Analog of the Method of Boundary Layer Function for the Solution of the Lighthill s Model Equation with the Regular Singular Point

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1 Aerca Joura of Maheacs a Sascs 3, 3(): 53-6 DOI: 593/as338 Aaog of he Meho of Bouary Layer Fuco for he Souo of he Lghh s Moe Equao wh he Reguar Sguar Po Kebay Ayuov Depare of Agebra a Geoery, Osh Sae Uversy, Osh, 7354, Kyrgyzsa Absrac The possby of appcao of he bouary ayer fuco for cosrucg he asypoc souo of he sguary perurbe of Lghh oe equao he case whe correspog o peru rbe equao have he poe of he ere orer o he reguar sguar po s prove Earer asypoc of hs probe was cosruce by he eho of uforzao a srucura achg The reaos bewee he ehos of he bouary ayer fu co, uforzao a srucura achg are aayze Keywors Sguary Po, Sguary Perurbe Equao, Asypoc of Souo, Moe Equao of Lghh, Meho of Bouary Layer Fuco (MBLF), Meho of Uforzao (MU), Meho of Srucura Machg (MSM) Irouco Faous Egsh echac a aheaca J M Lghh [] sue he foowg probe of he perurbe orary fferea equao u( x) ( x u( x)) q( x) u( x) r( x), u() u, () x () where - s a paraeer, u - s gve ae, x,, u( x) - uow fuco, ux u / x, q( x), r( x) - aayca fucos o he erva [,] He use he ea of Pocare eho he heory oear oscaos propose o see of asypoc of he souo of hs probe he for u( u ( u ( u (, x x ( x (, a here are o he rue o eere uow fucos u x Ths approach was ae afer h as he eho of Lghh Po x= s sguar po for uperurbe equao () ) u( x) Lu( x) : x q( x) u( x) r( x), x (3) u () u, * Correspog auhor: ebay@yahooco (Kebay Ayuov) Pubshe oe a hp://ourasapuborg/ as Copyrgh 3 Scefc & Acaec Pubshg A Rghs Reserve () We w se ha q () r () We oe ha he souo of he probe (3) has he vew: q u ( x) x w( x), (4) (4) here x q ( ) ( )[ ( ), w x p x u s p s s x (), ( ) exp{ ( ( ) ) } q q p s q s q s s If q, w(), he he souo (4) uboue fuco o he erva [,] a he po x s he poe of (4) The eho of Lghh eveope by G F Carrer, W A Wasow, H S Tse, G Tepe, M F Pruo, Sbuya a K J Tahahasy, H J Hoogsrae, C Coso, P Habes, K Ayuov a ohers I s possbe o rea hese hsorca revews [-5] Lghh s eho was spfe [4-5] The equvaece of he probe () o he foowg uforzao probe s prove here u( ) q x u r x u u () ( ( )) ( ) ( ( )), (), x( ) x( ) u( ), x(), [,], ( ) Now we ca see he souo of he probe () he vew () Ths eho was cae he eho of uforzao (MU) by suggeso of J Tepe[6], sce he sove a exape by hs eho (5)

