a L D a a a L D a a a L L L L M M M M ow L L L as saisfy (4) (44) hece L Frher by codiio 6 o R Ths L L i G o R imlies i G Ths osiive ow L M a M a M a M L M a M a M a M are L M a M a M a M By assmed codiio 6 LM am a M am am a a m L Similarly L M L a d L M L Hece by comariso ricile 4 alied o L L M i G M o R follows ha M i G Hece aai heir maimm o R REMARK 4: As maimm vales of deed o bodary vales if he bodary vales are chaged he he maimm ricile may or may o hold as he codiio 6 may be saisfied for cerai bodary vales may o be saisfied for oher bodary vales V COCLUDIG REMARKS Ths kow heory of arabolic ieqaliies is develoed frher by obaiig ew comariso riciles for sigle eqaios as well as for arabolic sysems showig heir alicaios for hermal elosio diffsio models ACKOWLEDGMET The ahor haks rof DYKasre for his gidace hrogho rearaio of his aer Ahor also haks Vidya raishhas College of Egieerig Baramai (Uiversiy of e) Idia for sor o his work REFERECES [] Aleer LLee James MHills Maimm riciles for liear sysems erveo i Redazioe i gigo [] Al Foso CasroMaya C Shivaji R o-liear eige vale roblems wih semiosioe srcre oliear differeial eqaios ElecJ Diffe Eqaios cof5-49 [] Arioli Gazzola F Gra H C Miidieri E A semiliear forh order elliic roblems wih eoeial olieariy mrs rojec"meodi variazioali ed Eqazioi Differeziali oliear" Oc -7 [4] C Amaore A Oleiick I Svir da Moa L Thoi Theoreical Modelig Oimizaio of he Deecio erformace: a ew Coce for Elecrochemical Deecio of roeis i Microflidic Chaels oliear Aalysis: Modellig Corol 6 Vol o [5] Dhaigde Kasre o-liear ime degeerae arabolic ieqaliies alicaios J of MahAalAlVol75 o ISB: ISS: (ri); ISS: (Olie) WCE 4

6 [6] Hadi K Ai Bolder arabolic oliear Secod Order Sli Reyolds Eqaio: Aroimaio Eisece oliear Aalysis: Modellig Corol 7 Vol o [7] Kasre D Y Alicaios of ieqaliies o Mechaics Lecre oes a I I T Madras 99 [8] Ladde G S Lakshmikaham V Vasala A S Moooe Ieraive Techiqes for oliear differeial eqaios ima Advaced b 985 [9] M ida F Ivaaskas I Igajev G Valiˇcis V Razmas Comaioal Modelig of he Ameromeric Bioaalyical Sysem for Liase Aciviy Assaya Time-Deede Resose oliear Aalysis: Modellig Corol 7 Vol o 45 5 [] McABB Comariso eisece heorems for mli comoe diffsio sysems JMAAVol o96-44 [] ao C V oliear arabolic elliic eqaios lemm ew York 99 [] roer M H Weiberger H F Maimm riciles i differeial eqaios reice Hall Ic (967) Taylor A B [] Taylor A B- Mahemaical models i Alied Mechaics Claredo ress Oford 986 [4] Waler W Differeial ieqaliies Fify years o from Hardy Lilewood olya (Edied by W Everi) ISB: ISS: (ri); ISS: (Olie) WCE 4