Green a0 a1 star a2a3a4a5a6a7a8a9a10 6 star a11a9 Green a12a13a14a15 (§20.5 a16 §20.6) a17 a18 a6a19a20 1 Green…? { 11 1 Green…? {{ 31 /–…?0￥§fi?— >LGreen…?§? ~ '' §Green …? ‰‰′!??5 9 ?{' UY? {K§? ' §‰)flKGreen…? Vg!??5 – 9~^ ?{' G~ '' §Green…? §A k /– …?0@ ' F Green1 ?“(‰{?Green?“) ZZZ V h u(r)r2v(r)?v(r)r2u(r) i dr= ZZ § h urv ?vru i ￠d ; ￥f(r) · f(x;y;z); dr= dxdydz; §·V >.?§? 5‰ { ' x20.1 Green…? Vg 12 x20.1 Green…? Vg k ?>| ~f' 3?. m￥k ‰ > ' §> ‰(r)'? §3 I r0 = (x0;y0;z0) N dr0S > = ‰(r0)dr0§§3 mr= (x;y;z): >?· 1 4…"0 ‰(r0) jr?r0jdr 0; >?U\ n§r m￥ > ) >?U\ 5§ 3r: o>? `(r) = 14…" 0 ZZZ ‰(r0) jr?r0jdr 0: ? (J‘?§ :> 3 m >?' §@o§ˇL> ' U\§ – ??> ' >>?' ?? { L·|^ '' § 5555 ' FXJ·k. m§ K E, –r mS > ? ' ' Fdu>.^ §3>.? ?k ‰ ( ‰ 4 ) a)?> ' § I ? ?> ? ' ' F /(‰(k. mS):> >?§ I ‰? >.^ ' 3k. m /e§flK · X ˇL(? >.^ e ):> >? U\§ ? ?> ' ??>.^ >?' ? ·‘§ ^‰)flK r2G(r;r0) = ? 1" 0 –(r?r0); r;r0 2 V ? >.^ )G(r;r0)U\ r2u(r) = ? 1" 0 ‰(r); r2 V uflfl§ = f(§) )u(r)§=ru(r)^‰(r); f(§)–9G(r;r0)L? 5' d§? G(r;r0) u(r) v §'O?–u(r) G(r;r0)§ ~§23 mVS ¨'§= ZZZ V ￡u(r)r2G(r;r0)?G(r;r0)r2u(r)?dr = ? 1" 0 ZZZ V ￡u(r)–(r?r0)?G(r;r0)‰(r)?dr = ? 1" 0 " u(r0)? ZZZ V G(r;r0)‰(r)dr # : x20.1 Green…? Vg 13 Green?“§ – “ N¨'z ?¨'§ZZ § ￡u(r)rG(r;r0)?G(r;r0)ru(r)?￠d : ?L￡ ! n§ k u(r0) = ZZZ V G(r;r0)‰(r)dr ?"0 ZZ § ￡u(r)rG(r;r0)?G(r;r0)ru(r)?￠d : 3 ? ?¨'￥§ F1 u(r)3>.?§ ? d>.^ §·fi ? G(r;r0) d‰)flK? § § F rG(r;r0)9 3>.? ? , ?? F1 ￥§ru(r)3>.? ? § ?–§ U ru(r)^‰(r); f(§) –9G(r;r0)L? 5§7L?G(r;r0)\ g>. ^ G(r;r0)flfl§ = 0: u·§ u(r0) = ZZZ V G(r;r0)‰(r)dr?"0 ZZ § f(§)rG(r;r0)flfl§ ￠d ; ‰ rr r0? e§ u(r) = ZZZ V0 G(r0;r)‰(r0)dr0 ?"0 ZZ §0 f(§0)r0G(r0;r)flfl§0 ￠d 0 = ZZZ V0 G(r0;r)‰(r0)dr0 ?"0 ZZ §0 f(§0) @G(r 0;r) @n0 flfl flfl §0 d§0; ￥ r0 @=@n0L??gC r0 ?§V0 §0 · 5 m? § >.?§ L· r§ IC U? r0' ? F G(r;r0)3r=r0: oY§ UA^Green?“? ? (J·? (” F ? " § – G(r;r0)? v §?U r2Gn(r;r0) = ? 1" 0 –n(r?r0); r;r0 2 V: m > …?–n(r?r0)·v — oY…?§3r0NC ‰” S?w 0§ o> 1 ' n !1 –n(r?r0) ! –(r?r0)' ? ? –A^Green?“' ? ?E ? {§, 2-n !1' x20.1 Green…? Vg 14 F \–…? —?TT 3u ??4 L§§TT 3u –r– …? ?oY…? 5?n' Fˇd§ ? (J· §· ( ' F, ? {·r:> ?3 r0: NC N¨§3? # m ? ￥A^Green?“(7L5?§y3 >.? 5 § § k3r0:? . ?)§, 2-? N¨“u0' – ˇL?>| ￠~ \ Poisson §31 a>.^ e({?Poisson § 11 > flK) Green…?' { §?￠Green…? · ::>> 3 g>.^ e >>?' ?u ?a. >.^ § KK –aq/? ' l?? ‘§ ? m(?‰flK) ' §(Laplace §, Poisson §, Helmholtz §￠￠￠￠￠￠)3 ‰>.^ e Green…? –‰′ Aˇ ‰)flK ) ? § 5‰)flK § § · g U –…?(: )? ? ?a. g>.^ ' ·§3, Aˇ /e§? ‰′ Green…? U?)' ~X?u ? Poisson §‰)flK§e>.^ U @u(r) @n flfl fl § = f(§); KU ? ? §Green…?G(r;r0)3>.? 7L v g 1 a>.^ @G(r;r0) @n flfl fl § = 0: (#) 3Green?“￥-u(r) = 1; v(r) = G(r;r0)§ATk ZZZ V r2G(r;r0)dr= ZZ § rG(r;r0)￠d = ZZ § @G(r;r0) @n d§; §¨'§ ZZZ V r2G(r;r0)dr= ? 