2 54 Kebay Ayuov : Aaog of he Meho of Bouary Layer Fuco for he Souo of he Lghh s Moe Equao wh he Reguar Sguar Po I s prove he foowg Theore [4] Theore Le q( x), r( x) - aayca fucos o he erva [,] If q a w() he he souo of he probe () exs o he erva [,] a hs asypoc w have he preseao he paraerc vew (4) The coe of hs heore: a) Here s o he coo of Wasov[7]: xu ( x), x, b) I s suffce for exsece of he souo probe () s ecessary o ow he souo uperurbe equao a o chec coos: q, w() I[8-9] he asypoc of he souo hs probe was receve by he eho of srucura achg Here he souo of Lghh s oe equao s cosruce by he bouary ayer fuco[-8] he case whe correspog o perurbe equao has he poe of he ere orer o he reguar sguar po I[7] cosere he case whe correspog o perurbe equao has he poe of he orer oe o he reguar sguar po a he souo cosruce by he eho of bouary ayer fuco Bu eho of he proof [7] s o suabe he case whe he orer of poe s ore ha oe (see beow he begg of he proof of he Theore ) Usuay he eho of bouary ayer fuco (MBLF) s appe for cosrucg he asypoc souo of he sguar perurbe equaos wh sa paraeer a hgher ervaves; ay arces a boos are wre o eaborae hs eho[-8] Now we w say a few wors abou MSM MSM s a spfe verso of he eho of Va De a was creae - We ca appy hs eho for cosrucg asypoc sguary perurbe equaos wh a sa paraeer a hgher ervave (ha s equaos Prae-Thoov ypes)[-] as sguary perurbe equaos ype of Lghh[8-9] Saee of he Probe Here we w coser he case whe q(): q, N, for spcy Therefore he souo (4) of he uperurbe equao (3) we ca rewre he vew u x x w x (6) ( ) ( ) A hs souo w have he poe of orer, whe w() To gve fucos we w pose he foowg coos U : q x, r x C, We us prove he coo of exsg of he souo of he probe () a cosruc asypoc of hs oe 3 Cosrucg he Souo of Ths Probe by he Meho of he Bouary Layer Fuco The souo of he probe () we w see he for u x u x u x, x /,, u x u x C [,], C [,/ ] here e () eoe, ha fuco ( ) (, ), (7) We w epe fro, bu hs epee o poe for brevy / b, b () u u u, Ia aa for fucos we w ae he for: u ( ),,,,,,, Subsug (3) o () we w have for efe of fucos,,,,,, a u x,, we have he foowg equaos: q, b, (8 ) D : q,, (8 ) D ( ) ( ),, (8 ), D 3 ( ) ( ),, (8 3) 3, 3 D ( ) ( ), 4 4, 3 4, (8 4) D ( ) ( ),, (8 ),

3 Aerca Joura of Maheacs a Sascs 3, 3(): D D, [ u ] ( ) ( ),, u( ) ] (8) Lu x : xu x q x u x r x, (9) ( )[ u ( ) ( )],, (8), ( u ( x), ),,, Lu x, (9) D [ u ] ( u ( ) ( ))[ u ( ) ( )] u ( ) u ( ), ( u ( ), ),,,, (8) Lu x, (9) D 3 [ u3] ( u ( ) ( ))[ u ( ) ( )] u( ) u ( ), ( u ( ), ), ( ) D,, 3 3, (83) Lu x (93) [ u ] ( u ( ) ( ))[ u ( ) ( )] u ( ) u ( ), ( u ( ), ),, u,, (8 ) D,, Lu x, (5 ) [ u ] ( u ( ) ( ))[ u ( ) ( )] u ( ) u ( ),, u,, (8 ) u Lu x u( x), u xc [,], (9 ) x [ u ] D f( ) : u u,, u u u,, (8 ),, Lu x u x u x,, u x C [,], (9 ) Now we w sove hese probes cosecuvey We are o prove he exsece of he souo of equaos (5), (5), (5) ha ee he foowg ea Lea The equao Lg( x) ( x), () () here ( x) c, ( ) C () have uque boue souo fro, a hs have he foowg vew x ( ) ( ) ( ) ( ), g x x p x x p s s s x q() s p( x) exp s s, () Reay, geera souo of he equao (6) has he vew x g( x) p( x) x g() s p ( s) ( s) s If we se g( ) s p ( s) ( s) s, he we have go

