BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS BY USING LIE GROUP

Transcription

1 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUATIONS BY USING LIE GROUP EMAN ALI HUSSAIN, ZAINAB MOHAMMED ALWAN Ass. Prof, Dr., Deparmen of Mahemas, College of Sene, Unversy of Al- Msansryah, Iraq Ass. Prof, Dr., Deparmen of Mahemas, College of Sene, Unversy of Al- Msansryah, Iraq E-mal: dreman@omsansryah.ed.q, lonwgh_9@omsansryah.ed.q ABSTRACT In hs as, we appled Le grop mehod for solvng bondary vale problems (BVPs) of paral dfferenal eqaons (PDEs) o obaned he fndamenal solons. Keywords: Le grop, Invarane,BVPs, Ea Solon, Bondary Condon. INTRODUCTION The mos ommon ehnqe for fndng he ea solon of he enormos varey of DEs omes from Le grop analyss of DEs. Many effen mehods for solvng DEs le separaon of varables, ravelng wave solons, self-smlar solons and eponenal self-smlar solons.bease he modern reamen of he lassal Le symmery nroded by [].Moreover nown as he lassal symmeres mehod, hs orgnally by Sophs Le [] over years ago. Hs mos seros wor n hs orenaon [],[]. In hs me he Le symmery mehod s eeedngly appled o sdy PDEs. Espeally for her redons o ODEs and onsrng ea solon,here are a major nmber of papers and many good boos devoed sh applaons [-7]. The heory of symmeres of DEs has been esablshed nensely and has vrally grown. A hge amon of lerare abo he lassal Le symmery heory s mplemenaon and s epanson s obanable n [8-,,-,-7,8-,]. In real world mplemenaon, mahemaal models are ypally based on PDEs wh pernen bondary and /or nal ondon. Conseqene one reqremens o nvesgae bondary vale problems (BVPs) and nal problems (Cahy problems). We observe ha he Le mehod has no been wdely sed for solvng BVPs and nal problems. A smmarzed Hsory frs aemps o apply Le symmeres for solvng BVPs are dsssed n he followng papers as [-]... Applaons o Bondary Vale Problem,[] In hs seon, we onsder he problem of sng nvarane o solve BVPs posed for PDEs. The applaon of he Le symmeres o BVPs for PDEs as follows : an nvaran solon arsng from an admed pon symmery solves a gven BVP provded ha he symmery leaves nvaran all bondary ondons. In oher words, ha he doman of he BVP or eqvalenly s bondary as well as he ondons (bondary ondons) mposed on he bondary ms be nvaran. In he ase of BVPs posed for lnear PDEs, he BVP need no be ompleely nvaran (nomplee nvarane)... Formlaon of Invarane BVPs for Saler PDEs,[] Le a BVP for a h order saler PDE ha an be wren n solved form : G g (,,, ) (,,, )...(.) Where g (,,, ) does no depend eplly on defned on a doman n -spae (,,..., n ) wh bondary ondons: B (,,, )...(.) Presrbed on bondary srfaes : ( ),,,..., s...(.) 67