1" 0 : ? §Green…?G(r;r0)3>.? ?¨'7L v(Gauss‰n) ZZ § @G(r;r0) @n d§ = ? 1 "0 6= 0: w, >.^ (#)g?'?‘?§3 g 1 a>.^ (#)e§ § ‰?)§ ?{ ‘§? Green…? ‰ 3'3?? /e§I ?2′ Green…?' x20.2 ?‰flKGreen…? 5 15 x20.2 ?‰flKGreen…? 5 ? ?‰flK Green…?Vg § I ? § 5 Green…?3: N C 1 –9Green…? ??5' 1. Green…?3: NC 1 E,^?>| 5￡aPoisson §1 > flK Green…?'l ! ' –w §3 mV￥ :> §7, 3>.? ) ‰ a)(?)> ' §l ? >.?? ?' >. / §q? k '> 6 ‰6\§? >.? >? / ( 0)'ˇd§?‰Green…? ‰)flKq – d(3VS d)/ ??. m￥ Poisson § r2G(r;r0) = ? 1" 0 ￡–(r?r0)+ (§)?; ￥ (§)·>.?§ a)?> ' A/§(‰′3VS )Green…?G(r;r0) AT ·? '> >? U\ :> –(r?r0) >?G0(r;r0) >.? a)> (§) >?g(r;r0)§ r2G0(r;r0) = ? 1" 0 –(r?r0); G0(r;r0) = 14…" 0 1 jr?r0j ?–§G0(r;r0)3r=r0 :· oY ' r2g(r;r0) = ? 1" 0 (§): ˇ a)> (§) ' 3??§ §?–§g(r;r0)9 ?3??§ (AO·§3VS)·????oY ' r? 'n 5§ k G(r;r0) = 14…" 0 1 jr?r0j +g(r;r 0): ?u1na>.^ § k (J' Lg(r;r0) NL “? k? ' ?u ?a. ?‰flK§~XHelmholtz § Green…?§ r2 ?G(r;r0)+k2 ?G(r;r0) = ? 1" 0 –(r?r0); r;r0 2 V; ?G(r;r0)flfl § = 0: y?§ Green…? k Poisson § Green…? oY5 ' r=r0: § ?G(r;r0)3VS·????oY '- ?g(r;r0) = ?G(r;r0)?G(r;r0); x20.2 ?‰flKGreen…? 5 16 G(r;r0)· APoisson § Green…?'d ?G(r;r0) G(r;r0) ? v ‰)fl K§ – r2?g(r;r0)+k2?g(r;r0) = k2G(r;r0); r;r0 2 V; ?g(r;r0)flfl§ = 0: d u? § m G(r;r0) 3r = r0 :·–1=jr?r0j /“u § ? –, ?g(r;r0) 3T: ‰oY(?Kr2?g(r;r0) ? y– …?)§? ‘? ?G(r;r0) G(r;r0) §3r =r0 : ·–1=jr?r0j /“u 'fl￠ §le ! ? §3r=r0 :NC§ ‰k ?G(r;r0) ? 1 4…"0 cos(kjr?r0j) jr?r0j : ? n m￥Green…?3: ? 1 § m￥Green…? ' ? m￥ Green…?·??oY § § ? oY' ? ?·N·n) §ˇ /: 0 5 ? § m￥ : ￠S ·n m ￥ ? ' ? J § m￥ Green…? ATLy 1 ' ?u m￥ Poisson §1 > flK§§ Green…?G(x;y;x0;y0)§·‰)flK h @2 @x2 + @2 @y2 i G(x;y;x0;y0) = ? 1" 0 –(x?x0)–(y ?y0); (x;y);(x0;y0) 2 S; G(x;y;x0;y0)flflC = 0 )§ ￥C·??? S >.'N·? § G(x;y;x0;y0) = ? 12…" 0 ln p (x?x0)2 +(y ?y0)2 +g(x;y;x0;y0); ￥1 · :> 3?. m￥ >?( –\ ~?§ ?u>?": )§3/: 0(￠S ·n m￥ )–(x ? x0)–(y ? y0)?·??u ?1 g(x;y;x0;y0)·>. a)> ) >?§3SS??oY' 2. Green…? ??5 k ec? )“ u(r) = ZZZ V0 G(r0;r)‰(r0)dr0 ?"0 ZZ §0 f(§0)r0G(r0;r)flfl§0 ￠d 0: ? (J3 n?′ k?) ? 3m N¨'￥§G(r0;r) Lr? :> 3r0? >?§§? 3* :r0? > ‰(r0)dr0§??* :¨'§ % r? >>? x20.2 ?‰flKGreen…? 5 17 ?? flK ￡ 9 Green…? ??5'ˇ §XJ ?. m Green…?@ §’X“ G(r0;r) = G(r;r0) (#) ?? {§@o§ “ UU ? u(r) = ZZZ V0 G(r;r0)‰(r0)dr0 ?"0 ZZ §0 f(§0)r0G(r;r0)flfl§0 ￠d 0; N¨' n?′ '1 ?¨' , ·5g>.? a) ?> z' y?(#)“' 1 ￥ { §2 ?G(r;r00)§§ v ‰)flK , · r2G(r;r00) = ? 1" 0 –(r?r00); r;r00 2 V; G(r;r00)flfl§ = 0: §'O?–G(r;r00) G(r;r0)§ ~§, 3? VS¨'§ ZZZ V ￡G(r;r00)r2G(r;r0)?G(r;r0)r2G(r;r00)?dr = ? 1" 0 ZZZ V ￡G(r;r00)–(r?r0)?G(r;r0)–(r?r00)?dr = ? 1" 0 ￡G(r0;r00)?G(r00;r0)?: Green?“§ “ N¨'z ?¨'§ k G(r0;r00)?G(r00;r0) = ?"0 ZZ § ￡G(r;r00)rG(r;r0)?G(r;r0)rG(r;r00)?