Abstract. 1 Introduction
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- Miles Francis
- 5 years ago
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Transcription
1 A on eleen ehnqe fo he nl of f n le f fonon ne vel long n oe on n el nfne o fne A. V. enonç V.. Ale & J. B. e v Deen of l Engneeng. o lo Unve o lo Bl. A In h le ole nvolvng xll-loe le f fonon e nle on eleen folon. In h oh he f e o hve lne el oee n eeene negl eqon e on Khhoff le oel. he ol oe o e n nfne lee e o fne ol. Nel nle e eene fo ffeen fonon onfgon n he el of hoe nle e nvege. Inoon In een ee ol-fonon neon ole hve een nvege n eehe n whh he ong e oe o hve el ehvo. he oel fo el eone of ol hve een eveloe long wo n lne. In he f he ol eeene ll-neenen ng e Wnle oel) n whh he ne e n en egon e no ne o nohe loon oe. In he eon oh he ol e o e onno el e o h he effe of long n lo e oeve w fo he long egon. n oel hve een eveloe ng no on he geoel n ehnl oee of he onn h he well-nown nln olon o he ole of nfo hoogeneo e-nfne oo lne el ol ne onene long. oe eehe hve eene folon fo he ol-fonon neon h o when he ong e e o e nln hlf e n hee nle n e ve no hee go: f-ol le-ol n f-le-ol neon ole. In he f go hee e nll olon o he olee f-ol ole e on vonl eho fo l geoe n long of he f []. Ohe ho hve ooe nel olon ng ngle ehnqe ehe he fne eleen eho FE) [] o he on eleen eho BE) [5] ) o xe FE/BE oel [6]. In olon o he eon e of ole exlong he neon eween le go n he ol he el onon of eh le o he glol efone h een nle
2 lll ng lo-nfe ve ) n e onn oh. In he oel [7-8] he neon of wo le wee nll one n hen le o he of le go eong he el fo ll le n he go [7-9]. he el-onn oel ee n nln [] w eloe o eeen he ol-le neon n n e h n hen e l. [] n whh he on of ol-le neon e oxe fne ee n n X n olo [] whee eh le onee D eleen fo he nefe on n lne eleen e fo nenl on. he lo-nfe ve hve een exene o oel he h e ole f-le-ol) oh fo he g f [-6] n he flexle [7-] nvong Khhoff le oel n nel lgoh e eevel on FE n BE. In Fe- An[8] n olo[] he le onon e eeene ng n oe og e o evle he ffne he f-ol neon foe e no en no on o on he fnl vle of he el onn of ng. Bee he oel he e lon of nln olon o f-le-ol neon ole h lo een eoe oe ho. In Befel n Bneee[] hen e l.[6] he neon eween g n le go wee oelle ng wo n ohe. In he f he le wee ve no lle n he ole w olve he Fne Dffeene eho FD) whle n he eon eh le w eeene fne ee n vonl eho wee e o oe ll onon fo ol le n g f. eenl n enonç n v [] folon h een gven n whh he ol he le n he f e e o e flexle n eeene ngle BE lgoh; oeove ll neon ong he l oonen of fonon n ol e lneol en no on. In ll le e o f he ol e o e hoogeneo oo lne el ol whh h n nfne. Howeve n oe e he nfne hne of oele e n le o oveee vle fo he el leen n ole ole. In he lee ehnqe ooe o oel fne le e fll ll no fo n e. In he f he onon of eh le e n eqvlen ng e [-]. Howeve hee ohe e ongl eenen on he oel on o evle he eqvlen ffne oeffen of he ng e. he eon lne of eeh h een eveloe fo Be oel [-8] whh w nll ooe fo lee el ol ne onene vel lo ng on he fee fe. In hn e l.[9] n[] he Be oh w exene o he onene foe ng whn lee hlf e. Howeve he govenng l ffeenl eqon DE) fo hee oel le o eoe oee o ge oxe nel olon n o he ho nowlege hee e no loe-fo olon o h fnenl el ole. he h oh o fne el he o-lle fne le ehnqe eoe n ll n Booe [] n n oho n ll []. In genel h ove no eeenon of he he ee long he hf on whn he le on o h hee ee e oe eqvlen onene foe ng he nol on loe on he le nefe n Beel fnon e eloe o evle he ol ffne x. Howeve olon o hee hegeoe fnon e vlle onl fo l on of long. In oe el e h he fne fhe lne of oellng h een eoe whh fo he nfne nln oel. enenne [] h eene folon fo he evlon of leen neo on of he el eoon of Boneq olon fo lo n fel on long he e vel n whh he f o he ee on n he vel oone of he eon on he eh
