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WuChong-shi ( ) §14.1 F Newton !#$%&’()* +, - .F/01* (2+, - 34567#893:;=?@A.BC DE67# FGHI=J2KLEM3NOG%P’(* .F =Q67#3R%&’()S* )TUV(2WX YZ[#.F DQ67# FG3I=J2KL\M.]^3Q67#∂2u(x,y)∂x2=0 FG_Pu(x,y)=c1(y)+xc2(y),‘Hc1(y)ac2(y)Py KL\M.b=#3OG$%’(.cYZdef3RPgh 3NiFjkl#m3 no5p+ qr3$s=nop+F[tua9v wxyz.N{3|}~e3 .F^ m = 3jkl#mR$s=nop+ .^ m ,3 3_ ~3 t0¡¢£⁄¥.i501TU(2W=* G YZ 3jM!d_P/ƒ§(2 G 3¤4567#893:;='v@Aa?@A.“«‹›“«‹›fifl –†‡“«⁄¥(t=0⁄) · ¶•¢‚£„¡”».F…‰ #e~3_P¿ ` ?m´ ˆ˜a¯˘(^ P! )3u t=0=φ(x,y,z),∂u∂t t=0=ψ(x,y,z),(x,y,z)∈V.F…‰˙¨l#3 ‰#H ˚¸ \Mu(x,y,z,t)…t (DQ6˝3˛ ˇ` WuChong-shi§14.1— 2 ?m´ ˘u t=0=φ(x,y,z),(x,y,z)∈V. ¶‹› 3/ WX HTÆ WX .ª¡ ¶‹›fifl –†‡ ¶•Ł„Ø¢£⁄¥(t≥0)¡”».Œ¡º ^ æ 3C'v@A_Pu x=0=0,u x=l=0,t≥0.ı¡ x=0æ 3Cx=0ł 'v@AøPu x=0=0. œ£æ(x=l)ßx •¡ 3 ˆu d PF(t). kl# 3jł,x=l ( p+3 ˘iε. Newton 3¤( p+˛ (9 q¿)3¿ ‰p+ + p+HV(, ¯˘3ρεS∂2u∂t2 x=l−αε=F(t)S−P(l−ε,t)S, 14.1 ‘H0≤α≤1. ε→03$ P=E∂u∂x,C=∂u∂x x=l=1EF(t).•^ 9i03x=læ !¡3C∂u∂x x=l=0.•^ 9F(t)%P(2* # \M3OP $%&’ $(3CF(t)=−ku(l,t)−u0,kP$% )˘*M3‰P3∂u∂x+kEux=l=kEu0.+,-./E0 'v@A=1 2345 F ¶•Ł„¡67893u Σ=φ(Σ,t).WuChong-shi:;=()?@ABCDEFG 3 ΣtH'vd I,3JmRtH¤K, LM.FNO⁄P SQRNOST¡ ¶SUV¡+W89.j'vqX ( YZp+3(2[ujp+ tu3\(2[ujp+qr.]X J[u w 3^˘_‰0. & ‘a 3bcdefghijhklmnopq3rstuvwxbcyz{|}~ mno. P3 p+ ^˘_‰0m3F[Xu ˙ ¿ _‰0(NiXu _‰0)3p+ ˙ _‰0(Nip+ + _‰0)3N{3F[p+tu ˙ 3¿ 1rF[YZ \([u p+qr.‰P3 U 'v@A∂u∂n Σ=1kψ(Σ,t), 14.2 ‘H∂∂n i 6˝3 P ˘ j9 nd 3∂∂n=n·∇∂u∂n=n·(∇u).• ¶ +3Cψ≡03∂u∂n Σ=0.F QR ¶ Newton + NO⁄PQRNOST S ¶ ¡+W S67u|Σ ¶67u0 ¡¢£.⁄ ]*MiH3−k∂u∂n Σ=H u Σ−u0,¥ƒU§∂u∂n+huΣ=hu0.du ˚ 'v@A=(2§J ¤, _¸ \MO'3 “P ( .«‹(›fi73: 7ifl4 F w–†‡· ¶•†‡‚„”m»…‰F –†‡· ¶•†‡‚„”»…m¿ `´‰F ˆ–†‡· ¶•†‡‚„”m»…‰˜¿ `´‰p¯m˘˙¨ WuChong-shi§14.1— 4 ˚¸f†‡h‚3„”m†‡· ˝w˛ˇ— wm 3 ˝ uw – . ‚h m ˙Æm†‡· 3 ª .F ¶ P¡./¤m 'v@A_¿ ŁØŒ9º Ø ¡ æ 3]^\M ı lMj ł =v.˚—‡ø¯mœß 3— • —‡· . 3 S 3 `´∂u/∂r˚ ” . ~ !#$%&’#$(WuChong-shi)*+,(-)./0123456758§14.29:;=9?