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A&15C5>H utt~ier~ hrsdsid yrdedsruhsid geujsdsrd jieioi {ieayfum u{rtussid j tr~~derzsi ie~d [VXV] 1 (p (x)y ) + q (x)y = f (x), − ρ(x) oyd ρ(x), p (x), p (x), q (x), f (x) ¤ uyussdm sd{edejsd su [ a, b ] gsf} rrm {er i~ j{ihsata gthijra ρ(x) ≥ ρ > 0, p (x) ≥ p > 0 X p¢dd eddsrd ufioi geujsdsra r~dd jry c¡ 0 0 0 0 0 y(x) = C1 y1 (x) + C2 y2 (x) + ỹ(x). xydt y (x) r y (x) ¤ yju hrsdsi sdujrtr~ eddsra iysieiysioi geuj} sdsram tiijdtjg¢doi geujsdsr [VXV]m u ỹ(x) ¤ zutsid eddsrd geuj} sdsra [VXV]m C m C ¤ {eirjihsd {itiassdX gt su fisu {ei~d|gfu [a, b] gsfra y(x) gyijhdjiead gthi} jra~ ( [VXc] R y (a) − S y(a) = t , R y (b) + S y(b) = t , oyd R m R m S m S m t m t ¤ uyussd {itiassdm |R | + |S | =6 0 m m R , R , S , S ≥ 0X |R | + |S | = 6 0 nufrd gthijra sujuta feudj~r rhr oeusrzs~rX xuyuzu sui|ydsra su rsdejuhd (a, b) eddsra yrdedsruhsioi geujsdsra [VXV]m gyijhdjiea¢doi j izfu x = a r x = b feudj~ gthi} jra~ [VXc]m sujudta feudji uyuzdX utt~ier~ sdfiied zutsd thgzur feudj gthijrX Rthijra R y (a) − S y(a) = 0, R y (b) + S y(b) = 0 sujuta iysieiys~rX (t =Rthijra t = 0) y(a) = t , y(b) =t sujuta feudj~r gthijra~r V}oi eiyu rhr gthijra~r (R = R = 0) SrerhdX Rthijra y (a) = t , y (b) = t sujuta feudj~r gthijra~r c}oi eiyu rhr gthijra~r (S = S = 0) vd~usuX Rthijra R y (a) − S y(a) = t , R y (b) + S y(b) = t sujuta feudj~r gthijra~r `}oi eiyuX ysieiys~r feudj~r gthijra~r ajhata uf|d gthijra {deri} yrzsitr y(a) = y(b), y (a) = y (b). ¬thr gsfrr ρ(x) m p(x) m p (x) m q(x) sd{edejs su ifei~ rsde} juhd (a, b) rhr dthr gsfrr ρ(x) rhr p(x) ipeu¢uta j sgh j oeusrzsi 1 2 1 1 2 2 2 1 2 1 1 1 2 1 2 0 1 2 0 2 1 2 2 1 0 1 1 2 1 2 0 2 2 1 1 2 0 1 2 0 1 2 2 1 0 1 0 1 0 0 2 0 2 2 1 izfdm i j ufr thgzua j fuzdtjd feudjioi gthijra rt{ihgdta gthi} jrd ioeusrzdssitr y(x) {er x → a + 0 rhr x → b − 0 X nufrd feudjd gthijra ajhata iysieiys~rX vu fisu {ei~d|gfu [a, b] ~iog p uyus feudjd gthijra eu} s r{ijX gt edpgdta edr feudjg uyuzg yha iysieiysioi hrsdsioi yrdedsruhsioi geujsdsra ( {er a < x < b, −y + qy = 0 00 R1 y 0 (a) − S1 y(a) = t1 , p¢dd eddsrd geujsdsra −y R2 y 0 (b) + S2 y(b) = t2 . r~dd jry 00 + qy = 0 y(x) = C1 y1 (x) + C2 y2 (x), oyd gsfrr y (x) m y (x) ¤ i yju hrsdsi sdujrtr~ eddsra geuj} sdsraX vdegysi u~drm zi gsfrr y (x − a) m y (x − a) uf|d fuf r gsfrr y (b − x) m y (b − x) pgyg yegor~r {ueu~r hrsdsi sdujrtr~ eddsrX ii~g ip¢dd eddsrd geujsdsra ~i|si {edytujr j jryd 1 2 1 1 2 2 y(x) = C y (x − a) + C y (x − a), rhr j jryd y(x) = C y (b − x) + C y (b − x). _pie jryu ip¢doi eddsra geujsdsra ujrtr i uyuss feudj gthi} jrX Teudjua uyuzu ~i|d r~d dyrstjdssid eddsrdm ~i|d r~d pdt} fisdzsi ~sioi eddsr rhr ~i|d sd r~d eddsrX vu {er~deu eupded~ sdfiied itipdssitrm fiied jisrfu {er eddsrr feudj uyuzX P®P® vur eddsrd feudji uyuzr ( {er a < x < b, −y − y = 0 1 1 2 2 1 1 2 2 00 y(a) = 0, y(b) = 2. suzuhu eutt~ier~ iysieiysid yrdedsruhsid geujsdsrdX ¬oi ueufdertrzdtfid geujsdsrd −p − 1 = 0 X qrthu p = i m p = −i ¤ fiesr geujsdsraX er i~ gsfrr y (x) = cos x r y (x) = sin x ipeug gs} yu~dsuhsg trtd~g eddsrX xsuzr ip¢dd eddsrd geujsdsra r~dd jry cos x + C sin x. {edydhr~ suzdsra C y(x) r C ={erC fiie suydssua gsfra gyijhdji} ead uyuss~ feudj~ gthijra~X 2 1 1 2 1 1 2 2 ( C1 cos a + C2 sin a = 0 C1 cos b + C2 sin b = 2. U 2 Sha sui|ydsra C r C thdygd edr trtd~g hrsds geujsdsrX bi|si g{eitr sui|ydsrd eddsra feudji uyuzrm dthr ip¢dd eddsrd geujsdsra rtfu j jryd 1 2 y(x) = C1 cos(x − a) + C2 sin(x − a). i gyipsim {itfihfg feudjid gthijrd j izfd x = a ¤ iysieiysidX {dejioi feudjioi gthijra {ihgzr~ eujdstji C cos 0 + C sin 0 = 0. xsuzr r y(x) = C sin(x − a) X Sha i gsfrr yih|si j{ihsata C = 0 jieid feudjid gthijrd 1 1 2 2 C sin(b − a) = 2. ddsrd ioi geujsdsra ujrtr i suzdsra b − a X ¬thr b − a = πk, k ∈ Z m i 2 C2 sin(πk) = 2 ⇔ C2 0 = 2. Reujsdsrd sd r~dd eddsrX xsuzr r feudjua uyuzu sd pgyd r~d ed} dsrX ¬thr b−a 6= πk (k ∈ Z) m ioyu sin(πk) 6= 0 r C = 2 X Teudjua sin(b − a) uyuzu r~dd dyrstjdssid eddsrd y(x) = 2 sin(x − a) X sin(b − a) r X utt~ier feudjd uyuzr {er eus suzdsra a b gt a = π m b = 7π X er i~ b − a = π r C = 4 X Teudjua uyuzu r~dd dyrstjdssid eddsrd 6 6 X y(x) =gt 4 sin(x − π)m X nioyu b − a = π m sin π = 0 X Teudjua uyuzu a = π b = 2 π eddsr sd r~ddX jd y(x) = 2 sin(x − a) {er b − a 6= πk (k ∈ Z) − a) eddsr sd {er b −sin(b X a = π k (k ∈ Z) P®N® vur eddsrd feudji uyuzr ( {er a < x < b, −y − y = 0 2 2 00 y(a) = 1, y 0 (b) = 0. Reujsdsrd j eutt~uerjud~i uyuzd ufid |dm zi r j {edyyg¢dX itfihfg iysieiysid feudjid gthijrd ydt uyusi j izfd x = b m i ip¢dd eddsrd geujsdsra dhdtiipeusi rtfu j jryd y(x) = C1 cos(b − x) + C2 sin(b − x). vuyd~ y (x) = C sin(b − x) − C cos(b − x) X t{ihga jieid feudjid gthijrdm {ihgzr~ C = 0 X xsuzr y(x) = C cos(b − x) X iytujr~ g ¥ 0 1 2 2 1 gsfr j {dejid feudjid gthijrd C1 cos(b − a) = 1. _ thgzua b − a = π + πk (k ∈ Z) i geujsdsrd sd r~dd eddsrX xsuzr r feudjua uyuzu2eddsr sd r~ddX ¬thr b − a 6= π + πk(k ∈ Z) m i cos(b − a) 6= 0 r C = 1 X 2 cos(b − a) Teudjua uyuzu r~dd dyrstjdssid eddsrd y(x) = cos(b − x) X cos(b − a) jd y(x) = cos(b − x) {er b − a 6= π + πk(k ∈ Z) − a) 2 eddsr sd {er b −cos(b X π a = + πk(k ∈ Z) 2eddsrd feudji uyuzr P®¯® vur ( {er 0 < x < π, −y − y = 0 1 00 y(0) = 0, y(π) = 0. Reujsdsrd j i~ {er~ded ufid |dm fuf r j g|d eutt~iedssX i} tfihfg feudjid gthijrd j izfd x = 0 ¤ iysieiysidm i ip¢dd eddsrd geujsdsra gyipsi rtfu j jryd y(x) = C1 cos x + C2 sin x. _it{ihgd~ta feudj~r gthijra~r ( C1 + C2 0 = 0 −C1 + C2 0 = 0. dua g trtd~gm {ihgzr~ C = 0 m C ¤ hpid zrthiX xsuzr fu|yua gsfra jryu y(x) = C sin x (C ∈ R) pgyd eddsrd~ uyussi feudji uyuzrX Teudjua uyuzu r~dd pdtfisdzsi ~sioi eddsrX jd y(x) = C sin x (C ∈ R) X P®Q® vur eddsrd feudji uyuzr j ujrtr~itr i l ( {er 0 < x < l, −y − y = 0 1 2 2 2 2 2 00 y 0 (0) − 2y(0) = 1, y(l) = 0. Reujsdsrd j uyuzd ufid |dm fuf r j {edyyg¢rX itfihfg iy} sieiysid feudjid gthijrd ydt uyusi j izfd x = l m i ip¢dd eddsrd geujsdsra pgyd~ rtfu j jryd y(x) = C1 cos(l − x) + C2 sin(l − x). W jieioi feudjioi gthijra thdygdm zi C 1 vuyd~ y (x) gthijrd 0 =0 X ii~g y(x) = C2 sin(l − x). = −C2 cos(l − x) r {iytujr~ gsfr j {dejid feudjid −C2 cos(l) − 2C2 sin(l) = 1 ⇔ ¬thr j{ihsadta eujdstji cos(l) + 2 sin(l) = 0 −C2 (cos(l) + 2 sin(l)) = 1. 1 l = arctg(− ) + πk (k ∈ Z), 2 ⇔ i feudjua uyuzu eddsr sd r~ddX _ thgzud cos(l) + 2 sin(l) 6= 0 m XdX {er l 6= arctg(− 1 ) + πk (k ∈ Z) 2 feudjua uyuzu r~dd dyrstjdssid eddsrd y(x) = − sin(l − x) X cos(l) + 2 sin(l) jd y(x) = − sin(l − x) {er l 6= arctg(− 1 ) + πk (k ∈ Z) cos(l) + 2 sin(l) 2 eddsr sd {er l = arctg(− 1 ) + πk (k ∈ Z) X 2 P®°® vur eddsrd feudji uyuzr  − 1 d ρy (ρ) + 1 n y (ρ) = 0 {er 0 < ρ < R (n > 0), ρ dρ ρ y (ρ) ioeusrzdsu {er ρ → 0 + 0, y (R) = A. xuyussid yrdedsruhsid geujsdsrd ¤ i geujsdsrd t {ded~dss} ~r firrdsu~rX edipeugd~ doim jzrthrj {eirjiysd 0 2 n n 2 n n 1 0 1 yn (ρ) + 2 n2 yn (ρ) = 0 ρ ρ 00 −yn (ρ) − ⇔ 0 00 −ρ2 yn (ρ) − ρ yn (ρ) + n2 yn (ρ) = 0. ihgzdssid geujsdsrd ajhadta geujsdsrd~ hdeuX {i~i¢ u~ds isi {edipeugdta f yrdedsruhsi~g geujsdsr t {itiass~r ρ = e firrdsu~rX er ufi u~dsd t{eujdyhrj eujdstju t yn (ρ) = yn (et ) = vn (t) = vn (ln(ρ)). er i~ v = y ρ m v = y ρ + y ρ X sitrdhsi gsfrr v {ihgzrta geujsdsrd t {itiass~r firrdsu~r jryu 0 n 0 n 00 n 00 2 n 0 n n 00 −vn (t) + n2 vn (t) = 0. Z xu{rd~ yha sdoi ueufdertrzdtfid geujsdsrd −p + n = 0 X ¬oi fiesr r p = −n X nioyu ip¢r~ eddsrd~ yrdedsruhsioi geujsdsra ppgyd = ngsfra 2 1 2 2 vn (t) = Cn ent + Dn e−nt . Rzrjuam zi t = ln(ρ) Dn yn (ρ) = Cn ρn + n ρ m suyd~ ip¢dd eddsrd rtiysioi geujsdsra X gthijra ioeusrzdssitr gsfrr {er ρ → 0+0 thdygdm zi D = 0 r y (ρ) = C ρ X t{ihga jieid feudjid gthijrdm {ihgzr~ C R = A X xsuzr C = A X ddsrd~ feudji uyuzr pgyd gsfra y (ρ) = Aρ X R jd y R(ρ) = Aρ X R P®±® vur eddsrd feudji uyuzr   1 d (ρy (ρ)) = 0 {er 0 < ρ < R, ρ dρ y(ρ) ioeusrzdsu {er ρ → 0 + 0, y(R) = A. suzuhum fuf r j {edyyg¢d~ {er~ded {edipeugd~ geujsdsrd n n n n n n n n n n n n n n 0 00 y (ρ) + 1 0 y (ρ) = 0 ρ 00 0 ρ2 y (ρ) + ρ y (ρ) = 0. ⇔ ihgzrhit geujsdsrd hdeuX _{ihsr~ u~dsg ρ = e X nioyu y(ρ) = {er i~ v = y ρ m v = y ρ + y ρ r suzr gsf} = y(e ) = v(t) = v(ln( ρ )) ra v(t) gyijhdjiead geujsdsr v = 0 X p¢r~ eddsrd~ ioi geuj} sdsra ajhadta gsfra v(t) = C + D t X Rzrjua eujdstji t = ln(ρ) m {ihgzr~ eddsrd uyussioi geujsdsra y(ρ) = C + D ln(ρ) X gthijra ioeusrzdssitr eddsra {er ρ → 0 + 0 thdygdm zi D = 0 X t{ihga jieid feudjid gthijrdm suyd~ C = A X nioyu eddsrd~ uyuzr ajhadta gsfra y(x) = A X jd y(x) = A X P®²® Teudjg uyuzg ( {er 0 < x < l, −y + 2y = 3x + 1 t 0 t 0 00 0 00 00 2 0 0 0 0 0 0 00 y 0 (2) − 3y(2) = 4, y 0 (4) = 1 tjdtr f uyuzd t iysieiys~r feudj~r gthijra~rX er {er~dsdsrr ~diyu ged feudjua uyuzu {edipeugdta f uyuzd t iysieiys~r feudj~r gthijra~rX gsfrm fiieua ajhadta eddsr} d~ uyussi uyuzr {edytujha j jryd tg~~ yjg y(x) = v(x) + w(x) X gsfr w(x) jpreu ufmzip isu gyijhdjieahu ufr~ |d feudj~ gthijra~m zi r gsfra y(x) X nioyu gsfra v(x) pgyd gyijhdjiea Y iysieiys~ feudj~ gthijra~X _i ~sior thgzua w(x) uyu fuf hr} sdsg gsfr w(x) = αx + β X Tirrds α r β suiya {ithd {iytusijfr w(x) j feudjd gthijraX jdyd~ uyussg feudjg uyuzg f uyuzd t iysieiys~r feudj~r gthijra~rX gt y(x) = v(x) + w(x) m oyd w(x) = αx + β X iypded~ α r β ufm zip yha gsfrr w(x, t) j{ihsahrt feudjd gthijra w0 (2) − 3w(2) = 4, w0 (4) = 1. vuyd~ w (x) = α rm rt{ihga feudjd gthijram {ihgzr~ trtd~g 0 ( α − 3(2α + β) = 4 α=1 ( α=1 ⇔ 1 − 3(2 + β) = 4 ( α=1 ⇔ β = −3. xsuzr w(x) = x − 3 X er i~ gsfra v(x) gyijhdjiead iysieiy} s~ feudj~ gthijra~X xu{rd~ feudjg uyuzg yha v(x) m gzrjuam zi m m y(x) = v(x) + x − 3 y (x) = v (x) + 1 y (x) = v (x) ( {er 0 < x < l, −v + 2(v + x − 3) = 3x + 1 0 0 00 00 00 v 0 (2) − 3v(2) = 0, v 0 (4) = 0. edipeugam {ihgzr~ rtfi~g feudjg uyuzg ( {er 0 < x < l, −v + 2v = x + 7 00 v 0 (2) − 3v(2) = 0, v 0 (4) = 0. jd y(x) = v(x) + x − 3 m {er ( −v 00 + 2v = x + 7 v 0 (2) − 3v(2) = 0, 0 < x < l, v (4) = 0. 0 ³´µ¶· ¸¹¸º» _ g{eu|sdsra VXV}VXU edr uyussd feudjd uyuzrX ( {er 0 < x < , −y − 4y = 0 1.1 00 π 4 y(0) = 0, y 0 ( π4 ) + 2y( π4 ) = 1. ( π −y 00 − y = 0 < x < π3 , 6 1.2 y 0 ( π6 ) − 2y( π6 ) = 0, y( π3 ) = 21 . ( −y 00 − 16y = 0 0 < x < π2 , 1.3 y 0 (0) = 2, y( π2 ) = 0. {er {er V\ {er ( −y 00 − 9y = 0 0 < x < π6 , 1.4 y(0) = 0, y 0 ( π6 ) = 0. ( 0 0 < ρ < R, ρ2 y (ρ) 0 − 6y = 0 1.5 y(ρ) ρ → 0 + 0, y(R) = 1. {er ioeusrzdsu {er _ g{eu|sdsra VX¥}VXZ edr uyussd feudjd uyuzr j ujrtr~itr i l X ( {er 0 < x < l, −y − 4y = 0 00 1.6 y(0) = 12 , y(l) = 0. ( 0 < x < l, −y 00 − 9y = 0 1.7 0 0 y (0) = 0, y (l) + 3y(l) = 1. ( 0 < x < l, −y 00 − 4y = 0 1.8 y 0 (0) = 0, y(l) = 0. {er {er _ g{eu|sdsra VXY}VXV\ tjdtr uyussg feudjg uyuzg f uyuzd t iysieiys~r feudj~r gthijra~rX ( {er 2 < x < 3, −y + 3y = −2x + 3 1.9 00 y(2) = 1, y 0 (3) + 4y(3) = 9. ( −y 00 − 5y = 4x 0 < x < 4, 1.10 y 0 (0) − 3y(0) = 2, y(4) = 9. {er ¼½¾¹½¿ VXV y(x) = 1 sin(2x) VXc y(x) = √ 1 (cos(x − π ) + 2 sin(x − π )) 2 6 6 3+2 VX` eddsr sd VX y(x) = C sin(3x) m C ∈ R VXU y(ρ) = ρ VX¥ y(x) = sin(l − x) {er l 6= πk m k ∈ Z eddsr sd {er lR= πk m k ∈ Z 2 sin(l) VXW y(x) = {er l 6= π + πk m k ∈ Z eddsr sd {er cos(3x) 3(cos(3l) − sin(3l)) 12 3 VXZ y(x) = C cos(2x), C ∈ R {er l = πk m k ∈ Z π πk m l = + k ∈ Z 12 3 2 {er VXY m πk m y(x) = 0 l 6= k∈Z y(x) = v(x) + x − 1 2 {er −v + 3v = −5x + 6 2 < x < 3, v(2) = 0, v (3) + 4v(3) = 0 VXV\ y(x) = v(x) + 5x + 1 m −v − 5v = 20x + 11 {er 0 < x < 4, X v (0) − 3v(0) = 0, v(4) = 0 2 2 2 1 00 0 2 1 0 00 VV À ' ACÁ C5>C <&= &6> Á 2?A) &Â(>? 