3 of he lne e g. Anohe lon of he nfne nln oel o el lee ol nvolve eon of he g lne ng FE o BE n ong vel onn fo leen nol on on he g ong e. In Bneee [] f-le-ol neon ole oelle ng h een fo he ol he f e o e g. In he een le BE folon fo f-le-ol ee n whh ll he l e n ol e e o e flexle. In on h folon h een eveloe fo long n fll on fe eween he l oonen n he ong e. he f of h e n exene evew of enonç n v [] n whh he negl n lge eeenon of f-le-ol neon wh he el e e o e n nfne ) e ee. In he eon he evo folon exne o nle le-f fonon oe on fne ng oh he enenne oel n e eon of he g lne nvong he nln oel. Benng le eeenon In h eon he negl eeenon e ee fo f ee o exenl long eve foe eveloe f-ol n f-le neon. he f e o e hn el le n eeene Khhoff le oel.. Inegl Eqon In fge e ee he foe ng on he en lne of he f n he effo on he le on. B efong he weghe el ehnqe o Be eo heoe he on foe n on effo of el enng le n e lne o he eeve l qne of he fnenl ole o h he nvee leen negl eeenon of he f wen w [ ] w ) w ) w) q ) θ ) ) V n n. n Ω Ω ) w ) n) θ ) g ) w ) Ω ) w ) Ω N Ω Ω w Nel ;... N ) whee efe o he oe on; n e he fel on oe wh he on n he on of he le eevel; n ) n V n ) e he on effo ele o he enng oen n Khhoff he foe eevel; ) he one eon; w θ ) e on qne efeng o he nvee leen n nol loe eevel; g) exenl long e ove he le on; ) he ol-f on foe; N N F e he ol ne of he one n onene foe eevel; Ω n Ω e egon of he on le he f-ol n fle neon on eevel. el
4 o ove ffen eqon fo he on nnown n h folon n onl negl eqon one ffeenon of he nvee leen eqn ) elng n Nel w w w) q θ Ω [ ] w n. [ Vn ) w n ) θ ] w g ) w ) ) w ) Ω ) w ) el N Ω n Ω Ω Ω... N. ) Fo eqn ) n ) he fee negl e n ) n he enel e e gven n Aenx I. x g g n V n x n x. Deon Fge : Bon Effo n Don Foe ng on he f. Genel nll olon o he negl eeenon eqn ) n eqn ) no vlle o h n lenve eg n hevng he olon o elo nel ehnqe n eon one of he ele ge n h. he BE eon on n vng he ole on no lle egon lle on eleen whle he on ve no lle e lle. Nol on e efne fo eh on eleen n he ole qne e oxe elong he nol vle n neolon fnon. Fo he on foe n nlogo ehnqe efoe on he. In he een wo he f on ve no lne eleen n lne neolon oe fo ll onnol e on qne of he f.e. nvee leen w nol loe θ enng oen n n Khhoff foe V n. he lne neolon fo hee qne n e wen w ϕ θ w ϕ θ ϕ w ϕ θ )
5 V n n ϕ V ϕ n ϕ n V n ) ϕ n whee w θ Vn n n e he nol vle fo he nvee leen nol loe Khhoff he foe n enng oen eevel whee he e gne he ne fo he f eleen noe n fo he eon. ϕ he lne neolon fnon whh n e gven n enonle fo ϕ ς )/ n ϕ ς )/ wh he oone ς efne whn nge ς. In on nol on fo he one eon n leen e loe he nonooh egon of he le. x x x Ω x x Fge : f Deon. Eh f-le neon foe ) oe o hve onn on ove l Ω n ngle nol on efne o. he f on ve no ngl n f-ol neon foe ) e o hve lne on ove eh Ω o h nol on e efne he ngle vee ee fge ). he lne neolon fo he f-ol neon foe ove ngle on gven n he fo ε ε ε 5) whee n e he nol vle of he f-ol eon ee fge ). ε he hoogeneo oone n vle e gven x x ) x x x ) x x x x x ) ε / D 5-) [ x x ) x x x ) x x x x x ) ] D ε 5-) / [ x x ) x x x ) x x x x x ) ] D / ε 5-)
