Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 ON A MEHOD FOR FINDING HE NUMERICAL SOLUION OF CAUCHY PROBLEM FOR D BURGERS EQUAION Mir Rsulo Prof. Been Uniersi Deprmen of Memis n Compuing Isnul ure Emine Born MS Suen Been Uniersi Insiue of Sienes n Engineering Isnul ure Asr In is pper ne meo is propose for fining e numeril soluion of e Cu prolem for D Burgers equion i iniil funion onsising of four pieeise onsns in lss of isoninuous funions. For is gol speil uilir prolem i s some nges oer e min prolem is inroue. Using ese nges of e uilir prolem e numeril soluion of e min prolem is oine. Some ompuer eperimens re rrie ou. Keors: Riemnn-pe prolem D-Burgers Equion Numeril soluion in lss of isoninuous funions Numeril e soluion Inrouion: As usull le R e n Eulien spe of poins. Le omin in R 3 R R s Q { 0 < }. Here n re gien onsns. In Q e onsier e Riemnn-pe iniil lue prolem for oimensionl slr equion i esries erin onserion l s 0 i e iniil 0 0. Here 0 is pieeise onsn on finie numers of eges enere e origin 0 0. In priulr is ineresing is e Q e 555
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 four-ege prolem i eges orresponing o four qurns 3 0 3 4 of e spil plne. e eisene n uniqueness of e soluion for e single onserion l in seerl imensions e meo of isosi ere suie in [4] [9] []. I soul e noe is meo gies lile informion ou e quliie sruure of e isoninui se of soluion. is prolem for e one spe imension s inesige in [] [5] [] e. ere e sruure of e soluion s reele in eil. e soluion of prolem oine using e meo of rerisis is 0 3 ere n re e speil oorines moing i e spee of respeiel. Firs inesigions of e o-imensionl Riemnn-pe iniil lue prolem s iniie Gueneimer [3]. Pper [7] is eoe o onsruion of e soluion of e D- Riemnn-pe iniil lue prolem for slr onserion l in e een of ree or more infleion poin in se funion f g nlzing su of e generlizion of one- imensionl Riemnn prolem o llo for iniil ing finie numer of ump isoninuiies i onsn or rrefion es eeen umps. In [6] i s een son e soluion of e D- Riemnn-pe iniil lue prolem n e lssifie n presen in erms of o-imensionl nonliner es in nlog i e nonliner rrefion n so es of e one imensionl Riemnn prolem i.e. eplii soluions re onsruile from ese es. Sine e soluions for D-Riemnn-pe iniil lue prolems e eplii sruure e lso sere s ousone for numeril lgorims. In [] e onep of e so n rrefion se poins re inlue n using e rerisi nlsis e nlil soluion for D-Riemnn-pe iniil lue prolem for e Burgers equion is onsrue. In iion for e numeril soluion e omposie seme eelope Lis-Wenroff in [8] is pplie. Auilir Prolems I is non glol oninuous soluions for D-Riemnn-pe iniil lue prolems ill no ie pproprie smoo iniil. e e soluion of prolem ill e efine s follos. Definiion : e funion sisfing iniil oniion is lle e soluion of prolem if e folloing inegrl relion 556
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 557 R R 0 0 0 R 4 ols for eer es funion efine in 3 R n ifferenile in e upper lf plne n nises for suffiienl lrge. Le Q e rengulr omin efine s } { Q su Q Q n Q Q 0. In orer o fin e e soluion of prolem in e sense of 4 e ill inroue e uilir prolem s oe. Inegring equion on e region Q e ge. 5 Here. I is lerl seen erm. Here. {} {} M We enoe e folloing epression 6 ere erm. From 6 i follos }. { M 7 ing ino onsierion 6 7 e ge 0. 8 e iniil oniion for 8 is. 0 0 9 Here e funion 0 is n ifferenile soluion of e equion }. { 0 0 M 0 o fin soluion of equion 8 i iniil oniion 9 e ill ll firs in uilir prolem. e uilir prolem 8 9 s folloing nges:
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 i In is se e funions n n e isoninuous oo ii e orer of iffereniili of e funion is greer n e orer of iffereniili of e funion iii e eriies n in lgorim for oining of e soluion of prolem 8 9 oes no our s ese eriies oes no eis. e folloing eorem is li. eorem : If e funion is soluion of uilir prolem 8 9 en e funion epresse M is e soluion of min prolem. o oin soluion of equion 5 i iniil oniion e ill ll seon in uilir prolem. Anlsis of Liner Prolem Before oining e soluion of nonliner prolems firs e inesige simple linerize prolem s A B 0 ere n B re gien onsns. e e soluion of i iniil oniion is 0 A B. In is pper e ill su e firs pe uilir prolem for. In is se equion 8 n e rerien in e folloing form A B 0. 3 For is se relion 7 is li oo. ing ino onsierion 7 e e A B 0. 4 e iniil oniion for 4 is 0 0. 5 A Here 0 is n oninuous ifferenile funion of equion 0. e e soluion of prolem 4 5 is 0 A B. 6 I is seen equion 4 oinies i equion u e iniil funion 0 is more smoo n e iniil funion of e min prolem. 558
