Some numerical methods for Schnackenberg model

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1 Ierol Jorl of Egeerg Ieos e-issn: p-isbn: 9-69 olme Isse Jry PP: 7-78 S. M Deprme of Mhemcs cly of Scece ersy of Zkho Dhok Krs Rego Irq src:- I hs pper Schckeerg moel hs ee sole merclly for fg pprome solo y e fferece meho om ecomposo meho. Emple showe h DM more ccre h DM more effce for hs k of prolems s show les fgres -. Keywors:- om Decomposo meho Prl Dfferel Eqos Schckeerg eqos. I. INTRODCTION My physcl chemcl egeerg prolems mhemclly c e moele he form of sysem of prl fferel eqos or sysem of orry fferel eqos. g he ec solo for he oe prolems whch ole prl fferel eqos s ffcl some cses. Here we he o f he mercl solo of hese prolems sg compers whch cme o esece ]. or oler prl fferel eqos howeer he ler sperposo prcple co e pple o geere ew solo. So ecse mos solo mehos for ler eqos co e pple o oler eqos here s o geerl meho of fg lycl solos of oler prl fferel eqo mercl echqes re slly reqre for her solo6].. MTHEMTIC MODE: geerl clss of oler-ffso sysem s he form f g f g Wh homogeos rchle or em ory coo o oe om Ω wh loclly lpschz coos ory. I s well kow h reco ffso of chemcl or ochemcl speces c proce rey of spl pers. Ths clss of reco ffso sysems cles some sgfc per formo eqos rsg from he moelg of kecs of chemcl or ochemcl recos from he ologcl per formo heory. I hs grop he followg for sysems re ypclly mpor sere s mhemcl moels physcl chemsry ology: Brsselor moel: f g g where re pose coss. Gry-Sco moel: k f g g where k re pose coss. Glycolyss moel: k k f g p g where k p re pose coss. Schckeerg moel: k f g g Where k re pose coss ]. The oe os he followg sysem of wo olerly cople reco-ffso eqos P g e 7

2 k Wh l ory coos: = = > Ω = = Ω wh Nem ory coos: Where k re pose coss ]. Reco-ffso RD sysems rse freqely he sy of chemcl ologcl pheome re rlly moele y prolc prl fferel eqos PDEs. The ymcs of RD sysems hs ee he sjec of ese reserch cy oer he ps eces. The reso s h RD sysem eh ery rch ymc ehor clg peroc qs-peroc solos ]. ros orers re self-orgze fr from he chemcl eqlrm. The heorecl proceres oos o escre he ymcs of pers formo he ee eelope for he ls hree eces ]. emps he lso ee me o ers morphologcl orers ology 5]. Clrfco of he mechsms of he formo of orers he reloshp mog hem hs ee oe of he fmel prolems o-eqlrm sscl physcs 9]. ros fe fferece lgorhms or schemes he ee presee for he solo of hyperolcprolc prolem or s smpler eres sch s he clsscl ffso eqo. I s well-kow h my of hese schemes re prlly ssfcory e o he formo of oscllos mercl ffso wh he solo ]. Solo y he fe fferece meho lhogh more geerl wll ole sly coergece prolems my reqre specl hlg of ory coos my reqre lrge comper sorge eeco me. The prolem of mercl sperso for fe fferece solos s lso ffcl o oercome 7]. om ecomposo hs ee pple o sole my fcol eqos so fr. I hs rcle we he se hs meho o sole he he eqo whch goers o meros scefc egeerg epermeos. Some specl cses of he eqo re sole s emples o llsre ly relly of he meho. Resrcos o pplyg om ecomposo meho for hese eqos re scsse ]. The ecomposo meho c e ce procere for lycl solo of we clss of ymcl sysems who lerzo or wek olery ssmpos closre ppromos perro heory or resrce ssmpos o sochscy ]. II. MTERIS ND METHODS. INITE DIERENCE PPROXIMTIONS The fe fferece Scheme geerlly reces ler oler prl fferel eqos o sysem of ler oler eqos ros mehos were eelope o f he mercl solo ccelero he coergece 8]. ssme h he recgle R { : } s se o - y m- recgle wh ses h k. Sr he oom row where = = he solo s f. The gr spcg s form eery row: + = + h = h s form eery colm: j + = j + k j = j k. se he ppromo for o o 8]. = j + j = j + j = j j + +j 7 j j + +j 8 Sse 5 8 he Schckeerg moel o ge 5 6 j j P g e 7