6 whee αβ D x x ; β ) x x ) x x x ) x x x ) 5-) α he oone of he nol on. x g x g g Fge : f-ol Foe Nol Vle n ngle ell. he negl nvolvng exenl long o he eve ge foe efne ove he h on Ω n e nfoe no eqvlen negl efne onl long he ono. he nfoon negl hee hown fo ln fo of he on. ng fo eqn ) n ) he on negl n e ee o ngle fo h Ω λ ) Ω ) 6) whee λ he on qn exenl long o ge eon) n veo gven he enel K K K w w ). 7) ) If lne neolon oe fo he on qn geoel eeenon el n ) A x ) A x ) A ) λ 8) whee x ) n x ) e he oone of on n he lol e x x ) B nlng he oone e he oone of n x x ) n e wen n new e x x ) n hen nfong he le e o he ol oone θ ) he on qne n e gven follow wh ) A oθ A nθ A ). λ 9)
7 A A x A x A ). ) he en-ol Jon Ω θ. ) ng eqn 7)-) no eqn 6) he on negl n e e-wen follow h K A o θ A n θ A ) θ ; θ Fo he geoel elon hown n fge follow h. ) oϕ θ ρ ; ) whee ϕ he ngle eween he n veo n he n veo γ nol o he ono he fel on ; ρ he ne eween he fel n lo on eeene eevel n ee fge ). ng eqn ) no eqn ) he on of he wen h K ρ oϕ ρ { } whee he veo oonen K ρ ) K ρ ) K ρ ) ) fo nvee leen n he loe negl eqon of he f e eevel gven ρ A ρ K ρ ) A ρ A ρ ) ln ρ 7) ln ρ ) 5) π D 8πD ρ A ρ K ρ) ρ A ρ A ρ ) ln ρ ) ρ ln ρ ) 6) 6πD 6πD whee D he flexl g of he f ρ he eon one of he l veo ρ n ρ he one eween he eon ρ n ee fge ). Fo he evlon of he f-le on negl l oee n e efoe n he l n he oeffen A n A hve eo vle ee he f-le neon foe e o hve onn on ove Ω ee fge ). I hol e noe h when he lo on loe he ene of he l ovol ϕ n heefoe
8 ρ θ. h nll negon n el e efoe on he negn of he eqn 5) n 6) elng n he followng fo A ρ K ρ ) ln ρ ) 7) 6D A ρ K ρ) ρ ln ρ ) 6D. 8) x _ x x ρ _ x ρ el el Ω ρ ρ el Fge : ell ono Inegon hee. he negon of enel ove eh on eleen o ono n e efoe nll o nel hee. he le ehnqe oe ol n hee e n wo n whh oe eloe een fo h ole e e; ee [5-7]. In genel he nel negon n e eeene n o fo K ) λ ) K ) ε ϕ ) ε J ε λ K ε ) ϕ ε ) Nq J W λ 9) whee J Jon oeo; ε n W e he oone n wegh qe fo on h n he on eleen o ono; N q he ne of on e o efo he negon.. Alge e Afe evlng he negl of he ee f eqon fo ll on eleen n n lge e n e ele follow
9 H H H I W W I W Q V Q Q G G G ) whee W W n W e he nol veo efeng eevel o he nvee leen he on long he f on he vee n he le o. V he nol veo fo he on effo n one eon of he f. n e he nol qne eevel oe wh he neon foe of he f-ol n he f-le ole. G G n G e he neenen veo ele o he exenl long. he nflene e n veo e wh e ' ' '' n '' e eevel oe wh he f on he n he le o. I n e he en n nll e eevel. le-ol eeenon In h eon he negl n lge eeenon of he le n he ol e eene. he ol e o hve ehe n nfne lee e o fne o h wo ohe e ee fo hee geoel onfgon.. Infne e he ol negl n lge eeenon fo n nfnel h e e hown f. In he followng eon he negl eqon e e o fne... Inegl Eqon In fge 5 e ee he neon foe ng on he fe o whn he nfne el hlf e e o e he el ole of he ol. B lng hel ehnqe h he weghe el eho l fo el fnon oe wh he eql eqon of he el hlf e n weghe fnon oe wh nln fnenl hlf e olon he followng negl eqon fo vel leen n e one n ) ) ) ) σ ) whee τ ) n ) of he le; n he ne of le n ) τ ) σ e eevel he he e on he hf n he nol e he he x eon lo n fel on gven nln fnenl olon fo
10 ) 6πG 5 ) ν 5 ) whee; x x ; n he nex ; x ) > ; x > x n x ) e he oone of he lo n fel on eevel ' he efleon of on n he hlf e fe lne x ) ) n oeffen gven ) ; 8 ν ) ν ); ν ) ) ν ; 5 ) 6 ) ;. -) In eqn ) ν n G e he oon o n he ol fo he ol. n e he one of he hf n he le e. he f-ol on fe n he f-ol neon foe. τ l x x ol fe eon x x ol le σ τ τ σ / / / l n τ τ n ) τ Fge 5: Ineon foe n he le-ol ole... Deon In he evo eon he lne neolon o oxe he ol-f neon foe w ee n h een nne fo he le-ol ole o h he fee on fe of nln hlf e ve no ngl on eleen whoe nol on one wh he noe of he f. Eh le geoell eeene one lne eleen n he he e long he hf neole wh q fnon ee fge 5) o h he hf he e eeene hell τ ) φ τ φ τ φ τ )
