a0 a1 star a2a3a4a5a6a7a8a9a10 6 1 o 'lC { 11 1 o 'lC { '' §‰)flK ~^){§'lC {' )~ ' §‰)flK §ˇ~o·k? ' § A)§d 5?’ A) U\ ˇ)§ ^‰)^ (~X—^ )‰ U\X?' 5 ' § ?)flK§? { ·=z 5~ ' §| ?)flK' ?u –9 p ' §‰)flK§ ?k =? –k? ' § ˇ)§duˇ)￥?k ‰…?§ ‘5§J– ‰)^ ‰ ' ?) '' § ‰‰)flK§ 7Lr?) ‰\–? ?U' 14.1 ‰u gd ? 12 14.1 ‰u gd ? ‰)flK ? l! ‰ u gd ?§ §9‰)^ @2u @t2 ?a 2@2u @x2 = 0; 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = `(x), @u@t flfl fl t=0 = ?(x), 0 ? x ? l: § >.^ · g § —'^ · g ' ? F"? A)) k'lC /“§= u(x;t) = X(x)T(t): F u(x;t) \ §§= X(x)T00(t) = a2X00(x)T(t): “ –X(x)T(t)§ 1 a2 T00(t) T(t) = X00(x) X(x) : 3? “￥§ ·t …? ( ?{‘§ x?’) m ·x …? ( ?{‘§ t?’) ˇd§ m § 7L u Q x?’!q t?’ ~?' ??§ ? (J –z? T00(t)+?a2T(t) = 0; X00(x)+?X(x) = 0: F u(x;t) \>.^ § X(0)T(t) = 0; X(l)T(t) = 0: ? 7Lk X(0) = 0; X(l) = 0: ? ? ^'lC {?) ' §‰)flK 1 'lC F8I 'lC /“ ")u(x;t) = X(x)T(t) F(J …?X(x) v ~ ' § >.^ –9T(t) v ~ ' § 14.1 ‰u gd ? 13 F^ ' § >.^ · g y3 y …?X(x) ~ ' §‰)flK§A:· ' §￥?k ‰~ ??§‰)^ · ? g>.^ '? ‰)flK u~ ' § — flK' ? ?u? ? § kQ v g~ ' §!q v g>.^ " )' k ? , A‰ § kQ v g~ ' §!q v g>.^ ")X(x)' ? ? A‰ ? § A ")? …?' …?X(x) ~ '' §‰)flK§? flK' 1 ?) flK Fe? = 0§ ' § ˇ)· X(x) = A0x+B0: \>.^ X(0) = 0; X(l) = 0; –‰ A0 = 0; B0 = 0: ?‘?? = 0 ' § k")' ?{‘§? = 0 · ' F ? 6= 0 §~ ' § X00(x)+?X(x) = 0 ˇ)· X(x) = Asin p ?x+Bcos p ?x; \>.^ § k B = 0; Asinp?l = 0: ˇ A 6= 0§ 7kp?l = n…§= ?n = ?n… l ·2 ; n = 1;2;3;￠￠￠ : A …? · Xn(x) = sin n…l x: 14.1 ‰u gd ? 14 ? ? k??? §§ –^ ?nIP§ˇd§3 ? (J￥§r A …? P ?n Xn(x)' 1n ?A)§?U\ ) 3?) flK §?uz ?n§d § T00(t)+?a2T(t) = 0 –? A Tn(t)§ Tn(t) = Cn sin n…l at+Dn cos n…l at: ˇd§ v ' § >.^ A) un(x;t) = ? Cn sin n…l at+Dn cos n…l at · sin n…l x (n = 1;2;3;￠￠￠): F? A)k??? Fz A) v g ' § g>.^ F ‘5§ ? A) U T— v‰)flK￥ —'^ §= ?{? ~?Cn Dn§ v Dn sin n…l x = `(x); Cnn…al sin n…l x = ?(x): F ' § >.^ · g §r§ (??k )A)U\ 5§E,· v g § g>.^ )'·? U v—'^ ” Fr ??? A)U\ 5 u(x;t) = 1X n=1 ? Cn sin n…l at+Dn cos n…l at · sin n…l x; ?? kv — ′?5(~X§ –ˉ ? ?)§@o§? u(x;t) E,· g ' §3 g>.^ e )' ??/“ )? )'§ u ' § ˇ)§ˇ ) · v ' §§ v g>.^ X J )￥ UU\X?Cn Dn” 1X n=1 Dn sin n…l x = `(x); (z) 1X n=1 Cnn…al sin n…l x = ?(x) (>) 14.1 ‰u gd ? 15 1o |^ …? 5‰U\X? n …? 5 Z l 0 Xn(x)Xm(x)dx = 0; n 6= m: 3(z)“ ?–sin m…l x§ˉ ¨'§ Z l 0 `(x)sin m…l xdx = Z l 0 1X n=1 Dn sin n…l xsin m…l xdx = 1X n=1 Dn Z l 0 sin n…l xsin m…l xdx = Dm ￠ l2: ?– Dn = 2l Z l 0 `(x)sin n…l xdx: §d(>)“§ – Cn = 2n…a Z l 0 ?(x)sin n…l xdx: ? § —'^ ￥ fi …?`(x) ?(x)§ – U\X?Cn Dn§l ? ‰)flK )' F …? 5 y? Xn(x) = sin n…l x Xm(x) = sin m…l x·'O?Au ?n ?m …?