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 Finie Differene Semes in Clss of Disoninuous Funions In is seion e inen o eelop e numeril meo for fining e soluion of prolem n inesige some of is properies. As i is se oe in e nonliner se e soluion of e min prolem s isoninuous poins ose loions re unnon eforen. ese properies o no permi us o ppl lssil numeril meos o is prolem irel. For is im e ill use uilir prolem 8. B using e nges of e suggese uilir prolem ne numeril lgorim ill e propose. In [0] e suggese numeril meo s suie for o-imensionl nonliner slr equion en f g n e Riemnn onsis of e o segmens pieeise onsn. e propose meo ill e eelope o fin e numeril soluion of e Cu prolem for equion in e folloing su. e Finie Differene Seme In orer o eelop e numeril lgorim e onsru e folloing gris. Le { i i i i 0... n} n n { 0... m} m i re oer of e segmens [ ] n [ ] respeiel. No e sll onsru o ne gris s { 0... np} n { 0... mq} i lso oer e segmens [ ] n [ ] respeiel ere n p. Ler e inroue e folloing noions q { ; 0...} n { ; 0...}. Sine p 0 n n 0 m i ler. Here n m q for n p q re e gien ineger numers i so nol poins on e segmens [ ] [ ] [ ] n [ ] respeiel. In orer o pproime equions 8 or 3 finie ifferene inegrls leing in 8 or 3 re pproime s follos: 559
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 i pi V 7 q V i 8 i... n... m. ing ino onsierion 7 n 8 equions 8 n 3 n poin of e gri re pproime e folloing eplii seme s i W q pi Wi V i V 9 i W q W A V B V 0 i i i pi i... n;... m; 0.... Compuer Eperimens ree pe ompuer ess on sis of e propose lgorims re rrie ou. ess ere me using e folloing : 0. 0.00 n m 500 n 4 3 3 4. A firs e onsier e iniil lue prolem for i pieeise onsns onneing four eges enere e origin 0 0 i.e > 0 > 0 < 0 > 0 0 3 < 0 < 0 4 > 0 < 0. We ill sole prolem 4 5 inse of. In is se e funion of ill e osen s oninuous soluion of 0 0 > 0 > 0 < 0 > 0 0 3 < 0 < 0 4 > 0 < 0. Prolem 4 5 is pproime folloing eplii finie ifferene semes s W g W [ g g ] W g W 3 i i i i W 0. 4 i 0 i 560
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 Here g A g n poin B n i W i of e gri enoe e pproime lue of. I is no iffiul o inie Wi Wi Wi Wi V i. Seon pe lulion e een me on sis of e folloing Vi Vi Vi Vi Vi i... n... m 0... Vi 0 0 i i 0... n 0... m ifferene semes. Finl using ifferene semes 9 0 e numeril soluion of e prolems n on e sme re foun. Oine soluions so goo oiniene i e soluions of e inesige prolem. Referenes: Dei Yoon Woone Hng o-dimensionl Riemnn Prolems for Burger s Equion Bull. Koren M. Sos. 45 008 No. pp. 9-05. Gelfn I. M. Some prolems in e eor of qusiliner equions. Usp M. Nu 4 87-58 959 Amer. M. so rnsl. 9 95-38 963. Gueneimer J. Sos n Rrefions in o Spe Dimensions Ar. Rionl Me. Anl. 59 975 No. 3 8-9. Kruzo S.N. Firs Orer Qusiliner Equion in Seerl Inepenen Vriles M. Sorni 8 970 p. 7-43. L P.D. Hperoli Ssems of Conserion Ls n e Memil eor of So Wes. Conf. Bor M. Si. Regionl Conf. Series in Appl. M. SIAM Pilelpi 97. Linquis W.B. e Slr Riemnn Prolem in o Spil Dimensions: Pieeise Smooness of Soluions n Is Breon SIAM J. M. Anl. 7 986 78-97. Linquis W.B. Consruion of Soluions for o-dimensionl Riemnn Prolems Compu. M. Appl. A 986 65-630. Lis R. Wenorff B. Composie Semes for Conserion Ls SIAM J. Num. Anl. 35 998 50-7. Oleini O.A. Disoninuous Soluions of Nonliner Differenil Equions Usp.M. Nu 957. 56
Europen Sienifi Journl Augus 05 /SPECIAL/ eiion ISSN: 857 788 Prin e - ISSN 857-743 Rsulo M.A. Cosun E. Sinsosl B. Finie Differenes Meo for o-dimensionl Nonliner Hperoli Equions in Clss of Disoninuous Funions. App. Memis n Compuion ol.40 Issue l Augus pp.79-95 003 USA. Smoller J.A. So We n Reion Diffusion Equions Springer- Verlg Ne Yor In. 983. Volper A.I. e Spes BV n Qusiliner Equions. M. USSR- Sorni 5-67 967. 56