11 whee he neolon fnon e gven ) 9 9 ) ξ ξ ξ φ ξ ξ ξ φ 6 9 ) n ) 9 ) ξ ξ ξ φ wh ) x ξ n nge ξ n τ he he e nol veo. In on he of he le he nol e e o hve onn on n ngle nol on loe hee. he ee negl eeenon fo eqn ) fo ll nol on of he h le n e wen [ ] ) Ω Ω N A σ τ τ τ φ φ φ [ ] Ω Ω N ε ε ε. ) he l e n eqn ) efe o he on eween he le n hlf e fe o h fhe negl eqon noong he -ol neon foe neee n oe o ge olvle lge e. One eqon whh fe h oe n e one wng n negl eeenon of he vel leen noe loe on he ol neon fe h [ ] Ω Ω N A w w w σ τ τ τ φ φ φ [ ] Ω Ω N w ε ε ε. 5) An nlogo oee fo he negl evlon hown n eon. n e efoe wh he ee negl eqon fo he le-ol neon ole. h he negon hee efl e n ve no wo e. he f efe o lo on loe whn o on he fe of he hlf e n fel on loe n he ee on gon level. he eon negon e oe wh ngl onfgon of he enel. If fl fe oe fo he on he veo n e eooe no eon nol n ngenl o he lne. I hol e noe h he veo nol o
12 he lne nvn ee fge 6) n h nln fnenl olon n e ewen he oeon of veo on he lne of he on. ) 6πG ν ) [ ] ) ρ [ ) ρ ] [ ] 5 ) ρ [ ) ρ ] ) ng eqn 6) no eqn ) gve he followng elon 5 [ ] ρ. 6) whee he fnon f ρ) gven f oϕ ) ) Ω f ρ) ρ Ω ρ ) g α ) g α ) g α ) g β ) 5 g β ) A [ h α ) h α ) h α ) h β ) h β )] [ ][ A ρ A ρ ] 5 7) 8) wh he ong fnon g ) n h ) α n β. gven n Aenx III; he gen e When he lo n fel on e loe long n le n h nl o evle he negl he hf he ee e e o long he longnl x of he le. In ohe wo he lo on ee ll fel on loe he feene whn he le ngle on oone he ene of h le. In genel fo h e he nel evlon of he negl n e efll efoe ng ol olnol qe. If he lo n fel on elong o he e le he el loon of he he e on he hf fe oe. Hene he enel en o e q-ngl he hn ngl. If nel evlon hoen o efo he elng q-ngl negl el ehnqe e eqe o heve e h hee ee n efeene gven n eon.. An lenve oh o elo n nll een of negl evlon ee elow. he q neolon fnon φ n eqn ) h he genel fo φ ξ ξ n n exl eeenon of he le on negl ) ) ) H H H ) τ τ 9)
13 whee H Ω Ω ξ H Ω Ω ξ n H Ω Ω e n Aenx IV. oe on lne ell lne ' h ρ γ ρ ϕ Fge 6: wo-denonl Inegon hee n D e... Alge e Afe evlng he negl hown n eqn ) n 5) fo ll n le noe he exl fo of he ow of he lge eeenon ele o he h le n o h n e wen follow w I σ ) o n o fo A A A A U W Q Q Q Q ) whee he veo W n U e he leen of he n le noe; e he eeve nol foe veo. he leen veo fo n le n e one oeng he e n )
14 B B B B U W wh A Q B. ) In eqn ) I Q le o enn o nnee) oeon n. Howeve hee hve een hown n h eon o gve nfo of eenon wh eon.. n whh e wh I Q o. he exl fo of eqn ) ele o he h le n o h n e gven w σ. ) If he l n n he le neglee he elongon nenl on of he h le N A E ) whee N he oeon foe ng on of he h le [ ] A N σ σ φ φ φ. 5) If en efeene leen he elongon of he on he o on he hf n he h le.e. n n e wen ng he eeve eh n / / no eqn ) elng n: E E σ / / 8 5 / / 8 / / 7 / / / 6 / 6 / 6)
15 whee E n e eevel lengh Yong ol n of he h le. ng eqn 6) n eqn 5) no eqn ) el n he followng eeenon w σ. 7) If he e of eqn 7) wen nveel he ow of lge eeenon efeng o he h le e gven N σ N el β β β β β β β β β β β β. 8) he nol e on he o of he he h le n e lle vng he ol nfee lo N eqn 5) wh he o-eonl e of le A : A N σ τ τ τ 9) whee ; ; n. If eh ow n eqn 8) lle he onn n n hen he fo ow e one lnel new lge e n e wen