2 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 We assme ha he BVP ((.)-(.)) has a nqe solon.consder an nfnesmal generaor of he form: ( )...(.) Whh defnes a pon symmery ang on boh (,)-spae as well as on s projeon o -spae. Defnon (.),[] The pon symmery of he form (.) s admed by he BVP((.)-(.)) f and only f - ( G(,,, )) when G(,,, )...(.) - ( ) when ( )...(.6) - B (,,, ) when B (,,, ) for,,..., s...(.7) Theorem (.),[] Sppose he BVP ((.)-(.)) adms he Le grop of pon ransformaons wh nfnesmal generaor(.).le y ( y ( ), y ( ),..., y n ( )) ben-fnonally ndependen grop nvarans of (.) ha depend only on. Le v(,) be a grop nvaran of (.) sh ha v. he BVP ((.)-(.)) rede o: G( y, v, v, v)...(.8) Defned on some doman y n y-spae wh bondary ondons: D ( y, v, v, v )...(.9) Presrbed on bondary srfaes V ( y )...(.) Forsome G( y, v, v, v), D ( y, v, v, v), V ( y),,,..., s. Moreover n he BVP ((.8)-(.)).The srfae y j ( ), j,,..., n, are nvarans srfae of he pon symmery (.). he nvarane ondon ((.)-(.7)) means ha eah bondary srfae ( y ) s an nvaran srfae V ( y ) of he projeed pon symmery....(.) j ( ) Gven by he resoraon of pon symmery (.) o -spae. From he nvarane of he BVP nder he pon symmery (.), he nmber of ndependen varables n ((.)-(.)) s reded by one. And he solon of he BVP ((.)-(.) s an nvaran solon. v ( y, y,..., y n )...(.) Of he PDE (.8) reslng from s nvarane nder pon symmery (.8).In erms of he dependen varable and ndependen varable appearng n PDE (.8) he orrespondng nvaran solon ( ) of PDE (.8) ms sasfy ( ( )) when ( )...(.) Tha s, ( )...(.) ( ) (, ( )) Theorem (.), [] If he nfnesmal generaor, gven by (.) s of he form:...(.) ( ) f ( ) he grop nvaran v(,) s of he form for some spef fnon g() v (, ) g ( ) and hene he nvaran form relaed o nvaran nder an be epressed n he separable form: ( ) g ( ) ( y )...(.6) In erms of an arbrary fnon ( y ) of y ( y ( ), y ( ),..., y n ( )). In [], he BVP ((.)-(.)) whh adms an r- parameer Le grop of pon ransformaons wh nfnesmal generaors of he form: j ( ) (, ),,,..., r j...(.7) he nqe solon ( ) of he BVP ((.)-(.)) s an nvaran solon sasfyng : ( ( )) when ( ),,,..., r...(.8) The nvarane of a BVP nder a mlparameer Le grop of pon ransformaons s gven by he followng heorem: Theorem (.), [] Sppose he BVP ((.)-(.)) adms an r- parameer Le grop of pon ransformaons wh nfnesmal generaors of he form: j ( ) (, ),,,..., r j...(.9) 67

3 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 Le R be he ran of he rn mar E ( ) n n r r r n...(.) q = n-r, and le z ( ), z ( ),..., z q ( ) be a omplee se of fnonally ndependen nvarans of (.9) sasfyng : To solve eqaon () we ms fnd he s prolongaon of () gven as: []...() Now, applyng formla gven n () o eqaon () as: [] ()...(6)...(7) he deermnng eqaon gven by : j Le v z ( ),,,..., r,,,..., q j g ( )...(.)...(.) Be an nvaran of (.9) sasfyng: v,,,..., r...(.) he BVP ((.)-(.)) redes o BVP wh q =n-r ndependen varables z ( z ( ), z ( ),..., z q ( )) and dependen varable v gven by (.). The solon of he BVP((.)-(.)) s an nvaran solon ha an be epressed n erms of a separable form: g ( ) ( z )...(.) Where he fnon ( z ) s o be deermned [].. APPLICATIONS In hs poron, we nsered some eamples for solvng BVPS for PDEs. Eample (): onsder he PDE gven n he form: (, ) (, ) (, )...() Wh bondary ondon (, ) (, ),...() And nal ondon (,) ( ),...() A frs le he veor feld n he syle: (,, ) (,, ) (,, )...() Immedaely, replang by oban:...(8) we...(9)...() Solve () by separaon of he oeffen of () we fnd he followng: :...() :...() :...() :...() :...() :...(6) :...(7) :...(8) :...(9) We fnd he general solon of he above sysem : (,, )...() 676