4 56 Kebay Ayuov : Aaog of he Meho of Bouary Layer Fuco for he Souo of he Lghh s Moe Equao wh he Reguar Sguar Po Fro hs Lea foow ha equaos (9), (9), w have uque souos a ( ) ( ), u x c и u ( x), o Theore If s ho: b he he probe (8-) have uque boue posve souo, I a ( ), ( ) ( ) Here a furher we w eoe by,,,, cosas, whch are o o epe fro Proof I orer o proof of exsg of he souo of hs equao [7] was appe he foowg approach We w rewre (8-) as equao z( ) z( ) q( ) z( ) z( ), z( ) ( ), z( ) b By sovg hs equao as hoogeeous equao we have go z( ) P(, ) b (, (, ( ) ( ), P s P s z s z s here P q s s s (, exp ( ( ) ) () If afer egrag by pars he equao () w reuce o z ( ) z( ) P(, ) b (, ) (, ), ( ) P P s s z s s : P(, ) T (, z ), s, s q( sb b b b (3) Le, he by sovg hs equao as quarac equao we have go z( ) F[, z], F[, z] P(, )( b T (, z )) I s prove[7] ha hs equao w have a uque souo I he cass z S z z z b p : ( ),, z ax I s possbe o appy such a approach whe Reay hs case he equao (3) w have he foowg vew: z( ) [ ( P(, ) b (, ) (, )( (, ))* P P s s s s z () s s s sguar egra equao a we ca Sce o o sove he prevous approach Now we w o sove he equao (8-) by eho of varao cosa of Lagrage Ths equao we w rewre he for of here Qz : ( z) z( ) z h(, z), z( ) b, h (, z) ( q( ( )) z The probe Qz : ( z) z( ) z, z( ) b w have he foowg souo where c : (, c), c b b, () c ( ), ( ) b :,, Thus, ( ) (4) (5), herefore exss a uque boue posve srcy ecreasg souo (, c ) : (, c),, Fro (5) we have c ( ) ( ) (6) The souo of he probe (4) we w see by he eho of varao paraeers of Lagrage z (, c), c c( ) The for c () we have he foowg equao Fro с h(, (, c)) c() ( ) (, c( )) ( q( )) (, c) ( (, c)) (, c( )) foow ( ) c c (, ) (, c) ( ), c c (8) (7)

5 Aerca Joura of Maheacs a Sascs 3, 3(): Therefore we ca (7) rewre he foowg vew c( ) ( q( )) (, c) c ( q ( )) Fro here we have go q( s) c c exp ( ) s ( ) s ( s, c( s)) : F(, c) I s eve, ha he fuco, c ),b Operaor (, c) aps, ( o F aps he sege c c q( ), we have go sef Usg J o s exp ( ) s ( ) s Now we w proof, ha operaor F s coracg J Sce ( q( s)) s F(, c ) F(, c) c exp{( ) } ( ) s ( s, c ( s)) ( q( s)) s c exp{( ) } ( ) s ( s, c ( s)) Fro here appyg ea vaue heore of Lagrage, we have go F(, c ) F(, c ) s ( s, c ( s)) ( s, c ( s)) s ( ) s ( s, c ( s)) ( ) s ( s, c ( s)) By usg (8) we have, s ( s, c) c ( s) c ( s) s F(, c ) F(, c ) ( ) s c ( S ( s)) Fro here by ve hs egra o wo a by usg (6) we have go s ( s, c) c( s) c( s) s ( ) s c ( s ( s)) s ( s, c) c( s) c( s) s ( ) s c ( s ( s)) c c ss c c s ( ) s c ( s b ) c c Therefore F(, c c c ) F(, c) I s show, ha operaor F coracg o J Now we w sove he probes (4+) (=,, ) For sovg hs probe we w use he foowg: Lea The equao D ( ) : ( ( )) ( ) ( ( ) q( )) ( ) has he a fuaea souo ( ) exp 3 ( ( ) ) q( s) ( s) s s () s c q( s) exp s ( ( )) s ( ) s c (, ) X (, ), ( ( )) here ( q( s)) s X (, ) exp, s () s, s (, ) exp ( ) () s s I s eve fro ers X (, ), (, ) : X (, ), X (, ), X(, ), () X (, ) (, ) X (, ), (, ) ( ) ( )

