"Science Stays True Here" Journal of Mathematics and Statistical Science, Volume 2016, Science Signpost Publishing

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"Scnc Says r Hr" Jornal of Mahmacs and Sascal Scnc Volm 6 343-356 Scnc Sgnpos Pblshng Mhod for a Solon o Som Class of Qas-Sac Problms n Lnar Vscolascy hory as Appld o Problms of Lnar orson of a

com Absrac wo horms ha rdc solons of h gnral qas-sac problm of lnar vscolascy hory o a solon of h corrspondng problm of lascy hory ar provd.

1 "Scnc Says r Hr" Jornal of Mahmacs and Sascal Scnc Volm Scnc Sgnpos Pblshng Mhod for a Solon o Som Class of Qas-Sac Problms n Lnar Vscolascy hory as Appld o Problms of Lnar orson of a Prsmac Sold Laf Kh. alybly Mhrban A. Mamdova Ins of Mahmacs and Mchancs Acadmy of Scncs of Azrbaan Bak Az 4 Azrbaan E-mal : lalybly@yahoo.com mr.mammadova@gmal.com Absrac wo horms ha rdc solons of h gnral qas-sac problm of lnar vscolascy hory o a solon of h corrspondng problm of lascy hory ar provd. hs horms hold f on of h followng condons s sasfd: ) h maral s clos o a mchancally ncomprssbl maral; ) h man srss s zro; 3) h shf and volm hrdary fncons ar qal. h horms provd fr drc and nvrs ransforms bwn solons of vscolascy and lascy problms whch maks hm convnn n applcaons. hy hav bn appld o solons of problms on h pr orson of a prsmac vscolasc sold wh an arbrary smply conncd cross scon. Som xampls dscrbng h oband rsls hav bn consdrd. Kywords: vscolascy qas-sac problms xac solons orson problms.. Inrodcon I s wll known ha solons o h qas-sac problms n lnar vscolascy hory n mos cass ar oband from h corrspondng problms of lascy hory by sng Volrra s prncpl ha s by h way of rplacng lasc consans by som opraors and sbsqn nrpraon of hs opraors []. hr ar also a nmbr of mhods for solvng h mnond problms by applyng Laplac s Laplac-Carson s Forr s ngral ransformaons [-4]. In h las cas an xac drmnaon of orgnal fncons from h oband mag fncon s no always possbl. In h prsn

2 344 papr ndr som condons horms rdcng solons of qas-sac problms of lnar vscolascy hory o solons of h corrspondng problms of lascy hory ar provd. Samn of h Gnral Qas-Sac Problm of Lnar Vscolascy W wll consdr an soropc homognos maral. Wr o h drmnng rlaons bwn componns of srss nsors and dformaon nsors ε n h followng form [4]: or G = s + Γ τ s τ dτ () Kθ = + U τ τ dτ () s = L ( τ) ( τ) dτ (3) G K = θ M τ θ τ dτ. (4) Hr s m; j = 3. Bsds = ε εδ s a dvaon of h dformaons ε ; ε= εδ / 3 s a man dformaon; δ s h Kronckr symbol; s = δ s a dvaon of h srsss ; = δ / 3 s a man srss; θ = 3ε s a rlav varaon of h volm; G = cons s an nsan lasc shf modl; K = cons s an nsan lasc modl of h volm dformaon; h fncons Γ ( ) U( ) L ( ) and M ( ) ar krnls of h shf crp volm crp shf rlaxaon and volm rlaxaon rspcvly. Rlaons ()-(4) ar h Volrra scond yp ngral qaons. Eqaons (3) and (4) ar drvd from () and () by solvng hm wh rspc o S and θ. In rn qaons (l) and () rsl from (3)

3 and (4) by solvng h las wo ons wh rspc o 345 ε and θ. In hs cas h fncons L ( ) s a rsolvn of h krnl Γ ( ) and M ( ) s a rsolvn of h krnl U( ). A h sam m h fncon Γ ( ) and U( ) ar rsolvns of h krnls L ( ) and M ( ) rspcvly. I s clar ha rlaons () () and (3) (4) ar qvaln. No ha bwn h krnls U( ) and M ( ) hr xs h followng ngral rlaons [4]: Γ = L+ L τ Γ τ dτ (5) U = M + M τ U τ dτ. (6) h componns of h srss sasfy h balanc qaon + = (7) j F whr F ar volm forcs. Sppos ha srfac forcs R ar gvn on a par S of h bondary srfac bondary dsplacmns ar gvn on h rmanng par S : l = R ; = (8) j S S whr l j ar drcon cosns. o solv h problm n h dsplacmns h Cachy gomrc rlaons ε = + / (9) j j shold b adjond o h rlaons ()() (or (3) (4)) (7) (8). Frhr o solv h problm n h srsss nsad of (9) s ncssary o s sx ndpndn qaons of dformaons compably. By [3]