§?n 6= ?m(=n 6= m)'§ 'O v X00n(x)+?nXn(x) = 0; Xn(0) = 0; Xn(l) = 0; X00m(x)+?mXm(x) = 0; Xm(0) = 0; Xm(l) = 0: ^Xm(x)?–Xn(x) §§^Xn(x)?–Xm(x) §§ ~§?3?m[0;l] ¨'§= (?n ??m) Z l 0 Xn(x)Xm(x)dx = Z l 0 ￡X n(x)X00m(x)?Xm(x)X00n(x) ?dx = ￡Xn(x)X0m(x)?Xm(x)X0n(x)? flfl fl l 0 = 0: ?^ Xn(x) Xm(x) v >.^ ' ? ?n 6= ?m§ y …? 5 Z l 0 Xn(x)Xm(x)dx = 0; n 6= m: 14.1 ‰u gd ? 16 4 3 ? y?￥ ^ 1. …? v ' § 2. …? v >.^ vk^ …? N…?/“ 4 ˇd§ …? v ' § X00(x)+?X(x) = 0; K(J (?n ??m) Z l 0 Xn(x)Xm(x)dx = ￡Xn(x)X0m(x)?Xm(x)X0n(x)? flfl fl l 0 E,??' 4 XJ …? v >.^ U fi1X(0)+fl1X0(0) = 0; fi2X(l)+fl2X0(l) = 0; ￥fi1 fl1!fi2 fl2 0§Kk fi1Xn(0)+fl1X0n(0) = 0; fi1Xm(0)+fl1X0m(0) = 0 fi2Xn(l)+fl2X0n(l) = 0; fi2Xm(l)+fl2X0m(l) = 0: ˇ fi1 fl1 0§?– flfl flfl fl Xn(0) X0n(0) Xm(0) X0m(0) flfl flfl fl = 0: qˇ fi2 fl2 0§?–qk flfl flfl fl Xn(l) X0n(l) Xm(l) X0m(l) flfl flfl fl = 0: F( ?u flK X00(x)+?X(x) = 0; fi1X(0)+fl1X0(0) = 0; fi2X(l)+fl2X0(l) = 0 14.1 ‰u gd ? 17 …? 5 Z l 0 Xn(x)Xm(x)dx = 0; n 6= m E,??' 4 ? >.^ ”X ! !nan?a. >.^ ' F …? kXnk2 · Z l 0 X2n(x)dx = l2: Fˉ?3 ‰u D′L§ { §E– Xd— ￡ ˉ? ~' t > 0 §— ￡ 3?.u 'O mD′§ ?· :x = 0‰x = l §7L ￡5§? k …(=3 :x = 0 x = l7L §?·d ‰? >.^ ?‰ )' u ?? :3?? ￡ §§ ·— ￡ 3 :m?g E U\ (J'?u— -u ˉ?§ , –aq/? ' Fu oU 3? t§u ?U U'O· 1 2 Z l 0 ‰ @u @t ?2 dx 12 Z l 0 T @u @x ?2 dx; oU E(t) = 12 Z l 0 ‰ @u @t ?2 dx+ 12 Z l 0 T @u @x ?2 dx: )“ \§|^ …? 8 5§ N·? E(t) = m… 2a2 4l2 1X n=1 n2￡jCnj2 +jDnj2?: “m w,·~?§ t?’§=u oU ˉ ' kXnk ?~? …? 8 ˇf'?·ˇ 1 kXnk2 Z l 0 X2n(x)dx = 1 = …?Xn(x)=kXnk 1', § – ? ?Z l 0 Xn(x)Xm(x)dx = l2–nm: ? …? 8 5' {· 13.6! {§ dE=dt = ?2§ 6u N ?) {(~X§'lC {)' 14.1 ‰u gd ? 18 F) 5 XJd‰)flKk )§u1(x;t) u2(x;t)§@o§v(x;t) · u1(x;t)?u2(x;t) ‰ v‰ )flK @2v @t2 ?a 2@2v @x2 = 0; 0 < x < l; t > 0; vflflx=0 = 0; vflflx=l = 0; t ? 0; vflflt=0 = 0; @v@t flfl fl t=0 = 0; 0 ? x ? l: U y?v(x;t) = 0= 'l n – §? ‰· ( 'lU ˉ ?5 w§ t = 0 u oU 0§ˇd– ? t§E(t) 0'?? X ‰k @v @x = 0; @v @t = 0; =v(x;t) ~?'d—'^ ‰>.^ § U‰ d~? 0' F|^'lC {?) ' §‰)flK ? ‰ 1. 1 §'lC ' ? ?–U ￠y§k?^ · ' § >.^ · g ' 'lC (J§· ( ‰? )?k ‰~? g~ ' § g>.^ §=( ‰? ) flK' 2. 1 §?) flK' 3. 1n §? A)§?? U\ )' w,flkvk? nd ￥ ? A)' 4. 1o §|^ …? 5‰U\X?' ‘5§ ? ·/“)'?u NflK§ 7L y 1. ? u(x;t)·? v ' §§ ?{‘§??)·? –ˉ ? ?? 2. ? u(x;t)·? v>.^ § ?{‘§??) ……??··?oY? 3. 3‰‰UU\X?? §ˉ ¨'·? {' 14.1 ‰u gd ? 19 ’u?n flK§ 9 ??) ′?5'duX?Cn Dn·d`(x) ?(x)?‰ § ˇ `(x) ?(x) 5 ?‰ ??n flK ￡ ' ln ‘§'lC { ??§ ?ue A ^ 1. flKk)? 2. ‰)flK ) ‰ –U …?—m§ ?{‘§ …? N· ? 3. …? ‰ k 5' – 3? ￡ ?A flK' ) n?′ kwA) un(x;t) = ? Cn sin n…l at+Dn cos n…l at · sin n…l x = An sin(!nt+–n)sinknx; ￥ !n = n…l a; kn = n…l ; An cos–n = Cn; An sin–n = Dn: F un(x;t) L 7ˉ F An sinknxL?u : ' F sin?!nt+–n￠L? ˇf F !n·7ˉ “˙§? ‰u k“˙‰ “˙§ —'^ ?’ F kn? ˉ?§· ˉ –ˇ? F –n·— §d—'^ ?‰ F3knx = m…§=x = m…=kn = (m=n)l; m = 0;1;2;3;￠￠￠ ;n : § ? 0§? ˉ!' )u :3S§ˉ!: kn+1 ' F3knx = (m+1=2)…§=x = (2m+1)…=2kn = (2m+1)l=2n; m = 0;1;2;3;￠￠￠ ;n?1 : § ? ? §? ˉ?'ˉ?: kn ' F flK )K·? 7ˉ U\' ·ˇ ? ˇ§??){ ? 7ˉ{' 14.1 ‰u gd ? 110 ‰ u5‘§ k“˙￥k §= !1 = …l a; ? ?“§ ? k“˙!n ·?“!1 ? § !n = n!1; n = 2;3;￠￠￠ ; ? “' Fu ?“ ?‰ ?u( N'3uW ￥§ u ‰(=‰ ‰) §ˇLU Cu ;§ (=UC T )§ –N!?“!1 ' F)“￥?“ “ U\X?fCng fDng ? ?‰ ( “ ' §=?‰ ( ' F ? 1X n=1 n2￡jCnj2 +jDnj2? u oU ? ’§?– ?‰ ( r ' F'lC { ) 1ˉ) ?X —'^ `(x) ?(x) '(x) = 8 < : ?`(?x); ?l ? x ? 0; `(x); 0 ? x ? l; “(x) = 8 < : ??(?x); ?l ? x ? 0; ?(x); 0 ? x ? l; , 2 –ˇ 2l –ˇ…?(EP '(x) “(x))'? (J y 3 :x = l · ' '(x) “(x)—m Fourier?? '(x) = 1X n=1 fin sin n…l x; “(x) = 1X n=1 fln sin n…l x; ￥ fin = 1l Z l ?l '(x)sin n…l xdx = 2l Z l 0 `(x)sin n…l xdx; fln = 1l Z l ?l “(x)sin n…l xdx = 2l Z l 0 ?(x)sin n…l xdx: c?‰ Cn Dn ’ § –w fin = Dn; fln = n…al Cn: 14.1 ‰u gd ? 111 ?– u(x;t) = 1X n=1 ? Cn sin n…l at+Dn cos n…l at · sin n…l x = 12 1X n=1 Dn h sin n…l (x?at)+sin n…l (x+at) i + 12 1X n=1 Cn h cos n…l (x?at)?cos n…l (x+at) i = 12 1X n=1 fin h sin n…l (x?at)+sin n…l (x+at) i + 12 1X n=1 fln n…a h cos n…l (x?at)?cos n…l (x+at) i = 12 ['(x?at)+'(x+at)]+ 12a Z x+at x?at “(x)dx: 1ˉ) /“ § L?p '(x) “(x)·d—'^ `(x) ?(x)U c? {K ' ? )“u(x;t)§ , ?^u?m0 ? x ? l￥' 14.2 /? S ?‰flK 112 14.2 /? S ?‰flK 'lC { ?^^uu9D § ?‰flK(~X§Laplace §) ‰‰)flK' k‰)flK @2u @x2 + @2u @y2 = 0; 0 < x < a;0 < y < b; uflflx=0 = 0, @u@x flfl fl x=a = 0, 0 ? y ? b; uflfly=0 = f(x),@u@y flfl fl y=b = 0, 0 ? x ? a: E^'lC {?)'- u(x;y) = X(x)Y(y); F \ §§'lC §= X00(x)Y(y) = ?X(x)Y00(y): u· X00(x) X(x) = ? Y00(y) Y(y) : 3? “￥§ ·x …?( y?’) m ·y …?( x?’) ˇd§ 7L u Q x?’!q y?’ ~?'-? ~? ??§ X00(x)+?X(x) = 0 Y00(y)??Y(y) = 0: F \’ux ? g>.^ X(0)Y(y) = 0; X0(a)Y(y) = 0; –'lC X(0) = 0; X0(a) = 0: ? §q flK X00(x)+?X(x) = 0; X(0) = 0; X0(a) = 0: F?) flK 14.2 /? S ?‰flK 113 e? = 0§~ ' § ˇ)· X(x) = A0x+B0: \( g)>.^ § A0 = 0; B0 = 0'ˇd ' § k")' ?{‘§? = 0 · ' e? 6= 0§~ ' § ˇ) · X(x) = Asin p ?x+Bcos p ?x: \( g)>.^ § B = 0; A 6= 0; cosp?a = 0'u·§ ? ?n = 2n+1 2a … ?2 ; n = 0;1;2;3;￠￠￠ …? Xn(x) = sin 2n+12a …x: A/§ Yn(y) = Cn sinh 2n+12a …y +Dn cosh 2n+12a …y: u·§ Q vLaplace §!q v g>.^ A) un(x;y) = Cn sinh 2n+12a …y +Dn cosh 2n+12a …y ? sin 2n+12a …x: ???? A)U\ 5§ ) u(x;y) = 1X n=0 Cn sinh 2n+12a …y +Dn cosh 2n+12a …y ? sin 2n+12a …x: \’uy ?( g)>.