4 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 (,, ) d, d...()...() (,, ) 6 (, ) Smlary, f e hen he nvarane of 8 e yelds :...() e, e The Le symmeres of () gven as: Conseqenly,f d, e he resl 8, and so here s no non...() raval Le grop of pon ransformaons admed by hea eqaon wh bondary ondon gven by eqaon (-), defned on he doman >, d<<e...() we fnd from (). I-If d and e, hen he - parameer...() 6...(6)...(7)...(8) (, )...(9) Now, o fnd he fndamenal solon of he hea eqaon () as follows: Consder he eqaon () defned on he doman >, a<<b, reall he PDE gven n () admed by nfnesmal generaor : (, ) ( ) (,, )...() Wh eqaons (-). he bondary rves of he doman are =, = d, = e, he nvarane of = leads o ()...() We oban, f d and e, hen here s no frher parameer redon reslng form nvarane of he bondary rves. f d hen he nvarane of d leads o ( d, ), for any >. ( d, ) d d...() From eqaon () Le grop of pon ransformaons s admed by he bondary of problem gven by (-) on he doman >, d<<e.hene he BVP (-) old admed mos -parameer,,,, 6 Le grop of pon ransformaons. II-If d (who damage of generaly d ) and e hen he - parameer Le grop of pon ransformaons s admed by he bondary of BVP (-). Hene he BVP old admed a mos a -parameer,,.le grop of pon ransformaons wh nfnesmals gven by: ( )...() (, )...() (,, ) (6) Now, derve he fndamenal solon for he hea eqaon, when (,) ( )...(7) Where ( ) s he Dra dela fnon enered a, d< <e, for an nfne doman d, e or a sem nfne doman d, e A- Infne doman d, e (, ) We see he fndamenal solon (,,, ) of he Cahy problem defned as follows :,,...(8) 677

5 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 Wh bondary ondon (, ), (, ) ( )...(9) Where ( ) s he Dra means a for nfne doman who loss of generaly, we an ae =. he bondary rves of he doman are leads o,,he nvarane of leads o. ()...() We oban, by he same way he nvarane of = leads o. (, )...() For any >, and hene. The Le grop of pon ransformaons wh he nfnesmals gven as ( -) s admed by he nal bondary vale problem (8) provded ha s : (,) (,) (,) ( ), where (,) ( )...() (,) ( ) (,) ( ) Here sed Theorem (.),he nfnesmals generaor: ( ) (, ) s of he form ( ) f (, ) From properes of he Dra dela fnon eqaon () s sasfed f : (,)...() (,) (,)...() Ths, n he nfnesmals eqaon (-) we have :...() (,)...(6) From (,) (,) we oban: 6...(7) And, hene -parameer,, Le grop of pon ransformaons s admed by he BVP (- ). Ths sb algebra of he Le algebra span by he generaor leaves nvaran he nal manfold, ha s, he lne =. And s resron on = onvers he nal ondon (, ) (, ), hs sb algebra -dmensonal algebra span by 8...(8) Correspondng o he nfnesmals :...(9)...()...()...() 8...() Le (, ) be an nvaran solon reslng from nvaran nder he nfnesmal generaor hen: 8...() Hene an nvaran solon orrespondng o and s also an nvaran solon orrespondng o. (, ) an nvaran solon orrespondng o and hen : (, )...() By sng he followng eqaon gven as: Q...(6) Q d d d...(7)...(8) 678

