Riemann Function and Methods of Group Analysis

Transcription

1 American Research Jornal of Mahemaics Original Aricle ISSN X Volme Isse 3 5 Riemann Fncion and Mehods of Grop Analsis Akimov Andre Chernov Igor Abdllina Rfina Serliamak Rssia Lenina sree 47A Deparmen of Phsics and Mahemaics Bashkir sae niversi Serliamak branch Absrac In his paper for he hperbolic eqaion was consrced he for-parameer grop and wih he help of he grop was fond he solion of he Cach problem b he Riemann mehod for a hperbolic eqaion Kewords: Problem Cach Riemann s Fncion Hperbolic Eqaion Grop Analsis I INTRODUCTION The general solion of he Cach problem for a second-order linear hperbolic parial differenial eqaion in wo variables is ofen given in erms of inegrals involving an ailiar solion called he Riemann fncion B eiher mehod compleion of he problem reqires he deerminaion of ailiar fncion which is a difficl ask becase here is no nified mehod for acall finding his fncion Mehods for finding Riemann fncions have been given b Riemann [] Copson [4] and Mackie [8] Regreabl here are sill onl a few eqaions wih known Riemann fncions This paper is of a snheic nare being a resl of combining Riemann s mehod [3] for inegraing second-order linear hperbolic eqaions wih Lie s classificaion [] of sch eqaions Using he resls for he grop classificaion of homogeneos hperbolic eqaion of he second order i was sggesed o find a fncion of Riemann sing he smmeries of he eqaion II PRELIMINARIES Le s consider he following hperbolic eqaion of he second order: Lv v v v R 4sin in an open domain D which is bonded b crves of AC CB secion AB Le s pose he problem of Cach: Find in he domain D fncion saisfing he condiions and wih he v C D C D AB C D Lv D 3 lim lim v v 4 5 Corresponding Ahor: andakm@ramblerr wwwarjonlineorg 54

2 American Research Jornal of Mahemaics Volme Isse 3 5 ISSN X where Appling real coordinae ransformaion given sfficienl smooh fncions eqaion leads o he canonical form: 6 sin r where lim lim r and bondar vales ~ ~ We shall define he operaor L b he ideni L a b c 7 The operaor L defined b he ideni L v v av bv is known as he adjoin of L cv To solve he problem we se he mehod of Riemann which is based on he following ideni: vl L v v v av v v bv 8 If he fncions v and are sch ha L Lv hrogho a domain G bonded b sfficienl smooh closed crve Γ Then an applicaion of Green s heorem ields G vl L v dd [ v v bv d v Riemann s mehod redces he problem of inegraion of he eqaion L f v av d] To ha of consrcing he ailiar fncion v R ; solving he ad join eqaion: L R and saisfing he following condiions on he characerisics: R R ar br R ; 9 wwwarjonlineorg 55

3 American Research Jornal of Mahemaics Volme Isse 3 5 ISSN X Provided ha he fncion v is known he solion of he Cach problem: L f n wih daa on an arbirar non-characerisic crve Γ is given b he formla R P R Q PQ R R ar d Rfdd G R R br d where he doble inegral is aken over he domain bonded b he characerisics and he crve Γ The fncion v R ; is called Riemann s fncion and he bondar-vale problem 9 is called he characerisic Cach problem or he Gorsa problem The solion of he Gorsa problem is niqe III MAIN RESULTS In or case he eqaion adjoin eqaion 5 has he form sin r Le s noe ha in or case he desired fncion of Riemann v R ; saisfies he following condiions on he characerisics: R R R ; 3 The smmer operaor of he homogeneos eqaion has he form [4]: X v w Ths as follows from [5] ms be done he following relaions: bv b w aw a v cv cw a b Sbsiing in his case a b sin r cos r c sin r w sin r w cos r The solion of his parial differenial eqaion of he firs order will fncions we ll obain he following relaions wwwarjonlineorg 56

4 American Research Jornal of Mahemaics Volme Isse 3 5 ISSN X Asin w Asin Bsin cos Ccos Bsin cos Ccos D where A B C D arbirar consans Ths eqaion admis he hree-parameer grop in addiion o srechings of he variable and he infinie grop consising of addiion o of an solion of he eqaion; his grop is common o all linear eqaions wih he generaors sin sin sin cos sin cos X X cos cos X X Le s find a linear combinaion of hese operaors X X X 3X3 4X 4 where 3 4 arbirar consans 4 Following [6] le s reqire firs he invariance of he characerisics he form: X = One can se resling operaor X X hen obain The invariance es has 3 One can readil verif ha he sin cos sin cos sin cos sin cos is admied b he Gorsa problem Invarian of his operaor have he form I if f is assmed o be a fncion of becomes z z f z z Wih he change of variable Now z he eqaion f z f z s z obain s s s f s s f s f s wwwarjonlineorg 57

5 American Research Jornal of Mahemaics Volme Isse 3 5 ISSN X wwwarjonlineorg 58 The solion of he obained eqaion is fncion Gass ; F f Then he Riemann s fncion in he will have he form ; F R Sbsiing in he formla a b f obain ~ d ~ d Rerning o he old variables and we ll ge he solion of he Cach s problem d ~ d ~ 4 THEOREM If he fncions ; C ; C hen he Cach s problem for eqaion has a niqe solion which is defined b 4 REFERENCES [] G F B Riemann Uber die Forpflanzng ebener Lfwellen von endlicher Schwingngsweie Abhandl Konigl Ges Wiss Goingen [] Akimov AA On niqeness Morawez problem for he Chaplgin eqaion IJPAM vol 97 No [3] Akimov AA Some Properies and Applicaions of he Riemann-Hadamard Fncion of Darbo Problem for Telegraph Eqaion American Research Jornal of Mahemaics Vol No [4] JS Papadakis DH Wood An addiion formla for Riemann fncions Jornal of differenial Eqaions [5] ET Copson On he Riemann-Green Fncion J RaMech Anal [6] DHWood Simple Riemann fncions Bll Amer Mah Soc vol 8 No [7] MM Smirnov Eqaions of mied pe American Mahemaical Socie Unied Saes 978 [8] Coran RHilber D Mehods of Mahemaical Phsics Inerscience Pblishers Inc New York 953 [9] KB Sabiov and RR Il asov Ill-posedness of bondar vale problems for one class of hperbolic eqaions Rssian MahemaicsIz VUZ vol 45 No [] AG Mackie Green's fncion and Riemann's mehod Proc Edinbrgh Mah Soc [] Lerner ME Qaliaive Properies of he Riemann Fncion Diff er Uravn vol 7 No 99 6

6 American Research Jornal of Mahemaics Volme Isse 3 5 ISSN X [] Bisadze AV Uravnenia maemaicheskoi fiziki Eqaions of Mahemaical Phsics Moscow:Naka 976 [3] N H Ibragimov Grop analsis of ordinar diff erenial eqaions and he invariance principle in mahemaical phsics for he 5h anniversar of Sophs Lie Uspekhi Ma Nak 47 No English ransl Rssian Mah Srves [4] P J Olver Applicaions of Lie grops o diff erenial eqaions Springer-Verlag New York 986 nd ed 993 wwwarjonlineorg 59