1>(( D gt D(L) ¤ i ~si|dtji yju|y yrdedsregd~ su {ei~d} |gfd (a, b) gsfrm gyijhdjiea¢r su fisu {ei~d|gfu iysieiy} s~ feudj~ gthijra~m su{er~de gthijra~ jryu R1 y 0 (a) − S1 y(a) = 0, R2 y 0 (b) + S2 y(b) = 0. bsi|dtji D(L) {edytujhad tipi hrsdsid {eiteustjiX Sha hd} ~dsij f (x) r g(x) ioi ~si|dtju i{edydhr~ tfuhaesid {eirjdydsrd {i ie~ghd (f, g) = Zb f (x)g(x)ρ(x)dx, oyd ρ(x) ¤ sd{edejsua su [a, b] gsfram ρ(x) ≥ ρ > 0 X su sujudta jdtiji gsfrdm rhr jdti~X gsfrr f (x) r g(x) sujuta ieioisuhs~rm dthr (f, g) = 0 X gt y ∈ D(L) X vie~u gsfrr y(x) m {iei|ydssua tfuhaes~ {ei} rjdydsrd~m suiyrta {i {eujrhg a 0 v u b uZ u kyk = t y 2 (x)ρ(x)dx. nufr~ ipeui~m hpua gsfra y ∈ D(L) ajhadta hd~dsi~ {eiteus} tju L [a, b; ρ(x)] [t~X `¡]X utt~ier~ hrsds yrdedsruhs i{deuiem ydtjg¢r r {eiteustju D(L) j hrsdsid {eiteustji sd{edejs su [a, b] gsf} rm jryu a 2 L(y) = − 1 (p(x)y 0 )0 + q(x)y, ρ(x) [a, b] p(x) ≥ p0 > 0 ρ(x) oyd ρ(x) m p(x) m q(x) ¤ sd{edejsd su gsfrrm m ¤ jdtijua gsfraX nufi i{deuie sujudta i{deuiei~ £ge~u¤lrg} jrhhaX Sha i{deuieu £ge~u¤lrgjrhha L(y) t{eujdyhrj thdyg¢rd tji} tju V] i{deuie tr~~derzdsm XdX ∀y, z, ∈ D(L) t{eujdyhrji eujdstji m (L(y), z) = (y, L(z)) c] i{deuie {ihi|rdhsi i{edydhdsm XdX yha ∀y ∈ D(L) t{eujdyhrji sdeujdstji (L(y), y) ≥ q kyk oyd q = min q(x) X Sifuudhtji r tjitj {erjiyrta j {itiprr V¡X Vc 0 2 0 x∈[a,b] J N®P ÃÄÅÆÇ λ ÈÉÊËÌÉÍÎ ÅÏ ÅÇÐÅ ÎÌÍÈÈËÑ Ò ÄÅÆÇÑ ÇÓ ÍÔ ÉÕ Å×Ø Í Å ÎÌ × ÍÎ ÈÍÈ ×Æ ÍÌÉÏ Ù× ÈÚÛ ÄÏ y(x) ∈ D(L) Ö Ü ÆÏ Ú Ç Î ÇÔÇÝ Î ÇÔ É L Ö Í ÅÆÄ ÔÄ à Î ÇÑ Ù× ÈÚÛ ÄÏ y(x) ÈÉÊËÌÉÍÎ ÅÏ ÅÇÐÅ ÎÌÍÈÈ ÇÝ Ù× ÈÚÛ Ä Í Ý L(y) = λy Þ ß ÇÓ ÍÔ ÉÎ ÇÔ ÉÞ xuyuzu sui|ydsra tiptjdss zrtdh r tiptjdss gsfr i{deu} ieu sujudta uyuzd £ge~u¤lrgjrhha [cXV] ( L(y) = λy, a < x < b R1 y 0 (a) − S1 y(a) = 0, R2 y 0 (b) + S2 y(b) = 0. bsi|dtji jtd tiptjdss zrtdh sujudta t{dfei~ i{deuieu rhr t{dfei~ uyuzr [cXV]X L(y) dedzrthr~ itsijsd tjitju tiptjdss zrtdh uyuzr £ge~u¤ lrgjrhha V] iptjdssd zrthu i{deuieu L(y) jd¢dtjdssdX c] iptjdssd zrthu i{deuieu L(y) yrtfedsdm XdX {edytujhad tipi {ithdyijudhsit {λ } X `] ithdyijudhsit {λ } ioeusrzdsu tsrg λ ≥ min q(x) r X lim λ = +∞ ] er sdfiie {ihi|rdhs A r B yha jtd yituizsi pihr t{eujdyhrj sdeujdstju An ≤ λ ≤ Bn X n Sha tiptjdss gsfr i{deuieu £ge~u¤lrgjrhha t{eujdyhr} j thdyg¢rd gjde|ydsra V] Tu|yi~g tiptjdssi~g zrthg tiijdtjgd ihfi iysu [t izsi} t yi {itiassioi ~si|rdha] tiptjdssua gsfraX c] nuf fuf tiptjdssd gsfrrm tiijdtjg¢rd euhrzs~ tip} tjdss~ zrthu~m ieioisuhsm suzr trtd~u tiptjdss gsfr i{deuieu {y (x)} ajhadta ieioisuhsi trtd~iX `] eioisuhsua trtd~u {y (x)} ajhadta {ihsi j {eiteustjd X i isuzudm zi hpua gsfra r ioi {eiteustju ~i} L [a, b; ρ (x)] |d p euhi|dsu j eay ged {i trtd~d {y (x)} r i eay pgyd tiyrta f gsfrr j sie~d {eiteustju L [a, b; ρ(x)] X Tei~d ioim yha gsfrr r iphutr i{edydhdsra i{deuieu L t{eu} jdyhrju died~u dfhiju % N®P ® ß ×Å Îá y ∈ D(L) Þ â ÅÆÄ Ù× ÈÚÛ Äã y(x) Ô ÉÊ ÆÇä Ä Îá Ì ÔÏ Ü å ×Ô áÍ ÓÇ ÇÔ Î ÇæÇ ÈÉÆ áÈ ÇÝ Ö ÓÇÆ È ÇÝ Ì ÓÔÇÅ Î Ô ÉÈ Å ÎÌÍ L [a, b; ρ(x)] ÅÄÅ ÎÍÕ Ç à Î Ç Î ÔÏ Ü Ð× ÜÍÎ É ÐÅÇÆã ÎÈ Ç ÅçÇ Ü Ä Î ÅÏ Ú y(x) ÑÍ Ù× ÈÚÛ ÄÝà {yÇ (x)} Ö Î ∀x ∈ [a, b] è Î Î ÔÏ Ü Ñ Çä È Ç ÓÇ Ò Æ ÍÈÈ Ç Ü ÄÙÙ ÍÔ ÍÈÛ ÄÔÇ ÌÉÎá 2 Ô ÉÊÉÞ é Ï Ü Ä Ê Ó ÍÔ ÌË ç ÓÔÇÄ ÊÌ Ç ÜÈË ç Ð× ÜÍÎ ÓÇ Î Ç ÒÍÒÈ Ç ÅçÇ Ü Ä Îá ÅÏ Ú y (x) Ö É ÔÏ Ü Ä Ê ÌÎ ÇÔ Ë ç ÓÔÇÄ ÊÌ Ç ÜÈË ç ÅçÇ Ü Ä Î ÅÏ Ú y (x) Ì È ÇÔ ÑÍ ÓÔÇÅ Î Ô ÉÈ Å ÎÌÉ L [a, b; ρ(x)] Þ V` +∞ n n=1 n n→+∞ n n 2 +∞ k=1 k 2 n k +∞ k=1 2 +∞ k=1 k 2 k 2 +∞ k=1 00 0 2 x∈[a,b] Sha euhrzs i{deuieij £ge~u¤lrgjrhha L(y) suyd~ tiptjdssd zrthu r tiptjdssd jdfieX N®P® Sha i{deuieu L (y) = − d y m a < x < b m i{edydhdssioi dx su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ gthijra~ Srerhd j izfu a r b m sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioi} suhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX Sha ioi i{deuieu edr~ uyuzg £ge~u¤lrgjrhha 2 x ( 2 −y 00 = λy, a < x < b, y(a) = 0, y(b) = 0. itfihfg q(x) ≡ 0 m i λ ≥ 0 X dr~ tsuzuhu yrdedsruhsid geuj} sdsrdX xu{rd~ yha sdoi ueufdertrzdtfid geujsdsrd −p = λ. ¬thr m i geujsdsrd r~dd yju iyrsufij fiesa p = p = 0 r gsfra λ=0 pgyd ip¢r~ eddsrd~ yrdedsruhsioi geujsdsraX y(x) = C + C x iytujr~ y(x) j feudjd gthijra 2 1 1 2 2 ( C1 a + C2 = 0, C1 b + C2 = 0, ifgyu C = C = 0 m XdX y ≡ 0 rm thdyijudhsim λ = 0 sd ajhadta tiptjdss~ zrthi~X gt λ > 0 m ipisuzr~ λ = µ X êueufdertrzdtfid geujsdsrd {er} ~d jry −p = µ . Tiesr ioi geujsdsra fi~{hdftsd p = µi m p = −µi X p¢dd eddsrd yrdedsruhsioi geujsdsra gyipsi u{rtu j jryd X Sha i{edydhdsra suzdsr C r y(x)jit{ihgd~ta = C cos(µ(x − a)) + C sin( µ (x − a)) feudj~r gthijra~r C 1 2 2 2 2 1 1 2 2 1 2 ( C1 = 0, C1 cos(µ(b − a)) + C2 sin(µ(b − a)) = 0. u trtd~u r~dd sdsghdjid eddsrd C = 0 m C 6= 0 m ihfi fioyu sin(µ(b− X pisuzr~ yhrsg {ei~d|gfu zded l = b − a m ioyu sin(µl) = 0 X −a)) = 0 ihi|rdhsd eddsra ioi geujsdsra µ = πk m k = 1, 2, . . . X nufr~ ipeui~m tiptjdss~r zrthu~r i{deuieu j thgzudl feudj gthijr Sr} erhd pgyg λ = µ = πk m k = 1, 2, . . . X ihi|rj {eirjihsg {itiassg C = 1 m {ihgzud~l trtd~g tiptjdss gsfr 1 2 k 2 k 2 k 2 yk (x) = sin(µk (x − a)) = sin V πk(x − a) , l k = 1, 2, . . . . Sha i{deuieu L (y) = −y jdtijua gsfra ρ(x) = 1 m suzr tiptjdssd gsfrr y (x) ipeug {ihsg ieioisuhsg trtd~g j {eiteustjd X L [a,vuyd~ b; 1] fjuyeu sie~ tiptjdss gsfr 00 x k 2 kyk (x)k2 = Zb yk2 (x)dx = a = Zb a Zb sin2 πk(x − a) dx = l a 2πk(x − a) 1 l 1 1 − cos dx = (b − a) = . 2 l 2 2 jd V] tX zX λ = πk m k = 1, 2, . . . l c] tX X y (x) = sin πk(x − a), k = 1, 2, . . . (l = b − a) l `] L [a, b; 1] ] ky (x)k = l X 2 N®N® Sha i{deuieu m i{edydhdssioi d ym L (y) = − a<x<b dx su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ gthijra~ vd~usu j izfu a r b m sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioi} suhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX dr~ uyuzg £ge~u¤lrgjrhha 2 k k 2 2 k 2 x ( 2 −y 00 = λy, a < x < b, y 0 (a) = 0, y 0 (b) = 0. u uyuzu ihrzudta i {edyyg¢d ihfi feudj~r gthijra~rX gt λ = 0 m ioyu ueufdertrzdtfid geujsdsrd −p = λ r~dd iyrsufijd fiesr p = p = 0 r ip¢r~ eddsrd~ yrdedsruhsioi geujsdsra pgyd gsfra y(x) = C + C x X iytujhaa y(x) j feudjd gthijram {ihgzr~ 2 1 2 1 ( 2 C2 = 0, C2 = 0. ihi|r~ C = 1 m ioyu gsfra y (x) = 1 ajhadta tiptjdssi gsfrd i{deuieu L (y) m tiijdtjg¢d tiptjdssi~g zrthg λ = 0 X er λ = µ > 0 m fiesa~r ueufdertrzdtfioi geujsdsra pgyg zrthu m r ip¢dd eddsrd yrdedsruhsioi geujsdsra ~i|si p = µi p = −µi 1 x 1 0 0 2 2 VU u{rtu j jryd y(x) = C j feudjd gthijra 1 cos(µ(x − a)) + C2 sin(µ(x − a)) X iytujr~ y(x) ( µC2 = 0, −µC1 sin(µ(b − a)) + µC2 cos(µ(b − a)) = 0. rtd~u r~dd sdsghdjid eddsrdm dthr sin(µ(b − a)) = 0 m XdX {er µ = πk m l X xsuzr yha i{deuieu L (y) = −y j thgzud feudj kgthijr = 1, 2, .vd~usu . . (l = b−a) tiptjdss~r zrthu~r uf|d pgyg zrthu λ = µ = X zruam zi C = 1 m {ihgzr~ tiptjdssd gsfrrm πk m k = 1, 2, . . . = l tiijdtjg¢rd r~ zrthu~ k x 00 k 2 2 k 1 yk (x) = cos(µk (x − a)) = cos πk(x − a) , k = 1, 2, . . . . l ρ(x) = 1 Sha i{deuieu L (y) = −y jdtijua gsfra m suzr tiptjdssd gsfrr y (x) ipeug {ihsg ieioisuhsg trtd~g j {eiteustjd X L [a,vuyd~ b; 1] fjuyeu sie~ gsfrr y (x) = 1 00 x k 2 0 ky0 (x)k2 = Zb y02 (x)dx = Zb 1dx = b − a = l. {edydhr~ fjuyeu sie~ tiptjdss gsfr y (x) m k = 1, 2, . . . a a k kyk (x)k2 = Zb yk2 (x)dx = a Zb cos2 πk(x − a) dx = l a Zb 1 2πk(x − a) 1 l = 1 + cos dx = (b − a) = . 2 l 2 2 a 2 πk λ0 = 0 λk = k = 1, 2, . . . (l = b − a) l πk(x − a) , k = 1, 2, . . . y0 (x) = 1 yk (x) = cos l l L2 [a, b; 1] ky0 (x)k2 = l kyk (x)k2 = 2 d2 y Lx (y) = − 2 a < x < b dx jd V] tX zX m m c] tX X m `] ] m X N®¯® Sha i{deuieu m m i{edydhdssioi su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~ Ry 0 (a) − Sy(a) = 0, V¥ y 0 (b) = 0. m m sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioi} suhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX utt~ier~ uyuzg £ge~u¤lrgjrhha (R 6= 0 S 6= 0) ( −y 00 = λy, a < x < b, Ry 0 (a) − Sy(a) = 0, y 0 (b) = 0. _ i uyuzd yrdedsruhs i{deuie i |dm zi r j {edyyg} ¢r {er~deuX xsuzr λ ≥ 0 X gt λ = 0 m ioyu ip¢dd eddsrd yrdedsruhsioi geujsdsra r~d} d jry y(x) = C + C x X t{ihga feudjd gthijram {ihgzr~ 1 2 ( RC2 − S(C1 + C2 a) = 0, C2 = 0. ddsrd i trtd~ C = 0 m C = 0 X xsuzr y(x) = 0 r zrthi λ = 0 sd ajhadta tiptjdss~ zrthi~ L (y) X ¬thr λ = µ > 0 m i zrthu p = µi m p = −µi ajhata fiesa~r ueufdertrzdtfioi geujsdsra −p = µ X p¢dd eddsrd yrdedsr} uhsioi geujsdsra yha eutt~uerjud~i uyuzr gyipsi u{rtu j jryd X iytujr~ gsfr y(x) j feu} y(x) = C cos( µ (b − x)) + C sin( µ (b − x)) djd gthijra 1 2 x 2 1 2 1 2 2 2   R (µC1 sin(µ(b − a)) − µC2 cos(µ(b − a))) − −S (C1 cos(µ(b − a)) + C2 sin(µ(b − a))) = 0,  −µC = 0. 2 pisuzr~ l = b − a X rtd~u eujsitrhsu geujsdsr C1 (Rµ sin(µl) − S cos(µl)) = 0. su r~dd sdsghdjd eddsram dthr C eujds sghX edipeugd~ doi f jryg 1 tg(µl) = 6= 0 S . Rµ X xsuzr jiei ~si|rdh iteir~ oeurfr {euji r hdji zutr geujsdsraX ertgsfu cXV jrysim zi geujsdsrd r~dd pdtfisdzsid ~si|dtji {ihi|rdhs eddsr µ m k = 1, 2, . . . X Sha sui|ydsra r eddsr thdygd rt{ihiju zrthdssd ~diyX nufr~ ipeui~m λ = µ (k = 1, 2, . . .) ¤ i tiptjdssd zrthu i{deu} ieu eutt~uerjud~i uyuzr £ge~u¤lrgjrhha X VW k k 2 k ëìíî ïîð ihi|r~ C = 1 m ioyu y (x) = cos(µ (b − x)) (k = 1, 2, . . .)¤ i tii} jdtjg¢rd r~ tiptjdssd gsfrr X sr ipeug {ihsg ieioisuh} sg trtd~g j {eiteustjd L [a, b; 1] X vuyd~ fjuyeu sie~ tiptjdss gsfr 1 k k 2 kyk (x)k2 = Zb yk2 (x)dx = Zb cos2 (µk (b − x)dx = a a Zb b 1 1 1 = (1 + cos(2µk (b − x))) dx = x+ sin(2µk (b − x)) = 2 2 2µk a a 1 1 1 1 (b − a) − sin(2µk (b − a)) = l− sin(2µk l) . = 2 2µk 2 2µk jd V] tX zX λ = µ m oyd µ ¤ i {ihi|rdhsd fiesr geujsdsra c] tX X y (x) = cos(µ (b − x)), k = S m tg(µl) = k = 1, 2, . . . (l = b − a) R µ = 1, 2, . . . `] L [a, b; 1] ] ky (x)k = 1 l − 1 sin(2µ l) X 2 2µ N®Q® Sha i{deuieu L (y) = − d y m 0 < ϕ < 2π m i{edydhds} dϕ {deriyrzdtfr~ feudj~ sioi su ~si|dtjd gsfrm gyijhdjiea¢r gthijra~ 2 k k k k 2 k 2 k k k ϕ 2 2 y 0 (0) = y 0 (2π) sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioisuhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX VZ y(0) = y(2π), dr~ uyuzg £ge~u¤lrgjrhha ( −y 00 = λy, 0 < ϕ < 2π, y(0) = y(2π), y 0 (0) = y 0 (2π). Tuf r j {edyyg¢r {er~deum yha i{deuieu −y tiptjdssd zrthu gyijhdjiea gthijr λ ≥ 0 X utt~ier~ thgzu λ = 0 X p¢r~ eddsrd~ yrdedsruhsioi geuj} sdsra pgyd gsfra y(ϕ) = C + C ϕ X iytujr~ dd j feudjd gthijra 00 1 2 ( C1 = C1 + C2 2π, C2 = C2 . ddsrd i trtd~ ( C1 = t, t ∈ R C2 = 0. ihi|r~ C = 1 m ioyu gsfra y (ϕ) = 1 pgyd tiptjdssi gsfrd i{deuieum tiijdtjg¢d tiptjdssi~g zrthg λ = 0 X gt d{de λ = µ > 0 X Tiesa~r ueufdertrzdtfioi geujsdsra pgyg zrthu p = µi m p = −µi X p¢dd eddsrd yrdedsruh} −p = µ sioi geujsdsra u{rd~ j jryd y(ϕ) = C cos(µϕ)) + C sin(µϕ) m {er i~ X _it{ihgd~ta feudj~r gthijra} y~r (ϕ) = −µC sin(µϕ)) + µC cos(µϕ) ( 1 0 0 2 2 2 1 2 1 0 1 2 2 C1 = C1 cos(2πµ) + C2 sin(2πµ), µC2 = −µC1 sin(2πµ) + µC2 cos(2πµ). xu{rd~ trtd~g j ~uerzsi~ jryd . 1 − cos(2πµ) − sin(2πµ) C1 0 = . µ sin(2πµ) µ − µ cos(2πµ) C2 0 ysieiysua trtd~u hrsds geujsdsr r~dd sdsghdjd eddsram dthr i{edydhrdh ~uer firrdsij eujds sgh µ(1 − cos(2πµ))2 + µ sin2 (2πµ) = 0. i eujdstji {edipeugdta f jryg 2µ(1 − cos(2πµ)) = 0. itfihfg µ 6= 0 m i cos(2πµ) = 1 X ihi|rdhsd fiesr geujsdsra ¤ i zrthu µ = k, k = 1, 2, . . . . xsuzr tiptjdss~r zrthu~r i{deuieu j thgzud {deriyrzdtfr feudj gthijr uf|d pgyg zrthu L (y) =m −y X {edydhr~ tiptjdssd gsfrrm tiijdtjg¢rd λ = k k = 1, 2, . . . VY ϕ k 2 k 00 r~ tiptjdss~ zrthu~X itfihfg yha hp suzdsr C r C gsf} rr 1 2 pgyg gyijhdjiea feudj~ gthijra~m i fu|yi~g tiptjdssi~g zrt} hg λ = k tiijdtjg yjd hrsdsi sdujrtr~d gsfrr cos(kϕ) r m k = 1, 2, . . . X Sha eutt~uerjud~i uyuzr {ihsi ieioisuhsi sin(k ϕ ) {eiteustjd L [a, b; 1] pgyd trtd~u gsfr yk (ϕ) = C1 cos(k ϕ) + C2 sin(k ϕ) k 2 2 {1, cos(k ϕ), sin(k ϕ)}, vuyd~ fjuyeu sie~ r gsfr k1k2 = Z2π k = 1, 2, . . . . 12 (x)dx = 2π, 0 k cos(k ϕ)k2 = Z2π 0 Z2π k sin(k ϕ)k2 = Z2π Z2π cos(k ϕ)2 (x)dx = 1 (1 + cos(2k ϕ)) dx = π, 2 1 (1 − cos(2k ϕ)) dx = π. 2 0 sin(k ϕ)2 (x)dx = jd V] tX zX λ = 0 m λ = k m k = 1, 2, . . . c] tX X {1, cos(kϕ), sin(kϕ)}, k = 1, 2, . . . `] L [0, 2π; 1] ] k1k = 2π m k cos(kϕ)k = π m k sin(kϕ)k = π X N®°® Sha i{deuieu ^dttdha B (y) = − 1 d (ρ dy ) m ρ dρ dρ m i{edydhdssioi su ~si|dtjd gsfrm gyijhdjiea¢r iysi} 0eiys~ <ρ<T feudj~ gthijra~ j izfu 0 r T ioeusrzdsu {er ρ → 0 + 0, y(T ) = 0, y(ρ) sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioisuhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX dr~ uyuzg £ge~u¤lrgjrhha 0 0 2 0 2 k 2 2 2 0  − 1 (ρy 0 )0 = λy, ρ y(ρ) 0 < ρ < T, ioeusrzdsu {er ρ → 0 + 0, y(T ) = 0. Sha i{deuieu ^dttdha B (y) gsfra q(ρ) eujsu sghX ii~g jtd tiptjdssd zrthu sdierudhsX c\ 0 utt~ier~ {ithdyijudhsi yju thgzua V] gt λ = 0 m ioyu {er 0 < ρ < T geujsdsrd 1 − (ρy 0 )0 = 0 ρ −y 0 − ρy 00 = 0 ⇔ ρ2 y 00 + ρy 0 = 0 ⇔ ajhadta geujsdsrd~ hdeuX _ {er~ded VX¥ suydsi ip¢dd eddsrd ioi geujsdsra y(ρ) = C1 + C2 ln(x). {dejioi feudjioi gthijra thdygdm zi C = 0 m u r jieioi ¤ C = 0 X itfihfg gsfra y(ρ) ≡ 0 m i zrthi λ = 0 sd ajhadta tiptjdss~ zrthi~ i{deuieu B (y) X c] gt λ = µ > 0 X utt~ier~ geujsdsrd 2 2 1 0 1 − (ρy 0 )0 = µ2 y ρ 1 ⇔ y 00 + y 0 + µ2 y = 0. ρ _{ihsr~ u~dsg {ded~dssi ρ = t X nioyu t{eujdyhrju d{izfu eujdstj X itfihfg v = y m v = y m i isitr} y( ρ ) = y( t) = v(t) = v( µρ ) dhsi {ded~dssi t geujsdsrd {er~d jry 1 µ 1 µ 1 µ 0 t 0 ρ 1 µ2 00 t 00 ρ 1 v 00 + v 0 + v = 0. t i geujsdsrd ^dttdha V¡X p¢r~ eddsrd~ ioi iysieiysioi yrd} edsruhsioi geujsdsra ajhadta gsfra v(t) = C J (t) + C N (t) m oyd r N (t) ¤ gsfrr ^dttdha r vd~usum tiijdtjdssiX er x → 0 + 0 J (t) gsfra N (t) sdioeusrzdssuX ii~g ioeusrzdssid j sghd ip¢dd edd} srd ioi geujsdsra r~dd jry v(t) = C J (t) X nioyu eddsrd~ rtiysioi geujsdsra pgyd gsfra y(ρ) = C J (µρ) X xuyuyr~ C = 1 X _it{ihgd~} ta jie~ feudj~ gthijrd~ 1 0 0 2 0 0 0 1 0 1 0 1 J0 (µT ) = 0. qrthu µ T = γ (k = 1, 2, . . .) m oyd γ ¤ i fiesr gsfrr J (ρ) X jdtsi V¡m zi u gsfra r~dd pdtfisdzsid ~si|dtji {eit fiesdX xsuzr zrthu k k 0 k λk = µ2k = γ 2 k , k = 1, 2, . . . ajhata tiptjdss~r zrthu~rm u gsfrr y (ρ) = J ( ρ) ¤ tiptjdss} ~r gsfra~r i{deuieu ^dttdha B (y) m eutt~uerjud~ioi su ~si|dtjd gsfrm gyijhdjiea¢r uyuss~ feudj~ gthijra~X rtd~u suydss tiptjdss gsfr pgyd {ihsu r ieioisuhsu j {eiteustjd L [0, T ; ρ] X cV T 0 2 k γk 0 T Sha sui|ydsra fjuyeuij sie~ tiptjdss gsfr jit{ihgd~ta rsdoeuhs~ i|ydtji~ V¡ ZT α T2 Jp2 ( x)xdx = T 2 2 p 2 Jp0 (α) + 1 − 2 Jp2 (α) , α _ eutt~uerjud~i~ thgzud p = 0 m {ii~g 0 γk kyk (ρ)k2 = kJ0 ( ρ)k2 = T ZT γk J02 ( ρ)ρdρ T α 6= 0. [cXc] i T2 h 0 2 2 = (J0 (γk )) + J0 (γk ) . 