￠d : \>.^ §?= m ?¨' 0'? y? G(r0;r00) = G(r00;r0); r00U r§? ·(#)“' XJ·1na>.^ § ? ( E, (' ?u ?a. ?‰flK§§ Green…?·?E,k??’X(#)§I N? 'l K ‘§? ?G(r;r0) G(r0;r) · § )§‰ ‘§ §3C r r0e· C ' x20.3 n ?. mHelmholtz § Green…? 18 x20.3 n ?. mHelmholtz § Green…? ?n ?. m￥Helmholtz § Green…?§=3n ?. m￥?) § r2G(r;r0)+k2G(r;r0) = ? 1" 0 –(r?r0); r;r0 2 V: ’u?? ? >.^ § ?2? ' ? §· g §§ˇd§ –U ?) g § IO {§ Fk? § A)§ § gz? F G(r;r0)U A gflK …?—m' ? ? {§AO·1 ? {§ KK vk o(J§?p N 0 ' F?q· Aˇ g § 3r=r0:§ g 0' F §du?·3?. m§ –? /S Ie§–?'u Laplace ? C 5§?flK ?' {z' ?k I?￡§ ? = x?x0; · = y?y0; ? = z ?z0; = :> ?3: # IX :'-G(r;r0) = g(?;·;?)§u·§g(?;·;?) v § r2?;·;?g(?;·;?)+k2g(?;·;?) = ? 1" 0 –(?)–(·)–(?); ￥ r2?;·;? · @ 2 @?2 + @2 @·2 + @2 @?2 ·– I?;·;? gC Laplace ?'N·w §C §·^= C , g(?;·;?) ·R = p?2 +·2 +?2 …?, g(?;·;?) = f(R). ˇd§XJ IX(?;·;?) = ￥ IX§K § C R 6= 0:? g § 1 R2 d dR h R2df(R)dR i +k2f(R) = 0 ( ˇ·33R = 0: 3 ?)–9R = 0:? >.^ (3R = 0:?k :> )'d §·" ￥Bessel §§§ ˇ)· f(R) = A(k)e ikR R +B(k) e?ikR R : R = 0 ?? ? >.^ ‰ ~?A(k) B(k)' XJ C f(R) = w(R)=R§K §z w00(R)+k2w(R) = 0: N· ˇ)' x20.3 n ?. mHelmholtz § Green…? 19 ?? ^ ‰B(k) ? Helmholtz § ￠S §’X‘§§·dˉ? §?L' lC ('l m ') ' ~f§b ? )3?? ? u ˉ' mˇf e?i!t§K)“￥ 1 u ˉ§1 ? ˉ'?–§ATkB(k) = 0' e ~?A(k) ATdR = 0? >.^ ?‰§=dR = 0?: r ?‰' R = 0? >.^ ‰A(k) ? ? U )“ \R = 0? >.^ § ˇ ·f(R)‰g(?;·;?)3R = 0? ?? 3', ?§? fi? ‰§ · 9–…? “ ATl¨'?′e n)'u·§?g,/§A §3R = 0NC N¨S¨'§ ZZZ r2?;·;?f(R)d?d·d? +k2 ZZZ f(R)d?d·d? = ? 1" 0 : (z) 1 N¨'A z ?¨' ZZZ r2?;·;?f(R)d?d·d? = ZZ h r?;·;?f(R) i ￠d ; ˇ ? –￡;K3R = 0: ? flK' ? N¨ –R = 0: ￥%§‰ ? ￥N§K ZZZ r2?;·;?f(R)d?d·d? = ZZ h r?;·;?f(R) i ￠d = ZZ df(R) dR R 2 sin d d` flfl fl R=‰ = ?4…A(k)(1?ik‰)eik‰: 1 N¨' – § ZZZ f(R)d?d·d? = 4…A(k) Z ‰ 0 eikRRdR = 4…A(k)k2 h (eik‰ ?1)?ik‰eik‰ i : ? (J ￡ (z)“§ k ?4…A(k) = ? 1" 0 ; ?–, A(k) = 1=4…"0, k ?’'? § ? n ?. mHelmholtz § Green …? g(?;·;?) = f(R) = 14…" 0 eikR R ; ‰ G(r;r0) = 14…" 0 eikjr?r0j jr?r0j : k = 0 §? (J ￡ Poisson § Green…?' §I ‘?§? (J·3?? ? u ˉ§? mˇf e?i!t ^ e ' – §XJ ??? ? ? ˉ§? E mˇf e?i!t§KGreen…?AT · G(r;r0) = 14…" 0 e?ikjr?r0j jr?r0j : XJ· ?/“ ?? ^ § , ? ?/“ )' x20.4 SPoisson §1 > flK Green…? 110 x20.4 SPoisson §1 > flK Green…? ! 8 ·ˇL?u SPoisson §1 > flKGreen…? ? §20 ?Green…? ~^ {{' SPoisson §1 > flKGreen…? ‰′· r22G(r;r0) = ? 1" 0 –(r?r0); jrj < a; jr0j < a; G(r;r0)flflr=a = 0; ￥ r2 = x2 +y2; r22 = @ 2 @x2 + @2 @y2: k0 ?IO {§= ? §· g §§?– Green…?U A g flK …?—m' ^??4 IX§ I : 3 %§ G(r;r0) = R0(r)+ 1X m=1 ￡R m1(r)cosm`+Rm2(r)sinm` ?: § –…? UT| …?—m§ –(r?r0) = –(x?x0)–(y?y0) = 1r0–(r?r0)–(`?`0) = 1r0–(r?r0) ‰1 2…+ 1 … 1X m=1 ￡cosm`cosm`0 +sinm`sinm`0? : y3 flK ·X ?)R0(r); Rm1(r) Rm2(r)' F?‰R0(r) ~ ' §‰)flK· 1 r d dr ? rdR0(r)dr ? = ? 