4 346 on of h forms has h form ε + ε ε ε = () kl kl k jl jl k whr jkl = 3. Solon of h Gnral Qas-Sac Problm of Lnar Vscolascy Undr h Condons Drmnd Frs formla h followng lmma [5] who proof. Lmma. For h sasfacon of h homognos qaon ( ) µ + r τ µ τ dτ = () whr r( ) has som rsolvn assocad wh r( ) by rlaons of h yp (5) (6) s ncssary and sffcn ha ( ) ( 3) µ = j =. () horm. L on of h followng hr condons hold ) K (maral s mchancally ncomprssbl) ) = (h man srss s zro b K < ) 3) ( ) U( ) shf and volm crps concd) or qvalnly L ( ) M ( ) Γ = (h krnls of h = (h krnls of h shf and volm rlaxaons concd). hn h xac solon of problms () () (or (3) (4)) (7)-() s rprsnd as = + Γ( τ) τ = d (3) whr and arc solons of h followng qas-sac problm of lascy hory: G s = (4) θ = ndr condons and (5)

5 347 or Kθ = ndr condon 3. (6) + F = l = R ; = (7) j S S h followng noaon has bn accpd: ε = + / ε + ε ε ε =. (8) j kl kl k jl jl k = ε εδ s = δ ε = ε δ / 3 = δ / 3 θ = 3ε (9) o o ( τ) o = L dτ. () h proof of horm s prformd by drc sbson of formla (3) no rlaons nvolvd n h samn of h nal problm. In dong so on shold s lmmas () () and also h accpd noaon (9) (). If n h samn of h vscolascy problm (3) and (4) ar sd as drmnng qaons hn s ncssary o ak advanag of rlaons (5) (6). No ha h dformaon componns ε ar dfnd by formla ( ) ( ) ε = ε + Γ τ ε τ dτ () whr ε s xprssd by n h frs formla of (8). horm. L on of condons 3 of horm hold. hn h xac solon of problm () () (or (3) (4)) (7)-() s rprsnd alrnavly o (3) by h formla = = L τ τ dτ () whr ar solons of h followng qas-sac problm of lnar lascy hory:

6 348 G s = (3) θ = ndr condons and (4) or Kθ = ndr condon 3. (5) + F = l = R ; = (6) j j S S o h followng noaon has bn accpd: ε = + / ε + ε ε ε =. (7) j j j kl lk k jl jl k = ε εδ s = δ ε = εδ / 3 = δ / 3 θ = 3ε (8) ( ) ( ) (9) F = F + Γ τ Fdτ; R = R + Γ τ Rdτ. h proof of horm s prformd by h drc sbson of formla () no all h ncssary rlaons. In hs cas noaon (8) (9) lmma () () and also rlaons (5) (6) bwn h rsolvn fncons. L now a solon of h lascy problm s known wh rspc o h volm forc F h srfac forc R and h bondary dsplacmn : = F R o ε = ε F R o = F R. By sng horm and changng F R and o o F R and o o rspcvly w drmn ε = F R whr s dfnd n (). o o : = F R o ε = ε F R o I s also ncssary o mak chang of h lascy modl G and K o G and K rspcvly. Afr drmnng h qans ε by formla (3) w fnd h sogh - for

7 349. Sogh - for componns of h dformaon nsor ε ar drmnd by formla (). If w mak s of h horm h qans o ε wll b: = F R o ε = ε F R o = F R whr F R ar dfnd by formla (9). In hs cas h chang of h o corrspondng maral consans s also ncssary. Afr fndng ε w drmn h sogh - for qans from formla (). h sogh for componns of h dformaon wll b: ε = ε.. Applcaon Solvng h Problm of Lnar orson of a Prsmac Vscolasc Sold wh an Arbrary Cross Scon h problm of lnar orson of a prsmac vscolasc sold srvs a good xampl for applcaons of h abov - formlad horms. Bcas n hs cas on of h condons of hs horms namly h condon ha h man srss s zro s flflld. L h forcs ladng o bradng copls r appld o h bas of a prsmac vscolasc sold wh an arbrary cross scon. W wll sppos ha h sd srfac of h sold s fr of xrnal forcs and volm forcs m abscn. Mchancal proprs of h maral of a prsmac sold ar characrzd by rlaons () () or (3) (4) of lnar vscolascy hory. W s h Carsan coordna sysm ( x x x ). Drc h axs x 3 paralll o h axs of h 3 prsmac sold. A no consrand (pr) orson of a prsmac vscolasc sold wh an arbrary cross scon accordng o San-Vnan w consdr ha n a fxd prod of m ) qally dsan cross scons ws a h sam angls; ) all h cross scons ar qally bn; dplanans ( 3 ) proporonally dpndng n m of orsonal angl ar mrgd whch s allowabl n lnar orson. Wr o mahmacally h mnond assmpons n h form: = γ x x ; = γ x x ; = γ ϕ x x. (3) 3 3 3

