APROXIMATE SOLUTION FOR A COUPLED SYSTEM OF DIFFERENTIAL EQUATIONS ARISING FROM A THERMAL IGNITION PROBLEM

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1 IJAS 3 Marc _0.pdf APOXIMATE SOLUTION FO A COUPLED SYSTEM OF DIFFEENTIAL EQUATIONS AISING FOM A THEMAL IGNITION POBLEM Sdra Ad Far Sa & Nad Salaa Skkr In of Bn Adnraon Skkr Karakora Inrnaonal Unry Gl ABSTAT T arcl prn approa r olon of a copld y of a paral dffrnal aon and ordnary dffrnal aon. T copl y ar fro fa n caalyc conrr n aoobl nnrn. T ooopy analy od HAM appld o oban r olon. T HAM olon conan an alary parar wc prod a connn way of conrolln conrnc ron of r olon. l ar prnd for dffrn ca d and copard for a ca wr conno olon plcly known. Kyword: Aoobl nnrn approa olon o y of PDE and ODE.. INTODUCTION T a y of aoobl locad bwn nn ol and a pp. T y a a dc a known a caalyc conrr. I fncon o conr pollan a flown o of nn no arl a. A proc na prr of o nl a a p conrr. Tr no nfcan ccal racon a a. Afr o paa of prar nd conrr ra and rac a nfcan ll for racon o occr. A racon ar p or a nra. T proc dd conrr no wo ron. A ron w low prar and l racon ra wl or ron w prar and lar racon ra. T ranon a bwn wo ron calld l-off fron. T loff fron o oward conrr nl. T wol proc affc pollon conrol of an aoobl. Hnc ndrandn of oor cl a on an poran apc n aoobl nnrn. T conrn pnona of caaly can b dcrbd n r of nrfac a-ranfr odl 045]. T dy bad on aacal odl prnd by Lon and Can 4]. T odl con of a copld y of paral dffrnal aon and an ordnary dffrnal aon. L T and T b cl prar and conrr prar rpcly. T y of aon n a follow a prnd n 4] a 6 T 6 T 6 a c p U ] a T T. n T T T r c p a 0 T T a r a A > 00 < n T 0 T > 0 0 T 0 T 0 T 0 < T pycal ann of fncon and or r n abo y can b fond n 7]. In ordr o plfy odl w l n n T T T T a U.. c a c b r c r a r c l. A wr T 6 n 0 T 6 p p p. Wc a c c > 00 < l.3 04

2 IJAS 3 Marc 05 Ad al. Proa Solon for a Copld Sy b b > 00 < l 0 > <. 0 0 l Sral aacal odl a bn prnd for aoobl on probl. Howr o of lrar rlad w cancal prnaon and nrcal laon 0 7. Anor dy ] bad on plfd ron wc a calar nal bondary al probl. A dffrn aacal odl conn of copld y of a aon and an ordnary dffrnal aon prnd n ]. T aor nad rann baor of a caalyc conrr n fr fw n of opraon. A aacal odl a dcrb lcrcally ad onol conrr wa propod n 3] n ordr o dy cold ar on prforanc. In arcl w condr copld y.. T papr prn r analyc olon for odl.. T ooopy analy od HAM appld o probl. T od wa fr propod by Lao n PH. D. 3]. A yac dcrpon on HAM prnd n 4]. In rcn yar od a bn ccflly ployd o ol any yp of nonlnar oono or nonoono aon and y of aon occrrd n cnc and nnrn probl 9 5 6]. T HAM conan a cran alary parar wc prod w a pl way o adj and conrol conrnc ron and ra of conrnc of r olon. In papr an alrna approac bad on HAM prnd o approa olon on nonlnar copld y of PDE. T arcl orand a follow: Scon II prn a brf nrodcon o ooopy analy od. In con III r analyc olon prnd for copld y. Scon IV copr of apl and rfac plo. A concldn rark n n fnal con.. BASIC IDEAS OF HAM W condr follown dffrnal aon wr N ] 0... n N ar nonlnar opraor a rprn wol aon and dno ndpndn arabl and b nknown fncon rpcly. By an of nraln radonal ooopy od Lao 3] conrcd o-calld ro-ordr dforaon aon L : 0 ] N : ]. wr 0] an bddn parar ar nonro alary fncon L an alary lnar opraor ar nal of and ar ; nknown fncon. I poran o no a on 0 a ra frdo o coo alary objc n HAM. Oboly wn 0 and bo ;0 0 and ; old. T a ncra fro 0 o olon ar fro nal o olon. Epandn n Taylor r w rpc o on a 0 ;. wr ;! 0.3 If alary lnar opraor nal alary parar and alary fncon and alary fncon ar proprly con n r aon. conr a and 05