6 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 Now, we solve he haraers eqaon (8) gve he resl as: r...(9) v e...(6) From (9) and (6) we oban: (, ) ( r) e, w here r...(6) By sng (6) we fnd he followng: Q....(6) Now, we solve he haraers eqaon we oban: d d d...(6) r...(6) v e...(6), from (6) and (6) we oban he followng: ( ), r e r...(66) From he nqeness of he solon of he BVP (- ) we ge: ( r ) e ( r ) e...(67) r r ( r) r e ( r ) e r r ( ) ( )...(68) r e r r e...(69) Hene he solon of he BVP (-) on he doman >,, wh bondary ondon : (, ),...(7) (,) ( )...(7) e...(7) By sng nal ondon (7) we ge, from we oban aomaally : e...(7) B- sem- nfne Doman d, e, onsder he BVP () on he doman, wh bondary ondon gven as: (, ),...(7) (,) ( ),...(7) The -parameer Le grop of pon ransformaons wh nfnesmals gve (-) s admed by PDE (). he bondary rves = and =, and he bondary ondon (), he nvarane of he nal ondon () leads o he resron. f (,) (,) (,) ( )...(76) Where (,) ( )...(77) f (,) ( ) (,) ( ) Hene (,) f (,) (,)...(78)...(79) Conseqenly, n he nfnesmals eqaon (- ) we ms have from (79) :, 6...(8) 8 Ths, he BVP of (-) adms he pon symmery : (8) The orrespondng nvaran solon has he nvaran form : Q...(8) (8) d d d (8) From solve (8) we oban : r...(8) e....(86) v e (,) () r, wherer...(87) 679

7 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 Where ( r) arbrarly fnon of he smlary varable r.afer ha sbsng he nvaran from (87) n (), we fnd ha ( r) sasfes he ODE: e (, ) ( ), where...(88) Now, sbsng (88) n () we resl: ( )...(89) By solvng eqaon (89) we oban: (,) sn 6 os 6...(9) For onsan and. From he bondary ondon (7) of B we fnd an, from he ondon (7) of B we ge hen. well- nown solon of he an BVP () an bondary ondon (7-7): (, ) sn 6 os 6 an...(9) [] ()...() We oban he deermnng eqaon as follows:...(6) By sng defnon of, and we ge: Immedaely, replang by...(7) we fnd:...(8) Eample (): onsder he PDE wh Transen ondon n sem-nfne sold wh onsan srfae emperare gven as : (, ) (, ) (, )...() Wh nal and bondary ondons:,,...() To solve () wre he veor feld as follows: (,, ) (,, ) (,, )...() Afer ha, we need he -prolongaon of as: []...() Now, apply he formla gve n () o eqaon () as:...(9) By separaon of he oeffen, and e we oban : :...() : : : : :...()...()...()...()...() 67

8 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 :...(6) :...(7) :...(8) he general solon of above sysem s: (,, ) e e...(9) (,, ) e e e e 6...() (,, ) e e (, )...() Where (, ) s arbrary fnon hen:...() e e e...()...()...()...(6) 6 e...(7) (, )...(8) Now, solon he IBVP: We onsder he general symmery operaor of he form: (9) of PDE () and for he operaor ha preserves he bondary and bondary ondons () he nvarane of he bondares =, = or eqvalenly:...() From eqaon () we fnd :...()...() From () we oban :...() Hene...() In addon o he resron mposed () he nvarane of he nal and bondary ondons : (, ), (,)...() (, ), (, )...(6) From eqaons () and (6) we resl, hene he IVP of () and () s nvaran nder symmery we have :...(7) e...(8) Where he nvaran solon of he problem s onsred by lzng he ransformaons hrogh smlary varables for. solvng he haraers sysem for I gves: Q...(9) Q e...() d d...() e From eqaon () we oban: (, ) e e...() v ( )...() Now, sbsng of smlary varables n () sasfes ha orrespondng smlary solon of PDE () of he form gven n () where v ( ) sasfes he ODE gven as: e e e d v d e d v d...() The above eqaon () an be solved by sbson dv w we ge: d e e e e v ( )...() 67