2 Rzrjuam zi J (x) = −J (x) rm zi J (γ ) = 0 m {ihgzr~ 0 0 0 2 0 1 k T2 γk 2 (J1 (γk ))2 . kyk (ρ)k = kJ0 ( ρ)k = T 2 γ 2 k λk = γk T 2 jd V] tX zX m oyd ¤ i i fiesr gsfrr J (ρ) m kc]=tX1,X2, . . . y (ρ) = J ( ρ), k = 1, 2, . . . `] L [0, T ; ρ] ] ky (ρ)k = T (J (γ )) X 2 N®±® Sha i{deuieu m i{ed} dy m 1 d (r ) 0<ρ<T L (y) = − r dr dr ydhdssioi su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~ j izfu 0 r T jryu ioeusrzdsu {er r → 0 + 0, Ry (T ) + Sy(T ) = 0 (R 6= 0, S 6= 0) y(r) sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioisuhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX dr~ uyuzg £ge~u¤lrgjrhha k 2 γk 0 T k 2 2 1 k 0 2 2 r 2 0   1 2 0 0 − 2 (r y ) = λy, r y(r) 0 < ρ < T, ioeusrzdsu {er r → 0 + 0, Ry (T ) + Sy(T ) = 0. Sha eutt~uerjud~ioi i{deuieu L (y) gsfra q(r) = 0 X ii~g jtd tiptjdssd zrthu sdierudhsX gt λ = 0 X edipeugd~ geujsdsrd − 0 r 1 1 2 0 0 (r y ) = 0 ⇔ (2ry 0 + r2 y 00 ) = 0 ⇔ 2y 0 + ry 00 = 0. 2 2 r r cc _jdyd~ sijg gsfr v(r) = y(r)r X Sha sdd ¤ v = y r+y m v nioyu isitrdhsi i gsfrr geujsdsrd {er~d jry 0 0 00 = y 00 r+2y 0 X v 00 = 0 p¢r~ eddsrd~ ioi geujsdsra pgyd gsfra v(r) = C + C r X xsuzr ip¢dd eddsrd rtiysioi geujsdsra ¤ y(r) = C 1 + C X _it{ihgd~ta r feudj~r gthijra~rX {dejioi gthijra thdygdm zi X iytujhaa C = 0 ji jieid feudjid gthijrdm gpd|yud~ta j i~m zi {er S 6= y(r) gsfra = C X ii~g λ = 0 sd ajhadta tiptjdss~ zrthi~ 6= 0 y(r) = 0 i{deuieuX gt λ = µ > 0 X dr~ geujsdsrd 1 2 2 1 1 2 2 − 1 2 0 0 1 2 (r y ) = µ y ⇔ (2ry 0 + r2 y 00 ) + µ2 y = 0 ⇔ 2y 0 + ry 00 + rµ2 y = 0, 2 2 r r v(r) = y(r)r v(r) v 00 + µ2 v = 0. t{ihgam fuf r eusddm gsfr trdhsi X xu{rd~ geujsdsrd isi} i geujsdsrd t {itiass~r firrdsu~rX ¬oi ueufdertrzdtfid geujsdsrd p = −µ r~dd fi~{hdftsd fiesr p = µi m p = −µi X p¢r~ eddsrd~ geujsdsra pgyd gsfra v(r) = C cos(µr) + C sin(µr) X nioyu ip¢dd eddsrd rtiysioi geujsdsra ¤ y(r) = C cos(µr) + C sin(µr) X itfihfg y(r) ioeusrzdsu {er r → 0 + 0 m i Cr = 0 Xxsuzrr C 6= 0 X xuyuyr~ C = 1 r {iytujr~ gsfrr y(r) = sin(µr) r y (r) = µ cos(µr) − r r sin(µr) ji jieid feudjid gthijrd − 2 2 1 2 1 2 1 2 1 2 0 2 r2 µ cos(µT ) sin(µT ) R − =0 T T2 ⇔ RT cos(µT ) − sin(µT )(R − ST ) = 0. ihgzdssid geujsdsrd {edipeugdta f jryg ctg(µT ) = R − ST . µRT iteir~ oeurfr gsfrm tia¢r j {euji r hdji zuta geujsdsraX usijrta atsim zi i geujsdsrd r~dd pdtzrthdssid ~si|dtji fiesdm {i{uesi tr~~derzs isitrdhsi suzuhu fiieyrsu [ert cXc]X Tiesr suiyam rt{ihga zrthdssd ~diym su{er~dem ~diy futudh} sX Tieds µ dhdtiipeusi suiyrm {edipeuga geujsdsrd f jryg 1 µRT tg(µT ) = . R − ST c` ëìíî ïîï pisuzr~ µ (k = 1, 2, . . .) {ihi|rdhsd fiesr ioi geujsdsraX nioyu zrthu λ = µ (k = 1, 2, . . .) ¤ tiptjdssd zrthum u gsfrr y (r) = sin(µ r) m tiijdtjg¢rd r~m tiptjdssd gsfrr i{deuieuX = r gsfrr ipeug {ihsg ieioisuhsg trtd~g j {eiteus} y (r) tjd X L [0,vuyd~ T ; r ] fjuyeu sie~ tiptjdss gsfr k k 2 k k k k 2 2 kyk (r)k2 = ZT yk2 (r)r2 dr = 0 ZT sin2 (µk r)dr = 0 0 1 1 r− sin(2µk r) = 2 2µk jd V] tX zX λ ZT T 1 (1 − cos(2µk r)) dr = 2 1 1 = T− sin(2µk T ) . 2 2µk m oyd µ ¤ i {ihi|rdhsd fiesr geujsdsra = c] tX X y (r) = sin(µ r), k = 1, 2, . . . R − ST m RT ctg(µT ) = k = 1, 2, . . . µ r `] L [0, T ; r ] ] ky (r)k = 1 T − 1 sin(2µ T ) X 2 2µ N®²® Sha i{deuieu L (y) = − 1 d (sin θ dy ) m sin θ dgyijhdjiea¢r θ dθ m i{edydhdssioi su ~si|dtjd gsfrm iysi} 0eiys~ <θ<π feudj~ gthijra~ j izfu 0 r π jryu ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0 y(θ) sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioisuhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX c k µ2k 0 k k k 2 2 k 2 k k θ dr~ uyuzg £ge~u¤lrgjrhha  1  (sin θy 0 )0 = λy, − sin θ y(θ) 0 < θ < π, ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0. itfihfg yha i{deuieu L (y) gsfra q(θ) = 0 m i tiptjdssd zrthu ioi i{deuieu sdierudhsdX utt~ier~ geujsdsrd 1 − (sin θy 0 )0 = λy sin θ ⇔ θ 1 d 1 d − − sin2 θ(− ) y = 0. sin θ dθ sin θ dθ _jdyd~ sijg {ded~dssg t = cos θ (−1 < t < 1) m ioyu pgyg j{ih} sata eujdstju y(θ) = y(arccos t) = v(t) = v(cos θ) X er i~ v (t) = −1 −1 X Rzrjua g jur~itja ~d|yg {eirjiys~rm =y √ y = sin θ 1 − t u uf|d eujdstji sin θ = 1 − t u{rd~ i geujsdsrd isitrdhsi gsfrr v(t) 0 t 0 θ 2 0 θ 2 2 d d − (1 − t2 ) v = λv. dt dt 0 L(v) = − (1 − t2 )v 0 {deuie ¤ i i{deuie ld|usyeum i{edydhds} s su ~si|dtjd gsfr yju|y yrdedsregd~ {er −1 < t < 1 X tiysua uyuzu £ge~u¤lrgjrhha {edipeuijuhut f uyuzd su tip} tjdssd suzdsra yha i{deuieu ld|usyeu ( 0 − (1 − t2 )v 0 = λv, v(t) <t<1 ioeusrzdsu {er −1 r {er t → 1 − 0. t → −1 + 0 _ {itiprr V¡ teX cY {iyeipsi i{rtusi eddsrd {ihgzdssi uyuzrX iptjdss~r gsfra~r i{deuieu ld|usyeu ajhata ~sioizhds ld} |usyeu vk (t) = Pk (t) = 1 2 k (k) (t − 1) . 2k k! i ~sioizhds td{dsr k P (t) = 1, P (t) = t m P (t) = 1 (3t − 1) . . . X sr tiijdtjg tiptjdss~ zrthu~ λ = k(k + 1) (k = 0,2 1, . . .) X iptjdss~r gsfra~r rtiysioi i{deuieu L (y) pgyg gsfrr m tiijdtjg¢r~r tiptjdss~ zrthu~ λ = k(k + 1) y (θ) = P (cos θ ) X (k = 0,ihsg 1, . . .) ieioisuhsg trtd~g gsfrr ipeug j {eiteus} y ( θ ) tjd X L [0, π; sin θ] cU 0 1 2 2 k θ k k k k 2 vuyd~ fjuyeu sie~ r gsfr kyk (θ)k2 = Zπ yk2 (θ) sin θdθ = Zπ Pk2 (cos θ) sin θdθ. er jzrthdsrr rsdoeuhu j{ihsr~ u~dsg {ded~dssi t = cos θ m dt = 0 0 = − sin θdθ kyk (θ)k2 = Z1 Pk2 (t)dt = 2 . 2k + 1 xsuzdsrd {ithdysdoi rsdoeuhu suydsi j [1] teX `VX jd V] tX zX λ = k(k + 1) m k = 0, 1, 2, . . . c] tX X y (θ) = P (cos θ), k = 0, 1, 2, . . . `] L [0, π; sin θ] ] ky (θ)k = 2 X 2k + 1 ³´µ¶· ¸¹¸º» _ g{eu|sdsra cXV}cXVV yha uyussioi i{deuieu L(y) m i{edydhdssioi su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~m sur tiptjdssd zrthu r tiptjdssd gsfrrm gfuu {eiteustji j fiiei~ tiptjdssd gsfrr ipeug {ihsg ieioisuhsg trtd~g r sur fjuyeu sie~ tiptjdss gsfrX ( L (y) = −y , a < x < b m 2.1. −1 k k k 2 k x 2 00 y(a) = 0, y 0 (b) = 0 ( Lx (y) = −y 00 , a < x < b 2.2. y 0 (a) = 0, y(b) = 0 ( Lx (y) = −y 00 , a < x < b 2.3. (R 6= 0, S 6= 0), Ry 0 (a) − Sy(a) = 0, y(b) = 0 ( Lx (y) = −y 00 , a < x < b 2.4. (R 6= 0, S 6= 0), y 0 (a) = 0, Ry 0 (b) + Sy(b) = 0 ( Lx (y) = −y 00 , a < x < b 2.5. R1 y 0 (a) − S1 y(a) = 0, R2 y 0 (b) + S2 y(b) = 0 (R1 6=  0, S1 6= 0 R2 6= 0, S2 6= 0), B (y) = − 1 d (ρ dy ), 0 < ρ < T 0 ρ dρ dρ 2.6. , y(ρ) 0 ρ → 0 + 0, y (T ) = 0  B (y) = − 1 d (ρ dy ), 0 < ρ < T 0 ρ dρ dρ 2.7. y(ρ) ρ → 0 + 0, Ry 0 (T ) + Sy(T ) = 0 m ioeusrzdsu {er ioeusrzdsu {er c¥ (R 6= 0,  S 6= 0), 1 d dy p2  Bp (y) = − (ρ + y), 0 < ρ < T 2.8. , ρ dρ dρ ρ2  ρ → 0 + 0, y(T ) = 0 y(ρ)  1 d dy p2  Bp (y) = − (ρ + y), 0 < % < T 2.9. ρ dρ dρ ρ2  ρ → 0 + 0, Ry 0 (T ) + Sy(T ) = 0 y(ρ) (R 6= 0, S 6= 0), 1 d dy  Lr (y) = − 2 (r2 ), 0 < r < T 2.10. , r dr dr y(ρ) ρ → 0 + 0, y(T ) = 0  1 d dy  Lr (y) = − 2 (r2 ), 0 < r < T 2.11. . r dr dr y(ρ) 0 ρ → 0 + 0, y (T ) = 0  1 d dy  Lr (y) = − 2 (r2 ), 0 < r < T 2.12. . r dr dr y(ρ) 0 ρ → 0 + 0, y (T ) + hy(T ) = 0 ioeusrzdsu {er ioeusrzdsu {er ioeusrzdsu {er ioeusrzdsu {er ioeusrzdsu {er ¼½¾¹½¿ cXVX V] tX zX λ = + πk m l = b − a m k = 0, 1, . . . l c] tX X y (x) = sin (π + 2πk)(x − a), k = 0, 1, . . . `] L [a, b; 1] ] ky (x)k = 2ll X 2 cXcX V] tX zX λ = + πk m l = b − a m k = 0, 1, . . . l c] tX X y (x) = cos (π + 2πk)(x − a), k = 0, 1, . . . `] L [a, b; 1] ] ky (x)k = 2ll X cX`X V] tX zX λ = µ m 2oyd µ ¤ i {ihi|rdhsd fiesr geujsdsra m S m ctg(µl) = − l = b − a k = 1, 2, . . . c] tX X y (x)Rµ= sin(µ (b − x)), k = 1, 2, . . . `] L [a, b; 1] ] ky (x)k = l − sin(2µ l) X cXX V] tX zX λ = µ m 2oyd µ 4¤µ i {ihi|rdhsd fiesr geujsdsra m S m tg(µl) = l = b − a k = 1, 2, . . . µ c] tX X yR(x) = cos(µ (x − a)), k = 1, 2, . . . 2 π 2 k k 2 2 k 2 π 2 k k 2 2 k 2 k k k 2 k k 2 k 2 k k k k k k k cW `] L [a, b; 1] ] ky (x)k = l + sin(2µ l) X 4µ sdierudhsd fiesr geujsdsra cXUX V] tX zX λ = µ m oyd2 µ ¤ i m µ(R S + R S ) m tg(µl) = − l = b − a k = 0, 1, 2, . . . SS c] tX X y (x)R =R Rµ µ− cos( µ (x − a)) + S sin(µ (x − a)), k = 0, 1, 2, . . . `] L [a, b; 1] ] ky (x)k = l (R µ + S ) + sin(2µ l) + S R sin (µ l) X 2 4µ cX¥X V] tX zX λ = γ m oyd γ ¤ i sdierudhsd fiesr geujsdsra T m J (γ) = 0 k = 0, 1,2, . . . c] tX X y (ρ) = J γ ρ , k = 0, 1, 2, . . . `] L [0, T ; ρ] ] ky (ρT)k = T J (γ ) X cXWX V] tX zX λ = γ m2oyd γ ¤ i {ihi|rdhsd fiesr geujsdsra T J (γ) ST m = k = 1, 2, . . . c]J (tXγ)X yR(γρ) = J γ ρ , k = 1, 2, . . . `] L [0, T ; ρ] ] ky (ρT)k = T J (γ ) + J (γ ) X cXZX V] tX zX λ = γ m2oyd γ ¤ i {ihi|rdhsd fiesr geujsdsra m T J (γ) = 0 k = 1, 2,. . . c] tX X y (ρ) = J γ ρ , k = 1, 2, . . . T `] L [0, T ; ρ] ] ky (ρ)k = T p J (γ ) − J (γ ) X 2 γ cXYX V] tX zX λ = γ m oyd γ ¤ i {ihi|rdhsd fiesr geujsdsra T J (γ) ST m k = 1, 2, . . . =− c]J (tXγ)X y (Rρ)γ= J γ ρ , k = 1, 2, . . . `] L [0, T ; ρ] ] ky (ρT)k = T J (γ ) 1 + S T X 2 Rγ cXV\X V] tX zX λ = µ m µ = πk m k = 1, 2, . . . c] tX X y (r) = sin(µ r) , k = 1,T 2, . . . 2 2 1 2 1 2 1 k 2 k 2 k 1 2 k k k 2 k k 1 2 1 k 2 2 1 k 2 k k k k 2 1 1 k 2 k k 1 k k 0 2 2 0 2 k k k 2 k k 1 0 k k 0 2 2 k k 2 1 2 0 k k 2 k k p k k p 2 2 2 k k p k p+1 k 2 k k 0 p p k k 2 p 2 k 2 k k 2 p 2 k 2 2 k k k k 2 r cZ k 1 2 k `] L [0, T ; r ] ] ky (r)k = T X 2 cXVVX V] tX zX λ = µ m oyf ¤ i {ihi|rdhsd fiesr geujsdsra µ 1 m ctg(µT ) = k = 1, 2, . . . µT c] tX X y (r) = sin(µ r) , k = 1, 2, . . . `] L [0, T ; r ] ] ky r(r)k = T sin (µ T ) X 2 cXVcX V] tX zX λ = µ m oyf ¤ i {ihi|rdhsd fiesr geujsdsra µ 1 − Thm ctg(µT ) = k = 1, 2, . . . µT c] tX X y (r) = sin(µ r) , k = 1, 2, . . . `] L [0, T ; r ] ] ky r(r)k = T − sin(2µ T ) = T T µ + T h − h X 2 2 2 k 2 k k k k k 2 2 2 2 k 2 k k k k k k 2 2 k k 2 2 4µk 2 2 2 k 2 2 T µk + 2 (1 − T h)2 ñ ' ?A&15 C5>C 2C@( E@A E1 E= 5 EF2> gt eutt~uerjudta suoedj dhu Ω t fgtizsi}ohuyfi oeusrd S X pisuzr~ u(M, t) [M ∈ Ω r t > 0] gsfrm i{rtju¢g r~dsdsrd d~{deuge ioi dhuX _ ip¢d~ ed~desi~ thgzud geujsdsrd d{hi{ei} jiysitr t gzdi~ rtizsrfij d{hu u{rtjudta j jryd òóô (k grad u) + q, oyd c ¤gydhsua d{hid~fitm ¤ {hisitm k ¤firrds d{hi{eijiy} sitrm q ¤ gsfram i{rtju¢ua ipõd~sg {hisit rtizsrfij d{huX _jiy ed~desioi geujsdsra {iyeipsi rhi|ds j ¡X ¬thr dhi Ω iysi} eiysid r riei{sid r j ~iydhr ~i|si tzrum zi firrds c m ρ r k ¤ i {itiassd jdhrzrsm i geujsdsrd d{hi{eijiysitr ipzsi {edipeug f jryg [`XV] ∂u = a ∆u + f, ∂t oyd zrthi a = k sujudta firrdsi~ d~{deugei{eijiysitrm gsfra f = qρcX {deuie ∆ ¤ i i{deuie lu{hutuX _ry i{deuieu ρc ujrtr i jpeussi trtd~ fiieyrsu ∂ u ∂ u ∂ u [j ydfueiji]m ∆u = + + ρc ∂u = ∂t ρ 2 2 2 2 2 ∂x2 ∂y 2 ∂z 2 cY [j rhrsyerzdtfi]m 1 ∂ ∂u 1 ∂ 2u ∂ 2u ∆u = ρ + + ρ ∂ ρ ∂ ρ ρ2 ∂ ϕ2 ∂z 2 ∂ 2u ∂ 1 ∂ ∂u 1 ∂u 1 2 ∆u = 2 ρ + 2 sin θ + 2 2 ρ ∂ρ ∂ρ ρ sin θ ∂ θ ∂θ ρ sin θ ∂ ϕ2 [j tderzdtfi]X T i~g geujsdsr yipujhadta suzuhsid gthijrd {er t = 0 oyd ϕ(M ) ¤ gsfram i{rtju¢ua suzuhsg d~{deugeg dhuX vu oeusrd iphutr S uyuta feudjd gthijra j tiijdtjrr t gthi} jra~r d{hiip~dsu dhu t ifeg|u¢d tedyiX i ~i|d p iysi r ed gthijr jryu u(M, 0) = ϕ(M ), u S = ν(M 0 , t), −k ∂u ∂~n = Q(M 0 , t), S −k ∂u ∂~n S = α(u − T0 ) S . xydt M ¤ izfu oeusr iphutr S m ∂u ¤ {eirjiysua {i su{eujhdsr ∂~n jsdsd sie~uhr f oeusrd S X dejid gthijrd uyudtam dthr su oeusrd {iyyde|rjudta uyussid eut{edydhdsrd d~{deuge ν(M , t) X i feudjid gthijrd sujudta gthi} jrd~ Srerhd rhr gthijrd~ V}oi eiyuX _ieid sujudta gthijrd~ vd~usu rhr gthijrd~ c}oi eiyuX xuyu} dta j thgzudm fioyu su oeusrd {iyyde|rjudta d{hiji {iif Q(M , t) [k ¤ firrds dhi{eijiysitr]X nedd gthijrd uyudtam dthr su oeusrd d{hiip~ds {eirtiyr {i ufisg visuX _ ie~ghd T ¤ i d~{deugeu ifeg|u¢d tedym ¤ firrds dhi{eijiysitrm ¤ firrds d{hiip~dsuX i kgthijrd sujudta edr~ feudj~αgthijrd~X vu eus zuta oeusr S ~iog p uyus oeusrzsd gthijra eusioi eiyuX ysi~desid geujsdsrd d{hi{eijiysitr {ihgzdsi j gzdp} si~ {itiprr V¡X nu~ |d i{rtusu {itusijfu suzuhsioi r feudj gthijrX Reujsdsrd jryu [`XV] {iajhadta sd ihfi {er eddsrr uyuzm tja} uss t eut{iteusdsrd~ d{huX vu{er~dem i geujsdsrd i{rtjud {ei} dtt yrgrr zutr j sdfiiei tedydX bdsadta ihfi t~th fi} rrdsijX Reujsdsrd yrgrr eutt~iedsi j ¡X ysr~ r surpihdd zuti {er~dsad~ ~diyij eddsra suzuhsi} feudj uyuz yha geujsdsra d{hi{eijiysitr ajhadta ~diy gedX khoier~ {er~dsdsra eayij ged yha eddsra feudj uyuz i{rtus j V¡ su {er~ded eddsra iysi~desi uyuzrX ddsrd feudji uyuzr suiya j jryd eayu gedm j{ihsaa eu} hi|dsrd rtfi~i gsfrr {i tiptjdss~ gsfra~ hrsdsioi tr~~d} erzsioi yrdedsruhsioi i{deuieu £ge~u¤lrgjrhha L m tjaussioi `\ 0 0 0 0 t edud~i uyuzdX {deuie L eutt~uerjudta su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~X ¬thr feudjd gthijra rtiysi feudji uyuzr sdiysieiysdm i tsuzuhu uyuzg tjiya f uyu} zd t iysieiys~r feudj~r gthijra~rX Sha ioi rtfi~g gsfr u {edytujha j jryd tg~~ yjg gsfr u = v + w X gsfr w [zuti hrsdsgm si sd jtdoyu] jpreu ufm zip isu gyijhdjieahu ufr~ |d feudj~ gthijra~m zi r gsfra u X nioyu