3 j j j j j k j j j j j j j j j j j j j j j k j ] j j j j j j j ] j j e r r he j j r j r j r j k j j j j j r j r j r j j j ] ] j r k j r j j j j ] j r ] j r j j j j rom he ory coos we he = j j = +j j lso for = j j = so j = j j = j = so +j = j j = j = so j = j j = j = +j j = so +j = j j = j 9 from he l coo: The resl eqo 8 s he fe fferece meho for he Schckeer moel.. DOMIN DECOMPOSITION METHOD Noler fferel eqos re slly rsg from mhemcl moelg of my froer physcl sysems. I mos cses lyc solos of hese fferel eqos re ery ffcl o chee. Commo lyc proceres lerze he sysem or ssme he oleres re relely sgfc. Sch proceres chge he cl prolem o mke rcle y he coeol mehos. Ths chges some mes serosly he solo. The oe rwcks of lerzo mercl mehos rse he ee o serch for lere echqes o sole he oler fferel eqos mely he lyc solo mehos sch s he perro meho he ero rol meho ] he om ecomposo meho. The ecomposo meho ws frs roce y om sce he egg of he 98s.The om ecomposo meho DM s se o sole we rge of physcl prolems. Ths meho proes rec scheme for solg ler oler eermsc sochsc eqos who he ee for lerzo hs yels coerge seres solos rply. ge of hs meho s h c proe lycl ppromo or pprome solo o rher we clss of oler sochsc eqos who lerzo perro closre ppromo or screzo mehos. lke he commo mehos.e. wek olery smll perro whch chge he physcs of he prolem e o smplfco DM ges he pprome solo of he prolem who y smplfco. Ths s resls re more relsc ]. We efe he operor. he P g e 7

4 P g e 7 sysem c e wre s: k By pplyg he erse of operor o we ge: k By l coos sysem he c e wre s: k By sg om ecomposo meho : k B Where re om polyomls. B = B ecse oh o-ler erms re Where! B here! ecse o-ler erm he wo fcos he y eqo : k k k k k k By eqo : k B k k k! B k 5 : ] k e k k k k 6 : ] e B ]!

5 P g e 75 ] ] ] 7 ] B K= ] ] k k k k ] ] : k e 8 B ] ] ] ] 9 : e ]!! ]! ] ]! ]!! 5 ] By eqs ] B

6 P g e 76 k k By eqos 8 ] ] ] ] k k k ] ] ] ] 9 By eqs B III. PPICTION NMERIC EXMPE We sole he followg emple merclly o llsre effcecy of he presee mehos. = k + + > Ω = + > Ω We he l coos = s +. s/ for = s. s/ for = s = s = s = s We wll ke = =. ==.9 k=-. s= s= I. IGRES ND TBES Tle Comprso ewee he DM DM for he les of cocero. = = = DM DM DM DM DM DM

7 Tle Comprso ewee he DM DM for he les of cocero. = = = DM DM DM DM DM DM g. DM for he les of cocero wh << << g. DM for he les of cocero wh << << g. DM for he les of cocero wh << << g. DM for he les of cocero wh << << P g e 77

8 . CONCSION The Schckeerg moel sole Nmerclly sg fe fferece meho om ecomposo meho we fo h s fe fferece meho s erler h DM DM s more ccre h DM more effce s show les - fgres -. CKNOWEDGEMENTS My eep feelg of gre mmese eeess o my wfe for her coos sppor ecorgeme o hs work. REERENCES om G. 988 Reew of he Decomposo Meho pple Mhemcs Cofer for pple Mhemcs ersy OJ Georg hes Georg 6 Sme y George om. om G. 99 Solg froer prolems of physcs: he Decomposo meho Boso: Klwer cemc Plshers. Bzr J. mrmoor.r. 5 lyc ppromo o he solo of he eqo y om ecomposo meho resrcos of he meho Deprme of Mhemcs Islmc z ersy Rsh rch Rsh P.O. Bo 5-56 Ir. Brer. Nohel J.. 97 Orry Dfferel Eqos: rs Corse eo W.. Bejm INC. oo. 5 Cross M.C. P.C. Hoheerg 99 Re. Mo. Phys Deh. 997 Noler Prl Dfferel Eqos for Scess Egeers Brkhser Boso. 7 Gymo G.. 97 e Eleme Solo Of The Oe Dmesol Dffso-Coeco Eqo Wer Resorces Reserch ol.6 No. pp.-. 8 Mhews J. H. k K. D. 999 Nmercl Mehos sg Ml Prece- Hll Ic. 9 Mrry J.D.99 Mhemcl Bology e. Sprger-erlg Berl. Ncols G. Prgoge I.977 Self-Orgzo Noeqlrm Sysems Wley New York. Shhkmr M. 989 Comper Bse Nmercl lyss Kh plsher Nesrc Delh- 6 I. Smh I. M. rry R.. O'Coer B.. 97 Rylegh-Rz Glerk e Elemes or Dffso-Coeco Prolems Wer Resorces Reserch ol. 9 No Temm R. 99 Ife mesol Dymcl sysems chcs physcs Sprger- erlgerl. 7] Ycheg Y. 7 Glol Dymcs of he Brsselor Eqos Dymcs of PDE ol. No Ycheg Y. 7 Glol Dymcs of he Brsselor Eqos Dymcs of PDE ol. No P g e 78