9 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 From eqaons () () and () we oban he ea solon of PDE () ha he nvaran nder (8) e e e e (, )...(6) Imposng he nal and bondary ondon deermnes when,...(7)...(7) Ye, replang by we oban: Now, when, we fnd : e erf e...(8) we wre he general solon for PDE () as: e e e e e (, ) erf e...(9) Eample (): onsder he nonlnear PDE gven as:...(8) (, ) (, ) (, ) (, )...() Wh nal bondary ondons,,, q,...() To solve problem () wre he veor feld as follows: (,, ) (,, ) (,, )...() We need he nd prolongaon of he form: []...() Now, applyng he prolongaon gve n eqaon () o eqaon () we fnd: []...() () he deermnng eqaon gves:...(6)...(9) By separaon he oeffen of varables, and e : : :...()...() : : :...()...()...() :...() 67

10 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 : : : : : :...(6)...(7)...(8)...(9)...()...() We oban he general solon of above sysem s:...()...()...() he Le symmeres gven as:...()...(6)...(7)...(8) We onsder he general symmery operaor of PDE () :...(9) Now, solve PDE () by sng he frs bondary ondon searh for he operaor ha preserves he bondary and bondary ondon (): he nvarane of he bondares =, = or denally :...()...() From () we oban:...() ms have:...() In addon o resron mposed by eqaon () he nvarane of nal and bondary ondons wre: (, ), (,)...() (, ), (, )...() From () we fnd. Hene he IBVP () and () s nvaran nder he symmery : Now, we have hosen, we wre he generaor:...(6) The nvaran solon of he problem s onsred by lzng he ransformaons hrogh smlary varables for. Solvng he haraers sysem for I, gves : d d d...(7) By solvng (7) we resl : (, )...(8) v ( )...(9) Afer ha, sbsng of smlary varables eqaon () sasfes he orrespondng smlary solon of PDE() s of he form v ( ) where v ( ) sasfes he ODE as: d v dv dv d v d v d...() he general solon of above sysem gven as: v ( ) ln erf...() Where erf denoes he error fnon. Hene he ea solon of PDE() ha s nvaran nder s : (, ) ln erf...() Imposng he nal and bondary ondons deermnes: When we oban :...() 67

11 Jornal of Theoreal and Appled Informaon Tehnology s Oober 8. Vol.96. No ongong JATIT & LLS ISSN: E-ISSN: 87-9 When we fnd: e...() Tang he general symmery operaor gven n eqaon (9) of PDE (), we deermnes he operaor ha leaves he bondary and he bondary ondons n seond par of eqaon () nvaran : Now, he nvarane of he bondares =, = or eqvalenly :...()...(6) From () and (6) yelds:...(7) ms be wre:...(8) Sne...(9) From (9) gves, he nvarane of he ondon dose noe mpose any resron on. Now, he nvarane of he bondary ondon s: q...() The () reqres he s prolongaon: q [] q,on...() q...() From () we ge :...() Choosng,provdes he symmery:...() Tha leaves he IBVP () and he seond par of ondon () nvaran, o fnd he smlary ransformaons ha wll lead o he solon he haraers sysem for I.s solved hs provdes he smlary varables : d d d...() By solved eqaon () we fnd: v ( )...(6) (, )...(7) Now, sbsng smlary varables n PDE () where v ( ) sasfes he ODE. v v v v( ) v( ) v v...(8) he general solon of above ODE eqaon (8) s: v e erf...(9), e erf...(6) When =, hen. When = hen q we oban q. q (, ) e erf...(6). CONCLUSION AND DISCUSSION We esablshed n hs sdy of Le grop heory s appled o PDEs o deermne symmeres. The one parameer Le grop whh leaves he PDEs nvaran, n eample () and () we wan o onsr a solon o bondary and nal vale problem for PDEs, n eample () we se ehnqe assmpon for onsan afer sbsng of nal bondary vale problem o fnd he fndamenal solon and nroded Dervave for ondon, we solved by sng s prolongaon a las we ge he general solon. REFRENCES: [] S. Le, Über De Inegraon Drh Besmme Inegrale Von Ener Klasse Lnearer Pareller Dfferenalglehngen. Arh. Mah, 8, (88). 67

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