gsfra v pgyd gyijhdjiea iysieiys~ oeusrzs~ gthijra~X xud~ jt feudjg uyuzg u{rtju isitrdhsi gsfrr v X ihgzdssg j edghud uyuzg edu ~di} yi~ gedX Sha ioi vuiya tiptjdssd zrthu λ r tiptjdssd gsfrr y hrsdsioi tr~} 1) ~derzsioi yrdedsruhsioi i{deuieu jieioi {ieayfu L m eutt~uer} jud~ioi su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~m tiijdtjg¢r~ feudj~ gthijra~ uyuzrX nX dX edu tii} jdtjg¢g uyuzg £ge~u¤lrgjrhhaX örtd~u tiptjdss gsfr ieioisuhsu r {ihsu j {eiteustjd L [a, b; ρ)] m oyd ρ ¤ i jdti} {y } jua gsfra i{deuieu L X gsfr v suiya j jryd eayu gedm euhuoua dd {i suydss~ tip} 2) tjdss~ gsfra~ y yrdedsruhsioi i{deuieuX Sha i{edydhdsra suzdsr firrdsij eayu ged i eay thdygd {iytujr j geuj} sdsrd d{hi{eijiysitr r j suzuhsid gthijrdX ¬thr j geujsdsrr rhr j suzuhsi~ gthijrr dt ihrzsd i sgha gsfrrm i r gsfrr uf|d thdygd euhi|r j eay gedX ithd {iytusijfr yha firrdsij rtfi~ioi eayu {ihgzuta yrdedsruhsd geujsdsraX edu r {ihgzu jeu|dsra yha firrdsijX tfi~id eddsrd {edytujha j jryd u = v + w m oyd gsfra v {edy} 3) tujhdsu j jryd eayu gedX bdiy euhi|dsra {i tiptjdss~ gsfra~ dtsi tjaus t ~diyi~ euydhdsra {ded~dssm fiie {edysusuzds yha i{edydhdsra zuts eddsr yrdedsruhs geujsdsr j zuts {eirjiysX er eddsrr sdfiie feudj uyuz dhdtiipeusi rt{ihiju ferjihrsdsg trtd~g fiieyrsum su{er~de rhrsyerzdtfg rhr tder} zdtfgX nioyu {er {er~dsdsrr ~diyu ged {iajata t{druhsd gsf} rrX iyeipsi tjitju r gsfr i{rtus j U¡m ¥¡m ¡m V¡X upded~ sdtfihfi {er~deijX ¯®P® de|ds yhrsi l t d{hirihreijussi {ijdesit suoed yi d~{deuge T X vuzrsua t ~i~dsu jed~dsr t = 0 iyrs r fisij tde|sa ihu|yum {iyyde|rju d~{deugeg su i~ fisd eujsg \ [su{er~dem fisd tde|sa i{gtfu j hdy]X vu yegoi~ fisd tde|sa d~{deugeg {iyyde|rju eujsi T X vur d~{deugeg j tedysd izfd tde|sa j ~i~ds jed~dsr t X k k +∞ k k=1 2 k 0 0 0 `V gt gsfra u(x, t) i{rtjud eut{edydhdsrd d~{deuge j tde|} sdX su gyijhdjiead geujsdsr ∂ u {er ∂u =a 0 < x < l, t > 0, ∂t ∂x suzuhsi~g gthijr u(x, 0) = T m feudj~ gthijra~ u(0, t) = 0, u(l, t) = T . bdiy ged sd{itedytjdssi {er~dsadta f uyuzu~m j fiie rt} fi~ua gsfra gyijhdjiead iysieiys~ feudj~ gthijra~X _ eutt~u} erjud~i uyuzd gthijrd su fisd tde|sa x = l sdiysieiysidX ii~g tsuzuhu tjdyd~ g uyuzg f uyuzd t iysieiys~r feudj~r gthijra} ~rX Sha ioi gsfr u(x, t) {edytujr~ j jryd tg~~ yjg gsfr X xuyuyr~ w(x, t) = αx + β r {iypded~ α r β ufm u(x, t) = v(x, t) + w(x, t) zip yha gsfrr w(x, t) j{ihsahrt feudjd gthijra 2 2 2 0 0 w(0, t) = β = 0, w(l, t) = αl + β = T0 . xsuzrm α = T m β = 0 r w(x, t) = T x X hdyijudhsi l l T X u(x, t) = v(x, t) + x sitrdhsi lgsfrr v(x, t) {ihgzrta uyuzu t iysieiys~r oeu} srzs~r gthijra~r  ∂v ∂ v {er   =a 0 < x < l, t > 0,  0 0 0 2 2 ∂x2 T0 x , v(x, 0) = T − 0   l   v(0, t) = 0, v(l, t) = 0.  ∂t g uyuzg pgyd~ edu ~diyi~ gedX V] gsfr v(x, t) pgyd~ rtfu j jryd eayu gedm euhuoua dd {i trtd~d tiptjdss gsfr hrsdsioi yrdedsruhsioi i{deuieu ∂ v X Sha sui|ydsra tiptjdss zrtdh r tiptjdss gsf} L (v) = − ∂x sdipiyr~i edr tiijdtjg¢g uyuzg £ge~u¤lr} r i{deuieu gjrhha 2 x 2 ( −y 00 (x) = λy(x), 0 < x < l, y(0) = 0, y(l) = 0. u uyuzu phu eutt~iedsu j euydhd c [{er~de cXV] r yha i{deui} eu L = −y m eutt~uerjud~ioi su ~si|dtjd gsfrm gyijhdjiea} ¢r feudj~ gthijra~ Srerhdm phr suyds tiptjdssd zrthu λ = r tiijdtjg¢rd r~ tiptjdssd gsfrr y (x) = πk m = k = 1, 2, . . . 00 x k 2 l k `c πkx , l m ipeug¢rd {ihsg ieioisuhsg trtd~g j {eiteustjd L [0, l; 1] X c] gsfr v(x, t) {edytujr~ j jryd eayu gedm j{ihsrj euhi} |dsrd {i suydssi trtd~d tiptjdss gsfr yrdedsruhsioi i{deuieu = sin k = 1, 2, . . . . 2 v(x, t) = +∞ X ck (t)yk (x). Teudjd gthijram {er i~m pgyg j{ihsataX Sha i{edydhdsra suzdsr firrdsij c (t) eay ged {iytujr~ j geujsdsrd d{hi{eijiysitr r j suzuhsid gthijrdX gsfra r suzuhsioi gthijra ϕ÷ (x) = T x ihrzsu l i sghaX ¬d uf|d euhi|r~ j eay ged k=1 k 0 ÷ ϕ (x) = +∞ X ϕk yk (x). k=1 Tirrds ged suyd~ {i rjdtsi~g {eujrhg ϕ = (ϕ÷ , y ) X Tjuy} ky k eu sie~ tiptjdss gsfr ky k = l m k = 1, 2, . . . phr {ihgzds j 2 {er~ded cXVX _zrthr~ (ϕ÷ , y ) k k k 2 2 k k ÷ (ϕ , yk ) = Zl 0 πkx T0 x T0 x ) sin dx = (T0 − ) (T0 − l l l T0 − πk Zl 0 −l πkx cos πk l πkx T0 l T0 l πkx cos dx = − sin l πk (πk)2 l l = 0 l 0 − T0 l . πk nioyu ϕ = 2T m k = 1, 2, ... . πk d{de eay ged gsfr iytujr~ r ϕ÷ (x) j geujsdsrd v(x, t) d{hi{eijiysitr r j suzuhsid gthijrd 0 k +∞ X c 0k (t)yk (x) 2 =a ck (t)y 00k (x), k=0 k=1 +∞ X +∞ X ck (0)yk (x) = k=1 +∞ X k=0 `` ϕk yk (x). itfihfg j{ihsata eujdstju y (x) = −λ y (x) r euhi|dsrd gsfrr j eay ged dyrstjdssim i yha firrdsij c (t) {ihgzr~ uyuzr Tir 00 k k k k ( c 0k (t) = −a2 λk ck (t), ck (0) = ϕk . p¢dd eddsrd yrdedsruhsioi geujsdsra c (t) = A e {ihga eujdstji c (0) = ϕ m {ihgzr~ A = ϕ m u suzrm k k k k k −a2 λk t X t} k 2T0 −a2 π22 k2 t e l . ck (t) = πk gsfra v(x, t) euhuoudta j eay ged 2 2 2t v(x, t) = +∞ −a π k 2T0 X e l2 π sin k k=1 πkx . l `] tfi~ua gsfra u(x, t) = v(x, t) + T x m i{rtju¢ua eut{edyd} hdsrd d~{deuge j tde|sdm {edytujr~u j ljryd 0 +∞ −a2 π2 k2 t T0 2T0 X e l2 u(x, t) = x + l π k sin k=1 πkx . l iytujr~ j {ihgzdssid jeu|dsrd x = l r t = t X ihgzrta zrthi} 2 ji eaym yha i{edydhdsra d~{deuge tde|sa j tedysd izfd j ~i~ds jed~dsr t X jd 0 0 +∞ l T0 2T0 X e u( , t0 ) = + 2 2 π −a2 π2 k2 t0 l2 k sin πkl . 2 ¯®N® ysu r {ijdesitd isfi {hutrs ih¢rsi l m su} oedi yi d~{deuge T m d{hirihreijusum u yegoua ti{erfutudta t jiygi~ d~{deuge T (T < T ) m X dX ihu|yudta {i ufisg visuX vur d~{deugeg {hutrs su i {ijdesitr j ~i~ds jed~dsr t X _ ~iydhr pgyd~ tzrum zi d~{deugeu {hutrs t dzdsrd~ jed} ~dsr ~dsadta ihfi {i ih¢rsdX pisuzr~ u(x, t) gsfrm fiieua i{rtjud r~dsdsrd d~{deugeX u gsfra gyijhdjiead geujsd} sr d{hi{eijiysitr ∂u ∂ u {er 0 < x < l, t > 0, =a k=1 1 0 0 1 0 2 ∂t 2 ∂x2 ` suzuhsi~g gthijr u(x, 0) = T m feudj~ gthijra~ ∂u(0, t) = 0, − ∂u(l, t) = h(u(l, t) − T ). ysi r feudj ∂xgthijr uyuzr∂xsdiysieiysidX ii~g sd{itedy} tjdssi {er~dsa ~diy ged f {ihgzdssi feudji uyuzd sdhaX edy} tujr~ gsfr u(x, t) j jryd tg~~ yjg gsfr u(x, t) = v(x, t)+ X gt w(x, t) = αx + β ¤ hrsdsua {i {ded~dssi x gsfraX +w(x, t) iypded~ α r β ufm zip gsfra w(x, t) gyijhdjieahu d~ |d feu} dj~ gthijra~m zi r gsfra u(x, t) m X dX {iytujr~ w(x, t) j feudjd gthijra 1 0 ( α = 0, −α = h(αl + β) − T0 . ihgzr~ α = 0 m β = T X xsuzr w(x, t) = T x. h h gsfra v(x, t) pgyd gyijhdjiea iysieiys~ feudj~ gthijra~X xu{rd~ feudjg uyuzg isitrdhsi i gsfrr  ∂ v {er ∂v   = a 0 < x < l, t > 0,  0 0 2 2  ∂x2  ∂t T0 v(x, 0) = T1 − x,  h     ∂v(0, t) = 0, ∂v(l, t) + hv(l, t) = 0. ∂x ∂x g uyuzg pgyd~ edu ~diyi~ gedX dr~ uyuzg £ge~u¤lrgjrhha yha i{deuieu L (v) = − ∂ v 1) 2 x ∂x2 ( −y 00 (x) = λy(x), 0 < x < l, y 0 (0) = 0, y 0 (l) + hy(l) = 0. i uyuzu r g{eu|sdsra cXX iptjdssd zrthu i{deuieu λ = µ m oyd ¤ i {ihi|rdhsd fiesr geujsdsra tg(µl) = h m k = 1, 2, . . . X ~ µ tiijdtjg tiptjdssd gsfrr y (x) = cos(µ x)µm k = 1, 2, . . . m ipeu} g¢rd {ihsg ieioisuhsg trtd~g j {eiteustjd L [0, l; 1] X Tjuy} eu sie~ tiptjdss gsfr i{edydhata {i {eujrhg ky (x)k = l sin(2µ l) X Rzrjuam zi 2 tg(α) m fjuyeu sie~ ~i|} = + sin(2α) = 2 4µ 1 + tg (α) si {edytujr j jryd ky (x)k = l + h X 2 2( ) gedm euhuoua dd {i c] gsfr v(x, t) pgyd~ rtfu j µjryd+ heayu 2 k k k k k 2 k k 2 k k 2 2 k Ù 2 2 trtd~d tiptjdss gsfrr yrdedsruhsioi i{deuieu v(x, t) = +∞ X ck (t)yk (x). Teudjd gthijra j{ihsataX gsfr ϕ÷ (x) = T − T x r suzuhsioi gthijra euhi|r~ j eay h ged {i i |d trtd~d gsfr k=1 0 1 +∞ X ÷ ϕ (x) = {er i~ ϕ k = ϕk yk (x), k=1 ÷ (ϕ , yk ) kyk k2 X _zrthr~ tfuhaesd {eirjdydsra ÷ (ϕ , yk ) = Zl 0 (T1 − T0 x) cos(µk x)dx = h T0 T0 l sin(µk l) − 2 (cos(µk l) − 1) = T1 − h µk hµk T0 2(µ2k + h2 ) T0 l sin(µk l) − 2 (cos(µk l) − 1) ϕk = 2 T1 − lµk + lh2 + h h µk hµk k = 1, 2, ... . v(x, t) ϕ (x) ioyu iytujr~ eay ged gsfr jiysitr r j suzuhsid gthijrd +∞ X c 0k (t)yk (x) m r ÷ j geujsdsrd d{hi{ei} 2 =a +∞ X ck (t)y 00k (x), k=1 k=1 +∞ X ck (0)yn (x) = +∞ X ϕk yk (x). _it{ihgd~ta eujdstju~r y (x) = −λ y (x) X _ trhg dyrstjdssi} tr euhi|dsra gsfrr j eay gedm yha firrdsij c (t) {ihgzr~ uyuzr Tir k=1 k=1 00 k k k k ( c 0k (t) = −a2 λk ck (t), ck (0) = ϕk . ddsra r uyuz {edytujhata j jryd c (t) = ϕ e k 2(µ2 + h2 ) ck (t) = 2 k 2 lµk + lh + h T0 l T1 − h k −a2 λk t X nioyu sin(µk l) T0 2 2 − 2 (cos(µk l) − 1) e−a µk t . µk hµk `¥ xu{rd~ eay ged gsfrr v(x, t) +∞ X T0 l sin(µk l) 2(µ2k + h2 ) v(x, t) = T1 − − lµ2k + lh2 + h h µk k=1 T0 2 2 − 2 (cos(µk l) − 1) e−a µk t cos(µk x). hµk `] gsfra u(x, t) = v(x, t) + w(x, t) +∞ T0 l sin(µk l) T0 x X 2(µ2k + h2 ) − + T1 − u(x, t) = h lµ2k + lh2 + h h µk k=1 T0 2 2 − 2 (cos(µk l) − 1) e−a µk t cos(µk x). hµk x =l t = t0 x = l t = t0 Sha i{edydhdsra d~{deuge {hutrs su {ijdesitr jed~dsr {iytujr~ j suydssid eddsrd r zrthiji eayX jd j ~i~ds X ihgzrta +∞ T0 l X 2(µ2k + h2 ) T0 l sin(µk l) − + u(l, t0 ) = T1 − h lµ2k + lh2 + h h µk k=1 T0 2 2 − 2 (cos(µk l) − 1) e−a µk t0 cos(µk l). hµk ¯®¯® nd~{deugeu su pifiji {ijdesitr yhrssioi rhrs} yerzdtfioi tde|sa euyrgtu R m suoedioi yi d~{deuge T m suzrsua t ~i~dsu jed~dsr t = 0 {iyyde|rjudta eujsi T [T > T ) X vur d~{d} eugeg j dsed tde|sa j ~i~ds jed~dsr t X Sha i{rtusra suoedju tde|sa gyipsi jjdtr rhrsyerzdtfg trtd} ~g fiieyrsuX i~dtr~ suzuhi fiieyrsu j dsed {eirjihsioi {i{d} edzsioi tdzdsra tde|saX t OZ su{eujr~ jyih itr tde|sam j {hit} fitr tdzdsra jjdyd~ {ihaesg trtd~g fiieyrsuX _ ~iydhr ~i|si tzr} u zi gsfra u m i{rtju¢ua r~dsdsrd d~{deuge j tde|sd sd ujrtr i {ded~dss z r ϕ m u ajhadta gsfrd ihfi {ihaesi fi} ieyrsu ρ r jed~dsr t u = u(ρ, t) X nioyu gsfra u(ρ, t) gyijhdjiead geujsdsr d{hi{eijiysitr {er 0 < ρ < R, t > 0, ∂u 1 ∂ ∂u =a ρ ∂t ρ ∂ρ ∂ρ suzuhsi~g gthijr u(ρ, 0) = T m feudj~ gthijra~ u(ρ, t) ioeusrzdsu {er ρ → 0 + 0 m u(R, t) = T . `W 0 1 1 0 0 2 0 1 Rthijrd ioeusrzdssitr u(ρ, t) {er r → 0 + 0 ajhadta iysieiys~X Teudjid gthijrd {er ρ = R ¤ sdiysieiysidX ii~g tsuzuhu u~dsi tjdyd~ eutt~uerjud~g uyuzg f uyuzd t iysi} u( ρ , t) = v( ρ , t) + w( ρ , t) eiys~r feudj~r gthijra~rX gt w(ρ, t) = T X _ i~ thgzud w(ρ, t) gyijhdjiead d~ |d feudj~ gthijra~m zi r gsfra u(ρ, t) w(ρ, t) ioeusrzdsu {er ρ → 0 + 0 m w(R, t) = T . nioyu gsfra v(ρ, t) ajhadta eddsrd~ uyuzr t iysieiys~r feu} dj~r gthijra~r  {er 0 < ρ < R, t > 0, ∂v 1 ∂ ∂v    =a ρ 1 1 2  ∂t ρ ∂ρ ∂ρ v(ρ, 0) = T0 − T1 ,    v(ρ, t) ioeusrzdsu {er ρ → 0 + 0, v(R, t) = 0. g uyuzg pgyd~ edu ~diyi~ gedX V] dr~ uyuzg £ge~u¤lrgjrhha yha tr~~derzsioi hrsdsioi yrdedsruhsioi i{deuieu ^dttdha B m eutt~uerjud~ioi su ~si|d} tjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~ edud~i uyuzr 0  − 1 (ρy 0 (ρ))0 = λy(ρ), 0 < ρ < R, ρ y(ρ) ρ → 0 + 0, y(R) = 0. ioeusrzdsu {er u uyuzu phu eddsu u j euydhd c [{er~de cXU]X iptjdssd zrthu i{deuieu λ = γ , k = 1, 2, . . . m oyd γ ¤ i {ihi|rdhsd ed} dsra geujsdsra JR(γ) = 0 X iptjdssd gsfrr i{deuieu y (ρ) = X rtd~u tiptjdss gsfr {ihsu r ieioi} = J ( ρ ) k = 1, 2, . . . suhsu j {eiteustjd L [0, R; ρ] X Tjuyeu sie~ tiptjdss gsfr i{edydhata {i ie~ghu~ ky (ρ)k = R (J (γ )) X c] uhi|r~ gsfr v(ρ, t) j eay2 ged {i trtd~d tiptjdss gsfr i{deuieu ^dttdha k 2 k γk 0 R k 0 k 2 k v(ρ, t) = 2 2 +∞ X 1 k 2 ck (t)yk (ρ). Teudjd gthijra {er i~ j{ihsataX gsfra r suzuhsioi gthijra ϕ÷ (ρ) = T j eay ged k=1 ÷ ϕ (ρ) = +∞ X k=1 `Z 0 ϕk yk (ρ). − T1 X ¬d uf|d euhi|r~ Tirrds ged suyd~ {i {eujrhg ϕ = (ϕ÷ , y ) X Tjuyeu sie~ tiptjdss gsfr g|d rjdtsX {edydhr~ (kyϕ÷k, y ) rt{ihga ie} ~ghg (cXc) m tzruam zi α = γ m β = 0 r gzrjuam zi J (γ) = −J (γ) k k k 0 0 k ZR ÷ (ϕ , yk ) = 0 k 2 1 γk R2 (T0 − T1 )J0 ( ρ)ρdρ = (T0 − T1 ) J1 (γk ). R γk _ riod {ihgzr~ ϕ = 2(T − T ) m k = 1, 2, ... . γ J (γged ) nd{de {iytujr~ eay gsfr v(ρ, t) r ϕ÷ (ρ) j geujsdsrd d{hi{eijiysitr r j suzuhsid gthijrd 0 1 k k 1 +∞ X k c0k (t)yk (ρ) 2 =a k=1 +∞ X k=1 +∞ X ck (0)yk (ρ) = Rzrjua eujdstju − 1 (ρy hgzr~ uyuzr Tir ρ 1 0 ck (t) (ρy 0k (ρ)) , ρ +∞ X ϕk yk (ρ). k=1 k=1 0 0 k (ρ)) = λyk (ρ) m yha firrdsij c (t) {i} k ( c 0k (t) = −a2 λk ck (t), ck (0) = ϕk . ddsra fiie c (t) = ϕ e k k −a2 λk t rhr 2(T0 − T1 ) −a22γ2k t ck (t) = e R . γk J1 (γk ) gsfra v(x, t) {edytujhadta j jryd eayu −a2 γ2 `] nioyu k +∞ X e R2 t γk v(ρ, t) = 2(T0 − T1 ) J0 ( ρ). γk J1 (γk ) R k=1 −a2 γ2 k +∞ X e R2 t γk J0 ( ρ). u(ρ, t) = T1 + 2(T0 − T1 ) γk J1 (γk ) R vuyd~ d~{deugeg j dsed tde|sa j ~i~ds jed~dsr t X Sha ioi {iytujr~ j eddsrd ρ = 0 r t = t X Rzrjuam zi J (0) = 1 m {ihgzr~ zrthiji eaym yha i{edydhdsra rtfi~i d~{deugeX `Y k=1 0 0 0 jd −a2 γ2 k +∞ X e R 2 t0 u(0, t0 ) = T1 + 2(T0 − T1 ) . γk J1 (γk ) ¯®Q® bduhhrzdtfr uerf euyrgtu R m suoed yi d~{deu} ge T m {i~dtrhr j |ryfit d~{deuge T (T < T ) X vur d~{deu} geg su {ijdesitr uerfu j ~i~ds jed~dsr t m dthr d{hiip~ds uerfu t |ryfit {eirtiyr {i ufisg visuX _ ~iydhr gyipsi jjdtr tderzdtfg trtd~g fiieyrsuX i~dtr~ suzuhi fiieyrsu j dsed uerfuX Rzrjua gthijra uyuzr ~i|si tzr} um zi gsfra u m i{rtju¢ua r~dsdsrd d~{deuge uerfu sd u} jrtr i gohij fiieyrsu θ r ϕ m u ~dsadta ihfi {i fiieyrsud r j ujrtr~itr i jed~dsr t u = u(r, t) X gsfra u(r, t) gyijhdjiead geuj} sdsr d{hi{eijiysitr {er 0 < r < R, t > 0, ∂u 1 ∂ ∂u =a r ∂t r ∂r ∂r suzuhsi~g gthijr u(ρ, 0) = T m feudj~ gthijra~ ioeusrzdsu {er r → 0 + 0 m − ∂u(R, t) = h(u(R, t) − T ). u(r, t) ∂r Teudjid gthijrd su oeusrd r = R ajhadta sdiysieiys~X ii~g tsuzuhu rt{ihga u~dsg u(r, t) = v(r, t) + w(r, t) tjdyd~ edud~g uyuzg f uyuzd t iysieiys~r feudj~r gthijra~rX xuyuyr~ w(r, t) = α X er gsfra w(r, t) gyijhdjiead d~ |d feudj~ gthijra~m zi r α = T gsfra u(r, t) ioeusrzdsu {er r → 0 + 0 r ∂w(R, t) + h(w(R, t) − T ) = 0. w(r, t) ∂r tiijdtjg¢r~ iysieiys~ er i~ gsfra v(r, t) gyijhdjiead feudj~ gthijra~X ie~ghregd~ uyuzg yha gsfrr v(r, t) X gsfra v(r, t) gyijhdjiead geujsdsr d{hi{eijiysitr {er 0 < r < R, t > 0, ∂v ∂v 1 ∂ r =a ∂t r ∂r ∂r suzuhsi~g gthijr v(r, 0) = T − T m feudj~ gthijra~ ioeusrzdsu {er r → 0 + 0 m ∂v(R, t) + hu(R, t) = 0. v(r, t) g uyuzg pgyd~ edu ~diyi~∂rgedX V] Sha tr~~derzsioi hrsdsioi yrdedsruhsioi i{deuieu L (y) m eutt~uerjud~ioi su ~si|dtjd gsfrm gyijhdjiea¢r iysieiys~ k=1 0 1 1 0 0 2 2 2 0 1 1 1 2 2 2 0 1 r \ feudj~ gthijra~ edud~i uyuzrm edr~ uyuzg £ge~u¤lrgjrhha   1 2 0 0 − 2 r y (r) = λy(r), 0 < r < R, r y(r) r → 0 + 0, y 0 (R) + hy(R) = 0. ioeusrzdsu {er i uyuzu r g{eu|sdsra cXVcX iptjdssd zrthu j i uyuzd λ = m oyd µ ¤ i {ihi|rdhsd fiesr geujsdsra ctg(µR) = 1 − Rh m = µ µR X iptjdssd gsfrr y (r) = sin(µ r) m k = 1, 2, . . . X rtd~u k = 1, 2, . . . tiptjdss gsfr {ihsu r ieioisuhsu jr {eiteustjd L [0, R; r ] X Tjuyeu sie~ tiptjdss gsfr suiyata {i ie~ghu~ ky (r)k = R sin(2µ R) R Rµ + Rh − h X = − = 2 c] gsfr 4µ 2 u{rd~ R µ + (1j−jryd Rh)eayu gedm euhuoua dd {i suyds} v(r, t) gsfr si trtd~d tiptjdss 2 k k k k k 2 2 2 k 2 2 k 2 k k 2 k 2 2 v(r, t) = +∞ X ck (t)yk (r). Teudjd gthijram {er i~m j{ihsataX utt~ier~ gsfr ϕ÷ (r) = T − T r suzuhsioi gthijraX uhi} |r~ dd j eay ged {i i |d trtd~d gsfr k=1 0 ÷ ϕ (r) = +∞ X 1 ϕk yk (r), k=1 oyd ϕ = (ϕ÷ , y ) X Tjuyeu sie~ tiptjdss gsfr g|d suydsX ky k {edydhr~ tfuhaesd {eirjdydsra k k ZR k 2 ÷ _it{ihgd~ta eujdstji~m fiiei~g gyijhdjiea zrthu µ ctg(µ R) = 1 − Rh m r {edipeugd~ {ithdysdd {ihgzdssid jeu|dsrd = (ϕ , yk ) = 0 sin(µk R) R cos(µk R) sin(µk r) 2 r dr = (T0 − T1 ) (T0 − T1 ) − . r µ2k µk k µk R ÷ (ϕ , yk ) = vuyd~ ϕ k = (T0 − T1 )R2 h cos(µk R) . µk (1 − Rh) (T0 − T1 )4R2 h cos(µk R) k = 1, 2, . . . . (1 − Rh)(2Rµk − sin(2µk R)) V k ay ged gsfr v(x, t) r ϕ÷ (r) {iytujr~ j geujsdsrd d{hi{ei} jiysitr r j suzuhsid gthijrd +∞ X c0k (t)yk (r) 2 =a +∞ X ck (t) k=1 k=1 +∞ X ck (0)yk (r) = k=1 +∞ X 1 2 0 0 r y k (r) , r2 ϕk yk (r). k=1 itfihfg j{ihsata eujdstju − 1 r y r rrdsij c (t) {ihgzuta uyuzr Tir 2 2 0 k (r) k 0 = λyk (r) m i yha fi} ( c 0k (t) = −a2 λk ck (t), ck (0) = ϕk . dr~ r −a2 λk t ck (t) = ϕk e (T0 − T1 )4R2 h cos(µk R) 2 2 ⇔ ck (t) = e−a µk t . (1 − Rh)(2Rµk − sin(2µk R)) gsfra v(x, t) euhuoudta j eay ged +∞ (T0 − T1 )4R2 h X cos(µk R) 2 2 sin(µk r) e−a µk t . v(r, t) = (1 − Rh) 2Rµk − sin(2µk R) r `] tfi~ua gsfra {edytujhadta j jryd k=1 +∞ (T0 − T1 )4R2 h X cos(µk R) 2 2 sin(µk r) u(r, t) = T1 + e−a µk t . (1 − Rh) 2Rµk − sin(2µk R) r ¬thr j g ie~ghg {iytujr r = R r t = t m i {ihgzrta zrthiji eay yha i{edydhdsra d~{deuge su {ijdesitr uerfu j ~i~ds jed~dsr X t jd k=1 0 0 +∞ (T0 − T1 )4R2 h X sin(µk r) cos(µk R) 2 2 u(R, t0 ) = T1 + e−a µk t0 . (1 − Rh) 2Rµk − sin(2µk R) r ³´µ¶· ¸¹¸º» `XVX vuzuhsua d~{deugeu iysieiysioi tde|sa yhrsi l t d{hirihr} eijussi {ijdesit eujsu T X vuzrsua t ~i~dsu jed~dsr t = 0 iyrs r fisij tde|sa {iyyde|rjudta {er d~{deuged T mu yegoi ¤ {er d~} {deuged T X vur d~{deugeg tde|saX c k=1 0 0 1 `XcX vur d~{deugeg iysieiysioi tde|sa yhrsi l t d{hirihreijus} si {ijdesitm dthr doi suzuhsua d~{deugeu i{rtjudta gsfrd fisd x = 0 d{hirihregdtam u T x r suzrsua t ~i~dsu jed~dsr t=0 l fisd {iyyde|rjudta {er d~{deugedm eujsi T X x = l `X`X ^dtfisdzsua {hutrsu ih¢rsi l suoedu yi d~{deuge T X vuzr} sua t ~i~dsu jed~dsr t = 0 oeus {hutrs x = 0 {iyyde|rjudta {er d~{deuged T m u yegoua x = l ihu|yudta |ryfitm d~{deugeu fi} iei T (T < T ) X nd{hiip~ds su i oeusr {hutrs {eirtiyr {i ufisg visuX vur d~{deugeg {hutrsX `XX ^dtfisdzsua {hutrsu ih¢rsi l suoedu yi d~{deuge T X vuzr} sua t ~i~dsu jed~dsr t = 0 oeus {hutrs x = 0 {iyyde|rjudta {er d~{deuged T cos(mt) u yegoua x = l ¤ {er d~{deuged T X vur d~{d} eugeg {hutrsX `XUX ^dtfisdzsua {hutrsu ih¢rsi l suoedu yi d~{deuge T X vuzr} sua t ~i~dsu jed~dsr t = 0 su oeus {hutrs x = 0 {iyudta d{hiji {iif {hisitr q m u yegoua oeus x = l {iyyde|rjudta {er d~{deuged eujsi T X vur d~{deugeg {hutrsX `X¥XShrss rhrsyerzdtfr tde|ds euyrgtu R suoed yi d~{deuge X vuzrsua t ~i~dsu jed~dsr t = 0 tde|ds {i~d¢udta j |ryfitm T d~{deugeu fiiei T (T < T ) X nd{hiip~ds tde|sa t |ryfit {ei} rtiyr {i ufisg visuX nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufijiX vur d~{deugeg tde|saX `XWXørhrsyerzdtfr {eijiysrf euyrgtu R suoedjudta jthdytjrd {eii|} ydsra {itiassioi ifuX põd~sua {hisit eut{edydhdsra d{hiji sdeorr ¤ Q X îfijua {ijdesit {eijiysrfu {iyyde|rjudta {er d~} {deuged eujsi T X vuzuhsua d~{deugeu {eijiysrfu uf|d eujsu T X vur d~{deugeg {eijiysrfum dthr j fu|yi~ doi {i{dedzsi~ tdzdsrr d~{deugeu tzrudta iyrsufijiX `XZX bduhhrzdtfr uerf euyrgtu R m suoed yi d~{deuge T X vur d~{deugeg uerfum dthr suzrsua t ~i~dsu jed~dsr t = 0 d~{deugeu su doi {ijdesitr {iyyde|rjudta eujsi T X `XYX vur d~{deugeg tderzdtfi ipihizfr R ≤ r ≤ R m dthr dd su} zuhsua d~{deugeu eujsi T m oeus r = R {iyyde|rjudta {er d~{d} euged T m u oeus r = R ¤ {er d~{deuged T X `XV\X Shrss ~duhhrzdtfr tde|ds t {ea~igoihs~ tdzdsrd~ (0 ≤ m 0 ≤ y ≤ B) suoed yi d~{deuge T X vuzrsua t ~i~dsu jed} ≤ x ≤ A ~dsr t = 0 oeusr tde|sa x = 0 m x = A r y = 0 d{hirihregtam u oeus y = B {iyyde|rjudta {er d~{deuged T X nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufijiX vur d~{deugeg tde|saX 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 2 1 0 1 ` 2 ¼½¾¹½¿ `XVX u(x, t) = T − T 1 0 x + T0 + l −a2 π2 k2 t +∞ 2(T1 − T0 ) X (−1)k e l2 πkx + sin . π k l k=1 `XcX u(x, t) = T 2 2t +∞ −a (π+2πk) 4l2 8T0 X e (π + 2πk)x cos . 0− π2 (1 + 2k)2 2l k=0 (T0 − T1 )h x− u(x, t) = T0 − 1 + hl +∞ X cos(µk l)(h2 + µ2k ) −a2 µ2k t −2(T0 − T1 ) e sin(µk x) µk (h2 l + µ2k l + h) k=1 h µk ctg(µl) = µ T0 (1 − cos(mt)) x + T0 cos(mt)+ u(x, t) = l +∞ 2 2 π kx 2T0 X a µ m2 2 2 k + sin(mt) − cos(mt) sin e−a µk t + 2 2 2 2 π k(m + (a µk ) ) m l k=1 πk µk = k = 1, 2 . . . l 2 2t +∞ −a (π+2πk) X 2 4l e (π + 2πk)x 8q cos u(x, t) = T0 + q(l − x) − 2 . lπ (1 + 2k)2 2l k=0 +∞ γ ρ X a2 γ2 t J1 (γk ) k − R2k u(ρ, t) = T1 + 2(T0 − T1 ) J0 e 2 2 γk (J0 (γk ) + J1 (γk )) R k=1 J1 (γ) hR γk = J0 (γ) γ t a2 γ2 k +∞ 2QR2 X (1 − e− R2 ) γk ρ J0 u(ρ, t) = T0 + a2 γ3k J1 (γk ) R k=1 γk J0 (γ) = 0 −a2 π2 k2 t +∞ 2R(T0 − T1 ) X (−1)k e R2 πkr u(r, t) = T1 + . sin πr k R k=1 T1 − T0 R 2 T0 − T1 R 1 u(r, t) = r+ + R2 − R1 R2 − R1 2 22 +∞ −a π k t 2(T1 − T0 ) X R2 (−1)k 2((−1)k − 1) (R2 −R1 )2 + − e + (R2 − R1 )r πk π2 k 2 `X`X m oyd ¤ i {ihi|rdhsd fiesr geujsdsra `XX oyd m X m X `XUX `X¥X oyd ¤ i {ihi|rdhsd fiesr geujsdsra `XWX oyd ¤ i {ihi|rdhsd fiesr geujsdsra `XZX `XYX k=1 X m X m X −a2 π2 k2 t π k(r − R ) 2(R2 − R1 )(1 − (−1)k ) 1 sin + 1 − e (R2 −R1 )2 a2 π3 k 3 R2 − R1 u(x, y, t) = u(y, t) = −a2 (π+2πk)2 t +∞ (π + 2πk)y 2B(T0 − T1 ) X (−1)k e 4B2 cos . = T1 + π 1 + 2k 2B `XV\X k=0 ù ' 1 E(5 E1 EC ?A&15C5>C er eddsrr ~sior rrzdtfr uyuzm tjaussm su{er~dem t rgzd} srd~ fihdpusrm tegsm ~d~peusm tde|sdm hdfei~uosrs fihdpusrm fihdpusr ouum suiya¢doita j ioeusrzdssi~ ipõd~dm jisrfu geujsd} sra jryu ∂ 2u = a2 ∆u − qu + f. 2 ∂t u = u(M, t) M ∈ Ω t > 0 S a2 q gsfra [ r ] i{rtjud fihdpusra j iphutr t oeusrd X Tirrds m i{edydhata tjitju~r tedy j Ω fiiei {eirtiyr {eidtt fihdpusrX gsfra f (M, t) jeu|ud rsds} trjsit jsdsdoi jiydtjraX {deuie ∆ ¤ i i{deuie lu{hutuX _ry i{deuieu ujrtr i jpeussi trtd~ fiieyrsum fiieua i{edydhadta ie~i iphutr Ω X pzsi tij~dtsi t geujsdsrd~ uyuta yju suzuh} s gthijra du(M, 0) = ψ(M ), dt u(M, 0) = ϕ(M ), i{rtju¢r suzuhsid {ihi|dsrd r suzuhsg tfieit izdf dhum ti} jdeu¢doi fihdpusraX vu oeusrd S iphutr Ω uyuta feudjd gthi} jram tiijdtjg¢rd eutt~uerjud~i rrzdtfi uyuzdX i gthijra Srerhdm vd~usu rhr `}oi eiyum tiijdtjdssi u S = ν(M 0 , t), ∂u ∂~n = Ψ(M 0 , t), S ∂u + σu ∂~n S = χ(M 0 , t). xydt M ¤ izfu oeusr S m ∂u ¤ {eirjiysua {i su{eujhdsr jsd} ∂~n sd sie~uhr f oeusrd S X vu eus zuta oeusr ~iog p uyus oeusrzsd gthijra eusioi eiyuX ¬thr j ~iydhr tzrudtam zi iphut Ω sdioeusrzdssum i {edy{ihuoum zi gsfra u(M, t) ioeusrzdsu su pdt} fisdzsitrX _jiy jihsijioi geujsdsra i{rtus j ¡X _ {itiprr V¡ {ihgzdsi geujsdsrd {i{dedzs fihdpusr tegs r {i} yeipsi i{rtusu {itusijfu feudj gthijrX Reujsdsrd fihdpusr tegs 0 U ¤ i iysi~desid jihsijid geujsdsrd jryu [XV] iyipsd geujsdsra {iajhata uf|dm su{er~dem {er rtthdyijusrr yhrss hrsr ¤ hdferzdtfr d{d t eut{edydhdss~r {ueu~deu~rX i yjg{eijiysd hrsrr tjarm fiuftruhsd fupdhsd hrsrr r X {XX vu{ea|dsrd u(x, t) r if i(x, t) j ufr hrsra tjaus yrdedsruh} s~r geujsdsra~rm fiied sujuta dhdoeus~r 2 ∂ 2u 2∂ u =a + f. ∂t2 ∂x2  ∂u ∂i   + L + Ri = 0  ∂x ∂t  ∂u ∂i   +C + Gu = 0. ∂x ∂t xydt R ¤ ti{eirjhdsrdm L ¤ rsygfrjsitm C ¤ d~fitm G ¤ {eijiyr} ~it ~d|yg {eijiyu~rm euttzrussd su dyrsrg yhrs {eijiyuX eiyrdedsregd~ {dejid geujsdsrd trtd~ {i {ded~dssi x r {iytujr~ j~dti ∂i jeu|dsrd r jieioi geujsdsram {ihgzrta iysi geujsdsrd ∂x ∂ 2u ∂u ∂ 2u LC 2 + (RC + LG) + RGu − 2 = 0, ∂t ∂t ∂x fiieid uf|d sujudta dhdoeus~X ¬thr ti{eirjhdsrd {eijiyij ~uhi r isr ieii rihreijusm i j ~iydhr ~i|si tzrum zi R = G = 0 X nufua hrsra sujudta hrsrd pd {ideX ndhdoeusid geujsdsrd {er i~ tusijrta geujsdsrd~ fihdpusr tegs [XV] [a = 1 m f = 0]X ¬thr {ueu~deLChrsrr tjaus tiisidsrd~ RC = LG m i ufua hr} sra sujudta hrsrd pd rtfu|dsraX ddsrd dhdoeusioi geujsdsra j i~ thgzud suiya j jryd u(x, t) = e v(x, t) m oyd v(x, t) ¤ jt{i~ioudh} sua gsfraX ¬thr gsfr u(x, t) m {edytujhdssg j gfuussi~ jrydm {iytujr j dhdoeusid geujsdsrdm i sdegysi {ifuum zi gsfra uf|d pgyd gyijhdjiea geujsdsr fihdpusr tegs [XV] {er v(x, t) X 1 m a = f =0 LC iyeipsi diera dhdoeus geujsdsr rhi|dsum su{er~dem j W¡X ysr~ r ~diyij eddsra feudji uyuzr yha jihsijioi geujsdsra ajhadta ~diy gedX er~dsadta is uf|dm fuf r yha geujsdsra d{} hi{eijiysitrX _ {itiprr V¡ euipeus {er~de {er~dsdsra ~diyu g} ed yha eddsra iysi~desioi jihsijioi geujsdsra t eus~r suzuhs~r r feudj~r gthijra~rX utt~ier~ d¢d sdtfihfi {er~deij {er~dsdsra ¥ 2 −R L 2 ioi ~diyu yha eddsra feudj uyuzm tjauss t rgzdsrd~ fihdpu} sr tegs r ~d~peusX Q®P® vur {i{dedzsd fihdpusra suasgi tegs t u} fed{hdss~r fisu~r x = 0 r x = l m dthr j suzuhs ~i~ds jed~dsr tegsg j yjg ~dtu iasghr r {eryuhr ie~gm i{rtjud~g gsfrd 2πx [ertgsif]X vuzuhsua tfieit izdf tegs eujsu sghX U sin l pisuzr~ gsfrm fiieua u(x, t) i{rtjud {i{dedzsd fihdpusra tegsX u gsfra gyijhdjiead jihsiji~g geujsdsr ∂ u ∂ u {er 0 < x < l, t > 0, =a ∂t ∂x suzuhs~ gthijra~ u(x, 0) = U sin 2πx m du(x, 0) = 0 m feudj~ gthijra~ u(0, t) = 0, u(l, t) =l 0. dt Teudjd gthijra j i uyuzd ¤ iysieiysdX gsfr u(x, t) pgyd~ rtfu j jryd eayu gedm j{ihsaa euhi|dsrd {i tiptjdss~ gsfr} a~ i{deuieu L (u) = − ∂ u X ∂x V] dr~ uyuzg £ge~u}lrgjrhha 0 2 2 2 2 2 0 2 x 2 ( −y 00 (x) = λy(x), 0 < x < l, y(0) = 0, y(l) = 0. u uyuzu eutt~uerjuhut j euydhd c [{er~de cXV]X iptjdssd zrthu i{deuieu L λ = πk m k = 1, 2, . . . m tiptjdssd gsfrr y (x) = l X sr ipeug {ihsg ieioisuhsg trtd~g j πkx , k = 1, 2, . . . = sin l {eiteustjd X er i~ ky (x)k = l X L [0, l; 1] 2 ged c] edytujr~ gsfr u(x, t) j jryd eayu 2 x k k 2 k u(x, t) = +∞ X 2 ck (t)yk (x). k=1 Teudjd gthijram {er i~m j{ihsataX gsfr ϕ(x) = U sin 2πx m l i{rtju¢g suzuhsid {ihi|dsrd izdf tegsm euhi|r~ j eay {i i |d trtd~d gsfr 0 ϕ(x) = +∞ X W k=1 ϕk yk (x). vuyd~ suzdsra firrdsij ged ϕ sd {eirjdydsra (ϕ, yk ) = Zl k = (ϕ, yk ) ||yk ||2 X _zrthr~ tfuhae} πkx 2πx sin dx. l l U0 sin 0 itfihfg eroisi~derzdtfrd gsfrr sin πkx r sin πmx ieioisuhs l l {er k 6= 2 X su {ei~d|gfd [0, l] m i tfuhaesd {eirjdydsra ( ϕ , y ) = 0 hrzs~ i sgha pgyd ihfi (ϕ, y ) k 2 (ϕ, y2 ) = Zl U0 sin2 l 2πx dx = U0 . l 2 nioyu ϕ = U m u {er k 6= 2 ϕ = 0 X Sha i{edydhdsra suzdsr firrdsij c (t) {iytujr~ eay g} ed gsfr u(x, t) r ϕ(x) j jihsijid geujsdsrd r j suzuhsd gthijra 0 2 0 k k +∞ X c00k (t)yk (x) 2 =a ck (0)yk (x) = ck (t)yk00 (x), k=0 k=1 +∞ X +∞ X +∞ X ϕk yk (x), +∞ X c0k (0)yk (x) = 0. t{ihga eujdstji y (x) = −λ y (x) m u uf|d tjitji hrsdsi sdu} jrtr~itr gsfr y (x) m yha firrdsij c (t) {ihgzr~ thdyg¢rd uyuzr Tir k=1 k=0 00 k k=1 k k k k ( c00k (t) = −a2 λk ck (t), ck (0) = ϕk , c0k (0) = 0. p¢rd eddsra r geujsdsr {edytujhata j jryd ck (t) = Ak cos aπkt aπkt + Bk sin . l l Ak = ϕ k B k = 0 Rzrjua suzuhsd gthijram {ihgzr~ m X usdd phi gtusijhdsim zi {er k 6= 2 ϕ = 0 r ϕ = U X ii~g c (t) = U cos a2πt l r c (t) = 0 {er k 6= 2 X xsuzr eay ged gsfrr u(x, t) tiyde|r ihfi iysi ihrzsid i sgha thuoud~idX tfi~id eddsrdm i{rtju¢dd {i{dedzsd fihdpusra tegsX ¤ i gsfra jryu 2 k 0 k u(x, t) = U0 cos Z a2πt 2πx sin . l l 2 0 jd u(x, t) = U cos a2πt sin 2πx . l l euyrgtu ufed{hdsu {i {der~degX Q®N® Tegohua ~d~peusu vur {i{dedzsd fihdpusra ~d~peusm dthrRj suzuhs ~i~ds jed} ~dsr dd ifhisrhr i {ihi|dsra eujsijdtra r {eryuhr ie~gm tr~~d} erzsg isitrdhsi dseu ~d~peusm i{rtjud~g ie~ghi ϕ(ρ) = ρ X vuzuhsua tfieit izdf ~d~peus eujsu sghX =h 1− R itfihfg eutt~uerjudta fegohua ~d~peusum i yha dd i{rtusra gyipsi {dedr f {ihaesi trtd~d fiieyrsuX gt gsfra u i{rt} jud {i{dedzsd fihdpusra ~d~peusX nioyu tiohutsi gthijra~ uyuzr u gsfra sd ujrtr i gohiji fiieyrsu ϕ r ajhadta gsfrd fiieyrsu ρ r jed~dsr t u = u(ρ, t) X gsfra u gyijhdjiead jihsiji~g geujsdsr {er 0 < ρ < R, t > 0, ∂ u ∂u 1 ∂ ρ =a 0 2 2 2 2 ∂t2 ρ ∂ρ ∂ρ suzuhs~ gthijra~ u(ρ, 0) = h 1 − ρ m du(ρ, 0) = 0 m R dt m feudj~ gthijra~ u(ρ, t) ioeusrzdsu {er ρ →ajhata 0+0 u(R, t) = 0. Teudjd gthijra eutt~uerjud~i uyuzr iysieiys~rX i} i~g ~diy ged sd{itedytjdssi {er~dsr~ f i uyuzdX V] dr~ uyuzg £ge~u¤lrgjrhha yha tr~~derzsioi hrsdsioi yrdedsruhsioi i{deuieu ^dttdha B m euttt~uerjud~ioi su ~si|d} tjd gsfrm gyijhdjiea¢r iysieiys~ feudj~ gthijra~ edud~i uyuzr 2 2 0  − 1 (ρy 0 (ρ))0 = λy(ρ), 0 < ρ < R, ρ y(ρ) ρ → 0 + 0, y(R) = 0. ioeusrzdsu {er u uyuzu phu eddsu u j euydhd c [{er~de VXU]X iptjdssd zrthu γ m oyd γ ¤ i {ihi|rdhsd eddsra geujsd} , k = 1, 2, . . . λ = sra J (γR) = 0 X iptjdssd gsfrr y (ρ) = J ( ρ) k = 1, 2, . . . X rtd} ~u tiptjdss gsfr {ihsu r ieioisuhsu j {eiteustjd L [0, R; ρ] X Tjuyeu sie~ tiptjdss gsfr i{edydhata {i ie~ghu~ X R ky (ρ)k = J (γ ) 2 c] gsfr euhi|r~ j eay ged {i trtd~d tiptjdss u( ρ , t) gsfr k 2 k k 0 k γk 0 R 2 k 2 2 2 1 k u(ρ, t) = +∞ X Y k=1 ck (t)yk (ρ). Teudjd gthijram {er i~m j{ihsataX Sha sui|ydsra suzdsr fi} rrdsij c (t) eay thdygd {iytujr j jihsijid geujsdsrd r j suzuh} sd gthijraX edyjuerdhsi gsfr ϕ(ρ) = h 1 − ρ r suzuhsioi R gthijra euhi|r~ j eay ged {i i |d trtd~d tiptjdss gsfr k 2 2 ϕ(ρ) = +∞ X ϕk yk (ρ). k=1 Tirrds ged i{edydhr~ {i {eujrhg ϕ = (ϕ, y ) X Tjuyeu sie~ k jit{ihgd~ta rs} tiptjdss gsfr suydsX Sha jzrthdsra (ϕky ,y ) doeuhs~r i|ydtju~r k k k 2 k ZT J0 (x)xdx = T J1 (T ), 0 ZT J0 (x)x3 dx = T 3 J1 (T )T 3 − 2T 2 J2 (T ). dejid i|ydtji {ihgzudtam dthr jit{ihijuta eujdstji~ [cXc]m u} yuj α = T m β = 0 X _ieid i|ydtji thdygd r {dejioim dthr {eirsdoer} eiju iyrs eu {i zuta~X vuyd~ 0 ZR γk ρ2 (ϕ, yk ) = h 1 − 2 J0 ( ρ)ρdρ. R R 0 _{ihsr~ u~dsg {ded~dssi t = γ ρ m ioyu k R Zγk t2 hR2 1 − 2 J0 (t)tdt = (ϕ, yk ) = 2 γk γk 0  hR2  = 2 γk _ riod {ihgzr~ ϕ Zγk 0 k 1 J0 (t)tdt − 2 γk = m Zγk 0 2J2 (γk )hR2  J0 (t)t dt = . γ2k 3 4J2 (γk )h k = 1, 2, ... . γ2k J12 (γk ) U\  ay ged gsfrr u(ρ, t) {iytujr~ j jihsijid geujsdsrd r j su} zuhsd gthijra +∞ X c00k (t)yk (ρ) 2 =a k=1 k=1 +∞ X +∞ X ck (0)yk (ρ) = k=1 +∞ X 1 0 ck (t) (ρyk0 (ρ)) , ρ +∞ X ϕk yk (ρ), k=1 c0k (0)yk (ρ) = 0. k=1 _it{ihgd~ta eujdstju~r − 1 (ρy (ρ)) = λy (ρ) itfihfg euhi|dsrd ρ gsfrr j eay ged dyrstjdssim i yha firrdsij c (t) {ihgzr~ uyuzr Tir 0 k 0 k k ( c00k (t) = −a2 λk ck (t), ck (0) = ϕk , c0k (0) = 0. dr~ yrdedsruhsid geujsdsrdm rt{ihga ~diy ueufdertr} zdtfioi geujsdsra r gzrjua eujdstji λ = γ X xud~ {iytujr~ {ihgzdssid eddsrd j suzuhsd gthijraX _ riod R{ihgzr~ k 2 k aγ aγ 4J (γ )h k k 2 k cos ck (t) = ϕk cos t = 2 2 t . R γk J1 (γk ) R nioyu eay ged rtfi~i gsfrr u(ρ, t) m i{rtju¢d {i{dedzsd fi} hdpusra ~d~peusm r~dd jry +∞ aγ X J2 (γk ) γk k u(ρ, t) = 4h cos t J0 ( ρ). 2 2 γk J1 (γk ) R R k=1 jd u(ρ, t) = 4h X J (γ ) cos aγ t J ( γ ρ). (γ ) R R Q®¯® bd~peusumγ Jr~d¢ua ie~g {ea~igoihsrfu ti tieisu} ~r A r B m ufed{hdsu {i {der~degX vur {i{dedzsd fihdpusra ~d~peu} sm dthr j suzuhs ~i~ds jed~dsr izfu~ ~d~peus {eryuhr tfieit X vuzuhsid ifhisdsrd izdf ~d~peus eujsi sghX V gt gsfra u = u(x, y, t) (0 ≤ x ≤ A, 0 ≤ y ≤ B, t ≥ 0) i{rtjud {i{dedzsd fihdpusra ~d~peusX u gsfra gyijhdjiead jihsiji~g geujsdsr {er 0 < x < A, 0 < y < B, t > 0, ∂ u ∂ u ∂ u =a + +∞ k=1 2 ∂t2 2 2 2 ∂x2 ∂y 2 2 k 2 2 k k 1 k k 0 UV suzuhs~ gthijra~ u(x, y, 0) = 0 m du(x, y, 0) = V m feudj~ gthijra~ u(0, y, t) = 0, u(A, y, t) =dt 0, u(x, 0, t) = 0, u(x, B, t) = 0. tfi~ua gsfra u = u(x, y, t) gyijhdjiead iysieiys~ feudj~ gthijra~X ii~g ~diy ged sd{itedytjdssi {er~dsr~ f i uyuzdX V] gsfr u(x, y, t) euhi|r~ j eay ged {i tiptjdss~ gsfr} a~ hrsdsioi yrdedsruhsioi i{deuieu L (u) = − ∂ u X Sha sui|} ∂x edr~ uyuzg ydsra tiptjdss zrtdh r tiptjdss gsfr i{deuieu £ge~u¤lrgjrhha 2 x 2 ( −y 00 (x) = λy(x), 0 < x < A, y(0) = 0, y(A) = 0. u uyuzu phu eddsu j euydhd c [{er~de cXV]X iptjdssd zrthu m tiijdtjg¢rd r~ tiptjdssd gsfrr πk m λ = k = 1, 2, . . . A X gsfrr ipeug {ihsg ieioisuhsg πkxm y (x) = sin k = 1, 2, . . . . A trtd~g j {eiteustjd X Tjuyeu sie~ tiptjdss gsfr L [0, A; 1] X Am ky k = k = 1, 2, . . . 2 c]_{ihsr~ euhi|dsrd {i suydssi trtd~d tiptjdss gsfr yrdedsruhsioi i{deuieu 2 k k 2 k 2 u(x, y, t) = +∞ X ck (y, t)yk (x). Teudjd gthijra {i {ded~dssi x pgyg j{ihsataX ysu r gsfr r suzuhs gthijr ihrzsu i sghaX uhi|r~ dd j eay ged {i i |s trtd~d tiptjdss gsfr k=1 V = +∞ X vk yk (x). k=1 Tirrds ged jzrthr~ {i {eujrhg v g|d suydsX {edydhr~ d{de ky k k k = 2 (V, yk ) = ZA πkx −V A πkx V sin dx = cos A πk A 0 Uc A 0 (V, yk ) kyk k2 X Tjuyeu sie~ V A(1 − (−1)k ) = . πk nioyu v = 2V (1 − (−1) ) m k = 1, 2, ... . πk jtd eay ged j jihsijid geujsdsrdm suzuhsd r iytujr~ d{de feudjd gthijra k k +∞ 2 X ∂ ck (y, t) ∂t2 k=1 +∞ X yk (x) = a2 +∞ X k=0 2 ∂ c (y, t) k ck (y, t)yk00 (x) + yk (x) , ∂y 2 +∞ X ck (y, 0)yk (x) = 0, k=1 ck (y, 0)yk (x) = k=1 +∞ X ck (0, t)yk (x) = 0, +∞ X vk yk (x), k=0 +∞ X ck (B, t)yk (x) = 0. itfihfg t{eujdyhrji eujdstji y (x) = −λ y (x) m i k=1 k=1 00 k +∞ 2 X ∂ ck (y, t) ∂t2 k k +∞ X ∂ 2 ck (y, t) −λk ck (y, t) + yk (x). yk (x) = a ∂y 2 2 uhi|dsrd gsfrr j eay ged dyrstjdssim {ii~g fu|yua gsfra gyijhdjiead geujsdsr c (y, t) k=1 k=0 k ∂ 2 ck ∂ 2 ck 2 , = a −λk ck + ∂t2 ∂y 2 suzuhs~ gthijra~ c (y, 0) = 0, dc (y, 0) = v m feudj~ gthijra~ c (0, t) = 0, c (B, t)dt= 0 X ihgzdssd yha gsfr c (y, t) suzuhsi}feudjd uyuzr tsiju ~i|} si edr ~diyi~ gedm rt{ihga euhi|dsrd rtfi~ gsfr j eay {i trtd~d tiptjdss gsfr hrsdsioi yrdedsruhsioi i{deu} ieu L (c ) = − ∂ c X Sha ioi suyi edr tiijdtjg¢g uyuzg £ge~u¤lrgjrhha∂y k k k k k k 2 y k k 2 ( −z 00 (y) = µz(y), 0 < y < B, z(0) = 0, z(B) = 0. iyipsua uyuzu g|d phu eddsuX iptjdssd zrthu µ = πm m B m tiijdtjg¢rd r~ tiptjdssd gsfrr πmy m m = 1, 2, . . . z (x) = sin B X gsfrr ipeug {ihsg ieioisuhsg trtd~g j {ei} m = 1, 2, . . . teustjd L [0, B; 1] X Tjuyeu sie~ tiptjdss gsfr kz k = B m 2 X m = 1, 2, . . . U` 2 m m 2 m 2 Tu|yua gsfra c (y, t) {edytujhadta j jryd eayu k ck (y, t) = +∞ X αkm (t)zm (y). er i~ gsfra v r suzuhsioi gthijra uf |d euhuoudta j eay {i i |d trtd~d gsfr m=1 k vk (y, t) = +∞ X wkm (t)zm (y). m=1 Tirrds w = (v , z ) jzrthata {i usuhiorr t firr} kz k dsu~r v X ii~g w = 2v (1 − (−1) ) = 4V (1 − (−1) )(1 − (−1) ) m πm π km k = 1, ihgzdssd 2, ... , m = 1, 2, ... . eay {iytujhad~ j yrdedsruhsid geujsdsrd r su} zuhsd gthijraX ithd {iytusijfr r rt{ihijusra eujdstj −z (y) = m u uf|d tjitju dyrstjdssitr euhi|dsra gsfrr j eay = µ z (y) gedm yha firrdsij α (t) {ihgzuta thdyg¢rd uyuzr Tir k km m m 2 k k m km k m 2 00 m m m km ( α00km (t) = −a2 (λk + µm )αkm (t), αkm (0) = 0, α0km (0) = wkm . t{ihga ueufdertrzdtfid geujsdsrd sdegysi {ihgzr ip¢dd edd} srd yrdedsruhsioi geujsdsra p p αkm (t) = C1km cos(a λk + µm t) + C2km sin(a λk + µm t). ithd {iytusijfr j suzuhsd gthijram {ihgzr~ xsuzr C1km = 0, wkm C2km = √ . a λk + µm p wkm αkm (t) = √ sin(a λk + µm t). a λk + µm ck (y, t) _ riod fu|yua gsfra ck (y, t) = +∞ X euhuoudta j eay p w πmy √ km . sin(a λk + µm t) sin B a λ + µ k m m=1 _it{ihgd~ta r~ euhi|dsrd~ r {ihgzr~ eay ged rtfi~i gsfrr X u(x, y, t) U jd u(x, y, t) = oyd w +∞ +∞ X X p w πkx πmy √ km sin(a λk + µm t) sin sin . B A a λ + µ k m k=1 m=1 m 4V (1 − (−1)k )(1 − (−1)m ) k = 1, 2, ... , m = 1, 2, ... , km = π2 km 2 πm 2 πk k = 1, 2, . . . µm = m = 1, 2, . . . λk = A B m m m X ³´µ¶· ¸¹¸º» XVX vur {i{dedzsd fihdpusra suasgi tegsm fisd x = 0 fiiei ufed{hdsm u fisd x = l ~i|d tjipiysi pd edsra {ded~d¢uta jyih jderfuhsi hrsrr [ ∂u(l, t) = 0]X _ suzuhs ~i~ds jed~dsr ie~u ∂x πx m suzuhsua tfieit i} tegs i{rtjudta gsfrd u(x, 0) = sin zdf tegs eujsu sghX l XcXvur {i{dedzsd fihdpusra suasgi tegsm fisd x = 0 fiiei ufed{hdsm u fisd x = l ~i|d tjipiysi pd edsra {ded~d¢uta jyih jderfuhsi hrsrr [ ∂u(l, t) = 0]X _ suzuhs ~i~ds jed~dsr {i tegsd ∂x gyuerhr r jjdhr r {ihi|dsra eujsijdtraX vuzuhsua tfieit jtd i} zdf tegs eujsu V X X`X vur {i{dedzsd fihdpusra suasgi tegsm jjussd jsdsd trhi t {hisit f (x, t) = F X _ izfu x = 0 r x = l tegsu ufed{hdsuX _ suzuhs ~i~ds jed~dsr isu suiyrta j {ifidX XX vur {i{dedzsd fihdpusra suasgi tegsm fis x = 0 r x = l fiiei ufed{hdsX _ suzuhs ~i~ds jed~dsr {i tegsd gyuerhr r jjdhr r {ihi|dsra eujsijdtraX vuzuhsua tfieit izdf tegs i{r} tjudta gsfrd ∂u(x, 0) = V (1 − x ) X l XUX vur {i{dedzsd ∂tfihdpusra suasgi tegsm fisd x = 0 fiiei ufed{hdsm u fisd x = l tijdeud fihdpusram i{rtjud~d gsfrd X _ suzuhs ~i~ds jed~dsr tegsu suiyrta j {ifidX [ er U sin( ω t) {dediyd f uyuzd t iysieiys~r feudj~r gthijra~r gsfr w(x, t) rtfu j jryd w(x, t) = sin(ωt)Φ(x)]X X¥ vur su{ea|dsrd j fupdhd yhrs l m dthr j suzuhs ~i~ds jed~dsr fisd x = 0 {iyfhzrhr f rtizsrfg {ded~dssi XyXtX E sin(ωt) m su fisd fupdh eui~fsg [if i(l, t) = 0]X edy{ihuoudtam zi {idea~r j x = l fupdhd ~i|si {edsdpedz [R = G = 0]X _ ~i~ds jfhzdsra su{ea|dsrd r if j fupdhd eujs sghX [ er {dediyd f uyuzd t iysieiys~r feudj~r gthijra~r gsfr w(x, t) rtfu j jryd w(x, t) = sin(ωt)Φ(x)]X XW vur su{ea|dsrd j fupdhd yhrs l m dthr j suzuhs ~i~ds jed} UU 0 ~dsr fisd x = 0 {iyfhzrhr f rtizsrfg {ded~dssi XyXtX E sin(ωt) m su fisd x = l fieifid u~fusrd [su{ea|dsrd u(l, t) = 0]X edy{ihu} oudtam zi {idea~r j fupdhd ~i|si {edsdpedz [R = G = 0]X _ ~i~ds jfhzdsra su{ea|dsrd r if j fupdhd eujs sghX [ er {dediyd f u} yuzd t iysieiys~r feudj~r gthijra~r gsfr w(x, t) rtfu j jryd ]X w(x, t) = sin( ω t)Φ(x) XZX Tegohua ~d~peusu euyrgtu R ufed{hdsu {i {der~degX vur {i{d} edzsd fihdpusra ~d~peusm dthr j suzuhs ~i~ds jed~dsr dd jjdhr r {ihi|dsra eujsijdtra gyuerj {i sdX vuzuhsua tfieit jtd izdf ~d~peus eujsu V X XYX Tegohua ~d~peusu euyrgtu R ufed{hdsu {i {der~degX vur {i} {dedzsd fihdpusra ~d~peusm jjussd jsdsd trhi t {hisit X _ suzuhs ~i~ds jed~dsr isu suiyrta j {ifidX fXV\X (ρ, t)bd~peusum =F r~d¢ua ie~g {ea~igoihsrfu ti tieisu~r A r B m ufed{hdsu {i {der~degX vur {i{dedzsd fihdpusra ~d~peusm dthr j suzuhs ~i~ds jed~dsr ~d~peusd {eryuhr ie~gm i{rtjud~g gsf} rd u(x, y, 0) = x(A − x)y(B − y) X vuzuhsua tfieit izdf ~d~peus eujsu sghX XVVX bd~peusum r~d¢ua ie~g {ea~igoihsrfu ti tieisu~r A r B m u} fed{hdsu {i {der~degX vur {i{dedzsd fihdpusra ~d~peusm jjus} sd jsdsd trhi t {hisit f (x, y, t) = F X _ suzuhs ~i~ds jed} ~dsr isu suiyrta j {ifidX XVcX vur {i{dedzsd fihdpusra ~d~peusm r~d¢d ie~g {ea~igoih} srfum jjussd jsdsd trhi t {hisit f (x, y, t) = F X ieis ~d~} peus x = 0 r x = A ufed{hdsm tieis y = 0 r y = B ~iog tjipiysi pd edsra {ded~d¢uta jyih jderfuhsi hrsrrX _ suzuhs ~i~ds jed~dsr ~d~peusu suiyrta j {ifidX ¼½¾¹½¿ XVX u(x, t) = 8 X (−1) cos(aµ t) sin(µ x), oyd µ = + πk . +∞ XcX X`X XX XUX k+1 k k π2 (−1 + 2k)(3 + 2k) k=0 +∞ 8V l X cos(aµk t) u(x, t) = µk = sin(µk x), aπ2 (1 + 2k)2 k=0 +∞ 2 X (1 − cos( aπlkt )) πkx 2F l sin u(x, t) = 2 3 aπ k3 l k=1 +∞ 2V l X sin( aπlkt ) πkx u(x, t) = sin aπ2 k2 l k=1 ωx sin( a ) u(x, t) = U0 sin(ωt) + sin( ωal ) oyd X X U¥ k π 2 π 2 + πk . l l oyd µ +∞ 2U0 ωa X (−1)k sin(µk at) sin(µk x) + , l µ2k a2 − ω2 k k=1 X¥X u(x, t) = E sin(ωt) cos( = πk l ω(l−x) a ) − cos( ωal ) +∞ oyd µ 2E ωa X sin(µk at) sin(µk x) − , l µ2k a2 − ω2 k k=0 XWX u(x, t) = E sin(ωt) sin( = π 2 + πk l ω(l−x) a ) − sin( ωal ) oyd +∞ X X X 2E ωa X sin(µk at) sin(µk x) πk , − µ = k 2 l µk a2 − ω2 l k=0 +∞ 2V R X sin( aγRk t ) γk ρ J0 u(ρ, t) = a γ2k J1 (γk ) R k=1 +∞ 2F R2 X (1 − cos( aγRk t ) γk ρ J0 u(ρ, t) = a2 γ3k J1 (γk ) R k=1 +∞ +∞ 16A2 B 2 X X (1 − (−1)k )(1 − (−1)m ) × u(x, y, t) = 3 m3 π6 k q k=1 m=1 πkx πmy πk 2 πm 2 sin . × cos a t sin + A B B A +∞ +∞ F AB X X (1 − (−1)k )(1 − (−1)m ) u(x, y, t) = 2 2 × aπ km(λ + µ ) k m k=1 m=1 √ πmy πkx × 1 − cos(a λk + µm )t sin sin , B A 2 πm 2 πk λk = µm = A B +∞ +∞ F AB X X (1 − (−1)k )(1 − (−1)m ) × u(x, y, t) = 2 2 aπ km(λ + µ ) k m k=1 m=0 √ πmy πkx × 1 − cos(a λk + µm )t cos sin , B A 2 πm 2 πk λk = µm = A B XZX X XZX X XV\X XVVX oyd m X m X XVcX oyd UW ú ' ) C2E= = &( &) ¨CA& utt~ier~ pdtfisdzsg tegsgm fiieg j suzuhs ~i~ds jed} ~dsr jjdhr r {ihi|dsra eujsijdtraX gt gsfra ϕ(x) i{rtjud suzuhsid {ihi|dsrdm u gsfra ψ(x) ¤ suzuhsg tfieit izdf teg} sX ¬thr jsdsrd trh itgtjgm ioyu gsfra u(x, t) m i{rtju¢ua tjipiysd {i{dedzsd fihdpusra tegsm gyijhdjiead jihsiji~g geuj} sdsr r suzuhs~ gthijra~ jryu  2 2   ∂ u = a2 ∂ u , −∞ < x < +∞, t > 0, ∂t2 ∂x2   u(x, 0) = ϕ(x), ∂u(x, 0) = ψ(x). ∂t ddsrd i uyuzr u(x, t) ~i|d p suydsi ~diyi~ Suhu~pdeu r {edytujhadta j jryd u(x, t) = 1 ϕ(x − at) + ϕ(x + at) + 2 2a x+at Z ψ(ξ). _ {itiprr V¡ {iyeipsi i{rtus uhoier~ t {i~i¢ fiieioi {ihgzdsu u ie~ghum sujud~ua ie~ghi Suhu~pdeuX ¬thr rgzudta {eidtt fihdpusr {ihgpdtfisdzsi tegs [x ≥ 0]m i eutt~uerjud~ua uyuzu tjiyrta f eddsr uyuzr i fihdpusrr pdt} fisdzsi tegsX er i~ gsfrr ϕ(x) r ψ(x) r suzuhs gthijr {eiyih|u su jt it OX sdzds~ ipeui~m dthr uyusi feudjid gthi} jrd u(x, 0) = 0 r zds~ ipeui~ yha feudjioi gthijra ∂u(x, 0) = 0 X °®P® vur {i{dedzsd fihdpusra pdtfisdzsi∂t tegsm dthr rjdtsim zi tegsu tijdeud fihdpusra u tzd suzuhsioi ifhisdsram i{rtjud~ioi gsfrd ϕ(x) X vuzuhsua tfieit izdf tegs eujsu sg} hX tfi~ua gsfra u(x, t) m ajhadta eddsrd~ thdyg¢d uyuzr x−at  2 2   ∂ u = ∂ u − ∞ < x < +∞, t > 0, ∂t2 ∂x2   u(x, 0) = ϕ(x), ∂u(x, 0) = 0. ∂t iohutsi ie~ghd Suhu~pdeu eddsrd r~dd jry u(x, t) = ϕ(x − at) + ϕ(x + at) . 2 UZ gsfra u(x, t) tfhuyjudta r yjg jihs {dejua 1 ϕ(x −at) eut{ei} teusadta j{euji r sujudta {ea~i jihsim jieua 21 ϕ(x + at) ¤ jhdji 2 r sujudta ipeusi jihsiX gt gsfra ϕ(x) uyusu {i {eujrhg [ert UXV] ϕ(x) = ( −x2 + l2 x ∈ [−l, l], 0 x 6∈ [−l, l]. vu ertX UXc thdju |resi hrsrd ripeu|ds oeurf gsfrr u(x, t) j ~i} ~ds jed~dsr t = 0 m t = l m t = 2a {ihgzds fuf lm 2l X eurf = t = u(x, t) 2 a tg~~u oeurfij gsfr 1 ϕ(x + at) r 2 [ripeu|ds {gsfresi hrsrd]X 1 ϕ(x − at) 0 1 u l 2 3 −l o ëìíî ûîðî l x 2 u −l l u(x,t 0) o u l x l u(x,t 1 ) −l ou l x l u(x,t 2) −l o u l x l u(x,t 3 ) −2l −l o ëìíî ûîï l 2l x jd u(x, t) = ϕ(x − at) + ϕ(x + at) . °®N® utt~ied 2{eidtt {i{dedzs fihdpusr tegs dthr UY j suzuhs ~i~ds jed~dsr {i suasgi tegsd gyuerhr r izfr tegs {ihgzrhr suzuhsg tfieitm i{rtjud~g gsfrd ψ(x) =   v x ∈ [−l, l],  0 x 6∈ [−l, l]. gt gsfra u(x, t) i{rtjud {i{dedzsd fihdpusra tegsX su gyi} jhdjiead jihsiji~g geujsdsr r suzuhs~ gthijra~ jryu  ∂ 2u ∂ 2u  2   =a   ∂x2  ∂t2    u(x, 0) = 0,    − ∞ < x < +∞, t > 0,   v x ∈ [−l, l], ∂u(x, 0) =  0 x 6∈ [−l, l]. ∂t pisuzr~ zded Ψ(x) = 1 Z a pdeum x ψ(t)dt m ioyu tiohutsi ie~ghd Suhu~} 0 _ yussi~ thgzud 1 u(x, t) = (Ψ(x + at) − Ψ(x − at)). 2 1 Ψ(x) = a Zx vdt = x vx =h , a l −l ≤ x ≤ l, Zl vdt = vl = h, a x>l Z−l vdt = − 0 1 Ψ(x) = a 0 1 Ψ(x) = a 0 vl = −h, x < −l, a oyd h = vl X nioyu Ψ(x) {edytujhadta j jryd a  x < −l,  −h, x Ψ(x) = h , −l ≤ x ≤ l,   l h, x > l. iteir~ oeurf gsfrr u(x, t) m ieu|u¢r {eirh tegsm j ¥\ ~i~ds jed~dsr t = 0 m t = l m t = l m t = 2l [ertX UX`]X eurf teirta fuf tg~~u oeurfij yjg2a gsfr2 1 Ψ(x +aat) r − 1 Ψ(x − at) X 0 1 2 3 2 2 u h o ll −l 0 u 0.5 Ψ ( x ) u(x , t 0) x −0.5Ψ ( x ) h 0.5 Ψ ( x + 0.5 l ) u (x , t 1) x l −0.5 Ψ ( x − 0.5 l ) h 0.5 Ψ ( x + l ) u (x , t 2) x l −0.5 Ψ ( x − l ) −l 0 u −l 0 u h u (x , t 3) −2l −l 0 0.5 Ψ ( x + 2 l ) l x 2l −0.5 Ψ ( x −2 l ) ëìíî ûîü iyips {er~de euipeus j {itiprr V¡X jd u(x, t) = 1 (Ψ(x + at) − Ψ(x − at)) X 2 °®¯® vur {i{dedzsd fihdpusra {ihgpdtfisdzsi tegsm fisd x = 0 fiiei sd{iyjr|si ufed{hdsX vuzuhsid ifhisdsrd izdf tegs i{rtjudta gsfrd U x {er x ≥ 0 X vuzuhsua tfieit izdf eujsu sghX pisuzr~ rtfi~g gsfr u (x, t) X su gyijhdjiead geujsdsr ∂ u ∂ u {er x > 0, t > 0, =a 2 0 1 2 1 2 ∂t 2 2 1 2 ∂x suzuhs~ gthijra~ u (x, 0) = U x , ∂u (x, 0) = 0, ∂t feudji~g gthijr u (0, t) = 0 X eiyih|r~ gsfr U x su ierudhsg {ihgit sdzds~ ip} 1 0 1 0 2 2 ¥V 1 eui~ ϕ1 (x) = ( U0 x2 , x ≥ 0, −U0 x2 , x < 0. ^gyd~ edu uyuzg su jtd itr tzruam zi ϕ(x) = ϕ (x) r ψ(x) = 0 X _ i~ thgzudm tiohutsi ie~ghd Suhu~pdeu 1 u(x, t) = 1 (ϕ(x − at) + ϕ(x + at)) . 2 tfi~ua gsfra u (x, t) = u(x, t) {er x ≥ 0 X rftregd~ sdfiieid su} zdsrd x ≥ 0 m ioyu 1 ϕ(x − at) = ( U0 (x − at)2 x − at ≥ 0, −U0 (x − at)2 x − at < 0. Rzrjua im {ihgzr~ eddsrd uyuzrX jd  U0   (x + at)2 + (x − at)2 ,  2 u1 (x, t) =    U0 (x + at)2 − (x − at)2 , 2 0≤t≤ t> x , a x . a ³´µ¶· ¸¹¸º» UXV vur {i{dedzsd fihdpusra pdtfisdzsi tegsm dthr j suzuhs ~i~ds jed~dsr tegsu r~dd ie~gm i{rtjud~g gsfrd U cos x X vu} zuhsua tfieit izdf tegs eujsu 0 X UXcX vur {i{dedzsd fihdpusra pdtfisdzsi tegsm dthr suzuhsua tfi} eit izdf tegs uyudta ie~ghi 0     0, | x| > h, ∂u(x, 0) = −v, −h ≤ x ≤ 0,  ∂t   v, 0 < x ≤ h, u suzuhsid ifhisdsrd izdf eujsi 0 X UX`X vur {i{dedzsd fihdpusra {ihgpdtfisdzsi tegsm dthr j suzuh} s ~i~ds tfieit jtd izdf tegs eujsu 0 m fisd tegs x = 0 ufed{hdsm u suzuhsua ie~u tegs i{rtjudta gsfrd  πx  sin , 0 ≤ x ≤ l, l u(x, 0) =  0, x > l. ¥c UXX ihgpdtfisdzsua tegsu t ufed{hdss~ fisi~ x = 0 j suzuhs ~i~ds r~dd ie~g u(x, 0) = 0 r suzuhsg tfieit ∂u(x, 0) = ∂t ( v, 0, 0 ≤ x ≤ l, x > l. vur ie~g tegs yha ~i~dsij jed~dsr t = l r t = 5l X a tegsm a UXUX vur {i{dedzsd fihdpusra {ihgpdtfisdzsi dthr dd fi} sd x = 0 ufed{hdsm suzuhsua ie~u tegs i{rtjudta gsfrd x m u suzuhsua tfieit jtd izdf tegs eujsu X u(x, 0) = +x UX¥X vur1 {i{dedzsd fihdpusra {ihgpdtfisdzsi tegsm dthr 0j suzuh} s ~i~ds jed~dsr t~d¢dsra izdf tegs eujs 0 m tfieit izdf i{r} tjudta gsfrd ∂u(x, 0) = sin x {er i~ fisd x = 0 tjipiysi pd ∂t edsra {ded~d¢udta jyih jderfuhsi hrsrr ∂u(0, t) = 0 X ∂x ¼½¾¹½¿ UXV u(x, t) = U (cos(x − at) + cos(x + at)) X 2 0 2 UXc u(x, t) = 1 (Ψ(x+at)−Ψ(x−at)) m oyd 2 UX` u(x, t) = ϕ(x − at) + ϕ(x + at) , oyd 2    v|x| , x ∈ [−h, h] a Ψ(x) = vh   , x∈ / [−h, h]. ( a sin πlx x ∈ [−l, l] ϕ(x) = 0, x ∈ / [−l, l]. UX u(x, t) = 1 (Ψ(x + at) − Ψ(x − at)) m oyd Ψ(x) ¤ i gsfra r 2 g{eu|sdsra UXc u(x, l 1 ) = (Ψ(x + l) − Ψ(x − l)) a 2 X 1 5l u(x, ) = (Ψ(x + 5l) − Ψ(x − 5l)) a 2 UXU X 1 x − at x + at u(x, t) = + UX¥ 2 1 + (x − at)2 1 + (x + at)2 ( 1 1 − cos x, u(x, t) = (Ψ(x+at)−Ψ(x−at)) Ψ(x) = 2 cos x − 1, m oyd ý ' ?A&15C5> D (&@(&F& > @?&FFE5& Reujsdsrd jryu ∆u = f, ¥` x≥0 x < 0. oyd ∆ ¤ i i{deuie lu{hutum sujudta geujsdsrd~ guttisuX ¬thr gsfra f ≡ 0 m i geujsdsrd sujudta geujsdsrd~ lu{hutuX nufrd geujsdsra jisrfum su{er~dem {er sui|ydsrr turisues eut{edydhdsr d~{deuge j jdey dhum {er rtthdyijusrr turisue} s hdfei~uosrs {ihd r ji ~sior yegor {erfhuys uyuzuX ¬thr rtfi~ua gsfra u eutt~uerjudta j ioeusrzdssi iphutr Ω t oeusrd S m i yi{ihsrdhsi su oeusrd uyuta feudjd gthijraX i ~iog p gthijra jryu u S ∂u ∂~n = ν(M 0 , t), = Ψ(M 0 , t), ∂u + σu ∂n S = χ(M 0 , t), ajha¢r~rta gthijra~r Srerhdm vd~usu r edr~ feudj~ gthijrd~m tiijdtjdssiX xydt M ¤ izfu oeusr iphutr S m ∂u ¤ {eirjiysua {i su{eujhdsr jsdsd sie~uhr f oeusrd S X vu eus∂~nzuta oeusr S ~iog p uyus oeusrzsd gthijra eusioi eiyuX ysr~ r ~diyij eddsra geujsdsr lu{hutu r guttisu ajhadta ~diy gedX upded~ sdtfihfi {er~deij {er~dsdsra ioi ~diyu yha eddsra uyuz i{edydhdsra turisues eut{edydhdsr d~{deuge j euhrzs dhuX ±®P® vur turisuesid eut{edydhdsrd d~{deuge j yhrs} si~ ~duhhrzdtfi~ tde|sd t {ea~igoihs~ tdzdsrd~ (0 ≤ x ≤ A m {er gthijrrm zi j tde|sd jydhada d{hi t ipõd~si {hi} 0sit ≤ y ≤ eut{edydhdsra B) d{hiji sdeorr Q X eusr tde|sa x = 0 r y = 0 d{hirihreijusm u oeussr x = A r y = B {iyyde|rjuta {er d~{d} euged T X nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufijiX gt u(x, y) ¤ i gsfram fiieua i{rtjud eut{edydhdsrd d~{d} euge j tdzdsrr tde|saX su gyijhdjiead geujsdsr {er 0 < x < A, 0 < y < B, ∂ u ∂ u + = −Q S 0 0 2 2 ∂x2 ∂y 2 feudj~ gthijra~ 0 ∂u(0, y) = 0, ∂x u(A, y) = T0 , ∂u(x, 0) = 0, ∂y u(x, B) = T0 , oyd Q = Q [K ¤ firrds d{hi{eijiysitr]X K gthijra {i ipdr~ {ded~dss~ sdiysieiysdX ii~g tsuzu} Teudjd hum rt{ihga u~dsg u(x, y) = v(x, y) + w(x, y) m tjdyd~ g feudjg uyuzg 0 ¥ f uyuzd t iysieiys~r feudj~r gthijra~r {i {ded~dssi x X Sha i} oi uyuyr~ w(x, y) = αx + β X qrthu α r β {iypded~ ufm zip w(x, y) gyijhdjieahu {i {ded~dssi x d~ |d feudj~ gthijra~m zi r gsfra u(x, y) ( α = 0, ⇔ β = T0 . ( α = 0, α A + β = T0 xsuzr w(x, y) = T X Rzrjuam zi u(x, y) = v(x, y) + T m u{rd~ feudjg uyuzg isitrdhsi gsfrr v(x, y)  {er 0 < x < A, 0 < y < B, ∂ v ∂ v   + = −Q   ∂y   ∂x [¥XV] ∂v(0, y) = 0, v(A, y) = 0, 0 2 0 2 2 0 2  ∂x    ∂v(x, 0)   = 0, ∂y v(x, B) = 0. gsfra v(x, y) gyijhdjiead iysieiys~ feudj~ gthijra~ su jtd oeusrdX ihgzdssg feudjg uyuzg pgyd~ edu ~diyi~ gedm j{ihsaa euhi|dsrd j eay {i trtd~d tiptjdss gsfr i{deuieu L = − d y X dx V] xu{rd~ uyuzg £ge~u}lrgjrhha 2 x 2 ( −y 00 = λy, 0 < x < A, y 0 (0) = 0, y(A) = 0. ddsrd i uyuzr suydsi [g{eu|sdsrd cXc]X iptjdssd zrthu m oyd µ = + πk m k = 0, 1, . . . tiptjdssd gsfrr y (x) = λ = µ A X sr ipeug {ihsg ieioisuhsg trtd~g j m = cos(µ x) k = 0, 1, . . . {eiteustjd L [0, A; 1] X er i~ ky (x)k = A X 2 gedm j{ihsrj euhi} c] gsfr v(x, y) pgyd~ rtfu j jryd eayu |dsrd {i trtd~d gsfr y (x) k 2 k π 2 k k k 2 2 k k v(x, y) = +∞ X ck (y)yk (x). gsfr Q r {euji zutr geujsdsra guttisu uf|d euhi|r~ j eay ged {i i |d trtd~d gsfr k=0 0 Q0 = +∞ X qk yk (x). ¥U k=0 Tirrds ged suyd~ {i {eujrhg q zi ky (x)k = A m {ihgzr~ k = 2 k (Q0 , yk (x)) kyk (x)k2 X Rzrjuam 2 2 qk = A ZA 2 2AQ0 (π + 2πk)x (π + 2πk)x dx = sin Q0 cos 2A A π + 2πk 2A 0 A = 0 4Q0 (−1)k = . π(1 + 2k) Sha i{edydhdsra suzdsr firrdsij c (y) {iytujr~ eay g} ed gsfr v(x, y) r Q j geujsdsrd guttisu r j feudjd gthijra {i {ded~dssi y X Rzrjua eujdstju −y (x) = λy (x) m yha firrdsij {ihgzr~ feudjd uyuzr c (y) k 0 k 00 k k ( −λk ck + c00k = −qk 0 < y < B, c0k (0) = 0, ck (B) = 0. p¢dd eddsrd sdiysieiysioi yrdedsruhsioi geujsdsra r~dd jry ck (y) = ck,0 (y) + c̃k (y), oyd c (y) ¤ i ip¢dd eddsrd iysieiysioi geujsdsra −λ c + c = 0 m ¤ zutsid eddsrd sdiysieiysioi geujsdsraX c̃ (y)vuyd~ tsuzuhu X Rzrjuam zi λ = µ m u{rd~ yha iy} c (y) sieiysioi yrdedsruhsioi geujsdsra ueufdertrzdtfid geujsdsrd X ¬oi fiesr p = µ r p = −µ X xsuzr −µ + p = 0 k,0 k k k k,0 2 k 2 2 k k 1 k 2 00 k k ck,0 (y) = C1 ch(µk y) + C2 sh(µk y). bdiyi~ sdi{edydhdss firrdsij `¡ suiyr~ c̃ (y) = q . p} λ ¢dd eddsrd sdiysieiysioi yrdedsruhsioi geujsdsra {edytujhad} ta j jryd k k k qk . λk ck (y) ck (y) = C1 ch(µk y) + C2 sh(µk y) + vuyd~ suzdsra C r C {er fiie gsfrr dj~ gthijra~ 1 ( µk C2 = 0 2 C1 ch(µk B) + C2 sh(µk B) + qk =0 λk ¥¥ ⇔ gyijhdjiea feu}  C2 = 0 C1 = −qk λk ch(µk B) Rzrjuam zi 16Q0 A2 (−1)k qk = 3 λk π (1 + 2k)3 m {ihgzr~ 16Q0 A2 (−1)k ck (y) = 3 π (1 + 2k)3 nioyu +∞ 16Q0 A2 X (−1)k v(x, y) = π3 (1 + 2k)3 ch(µk y) . 1− ch(µk B) ch(µk y) 1− cos(µk x). ch(µk B) `] tfi~ua gsfra {edytujhadta j jryd u(x, y) = T + v(x, y) X Teudjg uyuzg [¥XV] ~i|si edr yegor~ t{itipi~X gsfra v(x, y) gyijhdjiead iysieiys~ feudj~ gthijra~ su jtd oeusrdX ii~g ~i|si rtfu j jryd eayu gedm j{ihsrj euhi|dsrd {i tip} v(x, y) tjdss~ gsfra~ i{deuieu lu{hutuX V] utt~ier~ uyuzg su tiptjdssd suzdsra yha i{deuieu lu{hu} tu  {er 0 < x < A, 0 < y < B, ∂ U ∂ U   + = λU − k=0 0 2 2 ∂x2 ∂y 2 ∂U (0, y) = 0, U (A, y) = 0,  ∂x    ∂U (x, 0)   = 0, U (x, B) = 0. ∂y    ddsrd i uyuzr pgyd~ rtfu j jryd {eirjdydsra yjg gsfr m tzruam zi X(x) gyijhdjiead feudj~ gthijra~ U (x, y) = X(x)Y (y) uyuzr {i {ded~dssi x m u Y (y) ¤ feudj~ gthijra~ {i {ded~dssi y X 0 (0) = 0, X(A) = 0, Y 0 (0) = 0, Y (B) = 0. nioyu gsfra U (x, y) pgyd gyijhdjiea uyuss~ feudj~ gthijra~X iytujr~ gsfr U (x, y) = X(x)Y (y) j yrdedsruhsid geuj} sdsrd −(X 00 (x)Y (y) + X(x)Y 00 (y)) = λX(x)Y (y). uydhr~ ipd zutr ioi eujdstju su X(x)Y (y) X 00 (x) Y 00 (y) − + X(x) Y (y) = λ. ihgzdssid eujdstji j{ihsadta ihfi j i~ thgzudm dthr r − Y (y) = λ , X (x) − =λ X(x) Y (y) oyd λ r λ ¤ fistusX er i~ λ = λ + λ X ¥W 00 00 1 1 2 2 1 2 hdyijudhsim gsfrr X(x) r Y (y) ajhata eddsra~r yjg uyuz £ge~u¤lrgjrhha r ( −X 00 = λ1 X, 0 < x < A, X 0 (0) = 0, X(A) = 0 ( −Y 00 = λ2 Y, 0 < y < B, Y 0 (0) = 0, Y (B) = 0. r uyuzr g|d edds [g{eu|sdsrd cXc]X iptjdssd zrthu r tiijd} tjg¢rd r~ tiptjdssd gsfrr yha {deji r jiei uyuz r~d jry oyd µ = + πk , X (x) = cos(µ x), k = 0, 1, . . . ; λ =µ , 2 k 1,k π 2 k k A π + πm 2 λ2,m = νm , νm = 2 , 2B λkm = µ2k + ν2m k = 0, 1, . . . ; m = 0, 1, . . . k oyd Y (y) = cos(ν y), m = 0, 1, . . . . nioyu zrthu r gsfrr U (x, y) = cos(µ x) cos(ν y) pgyg tiptjdss~r zrthu~r r tiptjdss~r gsfra~r i{deuieu lu{hutu yha edud~i uyuzr j {ea~igoihsrfdX er i~ kUkm (x, y)k2 = ZA ZB m m km 2 Ukm (x, y)dxdy = ZA cos2 (µk x)dx k ZB cos2 (νm y)dy = m AB . 4 c] gsfr v(x, y) {edytujr~ j jryd eayu ged {i suydssi tr} td~d tiptjdss gsfr i{deuieu lu{hutu 0 0 0 v(x, y) = +∞ +∞ X X 0 ckm Ukm (x, y). gsfr Q euhi|r~ j eay ged {i i |d trtd~d gsfr k=0 m=0 0 Q0 = oyd q +∞ X +∞ X qkm Ukm (x, y), k=0 m=0 km = (Q0 , Ukm (x, y)) kUkm (x, y)k2 (Q0 , Ukm (x, y)) = ZAZB X _zrthr~ tfuhaesd {eirjdydsra Q0 Ukm (x, y)dxdy = Q0 0 0 ZA cos(µk x)dx 0 (−1)k (−1)m (−1)k+m 4AB = = 2 . µk νm π (1 + 2k)(1 + 2m) ¥Z ZB 0 cos(νm y)dy = Rzrjuam zi kU (x, y)k = AB m suyd~ q = (−1) 16 . π (1j+jryd 2k)(1 + 2m) iytujr~ gsfrr v(x, y) 4r Q m {edytujhdssd eayijm j geuj} sdsrd feudji uyuzr [¥XV]X k+m 2 km km 2 0 +∞ +∞ X X ckm ∂ 2 Ukm ∂ 2 Ukm + ∂x2 ∂y 2 =− +∞ +∞ X X qkm Ukm (x, y). itfihfg euhi|dsrd j eay ged dyrstjdssi r yha gsfr U j{ihsata eujdstju k=0 m=0 ∂ 2 Ukm ∂ 2 Ukm − + ∂x2 ∂y 2 i c = q yha jtd k = 0, 1, . . . uyuzr [¥XV]λ {edytujhadta j jryd km km ; k=0 m=0 km (x, y) = λkm Ukm , m = 0, 1, . . . r eddsrd feudji km +∞ X +∞ X qkm Ukm (x, y) = v(x, y) = λ km m=0 k=0 +∞ +∞ (−1)k+m 64Q0 X X = cos(µk x) cos(νm y). 1+2m 2 1+2k 2 π4 ) + ( ) ) (1 + 2k)(1 + 2m)(( A B m=0 k=0 +∞ 16Q0 A2 X (−1)k u(x, y) = T0 + π3 (1 + 2k)3 rhr k=0 ch(µk y) 1− cos(µk x) ch(µk B) u(x, y) = T0 + +∞ +∞ 64Q0 X X (−1)k+m cos(µk x) cos(νm y) + 4 1+2k 2 1+2m 2 π (1 + 2k)(1 + 2m)(( ) + ( ) ) A B k=0 m=0 π π + πk + πm Q µk = 2 νm = 2 Q0 = K A B K X xydt m m [ ¤ firrds d{hi{ei} jiysitr ~uderuhu]X ±®N® vur turisuesid eut{edydhdsrd d~{deuge j r} hrsyed (0 ≤ z ≤ H, 0 ≤ ρ ≤ R) m jdesdd itsijusrd fiieioi {iyyde|r} judta {er d~{deuged T m u sr|sdd itsijusrd r pifijua tieisu ¤ {er d~{deuged T X _jdyd~ rhrsyerzdtfg trtd~g fiieyrsum {i~dtrj dd suzuhi j ds} ed sr|sdoi itsijusra rhrsyeuX Rthijra uyuzr ufijm zi gsfram i{rtju¢ua turisuesid eut{edydhdsrd d~{deuge j rhrsyed sd u} 0 1 ¥Y jrtr i gohiji fiieyrsu ϕ m X dX u = u(ρ, z) X su gyijhdjiead geuj} sdsr lu{hutu {er 0 < ρ < R, 0 < z < H 1 ∂ ∂u ∂ u =0 ρ + ρ ∂ρ ∂ρ ∂z r feudj~ gthijra~ u(ρ, z) ioeusrzdsu {er ρ → 0 + 0 m u(R, z) = T m m u(ρ, H) = T X u(ρ, 0) = T dr~ g uyuzg rt{ihga euhi|dsrd gsfrr j eay ged {i tip} tjdss~ gsfra~ i{deuieu £ge~u}lrgjrhha L (u) = 1 ∂ ρ ∂u X ρ ∂ ρ rtfi~g ∂ρ itfihfg feudjd gthijra {i {ded~dssi ρ sdiysieiysdm gsfr {edytujr~ j jryd u(ρ, z) = v(ρ, z) + w(ρ, z) X xuyuyr~ w(ρ, z) = m ioyu gsfra v(ρ, z) pgyd gyijhdjiea iysieiys~ feudj~ = T gthijra~ {i ρ X xu{rd~ feudjg uyuzg isitrdhsi i gsfrr   {er 0 < ρ < R, 0 < z < H, 1 ∂v ∂ v ∂    ρ ∂ ρ ρ ∂ ρ + ∂z = 0 ioeusrzdsu {er ρ → 0 + 0, v(R, z) = 0, v(ρ, z) 2 2 1 1 0 ρ 1 2 2   v(ρ, 0) = 0, u(ρ, H) = T0 − T1 . g uyuzg pgyd~ edu ~diyi~ gedX V] utt~ier~ uyuzg £ge~u}lrgjrhha  − 1 (ρy 0 (ρ))0 = λy(ρ), 0 < ρ < R, ρ y(ρ) ρ → 0 + 0, y(R) = 0. ioeusrzdsu {er u uyuzu eddsu u j euydhd c [{er~de cXU]X iptjdssd zrthu i{deui} eu λ = γ , k = 1, 2, . . . m oyd γ ¤ i {ihi|rdhsd eddsra geuj} sdsra J (γR) = 0 X iptjdssd gsfrr i{deuieum tiijdtjg¢rd su} ydss~ tiptjdss~ zrthu~ y (ρ) = J ( ρ) k = 1, 2, . . . X rtd~u tip} tjdss gsfr {ihsu r ieioisuhsu j {eiteustjd L [0, R; ρ] X Tjuy} eu sie~ tiptjdss gsfr i{edydhata {i ie~ghu~ ky (ρ)k = X R (J (γ )) 2 c] gsfr euhi|r~ j eay ged v(ρ, z) k 2 k k 0 γk 0 R k 2 k 2 1 k 2 v(ρ, z) = +∞ X ck (z)yk (ρ). Teudjd gthijram {er i~m j{ihsataX k=1 W\ 2 gsfr T − T r feudjioi gthijra {er z = H uf|d euhi|r~ j eay ged {i trtd~d suydss tiptjdss gsfr 0 1 +∞ X T0 − T1 = ϕk yk (ρ). k=1 Tirrds ϕ = 2(T − T ) m k = 1, 2, ... . phr suyds j {er~ded `X`X γ J (γ ) firrdsij Sha i{edydhdsra suzdsr r eay {iytujr~ c (z) j geujsdsrd lu{hutu r j feudjd gthijra su {ijdesita rhrsyeu z = r z = H X ithd {iytusijfr r rt{ihijusra tjitj gsfr y (ρ) = 0 {ihgzr~ thdyg¢rd feudjd uyuzr ( {er 0 < z < H, −λ c (z) + c (z) = 0 0 1 k k 1 k k k 00 k k k ck (0) = 0, γ 2 ck (H) = ϕk . itfihfg λ = m i ip¢dd eddsrd yrdedsruhsioi geujsdsra {edytujhadta j jryd R k k ck (z) = Ak ch γ k R z + Bk sh t{ihga feudjd gthijram {ihgzr~ A m γ k R z . ϕk sh( γRk H) γ 2(T0 − T1 ) k z . sh ck (z) = γk γk J1 (γk )sh( R H) R k = 0 Bk = X xsuzr gsfra v(ρ, z) {edytujhadta j jryd eayu ged v(ρ, z) = +∞ X γ 2(T0 − T1 ) γk k sh z J0 ( ρ). γk γk J1 (γk )sh( R H) R R `] ihgzr~ rtfi~g gsfr k=1 u(ρ, z) = T1 + +∞ X k=1 γ 2(T0 − T1 ) γk k sh z J0 ( ρ). γk γk J1 (γk )sh( R H) R R jd u(ρ, z) = T + X 2(T − T ) sh γ z J ( γ ρ). γ J (γ )sh( H) R R ±®¯® vur turisuesid eut{edydhdsrd d~{deuge j ued euyrgtu R m zut {ijdesitr S = {(r, θ, ϕ) : r = R, 0 ≤ Θ ≤ α, fiieioi {iyyde|rjudta {er d~{deuged T m u ituhsua 0zut ≤ ϕ{ijdesitr < 2π} ¤ {er d~{deuged eujsi sghX WV +∞ 0 1 k=1 k 1 k 1 γk R k k 0 0 _ ~iydhr pgyd~ rt{ihiju tderzdtfg trtd~g fiieyrsuX Rthi} jra uyuzr ufijm zi gsfra u = u(r, θ) m i{rtju¢ua turisuesid eut{edydhdsrd d~{deuge j ued sd ujrtr i gohiji fiieyrsu ϕ X su gyijhdjiead geujsdsr lu{hutu 1 ∂ r2 ∂r r 2 ∂u ∂r ∂ 1 ∂u + 2 sin θ = 0, r sin θ ∂ θ ∂θ feudj~ gthijra~ u(r, θ) ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0 m ioeusrzdsu {er m T , 0≤θ≤α X u(r, θ) r → 0 + 0 u(R, θ) = 0 , α<θ≤π i {ded~dssi θ gsfra u = u(r, θ) gyijhdjiead dtdtjdss~ iysieiys~ feudj~ gthijra~X ii~g eddsrd uyuzr pgyd~ rtfu j jryd eayu gedm j{ihsrj euhi|dsrd {i tiptjdss~ gsfra~ i{deu} ieu L (y) = − 1 d (sin θ dy ) X sin θ dθ dθ V] xuyuzu £ge~u}lrgjrhha yha ioi i{deuieu 0 θ  1  (sin θy 0 )0 = λy, − sin θ y(θ) 0<θ<π ioeusrzdsu {er θ → 0 + 0 r {er θ → π − 0 phu eddsu j euydhd c [{er~de cXW]X iptjdssd zrthu i{deuieu λ = k(k + 1) m k = 0, 1, 2, . . . m tiptjds} sd gsfrr y (θ) = P (cos θ), k = 0, 1, 2, . . . m oyd P (x) ¤ i ~sioizhd} s ld|usyeu (P (x) = 1, P (x) = x, P (x) = (3x − 1) . . .) m fjuyeu sie~ tiptjdss gsfr ky (θ)k = 2 X iptjdssd gsfrr ip} +1 eug {ihsg ieioisuhsg trtd~g j2k{eiteustjd X L [0, π ; sin θ ] c] gsfr u(r, θ) pgyd~ rtfu j jryd eayu ged k k k 0 1 2 1 2 k 2 2 k 2 u(r, θ) = +∞ X ck (r)yk (θ). Teudjd gthijra {i {ded~dssi θ {er i~ j{ihsataX gsfr k=0 h(θ) = T0 , 0 , 0≤θ≤α α<θ≤π r feudjioi gthijra uf|d euhi|r~ j eay ged {i i |d trtd~d gsf} r h(θ) = +∞ X k=0 Wc hk yk (θ), {er i~ h k = (h, yk ) ||yk ||2 (h, yk ) = X _zrthr~ tfuhaesd {eirjdydsra Zπ h(θ)yk (θ) sin θdθ = Zα T0 Pk (cos θ) sin θdθ. _{ihsrj u~dsg {ded~dssi x = cos θ m {ihgzr~ 0 0 (h, yk ) = T0 Z1 Pk (x)dx. Sha ~sioizhdsij ld|usyeu j{ihsadta edfgeedssid tiisidsrd cos α 0 0 (2k + 1)Pk (x) = Pk+1 (x) − Pk−1 (x), k = 1, 2, . . . r t{eujdyhrj eujdstju P (1) = 1 r P (−1) = (−1) k = 0, 1, . . . X iyeip} si tjitju ~sioizhdsij ld|usyeu i{rtus j V¡X _it{ihgd~ta r~r tiisidsra~r yha jzrthdsra rsdoeuhu k T0 (h, yk ) = 2k + 1 Z1 cos α = k k 0 0 (Pk+1 (x) − Pk−1 (x))dx = T0 (Pk−1 (cos α) − Pk+1 (cos α)), 2k + 1 k = 1, 2, . . . . ydhsi suyd~ (h, y0 ) = T0 Z1 P0 (x)dx = T0 Z1 ihgzr~ d{de firrds ged h cos α cos α 1dx = T0 (1 − cos α). k h0 = T0 (1 − cos α), 2 hk = T0 (Pk−1 (cos α) − Pk+1 (cos α)), 2 k = 1, 2, . . . . Sha i{edydhdsra suzdsr firrdsij c (r) {iytujr~ eay g} ed gsfr u(r, θ) r h(θ) j geujsdsrd lu{hutu r j feudjd gthijram u} yussd {i {ded~dssi r X ithd {iytusijfr yha firrdsij c (r) {i} hgzuta feudjd uyuzr k k ( (r2 c0k (r))0 = k(k + 1)ck (r), r → 0 + 0, ck (r) ioeusrzdsu {er W` ck (R) = hk . Reujsdsrd (r c (r)) = k(k + 1)c (r) ⇔ r c (r) + 2rc (r) − k(k + 1)c (r) = 0 ajhadta geujsdsrd~ hdeuX Sha doi eddsra jjdyd~ sijg {ded~ds} sg t m rt{ihga {iytusijfg r = e X nioyu t{eujdyhrju d{izfu eujdstj X vdegysi {ihgzr geujsdsrdm fiiei~g cgyijhdjiead (r) = c (e ) =gsfra z (t) = z (ln r) z (t) 2 0 k 0 2 00 k k 0 k k t k k t k k k zk00 (t) + zk0 (t) − k(k + 1)zk (t) = 0. xu{rd~ yha sdoi ueufdertrzdtfid geujsdsrd λ + λ − k(k + 1) = 0 X Tiesr ueufdertrzdtfioi geujsdsra ¤ i zrthu λ = −(k + 1) m λ = k X ii~g ip¢r~ eddsrd~ yrdedsruhsioi geujsdsra pgyd gsfra X xsuzr z (t) = A e +B e 2 1 k k −(k+1) k 2 k ck (r) = Ak r−(k+1) + Bk rk , k = 1, 2, . . . . t{ihga {dejid feudjid gthijrdm {ihgzr~ A = 0 (k = 1, 2, . . .) X ji} eioi gthijra thdygdm zi B R = h m ioyu B = h X _ riod k k k k k Rk k r k T0 ck (r) = (Pk−1 (cos α) − Pk+1 (cos α)) , 2 R c0 (t) k = 1, 2, . . . . xsuzdsrd firrdsu suyd~ iydhsiX utt~ier~ {er k = 0 yr} dedsruhsid geujsdsrd (r2 c00 (r))0 = 0 ⇔ r2 c00 (r) = A0 . Z A A0 A0 0 dr = − +B0 c00 (r) = 2 c0 (t) = r r2 r T0 A0 = 0 B0 = (1 − cos α) 2 T0 c0 (r) = (1 − cos α). 2 u(r, θ) nioyu {ihgzr~ m X t{ihga feudjd gthijram m X xsuzr _td firrds suydsX tfi~ua gsfra eayu gedX jd {edytujr~u j jryd +∞ r k T0 T0 X u(r, θ) = (1 − cos α) + (Pk−1 (cos α) − Pk+1 (cos α)) Pk (cos θ). 2 2 R ³´µ¶· ¸¹¸º» ¥XVX vur turisuesid eut{edydhdsrd d~{deuge j yhrssi~ ~duhhr} zdtfi~ tde|sd t {ea~igoihs~ tdzdsrd~ (0 ≤ x ≤ A m 0 ≤ y ≤ B) m dthr oeusr tde|sa x = 0 r x = A {iyyde|rjuta {er d~{deuged eujsi mu oeusr y = 0 r y = B ¤ {er d~{deuged T X nd~{deugeu j fu|yi~ T W k=1 1 2 {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufijiX ¥XcX vur turisuesid eut{edydhdsrd d~{deuge j yhrssi~ ~duhhr} zdtfi~ tde|sd t {ea~igoihs~ tdzdsrd~ (0 ≤ x ≤ A m 0 ≤ y ≤ B) {er gthijrrm zi j tde|sd jydhadta d{hi t ipõd~si {hisit eut} {edydhdsra d{hiji sdeorr Q X nd~{deugeu su oeusa tde|sa {iyyde} |rjudta eujsi T X _ fu|yi~ {i{dedzsi~ tdzdsrr tde|sa d~{deugeu tzrudta iyrsufijiX ¥X`X vur turisuesid eut{edydhdsrd d~{deuge j {ea~igoihsi~ {u} euhhdhd{r{dyd (0 ≤ x ≤ A m 0 ≤ y ≤ B m 0 ≤ z ≤ C) m oeus fiieioi z = C {iyyde|rjudta {er d~{deuged T m u ituhsd oeusr {er d~{deuged X T ¥XX vur turisuesid eut{edydhdsrd d~{deuge j yhrssi~ ~duhhr} zdtfi~ tde|sd t fegoh~ tdzdsrd~ (0 ≤ ρ ≤ R, 0 ≤ ϕ ≤ 2π) m dthr zut {ijdesitr tde|sa ρ = R, 0 ≤ ϕ ≤ π {iyyde|rjudta {er d~{deuged T m u yegoua zut ρ = R, π < ϕ < 2π ¤ {er d~{deuged X nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsu} T fijiX ¥XUX vur turisuesid eut{edydhdsrd d~{deuge j yhrssi~ ~duhhr} zdtfi~ tde|sd t fegoh~ tdzdsrd~ (0 ≤ ρ ≤ R, 0 ≤ ϕ ≤ 2π) m dthr {i} jdesit tde|sa ρ = R {iyyde|rjudta {er d~{deuged T (1 + sin ϕ) X nd~{deugeu j fu|yi~ {i{dedzsi~ tdzdsrr tde|sa tzrudta iyrsufi} jiX ¥X¥X vur turisuesid eut{edydhdsrd d~{deuge j rhrsyed m su jdesdd itsijusrd fiieioi {iyudta d{hi} (0 ≤ z ≤ H, 0 ≤ ρ ≤ R) ji {iif {hisitr q m u sr|sdd itsijusrd r pifijua tieisu {iyyde|r} juta {er d~{deuged T X ¥XWX vur turisuesid eut{edydhdsrd d~{deuge j rhrsyed m sr|sdd itsijusrd fiieioi {iyyde|rjudta (0 ≤ z ≤ H, 0 ≤ ρ ≤ R) {er d~{deuged T m pifijua tieisu ¤ {er d~{deuged T m u jdesdd it} sijusrd d{hirihreijusiX ¥XZX vur turisuesid eut{edydhdsrd d~{deuge j rhrsyed m jdesdd itsijusrd fiieioi {iyyde|rjudta (0 ≤ z ≤ H, 0 ≤ ρ ≤ R) {er d~{deuged T m u sr|sdd itsijusrd r pifijua tieisu ihu|yuta jiygi~ d~{deugeu fiieioi T X nd{hiip~ds su i zutr {ijdesitr rhrsyeu {eirtiyr {i ufisg visuX ¥XYX vur turisuesid eut{edydhdsrd d~{deuge j ued m {ijdesit fiieioi r = R (0 ≤ r ≤ R, 0 ≤ θ ≤ π , 0 ≤ ϕ < 2 π ) {iyyde|rjudta {er d~{deuged T sin θ (0 ≤ θ ≤ π) yha hpioi ϕ X ¥XV\X vur turisuesid eut{edydhdsrd d~{deuge j ued m {ijdesit fiieioi r = R (0 ≤ r ≤ R, 0 ≤ θ ≤ π , 0 ≤ ϕ < 2 π ) {iyyde|rjudta {er d~{deuged T sin θ (0 ≤ θ ≤ π) yha hpioi ϕ r 0 1 0 1 0 0 0 1 0 1 0 0 0 WU 2 2 j ued jydhadta d{hi t ipõd~si {hisit eut{edydhdsra d{hiji sdeorr Q(1 − r ) X ¼½¾¹½¿ R ¥XVX u(x, y) = T + m 2(T − T ) X 1 − (−1) 1 − ch(µ B) sh(µ y) sin(µ x) + ch(µ y) + 1 +∞ 2 k 1 π oyd µ = πk X A ¥XcX u(x, y) = T k k=1 k k sh(µk B) k k k + (1 − (−1)k 1 − ch(µk B) 2Q0 A ch(µk y) + sh(µk y) sin(µk x) + π3 k3 sh(µk B) k=1 πk Q µk = Q0 = K A K +∞ +∞ 4Q0 X X (1 − (−1)k )(1 − (−1)m ) u(x, y) = T0 + 4 sin(µk x) sin(νm y) k 2 m 2 π km(( ) + ( ) )) A B k=1 m=1 πk πm µk = νm = A B Q Q0 = K K u(x, y, z) = T0 + p +∞ +∞ X k m sh( µ2 + ν2 z) X (1 − (−1) )(1 − (−1) ) 4(T1 − T0 ) m p k + × 2 2 + ν2 C) π km sh( µ m k k=1 m=1 πk πm µk = νm = × sin(µk x) sin(νm y) A B +∞ X sin(k ϕ) ρ k T1 + T0 2(T1 − T0 ) + u(ρ, ϕ) = 2 π k R k=1 ρ u(ρ, ϕ) = T0 (1 + sin ϕ) R +∞ X sh( γRk z ) 1 γk ρ), u(ρ, z) = T0 + 2q0 R J ( 0 γ2k J1 (γk ) ch( γkRH ) R k=1 q q0 = K K +∞ X ch( γk (z−H) ) γk 1 R u(ρ, z) = T0 + 2(T1 − T0 ) J0 ( ρ). γk J1 (γk ) ch( γkRH ) R k=1 u(ρ, z) = T0 + oyd rhr +∞ 2 X oyd ¥X`X 0 m [ ¤ firrds d{hi{eijiysitr ~uderuhu] m m [ ¤ firrds d{hi{eijiysitr ~uderuhu]X m oyd ¥XX ¥XUX ¥X¥X m m X X X [ ¤ firrds d{hi{eijiysitr ~uderuhu]X ¥XWX ¥XZX W¥ m +∞ X ( γRk ch( γRk z ) + hsh( γRk z )) γk RJ1 (γk ) + 2(T1 − T0 ) J ( ρ), 0 γk (J12 (γk ) + J02 (γk )) ( γRk ch( γkRH ) + hsh( γkRH )) R k=1 J1 (γ) hR γk = J ( γ ) γ 0 2 r 1 2 + 2 − cos2 θ u(r, θ) = T0 3 R 3 1 x2 = (2P2 (x) + P0 (x)) P0 (x), P2 (x) 3 2 r2 1 + − cos2 θ + u(r, θ) = T0 3 R2 3 Q0 Q + (2R3 + r3 − 3Rr2 ) Q0 = K 18R K oyd ¤ i {ihi|rdhsd fiesr geujsdsra X ¥XYX X er eddsrr gzdtm zi [ ¤ ~sioi} zhds ld|usyeu]X ¥XV\X m oyd [ ¤ firrds d{hi{eijiysitr ~uderuhu]X J VX bdefghij kXl Xm ueuusij m _Xm nedogp _Xl Xm qdejrstfua vX bX bdiy ~ud~urzdtfi rrfr RzdpX {itiprdX pX y}ji pnR lnm c\\¥X cX îygsij vXkXm rhors Xþ X Srdedsruhsd geujsdsra RzdpX {itiprdX pX y}ji pnR lnm c\VVX `X îedjrz ¬X xXm Tujeutfua TX _Xm qdhfuf XX ay ged RzdpX {itiprdX pX y} ji pnR lnm c\V\X X nrisij kXvXm u~uetfr kXkX Reujsdsra ~ud~urzdtfi rrfrX bX vugfum c\\X UX {eujizsrf {i t{druhs~ gsfra~ w iy edyX bXkpeu~ijru r XrousX bX vugfum VYWYX ¥X _utis X vX ndiera pdttdhdj gsfr _ c zX qX VX bX y}ji rsiteX hrXm VYYX WX keu~usijrz X Xm ldjrs _XX Reujsdsra ~ud~urzdtfi rrfrX bX vugfum VY¥YX lk_l¬v¬ _jdydsrd X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ` VX Tk¬_ÿ¬ xkSkq Sl ^ÿTv_¬vvÿê S¬¬vøkl vÿê Rk_v¬v X X X X X X X X X X X X X X X X X X X X X X X X cX ¬£ ¬v¬ xkSkq £nRbk¤lR_ll X X X X X X X X X X X X X X X X X Vc `X Rk_v¬v¬ n¬ l _Svn X X X X X X X X X X X X X X X X X X X X X X X X cY X _lv_¬ Rk_v¬v¬ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XU UX b¬nS Sklkb^¬k X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X UZ ¥X Rk_v¬v lk lkk Rkvk X X X X X X X X X X X X X X X X X X X X X X X¥` {rtif hrdeuge X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X WZ bdefghij khdftusye ljijrz nedogp _deu ldisryijsu qdejrstfua vrsu bruhijsu % % Rzdpsid {itiprd dyufie X TX Sihouij iy{rtusi j {dzu ie~u ¥\×Z VwV¥X ^g~uou itdsuaX dzu itdsuaX uesrgeu ó “ X dzX hX Um\X nreu| V\U fX xufu yudhtji pnR ln“ VYW`W¥m X} ddepgeom ghX ei X i{ijum U

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