16 el N N ) When he ow fo he enng le e ee n nlogo oee he fnl e of eqon n e wen n o fo follow W W ) whee W n W e he veo efeng o he nol vle of he n he le o.. Fne e he negl eqon fo he le-ol ole n whh he el e e o e fne wll e e f n e of he enenne oel []. An lenve negl folon wll hen e one fo nln oel... enenne oel he negl eqon fo leen n he le-ol e e on enenne hee e fnenl ole o h he vel leen gene on whn he fne ngle le e gven n σ τ ) whee τ n σ e efne n eqn ) n enenne leen enel fo he lo on n he fel on n he vel eon x ). In 9 enenne ooe n oxe olon fo onene vel foe ng on he fe of n el fne ng eoon of Boneq olon. Ohe eehe hve ofe he ognl wo of enenne o oel el ole ne onene lo ng whn fne lee ol ng eoon of he nln olon. he ofe enenne oh e h he leen n e one ng he ffeene eween nln enel fo lo on
17 n nohe fo olloon on A loe vell elow on on he g lne e o onn vel leen eh x h ) n fge 7 A) ) whee nln fnenl olon gven n eqn ). x x le τ τ ol h g e σ σ A ^ f Fge 7: enenne oel. o efo he nfoon of negl eqon no n lge eeenon fo he le-ol ole ng enenne hee few le on e e n he nfne-le oee ee n he evo eon. he ognl nflene x ele o he nfne le e oee he lge e whh one when he oeon of he ognl lo on e loe on he vell g lne. In ohe wo ll onon of negl e n he nfne le oel ele o nln enel e oee ng he lenve enel gven n eqn ). h he le-ol lge eeenon fo h enenne oel n e wen ll o eqn ) W W. ).. De nln hee In he evo eon he neon eween he fne le e n le go w e ng enenne oel o efo he le-ol nl. In he en eon h ole gong o e nvege ng nln fnenl olon o eeen he fne el eone.
18 In fge 7 hown geoel n long onfgon of le go eee n ngle fne-le el onn n he leen negl eqon fo he le-ol e n e wen N α ) ) ) ) σ τ 5) whee he e on n e ve no hee fe n whh he f he ; he eon fe he on of on-fee hlf e fe whee n he l fe he nln ole off he fe on whh ) vell g lne ee fge 7. he on fnenl olon - on he hoonl lne wh eo one ) - one ng nln olon fo e no he h fol ) σ n σ n σ n σ 8π ) ν ) whee oefen n eqn 6) gven ν ) ) ν ) ) ) 5 ) ) eqn 5) gne he followng vle α f.5 5. In on he fee e α n f o Ω f Ω he eon of negl eqn 5) efoe ng he e neolon fo he geoe n fo he neon foe of he le e n he evo eon. o on he lge eeenon fo negl eqn 5) he evlon of he negl of ) n ) eqe. he negon of he f enel n e e o ng eqn 7) n fo he eon enel gven n 5) n nlogo een of negl of ) eloe leng o he followng exeon Ω whee he fnon ρ) gven n e ) ) Ω ρ ) oϕ 7) ρ { [ ][ A ] ρ ) g α ) g α ) g α ) g β ) 5 g β ) 6 g β ) A ρ A [ h α ) h α ) h α ) h β ) h β )]} /[8π ν )] ρ 8) 5
19 Afe nfong ll he negl of eqn 5) he ow of he lge eeenon ele o he h le n e wen ll o eqn ) o W U q I q q o F σ 9) Alng l nfoon o eqn ) n eqn 9) he veo of leen n e one n hen he ffne of he le n e nee ng nlogo e ee fo eqn ) o eqn 6) elng n o o F W U σ. 5) If lge eeenon 5) wen nveel n lne onon efoe eween he fo ow of he h le fo h elng e n he onn n n eqon eqvlen o eqn ) one nf N N N F h g σ. 5) When h oe eee fo ll le n he fnl lge e n e eeene n o fo follow