12…" 0 1 r0–(r?r 0); R0(0)k.; R0(a) = 0: r 6= r0 § §· g §3 ? >.^ §k) R0(r) = 8 < : A0; r < r0; B0 ln ra; r > r0: 2 R0(r)3r = r0: oY5§=R0(r)3r = r0:oY§ R00(r) oY(§ –d §3r = r0: ¨' )§ dR0(r) dr flfl flfl r0+0 r0?0 = ? 12…" 0 1 r0; –‰ A0 B0§ A0 = ? 12…" 0 ln r 0 a ; B0 = ? 1 2…"0: x20.4 SPoisson §1 > flK Green…? 111 u· R0(r) = 8 >>< >>: ? 12…" 0 ln r 0 a ; r < r 0; ? 12…" 0 ln ra; r > r0: F?‰Rm1(r) ~ ' §‰)flK· h1 r d dr r ddr ? ? m 2 r2 i Rm1(r) = ?–(r?r 0) …"0r0 cosm` 0; Rm1(0)k.; Rm1(a) = 0: r 6= r0 § §· g §3 ? >.^ §k) Rm1(r) = 8 < : Am1 ?r a ·m ; r < r0; Bm1 h?r a ·m ? ?a r ·mi ; r > r0: Rm1(r)3r = r0: oY5§=Rm1(r)3r = r0 :oY§ R0m1(r) oY§ dRm1(r) dr flfl flfl r0+0 r0?0 = ? 1…" 0 1 r0 cosm` 0; ‰ Am1 Bm1§ Am1 = ? 12…" 0 1 m ??r0 a ·m ? ?a r0 ·m? cosm`0; Bm1 = ? 12…" 0 1 m ?r0 a ·m cosm`0: u· Rm1(r) = 8 >>< >>: ? 12…" 0 1 m h?rr0 a2 ·m ? ?r r0 ·mi cosm`0; r < r0; ? 12…" 0 1 m h?rr0 a2 ·m ? ?r0 r ·mi cosm`0; r > r0: F?‰Rm2(r) ~ ' §‰)flK· h1 r d dr r ddr ? ? m 2 r2 i Rm2(r) = ?–(r?r 0) …"0r0 sinm` 0; Rm2(0)k.; Rm2(a) = 0: § Rm1(r) v ~ ' §‰)flK /“A § ·r g ￥ cosm`0 ? sinm`0§?–§ Rm2(r) = 8> >< >>: ? 12…" 0 1 m h?rr0 a2 ·m ? ?r r0 ·mi sinm`0; r < r0; ? 12…" 0 1 m h?rr0 a2 ·m ? ?r0 r ·mi sinm`0; r > r0: x20.4 SPoisson §1 > flK Green…? 112 ? § ? SPoisson §1 > flK Green…?§ r < r0 § G(r;r0) = ? 12…" 0 ‰ ln r 0 a + 1X m=1 1 m h?rr0 a2 ·m ? ?r r0 ·mi cosm(`?`0) ; r > r0 § G(r;r0) = ? 12…" 0 ‰ ln ra + 1X m=1 1 m h?rr0 a2 ·m ? ?r0 r ·mi cosm(`?`0) ; ??? {§ Green…?U A gflK …?—m§ ‘5§ )“?·????' ,§ 3, Aˇ /e – ??? '~X§ y3 )“ ·Xd' L§?I ’ G??? E|' e?20 ? {§§ ) k /“' [ § 3 / ￥ :> §3 – 7, ya)> ' S?? : >?§ ·:> >? a)> >? U\'c 3:> ?3:·??u § 3 S·??oY 'XJ? U B/? a)> 3 S? ) >?§ , ? SPoisson §1 > flK Green…?' y3 0 ?? {(? > {)§ ? g · F >. a)> ^ d :> O' F ?{‘§ r / S :> flK d/=z ?. m￥ :> ( · ￠ :> §, · d /J0> ) flK' F? /J0> d5§ Ly3§ S ￠ :> §3 SU 5flK )' F S > ' C§ ? :> U ) –r = a /(>? 0) J§> flK) 3 5§ U y? ) 5flK )3 S ‰· ' F –?(/ §? d> XJ 3 {§§ ‰ u §?K S > ' 5 flK § U y d5'‰ ?‘{§dua)> >?3 S·??oY §3 S ? d(:)> U ) J' FA^> {?} ’ § 3uU?? ? d> > § m '?·? d> ·? 3 8￥Ny' F ??5 ?§ –? ‰§XJ? d> 3 {§§ ‰ u ￠> ?? ? ' x20.4 SPoisson §1 > flK Green…? 113 a20.1 > { ? d> r1 = (x1; y1)§> e§u·§§ ￠:> §3 S >? · G(r;r0) = ? 12…" 0 h lnjr?r0j+elnjr?r1j+C i ; (z) ￥~?C >?": Jk’'y3 flK · l ? –r = a >? 0§ ? 12…" 0 h lnjr?r0j+elnjr?r1j+C i r=a = 0; ? r1; e C'5?? §AT? – : ??'XJ ^??4 I§=- x = rcos`; x0 = r0cos`0; x1 = r1 cos`0; y = rsin`; y0 = r0sin`0; y1 = r1 sin`0; K §z ln￡a2 + r02 ?2ar0cos(`?`0)? + eln h a2 +r21 ?2ar1 cos(`?`0) i +2C = 0; §AT? ` ??'|^—m“ ln￡1+t2 ?2tcos`? = ln￡1?tei`?+ln￡1?te?i`? = ?2 1X m=1 1 mt m cosm`; jtj < 1; –? z 2lna+ln h 1+ ?r0 a ·2 ?2r 0 a cos(`?