6 58 Kebay Ayuov : Aaog of he Meho of Bouary Layer Fuco for he Souo of he Lghh s Moe Equao wh he Reguar Sguar Po Lea 3 The hoogeeous equao (8) D f ( ),, u, w have he uque souo a s s, I;, s, We w prove hs ea for he case s, oher cases are prove aaogousy (4-+) s hoogeeous equao wh zero org coos, herefore: ( ) 3( ) ( ) ( Aaogousy: We have for ) he foowg probe [ u ] D ( ), ( ) (9) The souo of he equao of (9) w represe of he vew: ( ) X (, ) (, )* () X ( s, ) ( s, ) s Afer egrag by pars we have [ u ( s) ( s)] s L ( ) M, J; ( ) ( ), L ( ) ( ) The case () () w prove aaogousy by vg egras o wo fro wo a fro o Therefore we prove he foowg: Lea The probe (9) has a uque boue souo, I a s va for evauaos () Aaogousy equaos (8к) (к=,, ) have uque boue souos, I ( ) C, a fro L L ( ) L, ( ), ( ) ( ) We prove he foowg: Theore Le s fufe coo () U : q( x), r( x) c,, q(), N, q() b u s exp s s The souo of he probe () w have uque souo a hs asypoc represe he vew (3) a ( ) (,, ); u ( x), o 4 The Esae of he Reaer Ter of Seres (4) Now we w proof he esae of he reaer er, ha s, he seres (4) reay s asypoc seres Lea 3 Le u x, u x u x u x u x U ( x, ) (, ) he () U( x, ), ( x[,]), (, ), [, ] We w prove hs ea for brevy for, ha s u x, U x,,, U ( x, ), ( x [,]), (, ), [, ] U x a, Proof Afer subsue () o () for he equao (8-), for, go: LU x, r( x), U ( x, ) C [,], () we have we have (, ) D (, ) [ U (, ) (, )] (3) U (, ) U (, ) (, ) ( ) U (, ) U (, ) ( ) U (, ), ( ) (4) The equao (3) has he souo U x, u ( x) C [,] I s fro (4) we w go o he egra equao (, ) (, ) () ( s, )[( U ( s, ) ( s, ) ( s, )) s U ( s, ) ( s) U (, ) ( ) s U ( s, ) U ( s, ) ] s s Afer egrae by pars he seco er we have he weey perurbe egra equao of Voerra ha w have J [, ] uque boue souo Lea 3 s prove

7 Aerca Joura of Maheacs a Sascs 3, 3(): The Exape Coparso of Three Mehos: Meho of Bouary Layer Fucos, Meho of Uforzao a Meho of Srucura Machg I s cosere a probe x u x u x u x x, u u Ths equao has he exac souo If a u x x x b x, b b [ u ], b [u ] () () b, he (6) exs o erva, u() b b (5) (6) (7) I A frs we w cosruc he souo of he probe (5) by eho of bouary ayer fucos, ha s u x, u x u x Ia aa for fucos b, b u u u u The for fucos, u ( x), (8) we w ae he for:,,,,, we have he foowg probes o, b x u( x) u( x) x, u( x) C [,], x ( ) [ u ( ) ], Fro here we have go b b, u ( x) x, b b [ b ( b b ] Therefore we w rewre (8) he for * x u x, [ b b ] [ b ( b b ) b b O( ), x (9) () Le b u Sce u (), he b b O( ) If we se x o (9), he u() b I w agree wh (7) II We w cosruc of he souo of he probe (5) by he eho of srucura achg 8 I s cosere sea of (5) he foowg uforzao equao (see (5)) u( ) x u u u () ( ) ( ), (), x( ) x( ) u( ), x(), [,], ( ) (3) souo of hs equao s represee he for u( ) b O( ), (3) b 3 x( ) [ b ] O( ) 4 4 Fro seco equay, afer sovg he equao x( we have go ( ) b If we se hs eag o he frs equay (3), he u( ) () b u, Tha w agree wh (7) oo III Now we w cosruc he souo of he probe (5) by he eho of srucura achg[8-] a) Frsy we w cosruc he ouer souo ( x -ouer varabe a x o epe fro ) of hs probe wh he a coo u u 5 We w have b x 3 uou ( x, ) x [ b x 8 x b ] x ) (3)