20 F o U W W o 5) whee W n W e he veo efeng o he nol vle of he n he le o. In eqn 5) he nol on he oo of he fne le hve he egee of feeo wh ee o leen ee n one wh he g ong n 5). o he eln e nole o h ll enng nnown elong onl o he f lne. Hene eqn 5) n e lfe o W W. 5) f-le-ol e olng o ele he fnl e of eqon fo he f-le-ol neon ole nee o ole he f lge eeenon n le-ol onon fo he eeve oel. In he e whee he ol oelle he nfne-le nln oel he lge e eqn ) hol e nee no eqn 6) gvng he followng elon H H H H H H H H H W W W V G G G. 5) If he fne enenne hee eqn ) o e nln oel eqn5) nee n eqn 6) he fnl e wll gve elon nlogo o eqn 5). 5 Evlon of Don Qne Afe he olon of he fnl e of eqon he nnown veo he on leen n loe of he f he leen he noe he le-o efleon n he foe ng on he g lne) eene. If he vle of h veo e e no eqn ) eqn ) o eqn 5) he f-ol n le-f neon foe n e evle n hen f hee foe n he noe leen e nee no eqn 7) o eqn 5) he neon he ee on he hf n nol e he of he le n lo e eene. In he e of e nln oel fo lee e he foe ng on he g e n e lle ng he exeon en fo eqn 5) h
21 F w w. 55) o Bee he qne he on of he f n neon foe of he ole hee e fhe fel whh n e eqe fo nne he on effo of he f. An exeon fo hee effo n e one ng he elon en fo ll le heo. Fo he enng oen he onve elon n e wen [ ν ] q Dν w q ) w q ;. 56) q On he ohe hn he negl eeenon fo he ve of he le n e wen efong ole ffeenon on he leen negl eqn ) w [ n n ] q ) w q w q ) ) q θ q N h [ n n θ ] q g ) w ) Ω V w ) ) Ω q q Nel ) w ) q w ) ) w ) ) Ωh Ωh h h Ω Ω ;... N ; 57) whee n q e eon one efeng o he eon of he ffeenon he lo on; he enel n eqn 57) e hown n Aenx II. he ee negl eeenon n lge eqon fo ve n e one lng l ehnqe o he leen n loe ee hogho he ele eon. o nfo he on negl o n eqvlen one efne long he on he followng eqon n e e whee q h w ) ) Ω q ρ) h Ω oϕ χ 58) ρ ρ A [ δ ρ ρ ] [ lnρ ) δ ρ ρ ] ρ χ ρ) A ρ A ρ ) lnρ ) 6πD 6 Nel nle 6πD h el. 59) In h eon e eene oe onfgon fo f-ol le-ol n le f neon ole n fo e long of he e.
22 6. ong on engl e of ogh g e In h exle he fe of ee o exenl long e ove qe egon ee fge 8 e ). he elng leen fo he enenne he e fne nln n he Be oel fo he el e e ne n le fo oe eh of he g e. le : Dleen of he on A W - ). η h Be [8] h wo De F. nln) 6. f oe on fne h wo enenne) one qe le e o nfo nvee lo g.n n eng on lne el hlf e ee fge 8 e ). he lengh of he e of he le hne.6 n el oee e Yong ol E n oon o ν. 5. he hlf e h Yong ol E 9. oon o ν. n hne h. he leen n enng oen fon he on A B n e hown n le. g g.6 le h η ) ) A B g e of he hlf e fe Fge 8: Fne ne nfo long. le : oe on n he h le. on of le [7] h wo De. F. nln) h wo enenne) A [Defleon w - )] B [ ].8.5. [ ] A [oen N/)]
23 6. ong on wo le In h exle f on wo le nle fo wo e ee fge ). In he f hee no on eween he g f n he ong e. he g e loe nfne eh h / ). he el e ne n fge 9 n exee n e of wo enonle ee. he f he o eween he Yong ol of he le n ol E E n he eon ee gven exeon whee he ol w E lo ng on he f w he vel leen of he o of he le n he le ee. he ng eween le.5 he f wh B.5 he oon E o.5 n G he he ol gven G. ν ) h wo [] h wo [] E /E /G w ) 8 6 h/ E /E / Fge 9: ffne of wo e-le go. / / g g e Fge : Geoel onfgon of f-le-ol ole.
24 In he eon e he le f eng on fne elow whh he g lne e eh h ). he el of h nl e ne n le fo evel vle of E E. le : Vel leen of he le f fo evel E E o. E E 6. engl f w E [9] h wo h wo enenne) De. F. nln) A engl le wh enon. x 5. x.5 Yong ol 7 E. N/ oon o ν. n ne nfo long g. N/ ) eng on nfnel-h e n nle wh o who ffenng of le-go ee fge ). In he le f e he go of le hve fo e of onfgon. he f on of one le he eon ooe of e o le he h eeene he le loe long he on of he f n he l e ll nne le. he le e e o e noele n he lengh n ee e 5 n D. eevel. he ehnl oee of he ol e Yong ol E. 5 N/ oon o ν. 5 n he f-ol nefe hown n fge. he el fo leen on long he lne x.75 ) e n fge n he enng oen long he lne x 7.5 e n fge.. N/ Fge : f-le-ol lo.