@ABCDE FGHIJGKL@J MNJ OPKQFJRFGJL@SFGHITG(UVWXYZ[\ ]^_‘ aWXbcdX(eU UVfghijklmnoW pfghqijrlnstuvWlwxyz({m|ij}~]^_‘l(FGJMNSFGHIJGKMNJ(UVWXYZ WXbclX} ~ fl( #$ (FGJQFS ¡FGHI¢J£⁄CD(¥ ƒ§¤FGCD¢J£⁄'“)T«‹›fifl G –T«‹J›fi(†‡·WXbc ¶•‚„”iy»…‰ ¿( ` ´ˆ†˜W¯˘˙¶l¨ ˚˘yl»…‰ ¿¸~ˆ ˝l(˛ˇFGHIGJL@S—MNS QFS FS(– HIJ K J CDJ KÆ ª— Fª fl (ŁØŒºt=0) æ ı NłJøœ CDJ KÆ ßP Fª ı Nł@t≥0Jøœ B JFGHI NFK FJ K GNFKL@J—MNJ OPKQFJ( ^ lbc~ CD CD¢ J£⁄'“ NFJ S ( bc e(UV YZ~u(x,y,z,t) Σ=f(Σ,t), YZ~u(x,y,z,t) t=0=φ(x,y,z), } ˆf(Σ,t) t=0=φ(x,y,z) Σ.ˆ WXbc fW {|ij(¶ ! #$ φ(x,y,z)lf%&’(uf|)*+(eU ) u0), - .&’/hl 0123u)u0 UVijl4 ˙¶ &’/h5678[9l:;¶ = }¶ »„? YZ@Au(x,y,z,t) Σ=u0.WuChong-shi§14.23456BC3D768{EFlGV HI‰4vl YZJˆKL M´i{NKL OP Xl˜W¯}QRST U VWlXlKX0~ OPl( ` UVST}~iYZ{N5678[9l:; {N ¿}~ ¶[l(WuChong-shi)*+,(\)0]^_‘778abcd(e)fghijklmno pqr s mtuvs( qlmno w xqy z {lmno | }~ | {x r #$( #$) { ( ~ klmno ns ~ qlmno ( klmno ! z {klmno x } x !¡ ¢ £⁄¥ ƒ §¤'“ #$ {( « klmno ‹›fi fl –† ‡·(§14.3 ¶•9‚=„”»…FGHI‰ ¿ l—mnoWlkl U`´ g9ˆWXYZ ∂2u∂t2−a2∂2u∂x2=0,0xl,t0,u x=0=0,u x=l=0,t≥0,u t=0=φ(x),∂u∂t t=0=ψ(x),0≤x≤l.g9‰ YZ˜~¯˘l YZ~˙¯˘l(¨ ˚¸ | !mtuv ˝˛ u(x,t)=X(x)T(t).F?u(x,t)ˇ—g9 X(x)T00(t)=a2X00(x)T(t). mn X(x)T(t) 1a2T00(t)T(t)=X00(x)X(x).†{| , n´~tl ( xd ) n´~xl ( td ) n‰ n^ } Æ ªf| xd —q td l ( −λ hlGV}¶ …AT00(t)+λa2T(t)=0,X00(x)+λX(x)=0.WuChong-shi§14.3 ŁØŒº ØŒ788F?u(x,t)ˇ— YZ X(0)T(t)=0,X(l)T(t)=0.{ ˆX(0)=0,X(l)=0.{E} A z# æ jX ı#g9WXbcl f łøfiœFß # æ l˙ Xu(x,t)=X(x)T(t)F ¡ X(x) l ı#g9‰ YZ ˆT(t) l ı#g9FCD ı#g9‰ YZ˜~¯˘l † l X(x)l ı#g9WXbc s~ ı#g9, ˆ W λ WXYZ~f ¯˘ YZ({ElWXbc ª ı#g9l bc( ˙ ª λ ˜ˆ ¯˘ ı#g9—q ¯˘ YZl˙ X(´ˆ λ[ W ¸ˆ ¯˘ ı#g9—q ¯˘ YZl˙ XX(x)(λl{ W ^ l˙ X '“( X(x)l ı#g9WXbc HI( G HIF λ=0 ı#g9l X~X(x)=A0x+B0.ˇ— YZX(0)=0,X(l)=0,}¶ W A0=0,B0=0.{ λ=0 ı#g9´ˆ X( λ=0 ~ (F λ6=0 ı#g9X00(x)+λX(x)=0WuChong-shi)*+,(\)0]^_‘798l X~X(x)=Asin√λx+Bcos√λx,ˇ— YZ }ˆB=0,Asin√λl=0. A6=0 ˆ√λl=nπ λn=nπl2,n=1,2,3,···.^ l }~Xn(x)=sinnπlx.{Ej l ˆd | T¶ z n !#$ % &’(! ˜ )λn&Xn(x)*