8 35 Hr ϕ ( x x ) s a fncon of dplanaon γ γ ( ) = s a rlav angl of orson a h nsan of m. In cas of ( ) [6]. γ cons rlaons (3) concd wh h corrspondng rlaons of San-Vnan Usng proprs of hrdy whch ar applcabl o vscolasc solds w rprsn h fncon γ ( ) n h form whr ϑ ( ) s som of m fncon o b drmnd. Consdrng (3) n (9) w hav γ = ϑ + Γ τ ϑ dτ (3) ε = ε = ε33 = ε = ; (3) ( ) ϕ γ ( ) γ ϕ ε = ; = +. 3 x ε 3 x x x (33) Usng rlaons (3) (33) n qaons (3) (4) drmn h qans : = = 33 = = = ; (34) ϕ 3 = G x γ ( ) R( τ) γ ( ) dτ ; x (35) ϕ 3 = G + x γ ( ) R( τ) γ ( ) dτ. x (36) akng no accon (3) and (5) rlaons (35) and (36) rn no h form ϕ ϕ = G x ϑ = G + x ϑ. 3 3 x x (37) From h balanc qaons only h followngs ar omd:

9 = = + =. x x x x (38) h frs wo qaons of (38) ar sasfd as an dny h hr4 f (37) s akn no accon ylds ϕ ϕ + ϕ =. x x (39) Formla (39) shows ha on h doman occpd by h cross scon of sold h dplanon fncon ( x x ) ϕ ms b a harmonc fncon of h varabls x and x. I follows from h las argmn ha h dplanaon slf shold also b a harmonc fncon. In h consdrd cas as n hory of lasc orson can b shown ha on conor Ω of h cross scon h dplanaon fncon p sasfs h condon or = n ϕ xcos ( nx) xcos ( nx ) Ω ϕ d x + x n ds = Ω (4) whr d dn d ds ar drvavs wh rspc o h normal n and h ar Ω rspcvly. hs h problm of vscolasc prsmac sold s orson n smlar way as h problm of lasc sold s orson s rdcd o h Nman problm (39) (4) for Lapla qaon. In hs cas can b shown ha xsnc condons for a solon of h Nman problm ϕ ds n = ar flflld. For srsss qally acng on h fac srfac w hav Ω 3d = 3d = (4) ω ω ω ω whr ω s an ara of h cross scon of a prsmac sold. akng (4) no accon w com o a conclson ha angn srsss appld o h cross scon ar rdcd o a par of forc whch has h momn

10 (4) M = ω x x dω. h balanc condon on h fac srfac gvs M ( ) = M ( ) whr M ( ) wsng momn. Consdrng hs and formlas (37) n rlaon (4) w oban ha ϕ ϕ whr = ω ( + + x x ) D >. ( ) M ( ) s h gvn ϑ = (43) D ω s a rgdy n orson. I can b show ha always D G x x x x d hrfor h problm of physcal lnar orson of a vscolasc prsmac sold s complly solvd f w fnd h dplanaon fncon ( x x ) ϕ. Now rprsn h solon of lnar orson problm n h form (3). In hs cas h qans ε and ε ar xprssd by formlas () and h frs formla of (8) rspcvly. h balanc qaons (38) n vw of h scond formla of (3) manan hr form: = = + =. x x x x 3 3 (44) Bsds from (34) and (37) w oban: = = = = = 33 ϕ ϕ = G x ϑ = G + x ϑ. 3 3 x x (45) Usng () and (3) formlas (3) and (33) ar convrd no h form ε ε ε ε = = 33 = = ( ) ϕ ϑ( ) ϑ ϕ ε = = +. 3 x ε 3 x x x (46)

11 353 From formla (3) for componns of h dsplacmn vcor n vw of h frs formla of (3) and also formla (3) follows ha = ϑ xx; = ϑ xx; = ϑ ϕ xx. (47) Rlaon (4) wh consdraon of formla (3) wll b wrn n h form ( ) = ( ) = 3 3 M M ω x x dω (48) Rlaon (43) dos no chang s form. As w s rlaons (43)-(48) ar rlaons of lasc qas-sac orson hory. hs mans ha as appld o h problm o orson h qans and ε nvolvd n formlas (3) () ar componns of dsplacmn vcor srss nsors and dformaons rspcvly whch appar n h consdrd prsmac sold whl s qas-sac lasc orson wh h orson momn M ( ). In hs cas plays h rol of only a paramr. hrfor f by any on of h xsng mhods h problm of lasc orson of a prsmac sold wh h gvn cross scon has bn solvd ndr h condon ha h dsplacmn modl G and h orson momn M ar known ha s h lasc dsplacmns dformaons ε srsss hav bn fond hn by changng G o G M o M ( ) n h xprssons of ε w fnd h qans ε. hrafr accordng o formlas (3) and () w drmn h sogh for solon of h corrspondng problm of vscolascy. A hs pon rmark ha by [6] whl solvng h consdrd problms of lasc and vscolasc orson nsad of h dplanaon fncon ϕ on can s hr h harmonc fncon ψ conjga o ϕ or h Prandl orson fncon Φ whch s assocad wh ψ by h rlaon ( x x ) ψ ( x x ) x x Φ = + /. In hs cas as s known from [6] h problms of drmnng h fncons ψ ( x x ) and ( x x ) Φ ar h Drchl problms for Laplac s qaon. Whn solvng h mnond problms h complx fncon of orson [6] can also b sd.