25 x Fge : f-ol eon. x - x - x - le le 6 le 8 le 9 le w ) x - x - x x ) Fge : Dleen long lne x. 75 ). 5 x N/) le le 6 le 8 le 9 le x ) Fge : Benng oen long lne x 7. 5 ). x
26 le : Foe on he le n on he f. Ne of ge % lo le eon onlon A on eleen folon h een ee fo el nl of he f n lef fonon e o vel long n oe on n nfne lee e o on fne. All he neon foe of he f-le-ol e e lneol en no on. he nel nle wee efoe fo he e long of he e n f-ol le-ol le-f-neon ole. In he e of he e long of he e he leen e oe oe he enenne oel hn he e lee nln oh when he eone of Be hee e en efeene vle. Alhogh he enenne oel n ve ehnqe e o he l of hel eeenon le o ooe vle when he g e en o he fee fe of he e ee of he eoon ehnqe eqe enenne oel h no en no on he nflene of he foe ole onn oe he g e fe. h f lo le o he f oe on fne n whh he el fo leen n enenne oel e lowe hn hoe one he e nln oel n Fe n Wle [7]. In he le-ol neon ole he nflene of he elve g eween he le n he ol e nvege. he ole on n nlng g f on wo le fo wo e. In he f he nfnel-h e no n on wh he f n he nl efoe fo ffeen o lengh/ee) of he le. In fge 9 hown goo geeen of eone of he een folon oe o he Befe n Bneee oh []. In he eon e ll e n on n he e onee fne-lee ol. he el of he een folon nvong he e nln oel loe o h eene Bneee [] e o he l of he wo ohe fo he fne lee e. he l ole nle on of engl f ffene o no le go. In fge e noe h he onenon of enng oen long lne x 7. 5 e ee o he vn of he le e o he hgh vle of f-le eon e whn he o-eon of he le. In on he el fo he leen long lne x. 75 ee fge ) how h hee vle e effevel no ofe when he ne of le n he go nee. l ehvo n e oeve fo le fo he lo nfee o he le n he f. Fo h ef exle n e een h he f g h l nflene on he ehvo of he le-f e. Anowlegen h wo h een fnnl oe FAE Bl).
27 efeene [] Zn.. I A. & Ke A.. Anl of l le-el hlf-e neon ng n eneg oh. Ale h.l oellng [] Ke A.. Zn.. & I A. Anl of fl oge n nlng fonon-ee neon. Ale hel oellng [] Hele J.A. El olon fo lge x ole n fonon neon nl. In: oeeng of he Ine vl Engnee Vol [] Hele J.A. Alon of lge x neon nl o f fonon. oeeng of he Ine vl Engnee Vol [5] v J.B. Folção o éoo o eleeno e onono nále neção olo-e Aoe ofeo Qlfng ex n ogee) ão lo Engneeng hool ão lo Unve Bl 99. [6] efe. oe.e. An lon of FE/BE olng o fonon nl. Avne n Bon eho. e..a. Be & onno oonl ehn lon: ohon n Boon [7] olo H.G. & Dv E.H. he eleen Behvo of le xll-loe noele le n e. Géoehnqe [8] olo H.G. & e N.. eleen n lo on nl of le go. Aln Geoehn Jonl G) [9] hen W.Y. howy.k. & Yong K.Y. A vonl oh fo vel efoon nl le go. Inenonl Jonl fo Nel n Anll eho n Geoehn; ) []nln.d. A foe he neo on of e-nfne ol. h []hen WY how Y.K & Yong K.Y. A vonl oh fo vell loe le go n n el hlf-e. Géoehnqe 9) []X K.J. & olo H.G. Genel el nl of le n le go. Inenonl Jonl fo Nel n Anll eho n Geoehn []Befel. & Bneee.K. he el nl of oele le n le go. Géoehnqe; ) []Befel. & Bneee.K. he ole of le go-le neon. Géoehnqe; ) [5]Kw E. An el nl fo le f fonon n hoogeneo ol. ol n Fonon; 9) [6]hen W.Y. how Y.K. & Yong K.Y. A vonl oh fo he nl of le gole neon. Géoehnqe; 5) [7]Bown.. & Wene.J. he ehvo of nfol loe le. Foong. ol n fonon 5) [8]Fe-An B. A onon o he nl of le-oe f fonon. h.d he Unve of ohon ohon U.K [9]Hn.J. & ee I.K. he nl of flexle le-f e. Géoehnqe 8) []olo H.G. An oxe nel nl of le-f neon. Inenonl Jonl fo Nel n Anll eho n Geoehn; []enonç A.V. & v J.B. A on eleen fo he nl of f fonon on le. Engneeng Anl wh Bon Eleen;. 7-7.
28 []noh.f. & Woh.. Anl of vel efoon of le go. Géoehnqe 9) []lon G. & Ge G. eleen n onl nenl foe of goe le n lee ol. Géoehnqe 8) []Be D.. heo of ee n leen n lon o he egn of o nw. o. Hghw eeh Bo [5]Be D.. he genel heo of ee n leen n lee ol e I. Jonl of Ale h 6) [6]Be D.. he genel heo of ee n leen n lee ol e II. Jonl of Ale h 6) [7]Be D.. he genel heo of ee n leen n lee ol e III. Jonl of Ale h 65) [8]Be D.. ee n leen he of wo le g e ol e. 5 h o. Hghw eeh Bo e. F. Bggf E.. W NA-N lon: Whngon [9]hn K.. Kh. & ee.. Foe on n he neo of lee el hlf e. Inenonl Jonl of ol n e []n E. geen fnon n llee hlf e. Ale hel oellng []ll J.. & Booe J.. Fne le nl of lee el el ng flexl oh - long. Inenonl Jonl of Nel eho n Engneeng []oho.h. & ll J.. Fne le nl of vell loe le n le go. oe n Geoehn 8) []enenne W. feln engeehnng. De e. 9. []Bneee.K. Effe of he le on he lo leen ehvo of le go when ee o een long. o. n Al-New Zeln onf. Geoeh [5]le V. & le J. ngl Inegl n Bon Eleen eho e..a. Be & Al oonl ehn lon: ohon n Boon 998. [6]H K. & Be.A. A new oone nfoon eho fo ngl n q-ngl negl ove genel ve on eleen. In: BE IX Vol. nge-velg: Beln 987. [7]elle J..F. A elf-ve oone nfoon fo effen nel evlon of genel on eleen negl. Inenonl Jonl of Nel eho n Engneeng [8]Fe.A. & Wle.J. Nel nl of engl f on lee fonon. Géoehnqe 6) [9]olo H.G. & Dv E.H. El olon n ol n o John Wle & on: New o Aenx I he fee e n negl eqn ) n ) e gven β ν β π nγ γ β ) π π n ν π K [ ] K [ o γ o γ β )] whee β he nenl ngle of he f; γ he ngle eween n ) n ) oone; ν he f oon o. he enel efee o eqn ) n evve ) e gven
29 w 8π D ) ln ) ln ) ν ) ln ν ) n ) n θ ;. πd [ ν ] n ;. π ν ) ) n ) n ;. π n [ ν V n ν ν ] ) n );. π π w ) ln ;. πd ) [ ) n ) n ) ln ] θ D ;. π n { ) ) [ ) ) )]} ν ν n n n ;. π ν n { ) ) ) ) )} n n n ;. π Vn { q )[ ) ) ) ) )] ν q n n n π ν ν ) [ n ) ) n )]} ) [ ) q q ) )] ; q. π whee D he flexl g of he f; he ol of veo n eo one n eon ; n e he n veo nol n ngenl o he on of he le n δ he Konee el n he ve of he on of he f. Aenx II: he enel efee o n he f ve negl eqon eqn 57) e gven ]} n { ) ) )[ ) n n n n n ν δ ν δ αnα ;. π ν n ) { [ n n ) n ) n α α δ )] n n )}; π α. V ) { ln } w δ ; πd θ ) {[ ] n n n } D δ ; π ) { ν ) ) [ n n n n )] ) n α α δ α α ν π ν ) n n ) ) ν ) [ δ ) n n n )]} n [ ] 8 α α ν π ) α α ; α whee he ol hve he e enng n Aenx I.
30 Aenx III: he ong fnon g h fo ρ f eqn 8) n ρ eqn 8) e wen g ] [ ln g ] [ ln g ]} [ ] {[5 5 5 g h h h h. Aenx IV he exeon elng fo he nll evlon of negl n eqn 9) e ] ] [ ln[ { o o H ξ ] ) [ ) ] ) [ ] ) [ ] )ln [ [ ] [ ] } { } { 6 [ ] } 5 ln 6 ] [ [ ] [ ] { [ ] ln ln ln H Ω Ω ξ [ ] ] [ ] ) [ ] ) [ [ ] [ ] } ln ] [ 8 ] [ ) 5 { Ω Ω ln ln H whee ) 6 G π ν ) 8 ν ν ν ) x ) x ; ; ) ) x x. ] ) ) ) ) [ ] [ ] [
On Fractional Operational Calculus pertaining to the product of H- functions
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