` 0) i +2elnr1 +eln h 1+ a r1 ?2 ?2 ar 1 cos(`?`0) i +2C = 2lna+2elnr1 ?2 1X m=1 1 m h?r0 a ·m +e ? a r1 ·mi cosm(`?`0)+2C = 0; u·§ lna+elnr1 +C = 0 (#) x20.4 SPoisson §1 > flK Green…? 114 ? r0 a ·m +e ? a r1 ·m = 0; m = 1;2;3;￠￠￠ ; = e = ? ?r1r0 a2 ·m ; m = 1;2;3;￠￠￠ ‰ e = ? ?r1r0 a2 ·1 = ? ?r1r0 a2 ·2 = ? ?r1r0 a2 ·3 = ￠￠￠ : ?–§ e = ?1 r1 = a 2 r0 ‰ r1 = ?a r0 ·2 r0: ? §? (? ? d> §§ u ￠> ?3 ? §? v r0r1 = a2: · v? ’X :§ ? ’u r = a :?' a(J‘?§ d> ￠> ??u r = a ::??§§ >> §45 ' e r1 (J \(#)“§q –? C = ?lna+lnr1 = ln ar0: 2 e; r1 C (J ￡(z)“§ ? SPoisson §1 > flK Green…? G(r;r0) = ? 12…" 0 h lnjr?r0j?ln flfl flr? ?a r0 ·2 r0 flfl fl+ln ar0 i ; ‰ 34 IX￥ L “§ G(r;r0) = ? 14…" 0 ( ln h r2 +r02 ?2rr0cos(`?`0) i ? ln h r2 + ?a2 r0 ·2 ?2ra 2 r0 cos(`?` 0)i+2ln a r0 ) : ??…? —m§ –w §? ·c ? {{ (J' 3? SPoisson §1 > flK Green…? § , – ‰)flK r22u(r) = ? 1" 0 ‰(r); jrj < a; u(r)flflr=a = f(`) )' d§ §￥ gC U ?r0§ r022u(r0) = ? 1" 0 ‰(r0); jr0j < a; u(r0)flflr0=a = f(`0); x20.4 SPoisson §1 > flK Green…? 115 , ?§ – G(r0;r)?AT v ‰)flK§ r022G(r0;r) = ? 1" 0 –(r?r0); jrj < a; jr0j < a; G(r0;r)flflr0=a = 0; 2|^Green…? ??5(§ –w?·20.2! Aˇ /§ Ul ?? G(r;r0) NL “ w )§ G(r;r0) = G(r0;r); ? U ? r022G(r;r0) = ? 1" 0 –(r?r0); jrj < a; jr0j < a; G(r;r0)flflr0=a = 0: §'O?–G(r;r0) u(r0)§ ~§23 S¨'§ ZZ r0<a ‰(r0)G(r;r0)dr0 ?u(r) = ?"0 ZZ r0<a ￡G(r;r0)r0 2 2u(r0)?u(r0)r0 2 2G(r;r0)?dr0: r ? ?¨'z –r = a ¨'§? \>.^ § k u(r) = ZZ r0<a ‰(r0)G(r;r0)dr0 +"0 Z 2… 0 ￡G(r;r0)r0u(r0)?u(r0)r0G(r;r0)? r0=aad` 0 = ZZ r0<a ‰(r0)G(r;r0)dr0 ?"0 Z 2… 0 f(`0)@G(r;r 0) @r0 flfl fl r0=a ad`0: w,§m 1 L? S> ' z?1 K·5g – a)> ) > ?§a)> ' , ‰ >.^ ( – >? ' )k’' /w – > ' § – 1 ￥ ¨'2U ? Z 2… 0 f(`0)@G(r;r 0) @r0 flfl fl r0=a ad`0 = ? Z 2… 0 f(`0) lim ?r!0 1 ?r h G(r;r0)flflr0=a??r ?G(r;r0)flflr0=a i ad`0 = ? ZZ r0<a f(`0) lim ?r!0 G(r;r0) ?r h –(r0 ?a+?r)?–(r0 ?a) i r0dr0d`0 = ? ZZ r0<a f(`0)G(r;r0)–0(r0 ?a)r0dr0d`0; ? § –r ? (J ? u(r) = ZZ r0<a ￡‰(r0)+" 0f(`0)–0(r0 ?a) ?G(r;r0)dr0: x20.4 SPoisson §1 > flK Green…? 116 ?L?§ – a)> ·"0f(`0)–0(r0 ?a)'…?–0(r0 ?a) y§‘?3 – a)> 4 ' ?p – ? ( ' k, ‰)flK r22u(r) = ? 1" 0 h ‰(r)+"0f(`)–0(r?a) i ; jrj < a; u(r)flflr=a = 0; w,§§ ?k )' F?‘?§3 ?–…?9 ? cJe§ g>.^ ‰)flK§ –U g >.^ § L3 §￥ A/O\ Aˇ g §3? S ?? 0§ 3>. 0 (￠S? ??) g ' ? ' J? ?? g {' F ,§ g>.^ –=z § Aˇ/“ g §j?? ? X –? g>.^ (￡ >.? ' ) § g (? S ' ) ?O'=?r g>.^ U ? § g §§￡ E,· 3u>.? ' – 0 Green…? ?){' 1 ?){·U A gflK …?—m'??){?^ 2§":· ) ·????' 1 ?){·> {§ ￥%g ·r>. a)> ^ d :> (? > ) O'? k3, ~Aˇ A /G(~X￥/§ ?. m§ )e U￠y'=? ?^A ( · ) > 5 / O>. a)> §? m A /GE,k '?–‘§> { ‘:· – k /“ )§":··??^ k ' x20.5 ˉ? § Green…? 117 x20.5 ˉ? § Green…? (‰ §–k.u ˉ?flK ~' ‰)flK · @2u(x;t) @t2 ?a 2@ 2u(x;t) @x2 = f(x;t); 0 < x < l; t > 0; u(x;t)flflx=0 = ?