3 IJAS 3 Marc 05 Ad al. Proa Solon for a Copld Sy ; 0 wc b on of olon of ornal nonlnar aon a prod by Lao ]. A E. bco L : 0 ] N : ].4 wc ar frnly d n ooopy-prrbaon od 9]. Accordn o.3 ornn aon can b ddcd fro ro-ordr dforaon aon.. Dfn cor {. }. n o n Dffrnan. w rpc o bddn parar and n n 0 and fnally ddn by! w a o-calld -ordr dforaon aon L ]..5 wr and! 0 N ; ].6 0 >.. APPLICATIONS W wll apply HAM o nonlnar copld y of paral dffrnal aon and ordnary dffrnal aon o llra rn of od and o abl approa r olon for probl.. A. An Eapl w Solon Known Eplcly.. Eapl Condr follown dffrnal y: wr > 0 a parar and 0 < < T0 < 0 < < T0 < 0 0 < < T < To ol y 3. by an of ooopy analy od l coo nal approaon 0 and lnar opraor 0 ; L ; ] w propry L c ] 0 wr c ar nral conan. Un dfnon ro-ordr dforaon aon ar 06

4 IJAS 3 Marc 05 Ad al. Proa Solon for a Copld Sy 07. ] : ] : 0 N L For 0 and w a ;0 0 0 ; ; ; T a ncra fro 0 o olon ar fro nal o olon. Epandn n Taylor r w rpc o on a ; 0 wr 0 ;! If alary lnar opraor nal alary parar and alary fncon and alary fncon ar proprly con n r aon. conr a and w a 0 0 wc b on of olon of y. Condr a cor wc dfnd a }.. { n o n So -ordr dforaon aon ar. ] L 3.3 w nal condon and T olon of -ordr dforaon aon 3.3 for n a 0 c d wr nraon conan c ar oband by nal condon 3.4. T r olon pron by HAM can b wrn n for For approad r olon ar copd a follow. 3.5 ] ] For rap of and ar prnd n Fr blow.

5 IJAS 3 Marc 05 Ad al. Proa Solon for a Copld Sy.. Eapl To onra applcaon of HAM for a larr cla of racon fncon w condr follown or nral y. a c a a 0 < < T0 < c b b 0 < < T0 <

6 IJAS 3 Marc 05 Ad al. Proa Solon for a Copld Sy 0 0 < < T <. wr a and ar nonna conan Sr olon for ca b b c 0 a a b b3 3/ 3/ a / and HAM a follow. ar copd by b a. b b b cb / b / /6]]. T aldy of od bad on apon a r conr a. I alary parar wc nr a apon can b afd. In nral by an of o-calld -cr raforward o coo a propr al of wc nr a olon r conrn. 3. CONCLUSION In papr own a ow HAM can b appld o copld y of aon rlad o aoobl nnrn. T r olon ar oband by applyn HAM. T analycal approa olon for or nral probl ar prnd fr. T plod rl ar w o nrcal rl alrady prnd n lrar. T adana of HAM alary parar wc prod a connn way of conrolln conrnc ron of r olon wc no pobl n or analycal od. EFEENCES ]. A. Sa Baan M. S. M. Nooran I. Ha Nrcal laon of y of PDE b araonal raon od Py L. A ]. C. P. Pla P. S. Haan and D. W. Scwndan L-off baor of caalyc conrr SIAM J Appl Ma ]. C. V. Pao Nrcal od for lnar Bolann ranpor aon n lab oory J Nr Mod Par Dffr E ]. D. T. Lon and H. C. Can A ory for fa-nn caalyc conrr AICHE J ]. D. M. Yon Ira olon of lar lnar y Acadc Pr Nw York 97. 6]. E. J. B and S. H. O Elcrcally ad conrr for aoo on conrol: drnaon of b r for ad ln Cn En Sc ]. G. P. Crpano Two dnonal conc a/a ranfr for low Prandl and any pcl nbr SIAM J Appl Ma ]. J. D. Hornl A nonaonary probl copln PDE and ODE dlln aoobl caalyc conrr Appl Anal ]. S. C. Caan and V. K. Sraaa Modlln a a pollon aban: Par I nl ydrocarbon propyln Cop Ma Appl ]. S. N. Ha S. W. o and J. Park Nrcal dy of an opal conrol probl for a caalyc conrr fncon Cop a Appl ]. S. Snd J. P. Lclrc D. Scwc M. prn and F. Caana Tr-way onolo conrr laon r prn Cn En Sc ]. S. H. O and C. J Cand Trann of onol caalyc conrr: rpon p can n fdra prar a rlad o conrolln aoobl on Ind En C Prod D ]. S. J.Lao T propod ooopy analy cn for olon of nonlnar probl P. D. draon Sana Jao Tan Unry Snaa 99. 4]. S. J. Lao Byond prrbaon: Inrodcon o ooopy analy od CC Pr Boca aon Capan and Hall ]. S. J. Lao An approa olon cn wc do no dpnd pon all parar Par : An applcaon n fld canc Inrn. J. Non-lnar Mc ]. Y. H. Can G. C. Ja and C. V. Pao Blow p and lobal anc of olon for a caalyc conrr n nrfa a ranfr Nonlnar Anal al World Anal