^ § uflfly=0 = 1X n=0 Dn sin 2n+12a …x = f(x); @u @y flfl flfl y=b = 1X n=0 2n+1 2a … ? Cn cosh 2n+12a …b +Dn sinh 2n+12a …b · sin 2n+12a …x = 0; 2g …? 8 5§Z a 0 sin 2n+12a …xsin 2m+12a …xdx = a2–nm; –? Dn = 2a Z a 0 f(x)sin 2n+12a …xdx Cn cosh 2n+12a …b+Dn sinh 2n+12a …b = 0; dd Cn = ?Dn tanh 2n+12a …b: ? § ? /? SLaplace §> flK ??)'XJ f(x) N/ “§ –? ? U\X?Cn Dn N/“' 14.2 /? S ?‰flK 114 F? flK·?‰flK§ mt?’§ˇd y—'^ ' F^'lC {?) § ^ g>.^ ? flK§ ^ g>.^ ‰UX?' 14.3 ?u gC ‰)flK 115 14.3 ?u gC ‰)flK ‰)flK @u @t ?? ?@2u @x2 + @2u @y2 · = 0; 0 < x < a; 0 < y < b; t > 0; @u @x flfl fl x=0 = 0, @u@x flfl fl x=a = 0, 0 ? y ? b; t ? 0; @u @y flfl fl y=0 = 0, @u@y flfl fl y=b = 0, 0 ? x ? a; t ? 0; uflflt=0 = `(x;y); 0 ? x ? a; 0 ? y ? b: u(x;y;t) = v(x;y)T(t); \ §§'lC § @2v @x2 + @2v @y2 +?v(x;y) = 0; T0(t)+??T(t) = 0; ￥?·'lC ? ‰~?'2- v(x;y) = X(x)Y(y); ? 'lC § X00(x)+?X(x) = 0; X0(0) = 0; X0(a) = 0 Y00(y)+”Y(y) = 0; Y0(0) = 0; Y0(b) = 0: ?pq ? ~?? ”§ ?; ” ?￥ k · ? §§ 7L v?+” = ?' B§? //?? ~?' ?)’uX(x) flK F ? = 0 §~ ' § ˇ)· X(x) = A0x+B0: \( g)>.^ § A0 = 0; B0??: ?‘?? = 0· § …? X(x) = 1: 14.3 ?u gC ‰)flK 116 c ! ·?p ? = 0· §?·ˇ ? = 0 § flKk ")X(x) = B0; B0·??~?' F ? 6= 0 §~ ' § ˇ)· X(x) = Asinp?x+Bcosp?x: \( g)>.^ §q A = 0; p?sinp?a = 0: ?–§p?a = n…§= ?n = ?n… a ·2 ; n = 1;2;3;￠￠￠ : A/§ …?Xn(x) = cos n…a x: r? = 0 ? > 0 (J ? 5§ – ? ?n = ?n… a ·2 ; n = 0;1;2;3;￠￠￠ ; …? Xn(x) = cos n…a x: –) ’uY(y) flK ) ”m = ?m… b ·2 ; m = 0;1;2;3;￠￠￠ ; …? Ym(x) = cos m…b y: ?u ‰ n m§2? ? T00(t) = A00; n = m = 0; Tnm(t) = Anm e??nm?t; ? /; – ? /“ Tnm(t) = Anm e??nm?t; n = 0;1;2;3;￠￠￠ ; m = 0;1;2;3;￠￠￠ ; ?nm = ?n +”m = ?n… a ·2 + ?m… b ·2 : ˇd§ ? ‰)flK A) unm(x;y;t) = Xn(x)Ym(y)Tnm(t) = Anm cos n…a xcos m…b ye??nm?t 14.3 ?u gC ‰)flK 117 ) u(x;y;t) = 1X n=0 1X m=0 unm(x;y;t) = 1X n=0 1X m=0 Anm cos n…a xcos m…b ye??nm?t = 1X n=0 1X m=0 Anm cos n…a xcos m…b y ￡exp ‰ ? ??n… a ·2 + ?m… b ·2? ?t : \—'^ §k u(x;y;t)flflt=0 = 1X n=0 1X m=0 Anm cos n…a xcos m…b y = `(x;y): e A …? 5‰ U\X?'y3Q ^ fXn(x); n = 0;1;2;￠￠￠g 5§q ^ fYm(y);m = 0;1;2;￠￠￠g 5§" ' g§ ? § 8 5 Z a 0 Xn(x)Xn0(x)dx = a2 (1+–n0)–nn0; Z b 0 Ym(y)Ym0(y)dy = b2 (1+–m0)–mm0: O ￥ I 3%?'n = 0 n 6= 0 m = 0 m 6= 0 /'O (J· Anm = 4ab 1(1+– n0)(1+–m0) ￡ Z a 0 Z b 0 `(x;y)cos n…a xcos m…b ydxdy: 14.4 ‰u r‰ ? 118 14.4 ‰u r‰ ? g ' § g>.^ 3'lC {￥ X’ ^ ˇ § >. ^ · g §'lC –￠y' XJ‰)flK￥ § >.^ · g § kvk UAA^^'lC {” ‰)flK @2u @t2 ?a 2@2u @x2 = f(x;t); 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l: ?u § g ?n§?p ?X{d ‰u r‰ ?§u — ￡ — 0' ? ){ § >.^ gz' u(x;t) = v(x;t)+w(x;t); 3 g § gz §7L – k g>.^ C' ){ ’ 3u? A)v(x;t)'?^uf(x;t)/“’ { /' ) U ?) g § 0 {§k? g § A)v(x;t)§ @2v @t2 ?a 2@2v @x2 = f(x;t): ? §XJ u(x;t) = v(x;t)+w(x;t); Kw(x;t) ‰· A g § )§ @2w @t2 ?a 2@2w @x2 = 0: UAA^^'lC {§w(x;t)7L v g>.^ w(x;t)flflx=0 = 0; w(x;t)flflx=l = 0: ˇd§? ? ˇ? A)v(x;t) AT v g>.^ ' u(x;t)flflx=0 = 0; u(x;t)flflx=l = 0: 14.4 ‰u r‰ ? 119 ? ? A)§ –? w(x;t) ) w(x;t) = 1X n=1 ? Cn sin n…l at+Dn cos n…l at · sin n…l x; ?– u(x;t) = v(x;t)+ 1X n=1 ? Cn sin n…l at+Dn cos n…l at · sin n…l x; \—'^ § 1X n=1 Dn sin n…l x = ?v(x;t)flflt=0; 1X n=1 Cnn…al sin n…l x = ?@v(x;t)@t flfl fl t=0 ; |^ …? 8 5§‰ U\X? Cn = ? 2n…a Z l 0 @v(x;t) @t flfl fl t=0 sin n…l xdx; Dn = ? 2l Z l 0 v(x;0)sin n…l xdx: ??){? § >.^ gz' 3 g § gz §7L – k g>.^ CC' ){ ’ 3u? A)v(x;t)'?^^uuf(x;t)/“’ { /' g—'^ – ' ~1 ?)‰)flK @2u @t2 ?a 2@2u @x2 = f(x); 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l; ￥f(x) fi …?' ) )K g·' du § g ·x …?§ –r gz…? ·x …?§= u(x;t) = v(x)+w(x;t); ￥v(x) v~ ' § > flK v00(x) = ? 1a2f(x); v(0) = 0; v(l) = 0; 14.4 ‰u r‰ ? 120 w(x;t)K v‰)flK @2w @t2 ?a 2@2w @x2 = 0; 0 < x < l; t > 0; wflflx=0 = 0, wflflx=l = 0, t ? 0; wflflt=0 = ?v(x), @w@t flfl flfl t=0 = 0, 0 ? x ? l: ~2 ?)‰)flK @2u @t2 ?a 2@2u @x2 = A0 sin!t; 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l; ￥a; A09! fi ~?' ) u(x;t) = v(x;t)+w(x;t); ? g N/“§ gz…?v(x;t) v(x;t) = f(x)sin!t: ? v(x;t) v g §9 g>.^ § @2v @t2 ?a 2@2v @x2 = A0 sin!t; 0 < x < l; t > 0; vflflx=0 = 0, vflflx=l = 0, t ? 0; · Jf(x)§? ?!2f(x)?a2f00(x) = A0; f(0) = 0; f(l) = 0: ? g~ ' § ˇ) f(x) = ?A0!2 +Asin !ax+Bcos !ax: \ g>.^ –‰ B = A0!2 ; A = A0!2 tan !l2a: u· f(x) = ?A0!2 ?? 1?cos !ax · ?tan !l2a sin !ax ? = ?A0!2 ? 1? cos(!(x?l=2)=a)cos(!l=2a) ? : 14.4 ‰u r‰ ? 121 ? U w(x;t)? v ‰)flK§ @2w @t2 ?a 2@2w @x2 = 0; 0 < x < l; t > 0; w flfl fl x=0 = 0, wflflx=l = 0, t ? 0; wflflt=0 = 0, @w@t flfl flfl t=0 = ?!f(x), 0 ? x ? l: § ) w(x;t) = 1X n=1 h Cn sin n…l at+Dn cos n…l at i sin n…l x: |^ ? —'^ –‰ Dn = 0; Cn = ? 2!n…a Z l 0 f(x)sin n…l xdx = ?2A0!l 3 …2a 1?(?)n n2 1 (n…a)2 ?(!l)2: kn = ? §Cn 0'? § ? w(x;t) = ?4A0!l 3 …2a 1X n=0 ? 1 (2n+1)2 1 [(2n+1)…a]2 ?(!l)2 ￡sin 2n+1l …x sin 2n+1l …at ? u(x;t) = ? A0!2 ? 1? cos!(x?l=2)=acos(!l=2a) ? sin!t ? 4A0!l 3 …2a 1X n=0 ? 1 (2n+1)2 1 [(2n+1)…a]2 ?(!l)2 ￡sin 2n+1l …x sin 2n+1l …at ? : Aˇ / r‰ “˙! —·u , k“˙§ ! = (2k +1)…a=l; k , (‰ K ? u3r‰ ^e?u) y ' ~3 ?)‰)flK @2u @x2 + @2u @y2 = xy; 0 < x < a; 0 < y < b; uflflx=0 = 0, uflflx=a = 0, 0 ? y ? b; uflfly=0 = `(x), uflfly=b = ?(x), 0 ? x ? a: 14.4 ‰u r‰ ? 122 ) N·? § ˇ) 1 6x 3y +f(x+iy)+g(x?iy): ? J…?f g§~X§ f(x+iy)+g(x?iy) = ? a 2 24i h (x+iy)2 ?(x?iy)2 i = ?16a2xy; ? ) v(x;y) = 16 ? x2 ?a2 · xy v g>.^ v(x;y)flflx=0 = 0; v(x;y)flflx=a = 0: - u(x;y) = v(x;y)+w(x;y); – w(x;t)?A v ‰)flK§ @2w @x2 + @2w @y2 = 0; 0 < x < a; 0 < y < b; wflflx=0 = 0; wflflx=a = 0; 0 ? y ? b; wflfly=0 = `(x); wflfly=b = ?(x)? b6 ?x2 ?a2￠x; 0 ? x ? a: ￥ § ?>.^ · g §ˇd§?N·?)' XJ § g f(x;t) /“’ E,§J–? g § A)§ – ^e? ?? ){' ? ){ ‰n{ ? g·· ‰ ? flK=z (??? )gd ?flK U\' ~X§?uu r‰ ?flK @2u @t2 ?a 2@2u @x2 = f(x;t); 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l; –r g ( ? )f(x;t)L? (??? )] ( ) U\§ f(x;t) = Z 1 0 f(x;?)–(t??)d?; A/§r ￡u(x;t) L? u(x;t) = Z 1 0 v(x;t;?)d?; 14.4 ‰u r‰ ? 123 Kv(x;t;?)d? A ·] f(x;?)–(t??)d?? ) ￡'O(/‘§v(x;t;?)A ·‰) flK @2v @t2 ?a 2@2v @x2 = f(x;?)–(t??); 0 < x < l; t > 0; vflflx=0 = 0, vflflx=l = 0, t ? 0; vflflt=0 = 0, @v@t flfl flfl t=0 = 0, 0 ? x ? l )' g ( 3 mS? )f(x;?)–(t??) 3u? § J ·? u3? … ] '? l ' § ¨' Z ?+0 ??0 @2v @t2 dt?a 2 Z ?+0 ??0 @2v @x2dt = Z ?+0 ??0 f(x;?)–(t??)dt 'ˇ v(x;t;?)·t oY…?§@2v(x;t;?)=@x2 A ·t oY…?§ “ z @v(x;t;?) @t flfl fl t=?+0 t=??0 = f(x;?): =@v(x;t;?)=@t3t = ? oY'dut < ? u ] ^§E?u? G § @v(x;t;?) @t flfl fl t=??0 = 0; ?– § @v(x;t;?) @t flfl fl t=?+0 = f(x;?): ? § ‰)flK￥ ' § g =￡ —^ §= ‰)flKU ?(–e t = ? +0{ / t = ?) @2v @t2 ?a 2@2v @x2 = 0; 0 < x < l; t > 0; vflflx=0 = 0, vflflx=l = 0, t ? 0; vflflt=? = 0, @v@t flfl flfl t=? = f(x;?), 0 ? x ? l; kv(x;t;?)flflt<? = 0'?p 5? · ?EO § ] ^QO\ § g §qO\—— ' n ?a§ ‰n r?) g §! g>^ g—^ ‰)flK=z ? ) g §! g>^ ! g—^ ‰)flK§ U\= § u(x;t) = Z 1 0 v(x;t;?)d? = Z t 0 v(x;t;?)d?: y3$^ {?#?) ? ~2'? v(x;t;?)A v‰)flK @2v @t2 ?a 2@2v @x2 = 0; 0 < x < l; t > 0; vflflx=0 = 0, vflflx=l = 0, t ? 0; vflflt=? = 0, @v@t flfl flfl t=? = A0 sin!?, 0 ? x ? l; ' u3? ?… §l ? UC§ – ? ( ' 14.4 ‰u r‰ ? 124 N· ) v(x;t;?) = 1X n=1 h Cn sin n…l a(t??)+Dn cos n…l a(t??) i sin n…l x·(t??): d—^ –‰ Dn = 0; Cn = 2n…aA0 sin!? Z l 0 sin n…l xdx = 2A0l(n…)2a￡1?(?)n?sin!?: ?–§ = u(x;t) = Z t 0 v(x;t;?)d? = 4A0l…2a 1X n=0 1 (2n+1)2 sin 2n+1 l …x ￡ Z t 0 sin!? sin 2n+1l …a(t??)d? = 4A0l 2 …2a 1X n=0 1 (2n+1)2 1 [(2n+1)…a]2 ?(!l)2 sin 2n+1 l …x ￡ ? (2n+1)…a sin!t?(!l)sin 2n+1l …at ? : { –A^u?) g9D §! g>^ —^ ‰)flK§~X @u @t ?? @2u @x2 = f(x;t); 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = 0; 0 ? x ? l: ? A ?] 9 9D flK v g §! g>^ g —^ ' ? { ’ ‰)flK' ? ){n ￥%g · {? | …?fXn(x); n = 1;2;3;￠￠￠g§ ?| …?· §@o§ – )u(x;t)9 g § g f(x;t) U …?—m u(x;t) = 1X n=1 Tn(t)Xn(x); f(x;t) = 1X n=1 gn(t)Xn(x); , 2 {? Tn(t)= 'duTn(t)· …?§§ v ·~ ' §(|)§k U’? ) ' §5 { ' …?|fXn(x)g { {· JfXn(x)g A g‰)flK … 14.4 ‰u r‰ ? 125 ?§= vd g ' § g>.^ @2u @t2 ?a 2@2u @x2 = 0; 0 < x < l; t > 0; uflflx=0 = 0; uflflx=l = 0; t ? 0 'lC flK X00n(x)+?nXn(x) = 0; Xn(0) = 0; Xn(l) = 0: ru(x;t) f(x;t) —m“ \ ' §§?ˉ ?§ 1X n=1 T00n(t)Xn(x)?a2 1X n=1 Tn(t)X00n(x) = 1X n=1 gn(t)Xn(x): |^Xn(x)? v ~ ' §§qz? 1X n=1 T00n(t)Xn(x)+a2 1X n=1 ?nTn(t)Xn(x) = 1X n=1 gn(t)Xn(x): 2 …? 5§ Tn(t)? v ~ ' § T00n(t)+?na2Tn(t) = gn(t): § u(x;t) —m“ \—'^ § 1X n=1 Tn(0)Xn(x) = 0; 1X n=1 T0n(0)Xn(0) = 0: …? 5§=U Tn(0) = 0; T0n(0) = 0: ^) g~ ' § ~?C·{§‰ ^LaplaceC § –? Tn(t) = ln…a Z t 0 gn(?)sin n…l a(t??)d?: ?1n?){§? U A gflK …?—m{' 2^?? {?)~2￥ ‰)flK @2u @t2 ?a 2@2u @x2 = A0 sin!t; 0 < x < l; t > 0; uflflx=0 = 0, uflflx=l = 0, t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l: ) A gflK …?fi314.1!￥ §ˇd u(x;t) = 1X n=1 Tn(t)sin n…l x; 14.4 ‰u r‰ ? 126 g A0 sin!t U? | …?—m§ A0 sin!t = 2A0… 1X n=1 1?(?1)n n sin n… l xsin!t; \ § —'^ § T00(t)+ ?n… l a ·2 Tn(t) = 2A0… 1?(?1) n n sin!t; T(0) = 0; T0(0) = 0: ) = Tn(t) = 2A0l 2 … 1?(?1)n n 1 (n…a)2 ?(!l)2 sin!t ?2A0!l 3 …2a 1?(?1)n n2 1 (n…a)2 ?(!l)2 sin n… l at: ˇdq –? ~2 , ?/“ ) u(x;t) = 4A0l 2 … 1X n=0 1 2n+1 1 [(2n+1)…a]2 ?(!l)2 sin 2n+1 l …x sin!t ? 4A0!l 3 …2a 1X n=0 ? 1 (2n+1)2 1 [(2n+1)…a]2 ?(!l)2 ￡sin 2n+1l …x sin 2n+1l …at ? : ?u?‰flK§~X§Poisson § 1 a> flK @2u @x2 + @2u @xy2 = f(x;y); 0 < x < a; 0 < y < b; uflflx=0 = 0, uflflx=a = 0, 0 ? y ? b; uflfly=0 = 0, uflfly=b = 0, 0 ? x ? a; , ^U A gflK …?—m {?)'~X§ u(x;y) = 1X n=1 Yn(y)sin n…a x; f(x;y) = 1X n=1 gn(y)sin n…a x: \ § >.^ § Y00n (y)? ?n… a ·2 Yn(y) = gn(y); Yn(0) = 0; Yn(b) = 0; ? Yn(y)§ )u(x;y)' d/§ – u(x;y) = 1X m=1 Xm(x)sin m…b y; f(x;y) = 1X m=1 hm(x)sin m…b y; 14.4 ‰u r‰ ? 127 Xm(x) v g~ ' §> flK X00m(x)? ?m… b ·2 Xm(x) = hm(x); Xm(0) = 0; Xm(a) = 0; ? Xm(x)= ' ? ? {vk? O' · g gn(y) hm(x) …?/“ U §ˇ 3’uYn(y) Xm(x) g ~ ' §￥k ·u? )' – ? ? {§= u(x;y) f(x;y) QU …?fXn(x)g!qU … ?fYm(y)g—m( ???) u(x;y) = 1X n=1 1X m=1 cnm sin n…a xsin m…b y; f(x;y) = 1X n=1 1X m=1 dnm sin n…a xsin m…b y; —mX?cnm ?'ˇ f(x;y)·fi …?§?–cnm ·fi '3 ???—m § ,fi? ? >.^ ' ? —m“ \ §§= ? 1X n=1 1X m=1 cnm ??n… a ·2 + ?m… b ·2? sin n…a xsin m…b y = 1X n=1 1X m=1 dnm sin n…a xsin m…b y: …? 5§’ X?§= ?cnm ??n… a ·2 + ?m… b ·2? = dnm: u· cnm = ? dnm?n… a ·2 + ?m… b ·2: §? ) u(x;y) = ? 1X n=1 1X m=1 dnm? n… a ·2 + ?m… b ·2 sin n…a xsin m…b y: ?? { —?· ; ?) g~ ' §' 14.5 g>.^ gz 128 14.5 g>.^ gz 8c § 3?‰flK￥I k '>.^ ^u‰U\X?!ˇ # N· g – §? o· ?>.^ · g ' o>.^ 7L· g ” ? g>.^ U'lC ? k v g § g>.^ A)U \ 5 EU v g § g>.^ ? ˇ 9 …? 5 g>.^ X ?n” E–ˉ? § ‰)flK ~' g>.^ ?n§b‰ § —'^ · g ' @2u @t2 ?a 2@2u @x2 = 0; 0 < x < l; t > 0; uflflx=0 = ?(t), uflflx=l = ”(t), t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l: A^'lC {§O? J§ kk g>.^ gz§=- u(x;t) = v(x;t)+w(x;t); ? Jv(x;t)§? v v(x;t)flflx=0 = ?(t); v(x;t)flflx=l = ”(t): ? §w(x;t) , ‰ v g>.^ w(x;t)flflx=0 = 0; w(x;t)flflx=l = 0: ‘5§w(x;t)? v § —'^ · g § @2w @t2 ?a 2@2w @x2 = ? @2v @t2 ?a 2@2v @x2 ? ; wflflt=0 = ?vflflt=0; @w@t flfl flfl t=0 = ? @v@t flfl flfl t=0 : ^114.4! {§ –? w(x;t)§ )u(x;t)' FX gz…?v(x;t)” ˇ = ?v(x;t) v>.^ v(x;t)flflx=0 = ?(t); v(x;t)flflx=l = ”(t); 14.5 g>.^ gz 129 ?–k J{/' XJrtw?·o?§? ?3(x;y)?? ? y = v(x;t) ˇL ‰ :(0;?(t)) (l;”(t))= ' ~X§ v(x;t) = A(t)x+B(t); \>.^ §= ‰ B(t) = ?(t); A(t) = 1l￡”(t)??(t)?: v(x;t) = A(t)x2 +B(t); A(t) = 1l2￡”(t)??(t)?; B(t) = ?(t); ‰ v(x;t) = A(t)(l?x)2 +B(t)x2; A(t) = 1l2?(t); B(t) = 1l2”(t): ~4 ?)‰)flK @u @t ?? @2u @x2 = 0; 0 < x < l; t > 0; uflflx=0 = Asin!t, uflflx=l = 0, t ? 0; uflflt=0 = 0; 0 ? x ? l: ) ? g>.^ N/“§ gz…? v(x;t) = A ? 1? xl · sin!t: u·- u(x;t) = A ? 1? xl · sin!t+w(x;t); Kw(x;t) v‰)flK @w @t ?? @2w @x2 = ?A! ? 1? xl · cos!t; 0 < x < l; t > 0, wflflx=0 = 0, wflflx=l = 0, t ? 0; wflflt=0 = 0; 0 ? x ? l: w(x;t) § g 1?x=l U A gflK …?—m§k w(x;t) = 1X n=1 Tn(t)sin n…l x; 1? xl = 1X n=1 2 n… sin n… l x: 14.5 g>.^ gz 130 Tn(t)?AT v g ~ ' § T0n(t)+? ?n… l ·2 Tn(t) = ?2A!n… cos!t —'^ Tn(0) = 0; N·? Tn(t) = 2A!l 2 n… 1 ?2(n…)4 +!2l4 ‰ ?(n…)2 exp h ? ?n… l ·2 ?t i ??(n…)2 cos!t?!l2 sin!t : ? ? w(x;t)§2 ￡ § ‰)flK )u(x;t)' J gz…?v(x;t)§ w(x;t) ‰)flK , §? w(x;t) ' ·§‰)flK ) 3 5§ y u(x;t) ‰· §?+L “ /“ Uk? ' ? –J p ? J ? gz…?v(x;t)§?w(x;t)? v ‰) flK? U{ ' n ?§ , · 5u(x;t) §· · g § “w(x;t) §· g ' ? ‰)flK §?? X ? gz…?v(x;t) · § )§ @2v @t2 ?a 2@2v @x2 = 0: ?u, Aˇ ?(t) ”(t)§· – ? : ' 5 §· · g §? r?? { ? § >.^ gz' ~5 ?)‰)flK @2u @t2 ?a 2@2u @x2 = 0; 0 < x < l; t > 0; uflflx=0 = 0, @u@x flfl flfl x=l = Asin!t, t ? 0; uflflt=0 = 0, @u@t flfl flfl t=0 = 0, 0 ? x ? l: ) y3 `a? gz…?§ § >.^ gz' d§ u(x;t) = v(x;t)+w(x;t)§ ? g>.^ N…?/“§ gz …?v(x;t) v(x;t) = f(x)sin!t; f(x)·e ~ ' §> flK f00(x)+ ?! a ·2 f(x) = 0; f(0) = 0; f0(l) = A 14.5 g>.^ gz 131 )§ f(x) = Aa! 1 cos !la sin !ax: w(x;t)? v ‰)flK· @2w @t2 ?a 2@2w @x2 = 0; 0 < x < l; t > 0; wflflx=0 = 0, @w@x flfl flfl x=l = 0, t ? 0; wflflt=0 = 0, @w@t flfl flfl t=0 = ? Aa cos !la sin !ax, 0 ? x ? l: ) w(x;t) = 1X n=0 Cn sin 2n+12l …at+Dn cos 2n+12l …at ? sin 2n+12l …x: —'^ § –‰ Cn = ? 4A …cos !la 1 2n+1 Z l 0 sin !axsin 2n+12l …xdx = (?)n 4A!(2n+1)…a 1?! a ·2 ? 2n+1 2l … ?2; Dn = 0: ? ? v(x;t) w(x;t) \§ )u(x;t)'