(t);u(x;t)flflx=l = ”(t); t > 0; u(x;t)flflt=0 = `(x); @u(x;t)@t flfl fl t=0 = ?(x);0 < x < l: – § A Green…?G(x;t;x0;t0)AT·] (= 3u, ):(= 3u m, :) flK h @2 @t2 ?a 2 @2 @x2 i G(x;t; x0;t0) = –(x?x0)–(t?t0); 0 < x;x0 < l; t;t0 > 0 3 g‰)^ G(x;t;x0;t0)flflx=0 = 0; G(x;t;x0;t0)flflx=l = 0; t;t0 > 0; G(x;t;x0;t0)flflt<t0 = 0; @G(x;t;x 0;t0) @t flfl fl t<t0 = 0; 0 < x;x0 < l e )'?p—'^ n?′·? ˇ r‰ ·3t = t0 y §?–§3 d–c§u , ‰ –? ' flK §y3I ? n flK Green…?G(x;t;x0;t0) ??5 X ^Green…?9fi ^ f(x;t), ?(t); ”(t) `(x); ?(x) ‰)flK )u(x;t)L? 5 n X ? Green…? F?k§’uGreen…? ??5§ ( /‘§Green…?3 m ??5 m ·5' d§2 ’uGreen…?G(x;?t;x00;?t00) ‰)flK ? @2 @t2 ?a 2 @2 @x2 ? G(x;?t;x00;?t00) = –(x?x00)–(t?t00); 0 < x;x00 < l; t;t00 > 0; G(x;?t;x00;?t00)flflx=0 = 0; G(x;?t;x00;?t00)flflx=l = 0; t;t00 > 0; G(x;?t;x00;?t00)flfl?t<?t00 = 0; @G(x;?t;x00;?t00) @t flfl fl ?t<?t00 = 0; 0 < x;x00 < l: x20.5 ˉ? § Green…? 118 §'O?–Green…?G(x;?t;x00;?t00) G(x;t;x0;t0), ~,23?m[0; l] [0; 1) ?x t¨',= G(x0;?t0;x00;?t00)?G(x00;t00;x0;t0) = Z l 0 dx Z 1 0 ? G(x;?t;x00;?t00)@ 2G(x;t;x0;t0) @t2 ?G(x;t;x0;t0)@ 2G(x;?t;x00;?t00) @t2 ? dt ? Z 1 0 dt Z l 0 ? G(x;?t;x00;?t00)@ 2G(x;t;x0;t0) @x2 ?G(x;t;x0;t0)@ 2G(x;?t;x00;?t00) @x2 ? dx = Z l 0 ? G(x;?t;x00;?t00)@G(x;t;x 0;t0) @t ?G(x;t;x0;t0)@G(x;?t;x 00;?t00) @t ?1 0 dx ? Z 1 0 ? G(x;?t;x00;?t00)@G(x;t;x 0;t0) @x ?G(x;t;x0;t0)@G(x;?t;x 00;?t00) @x ?l 0 dt; \k’ >.^ —'^ § –w §m ¨' 0§?– Green…?3 m ??5 m ·5§ G(x00;t00;x0;t0) = G(x0;?t0;x00;?t00); ‰ x00 t00U ?x t§ G(x;t;x0;t0) = G(x0;?t0;x;?t): 3? ’X“￥§ t t0? y K § — y m k gS C§?K ? k uˇJ? ?' F^Green…?9fi ^ f(x;t), ?(t); ”(t) `(x); ?(x) ‰)flK )u(x;t)L? 5' d§ ‰)flK￥ gC U ?x0 t0§ @2u(x0;t0) @t02 ?a 2@2u(x0;t0) @x02 = f(x 0;t0); 0 < x0 < l; t0 > 0; u(x0;t0)flflx0=0 = ?(t0); u(x0;t0)flflx0=l = ”(t0); t0 > 0; u(x0;t0)flflt0=0 = `(x0); @u(x 0;t0) @t0 flfl fl t0=0 = ?(x0); 0 < x0 < l: x20.5 ˉ? § Green…? 119 2 Green…? ‰)flK ? @2 @(?t0)2 ?a 2 @2 @x02 ? G(x0;?t0;x;?t) = –(x?x0)–(t?t0); 0 < x;x0 < l; t;t0 > 0; G(x0;?t0;x;?t)flflx0=0 = 0; G(x0;?t0;x;?t)flflx0=l = 0; t;t0 > 0; G(x0;?t0;x;?t)flfl?t0<?t = 0; @G(x0;?t0;x;?t) @t flfl fl ?t0<?t = 0; 0 < x;x0 < l: |^Green…? ??5 ·5’X§ –U ? ? @2 @t02 ?a 2 @2 @x02 ? G(x;t;x0;t0) = –(x?x0)–(t?t0); 0 < x;x0 < l; t;t0 > 0; G(x;t;x0;t0)flflx0=0 = 0; G(x;t;x0;t0)flflx0=l = 0; t;t0 > 0; G(x;t;x0;t0)flflt0>t = 0; @G(x;t;x0;t0) @t flfl fl t0>t = 0; 0 < x;x0 < l: §'O?–G(x;t;x0;t0) u(x0;t0)§ ~§2¨'§ Z l 0 dx0 Z 1 0 G(x;t;x0;t0)f(x0;t0)dt0 ?u(x;t) = Z l 0 dx0 Z 1 0 ? G(x;t;x0;t0)@ 2u(x0;t0) @t02 ?u(x 0;t0)@2G(x;t;x0;t0) @t02 ? dt0 ?a2 Z 1 0 dt0 Z l 0 ? G(x;t;x0;t0)@ 2u(x0;t0) @x02 ?u(x 0;t0)@2G(x;t;x0;t0) @x02 ? dx0: x20.5 ˉ? § Green…? 120 \>.^ —'^ § –z{ u(x;t) = Z l 0 dx0 Z 1 0 G(x;t;x0;t0)f(x0;t0)dt0 ? Z l 0 ? G(x;t;x0;t0)@u(x 0;t0) @t0 ?u(x 0;t0)@G(x;t;x0;t0) @t0 ?1 0 dx0 + a2 Z 1 0 ? G(x;t;x0;t0)@u(x 0;t0) @x0 ?u(x 0;t0)@G(x;t;x0;t0) @x0 ?l 0 dt0 = Z l 0 dx0 Z t 0 G(x;t;x0;t0)f(x0;t0)dt0 ? Z l 0 ? G(x;t;x0;0)?(x0)?`(x0) @G(x;t;x 0;t0) @t0 flfl flfl t0=0 ? dx0 ? a2 Z t 0 ? ”(t0) @G(x;t;x 0;t0) @x0 flfl flfl x0=l ??(t0) @G(x;t;x 0;t0) @x0 flfl flfl x0=0 ? dt0: F? Green…? N/“' ? 1 ? {E,·U A gflK …?—m§ G(x;t;x0;t0) = 1X n=1 Tn(t)sin n…l x; § –…? UT| …?—m§ –(x?x0) = 2l 1X n=1 sin n…l x0sin n…l x; u·§Tn(t) v~ ' § — flK T00(t)+ ?n…a l ·2 Tn(t) = 2l sin n…l x0–(t?t0); Tn(t < t0) = 0; T0n(t < t0) = 0: ) = Tn(t) = 2n…a sin n…l x0 sin n…l a(t?t0)·(t?t0): ?–§Green…?G(x;t;x0;t0) · G(x;t;x0;t0) = 2…a 1X n=1 1 n sin n… l x 0 sin n… l x sin n… l a(t?t 0)·(t?t0): ? 1 ? {· ‰)flK LaplaceC ' - g(x;p;x0;t0) = Z 1 0 G(x;t;x0;t0)e?pt dt; x20.5 ˉ? § Green…? 121 Kg(x;p;x0;t0) v~ ' § > flK h d2 dx2 ? ?p a ·2i g(x;p;x0;t0) = ? 1a2e?pt0 –(x?x0); g(x;p;x0;t0)flflx=0 = 0; g(x;p;x0;t0)flflx=l = 0: x 6= x0§ d2g(x;p;x0;t0) dx2 ? ?p a ·2 g(x;p;x0;t0) = 0: ? >.^ § k) g(x;p;x0;t0) = 8> < >: Asinh pax; 0 ? x < x0; Bsinh pa(l?x); x0 < x ? l: y3§x = x0? g KNy o ^ g(x;p;x0;t0) flfl fl x=x0+0 x=x0?0 = 0; dg(x;p;x 0;t0) dx flfl fl x=x0+0 x=x0?0 = ? 1a2e?pt0: u· Bsinh pa(l?x0)?Asinh pax0 = 0; Bcosh pa(l?x0)+Acosh pax0 = 1pae?pt0: ) A = sinh pa(l?x0) pa sinh pal e?pt0; B = sinh pax0 pa sinh pal e?pt0; ? § ? 0 ? x < x0 § g(x;p;x0;t0) = sinh pa(l?x0) sinh pax pa sinh pal e?pt0; x0 < x ? l § g(x;p;x0;t0) = sinh pax0 sinh pa(l?x) pa sinh pal e?pt0: §? § G(x;t;x0;t0) = 12…i Z L g(x;p;x0;t0)ept dp; A^3?‰nO ? ¨'§ – 1 ? { (J' 2? n m ~f'? Green…?G(r;t;r0;t0) v‰)flK h @2 @t2 ?a 2r2 i G(r;t;r0;t0) = –(r?r0)–(t?t0); t;t0 > 0; G(r;t;r0;t0)flflt<t0 = 0; @G(r;t;r 0;t0) @t flfl flfl t<t0 = 0: x20.5 ˉ? § Green…? 122 FourierC g(r;!;r0;t0) = 1p2… Z 1 ?1 G(r;t;r0;t0)ei!t dt; u· ‰)flKz h (?i!)2 ?a2r2 i g(r;!;r0;t0) = 1p2… ei!t0 –(r?r0) = h r2 + ?! a ·2i g(r;!;r0;t0) = ? 1p2…a2 ei!t0 –(r?r0); 20.3! (J§ – g(r;!;r0;t0) = 1p2…a2 ei!t0 14…jr?r0jei(!=a)jr?r0j: _C § k G(r;t;r0;t0) = 1p2… Z 1 ?1 g(r;!;r0;t0)e?i!td! = 14…a2 1jr?r0j 12… Z 1 ?1 e?i!(t?t0) ￠ei(!=a)jr?r0jd! = 14…a2 1jr?r0j– jr?r0j a ?(t?t 0) ? = 14…a 1jr?r0j–?jr?r0j?a(t?t0)￠: ? )“ n?′??( t0 3r0?u & §t ‰ r0: a(t?t0) ￥ ? ' g K1 ? )KL§￥§—'^ ^^uu ?” g K2 ? )KL§￥§ UAA^^20.3! (J” ? Green…?§ , – n ?. m￥ˉ? § — flK @2u(r;t) @t2 ?a 2r2u(r;t) = f(r;t); t > 0; u(r;t)flflt=0 = `(r); @u(r;t)@t flfl fl t=0 = ?(r) )§ u(r;t) = 14…a2 ZZZ jr0?rj<at f(r0;t?jr0 ?rj=a) jr0 ?rj dr 0 + 14…a "ZZ §0 ?(r0) jr0 ?rjd§ 0 + @ @t ZZ §0 `(r0) jr0 ?rjd§ 0 # ; ￥§0·–r: ￥%!at ? ￥?jr0 ?rj = at' x20.6 9D § Green…? 123 x20.6 9D § Green…? n k. m￥ 9D flK§ @u(r;t) @t ??r 2u(r;t) = f(r;t); r2 V; t > 0; u(r;t)flfl§ = ?(§;t); t > 0; u(r;t)flflt=0 = `(r); r2 V; A Green…?G(r;t;r0;t0) ·t0 3r0?k ( r )] :9 ? ) § ' § ?{‘§ ·‰)flK ? @ @t ??r 2 ? G(r;t;r0;t0) = –(r?r0)–(t?t0); r;r0 2 V; t;t0 > 0; G(r;t;r0;t0)flfl§ = 0; t;t0 > 0; G(r;t;r0;t0)flflt=0 = 0; r;r0 2 V )'N·n)§—'^ – ?G(r;t;r0;t0)flflt<t0 = 0' ˉ?flK {§ –y?? Green…? k?u m ??5 m · 5§ G(r;t;r0;t0) = G(r0;?t0;r;?t): ^Green…? {)9D §‰)flK§ A !￥ IO { ?k ‰)flK gC U ?r0 t0§ @u(r0;t0) @t0 ??r 02u(r0;t0) = f(r0;t0); r0 2 V; t0 > 0; u(r0;t0)flfl§0 = ?(§0;t0); t0 > 0; u(r0;t0)flflt0=0 = `(r0); r0 2 V: , Green…?G(r0;?t0;r;?t) v ‰)flK§ h @ @(?t0) ??r 02iG(r0;?t0;r;?t) = –(r?r0)–(t?t0); r;r0 2 V; t;t0 > 0; G(r0;?t0;r;?t)flfl§0 = 0; t;t0 > 0; G(r0;?t0;r;?t)flfl?t0<?t = 0; r;r0 2 V; ? 2|^Green…??u m ??5 m ·5’X§U ? ? ? @@t0 ??r02 ? G(r;t;r0;t0) = –(r?r0)–(t?t0); r;r0 2 V; t;t0 > 0; G(r;t;r0;t0)flfl§0 = 0; t;t0 > 0; G(r;t;r0;t0)flflt0>t = 0; r;r0 2 V: x20.6 9D § Green…? 124 §'O?–G(r;t;r0;t0) u(r0;t0)§ ~§?¨'§ Z 1 0 dt0 ZZZ V f(r0;t0)G(r;t;r0;t0)dr0 ?u(r;t) = ZZZ V dr0 Z 1 0 ? G(r;t;r0;t0)@u(r 0;t0) @t0 +u(r0;t0)@G(r;t;r 0;t0) @t0 ? dt0 ?? Z 1 0 dt0 ZZZ V ? G(r;t;r0;t0)r02u(r0;t0) ?u(r0;t0)r02G(r;t;r0;t0) ? dr0 = ZZZ V G(r;t;r0;t0)u(r0;t0) flfl fl t0=1 t0=0 dr0 ?? Z 1 0 dt0 ZZ §0 ￡G(r;t;r0;t0)r0u(r0;t0) ?u(r0;t0)r0G(r;t;r0;t0)?￠d 0; \>.^ —'^ § u(r;t) = Z t 0 dt0 ZZZ V f(r0;t0)G(r;t;r0;t0)dr0 + ZZZ V `(r0)G(r;t;r0;0)dr0 ? ? Z t 0 dt0 ZZ §0 ?(§0;t0) @G(r;t;r 0;t0) @n0 flfl flfl §0 d§0: y35?)1 n ￥ ￠3e ‰)flK§=’u ?. m9D § Green…?flK§ ? @ @t ?? @2 @x2 ? G(x;t;x0;t0) = –(x?x0)–(t?t0); t0 > 0; G(x;t;x0;t0)flflx!§1k.; G(x;t;x0;t0)flflt=0 = 0: LaplaceC §=- g(p;x;x0) = Z 1 0 G(x;t;x0;t0)e?ptdt; u·§‰)flKz pg(p;x;x0)??d 2g(p;x;x0) dx2 = –(x?x 0)e?pt0; g(p;x;x0)flflx!§1k.: x20.6 9D § Green…? 125 x 6= x0 § §· g ' ? >.^ § – § ) g(p;x;x0) = 8> >>< >>> : Aexp ‰rp ?(x?x 0) ; x < x0; Bexp ‰ ? rp ?(x?x 0) ; x > x0: 2 §￥ g § – o ^ g(p;x;x0) flfl fl x=x0+0 x=x0?0 = 0; ?? dg(p;x;x 0) dx flfl flfl x=x0+0 x=x0?0 = e?pt0; = B ?A = 0; ?? ? ? rp ?B ? rp ?A ? = e?pt0; dd= ‰ A = B = 12p?pe?pt0: u· G(p;x;x0) = 12p?pe?pt0 exp ‰ ? rp ?jx?x 0j : 2|^1 (J§ –? G(x;t;x0;t0) = 12p?…(t?t0) exp ‰ ?(x?x 0)2 4?(t?t0) ·(t?t0): ? ·1 n ￥ (J' x20.6 9D § Green…? 126 ? S K '? fzHall AflK(1999cNobel n?) O NX￥N >f ! > (oN –>￥5) m Coulomb p ^U' >f :> 'N·? § ( ):> ei ej('O uri rj?) m Coulomb p ^U U = ?2keiej ln jri ?rjja ; ￥k a ~?§c k’§ · ? ? ~? a ” 'ˇd§> fNX ‰0 ! > m Coulomb p ^U – ? U = 2ke‰0 NX l=1 ZZ ln jr?rlja d2r+~?: b NX ?R §‰0 = Ne=…R2§y? U = k2 Ne 2 …R2 NX l=1 jrlj2 +~?: