NUMERICAL SOLUTION OF THIN FILM EQUATION IN A CLASS OF DISCONTINUOUS FUNCTIONS

Transcription

1 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN NMERCAL SOLON OF HN FLM EQAON N A CLASS OF DSCONNOS FNCONS Bn Snsoysl Assoc Prof Mr Rslov Prof Beyen nversy Deprmen of Memcs n Compng snbl rey Absrc n s pper n orgnl meo s been sggese o fn nmercl solon of nl vle problem for for orer egenere ffson eqon wc moels e n flm flow For s n lry problem esblse n specl wy n vng some vnges over e mn problem s been nroce Avnges of e lry problem llow s o pply one of e well-nown meos n lerre n s e nmercl solon of e mn problem cn be clcle by sng e obne solon Keywors: n flm eqon We solon Nmercl solon n clss of sconnos fncons nrocon Le be n Eclen spce of pons n le G R be recnglr regon s G [ were [ ] n re gven consns n G we conser e for-orer oble egenere nonlner n flm eqon s w e followng nl n bonry R 8

2 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN conons Here e fncon escrbes e fne mss erefore n e bonry conons sow e fl s perme o rn over e eges An nlyss of e solon obne n [5] sows for n b On e bss of ese esmes we cn sy problem - oes no ve clsscl solon Snce eqon s egenere followng [] we conser e ppromng eqon s n Q 4 were s posve prmeer n s son s lso necessry o pprome n H norm by fncons ssfyng e conons n replcng 5 Q C ner e ssmpon s solon of problem 4 5 n for some < < we erve esmes o be se ler Png nse of e we begn w snce for ny < < 4 on e bonry Dvng by n leng we ge < Mlplyng 4 by n negrng over Q < < n sng e ls eny we obn 8

3 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN Now negrng 4 over we lso obn 6 negrng 4 over we obn 7 Alry Problem negrng eqon w respec o from nl o n sng conon we obn 8 ng no conseron e eqly n once more negrng eqon 8 w respec o from nl o n compensng relon of negron o zero we ge 9 n ccornce w from 9 follows negrng gn n ls forml we ve

4 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN sng e Ccy forml we ge! s cler f e fncons n re fferenble connos en e eqons or n re eqvlen By fferenng for mes e ls eqon w respec o we prove s clm Nmercl Algorm o pprome of eqon by e fne fference formls frs we cover e omn G by e gr Here n n e nmber n > n wll be obne from e conon of sbly of e fference sceme Now we consrc sb gr n For ny we cover e nervl w np Here p s ny consn As s seen p s we ge We now my presen fne fference sceme o eqon A frs sng cbre formls for emple e meo of recngles e negrls n re pprome s follows p p 4

5 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN were ng no conseron epressons n 4 snce for ny negro-fferenl eqon cn be p pprome by e fne fference s follows p p p p 5 p 9 were fncon respecvely ny pon n ˆ re pprome vles of e e nl n bonry conons re n n Noe: o relze of or lgorm 4 frs ˆ of e gr s fon pon by sng Eler s meo en e nnown vles re fon me level from lgorm 5 e coeffcens n 5 n oer nl fncons re clcle n me level Now we wll nvesge e conssence n convergence of fference sceme 4 o solon Le re e errors of ppromon by e cbre forml of e negrls nvolvng eqon by fne fference formls w Oerwse le re errors of ppromon of n by fne fference formls respecvely Here n w m w 86

6 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN ˆ W W w De o e fc fncons n re connos e * 6 * were f s mols conny of ny fncon f on ny nervl ] [ s s < f f p f Le s esy o see e fncons n re connos en for we ve z 7 for some ] [ z Le De o e fc e fncon s connos for we ge z 8 W W w ˆ!!

7 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN m 9 As s seen from 6-9 follows fference sceme 4 s conss o Nmercl Epermens n orer o es e propose meo we ve se e from pper [5] e negrl of s clcle s follows > lm 7 7 sng lgorm 5 wn e lm of nl conon some comper epermens re crre o As s seen obne resls pproc sffcenly enog o ec solon gvng n pper [5] eorecl nvesgon of convergence n sbly of fne fference sceme 5 wll be mer of ne reserc References: Berns F Fremn A Hger Orer Nonlner Degenere Prbolc Eqons Jornl of Dfferenl Eqons Kng JR negrl Resls for Nonlner Dffson Eqons Jornl of Engneerng Memcs O'Bren SBG Scwz LW eory n Moelng of e n Flm Flows Encyclope of Srfce n Collo Scence Aceson D Elemenry Fl Dynmcs Vlrenon Press Ofor 99 BowenM Kng JR Asympoc Bevor of e n Flm Eqon n Bone Domns Eropen J Appl M pp -56 Gonosov Cyclc Operor Decomposon for Solvng e Dfferenl eqons Avnces n Pre Memcs pp 78-8 L Ansn Lorenzo Gcomell Dobly Nonlner n-flm Eqons n One Spce Dmenson Arcves for Ronl Mecncs n Anlyss 7 pp 89-4 Rslov MA On Meo of Solvng e Ccy Problem for Frs Orer Nonlner Eqon of Hyperbolc ype w Smoo nl Conon Sove M Do 4 No 99 88

8 Eropen Scenfc Jornl Ags 5 /SPECAL/ eon SSN: Prn e - SSN Rslov MA Rgmov A A Nmercl Meo of e Solon of Nonlner Eqon of Hyperbolc ype of e Frs Orer Dfferenl Eqons Mns Vol 8 No7 pp