8 6 Kebay Ayuov : Aaog of he Meho of Bouary Layer Fuco for he Souo of he Lghh s Moe Equao wh he Reguar Sguar Po Tha s ho he erva [,], b) Secoy we w cosruc he er souo of he probe (5), ha sasfy hs equao of he sguar po x ear For hs rouce er varabe by forua: x ( ) The equao (5) we w rewre he for U [ U ( )] U ( ) (33) The souo of hs equao has he vew c Uer (, ) ( c ) c (34) c ( c ) O( ) c c, c, c - arbrary cosas here If he ouer souo (3) we w rewre he er varabe, he b b ( b ) u( x, ) [ ] [ ] x ) ], If we w seec cosas (35) c, c, c such c b, c, c c, he he ouer souo a er souo agree a he er souo w have he for Uer (, ) ( в ) c ( c ) c O( ) (36) Now he ufor souo of he probe (5) w have he foowg for x u( x, ) ufor ( ( x в ( x ) b ( x ( ( x b ( x b ( x If we w se here x agrees wh (7) 6 Cocusos, he u() O( ) b, hs Fro hs exape ca be see ha he eho of bouary ayer fuco s a abour-esve ha he eho of uforzao a he eho of srucura achg Bu he eho of srucura achg we ca appy o cosruc asypoc souo of o aos a sguary perurbe equaos ACKNOWLEDGMENTS I scerey ha he uow o e revewer for hs beevoe revew of y arce REFERENCES [] M G Lghh, A echque for reerg approxae souo o physca probes ufory va Ph Mag, 4 (949), 79- [] H S Tse The Pocare-Lghh-Kuo Meho Av Ap Mah, v 4, 956, pp [3] C Coso The Pocare-Lghh perurbao echque a s geerazaos SIAM Revew, Vo 4, No3, 97 [4] К Аyuov, Meho of uforzao a he vay of he Lghh eho, Izvesa of Acaey Scece Kyrgyz SSR ( Russa), 98, No, pp [5] К Аyuov Perurbe fferea equaos wh sguar pos a soe probes of bfurcaos ass ( Russa), Bshe, I, 99 [6] G Tepe Learzao a eearzao Proc Mah Cogress, Eburgh, 958, pp [7] W A Wasov O he covergece of a approxao eho of M J Lghh J Raoa Mech Aa, Vo 4, 955, pp [8] K Ayuov, Jeeaeva J K Meho of srucura achg he souo o he Lghh oe equao wh a reguar sguar po Repors Mahs, Vo 7, No, 4, p -6 [9] K Ayuov, J K Jeeaeva Meho of srucura achg of he oe of Lghh equao wh he reguar crca po Mah Noes, Vo 79, No 5, 6, [] Ayuov K, Zupuarov A Z Ufor asypoc of he souo of he bouary - vaue probe of he sguary perurbe equao wh wea sguar po ( Russa) Repors of Russa Acaey of Sceces, Vo 398, No5, 4, pp -4 [] A Z Zupuarov Meho of srucura achg of souos bouary vaue probe of sguary perurbe equao wh sguar pos ( Russa) Auhor's absrac of sserao, Osh, 9 [] A B Vas'eva, V F Buuzov Asypoca expasos of he souo of sguary perurbe equaos ( Russa), Moscow, Naua, 973 [3] A B Vas'eva, V F Buuzov, L V Kaachev The Bouary Fuco Meho for Sguar Perurbe Probes, SIAM (Cabrge Uversy Press), 987 [4] M I Iaaev Asypoca Mehos he Theory of Sguary Perurbe Iegro-Dfferea Syses (

9 Aerca Joura of Maheacs a Sascs 3, 3(): Russa), Bshe, I, 97 [5] V A Treog, Deveope a appcao of he asypoca Luser-Vsh eho, Russa Mah Surveys, 5:4 (97), 9-56 [6] O Maey R E, Sguar Perurbao Mehos for Orary Dfferea Equaos, Sprger-Verag, 99 [7] К Аyuov, A Khaaov Bouary Fuco Meho for sovg he oe Lghh Equao wh a Reguar Sguar Po Mahe Noes, Vo 9, No6,, pp (Tras fro Maheachese zae,, Vo9, No 6 pp 89-84) [8] КАyuov, T D Asybeov, S F Dobeeva Geerazao of he bouary fuco eho for he bouary probe of he bsguary perurbe fferea equao ( Russa) Maheachese zae, 3, pr)