12 Exampls. orson of a Prsmac Bam wh an Ellpc Cross Scon L a and b b smaxs of an llps. W draw on h solon of h corrspondng problm of lascy [6]: M a + b x x M a + b x x 3 3 = 3 3 = 3 3 Gπab Gπab M b a xx M M 3 = ; x 3 3 x 3 G ab = ; = π πab πab whr G s a modl of h maral shf M s a momn of orson. Rplacng n h las xprssons G by G M by M ( ) w wll hav xprssons for h qans Consdrng h oband xprsson n ransonal formla (3) wr o h solon o h problm of orson of an vscolasc prsmac sold wh an llpc cross scon: Hr a b x x a b x x = + M ; = + M ; Gπab Gπab b a xx M M ( ) G ab π πab πab = M ; = x ; = x. In cas of a M = M + Γ τ M τ dτ. (49) = b h oband solon corrsponds o h solon of h orson problm rlad o a vscolasc prsmac sold wh a crclar cross scon. In hs cas 3 = whch shows an absnc of dplanaon.

13 . Rond Prsmac Bam wh a Smcrclar Longdnal Bor 355 Accordng o [6] h solon o h problm of lascy s rprsnd n h form Mx x Mx x Mb x = ; = ; = ; GDa GDa GDa x + x M ab xx M ab ( x x ) 3 = 4 x ; 3 = + x 4 a. Da Da x + x x x + Hr as n h prvos problm G s a modl of h maral shf M s a momn of orson a s a rads of h bam dsk b s a rads of h bor dsk. Bsds whr arccos ( ) 3 b 4 b D ( sn 4 8sn ) ( sn ) sn 4 α α a a α α = a α α =. b a Now o oban a solon o h problm of orson of a crclar vscolasc bam wh a smcrclar longdnal bor rplac G by lascy and s formla (3). hn w oban: G M by M ( ) n h rprsnd solon of h problm of xx xx bx M M M 3 3 = 4 ; = 4 ; 3 = 3 GDa GDa GDa x + x M ( ) M ( ab x x ) ab x x 3 = x 4 ; 3 = + x 4 a. Da Da x + x x x + h opraor 3 nvolvd n hs rlaons has h form (49).

14 Rmarks h problm of orson of a prsmac vscolasc sold wh an arbrary smply conncd cross scon has bn solvd. Problms of vscolascy for prsmac solds wh mlply conncd cross scons can b solvd n h smlar way. 5. Conclson l. Formlas rdcng solons of h gnral qas-sac problm of lnar vscolascy for an soropc and homognos sold o a solon of h corrspondng problm of lascy hory ar prsnd. hs formlas ar vald f on of h followng condons holds: l) h maral s clos o a mchancally ncomprssbl manal; ) h man srss s zro; 3) h shf and volm hrdary fncons ar qal. hy provd a fr convrson from h problm of vscolascy o h problm of lascy and vc vrsa. hs qaly maks hm convnn n applcaons.. h oband rsl has bn appld o a solon of h problm of pr orson for a prsmac vscolasc sold wh an arbrary smply conncd cross scon. Exampls dscrbng h consrcon procdr for a solon of h problm of vscolascy from h known solons of h corrspondng problm of lascy hav bn prsnd. Rfrncs []. Y.N. Rabonov 977 Elmns of hrdary mchancs of solds. Naka Moscow p.384. []. A.A. Ilyshn and B.E. Pobdrya 97 Bascs of h mahmacal hory of hrmovscolascy. Nak Moscow p.8. [3]. R.M. Chrsnsn 97 hory of vscolascy. Acadmc prss Nw- York-London p.338. [4]. V.V. Moscwn 97. Rssanc of vscolasc marals. Naka Moscow p.37. [5]. F.G. rcom 957 Ingral qaons. Inrscnc pblshrs Inc. Nw- York; Inrscnc pblshr LD London p.99. [6]. HG. Hahn Elaszashor. B.G.bnr Sgr 985 p.343. Pblshd: Volm 6 Iss 7 / Jly 5 6

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm