Wu Chong-shi a0a1a2a3 Green a4a5 (a1) §30.1 a6a7 Green a8a9 a10a11a12a13a14a15a16a17 a15a18a19a20a21a22a23a24a25a26a27a28a29a11a12a30a31a17 a32a33a34a29a20a35a30a31a36a37 ?2u(x,t) ?t2 ?a 2? 2u(x,t) ?x2 = f(x,t), 0 <x<l, t> 0, u(x,t)vextendsinglevextendsinglex=0 = μ(t), u(x,t)vextendsinglevextendsinglex=l = ν(t), t> 0, u(x,t)vextendsinglevextendsinglet=0 = φ(x), ?u(x,t)?t vextendsinglevextendsingle vextendsingle t=0 = ψ(x), 0 <x<l. a38a10a39a40a23a41a42a29Green a43a44G(x,t;xprime,tprime)a42a45a37a46a47(a48a49a50a51a52 a33a47a53) a54(a48a49a50a51a55a56a52a54) a57a30a31 bracketleftBig ?2 ?t2 ?a 2 ?2 ?x2 bracketrightBig G(x,t; xprime,tprime) = δ(x?xprime)δ(t?tprime), 0 <x,xprime <l, t,tprime > 0 a50a58a59 a20a35a60a61 G(x,t;xprime,tprime)vextendsinglevextendsinglex=0 = 0, G(x,t;xprime,tprime)vextendsinglevextendsinglex=l = 0, t,tprime > 0, G(x,t;xprime,tprime)vextendsinglevextendsinglet<tprime = 0, ?G(x,t;x prime,tprime) ?t vextendsinglevextendsingle vextendsingle t<tprime = 0, 0 <x,xprime <l a62a29a35a17a63a64a65a66a60a61a29a67a68a69a70a37a71a72a73a29a74a75a15a76a12a77a37 a50t = tprime a47a53a78a79a29a23a80a10a23a50a81 a10 a82a23a28a83a84a33a20a85a86a87a88a17 a89a33a34a29a30a31a33a90a23a79 a50a91a92 a24a25a93a94a30a31a74 a33 Green a43a44G(x,t;xprime,tprime)a29a95a96a97 a98 a99a100a101Green a43a44a102a103 a104a60a61f(x,t), μ(t), ν(t)a89φ(x), ψ(x) a105 a20a35a30a31a29a35u(x,t) a106a107 a78a108 a93 a99a100a109a78 Green a43a44 Wu Chong-shi §30.1 a110a111 Greena112a113 a1142a115 Green a116a117a118a119a120a121 (Greena43a44a50a122a123a124a125a126a127a128a129a130a123a124a125a131a132a128) a15 a81 a23a133a134a78a135 a51 Greena43a44G(x,?t;x primeprime,?tprimeprime)a29a20a35a30a31 bracketleftbigg ?2 ?t2 ?a 2 ?2 ?x2 bracketrightbigg G(x,?t;xprimeprime,?tprimeprime) = δ(x?xprimeprime)δ(t?tprimeprime), 0 <x,xprimeprime <l, t,tprimeprime > 0, G(x,?t;xprimeprime,?tprimeprime)vextendsinglevextendsinglex=0 = 0, G(x,?t;xprimeprime,?tprimeprime)vextendsinglevextendsinglex=l = 0, t,tprimeprime > 0, G(x,?t;xprimeprime,?tprimeprime)vextendsinglevextendsingle?t<?tprimeprime = 0, ?G(x,?t;x primeprime,?tprimeprime) ?t vextendsinglevextendsingle vextendsingle ?t<?tprimeprime = 0, 0 <x,xprimeprime <l. a105a136 a94a13a14a137a138a139a10Green a43a44G(x,?t;xprimeprime,?tprimeprime) a89 G(x,t;xprime,tprime) a23a41a140a23a133a50a141a56[0, l]a89 [0, ∞) a142a95xa89t a143 a137a23a144a145 G(xprime,?tprime;xprimeprime,?tprimeprime)?G(xprimeprime,tprimeprime;xprime,tprime) = integraldisplay l 0 dx integraldisplay ∞ 0 bracketleftbigg G(x,?t;xprimeprime,?tprimeprime)? 2G(x,t;xprime,tprime) ?t2 ?G(x,t;x prime,tprime)?2G(x,?t;xprimeprime,?tprimeprime) ?t2 bracketrightbigg dt ? integraldisplay ∞ 0 dt integraldisplay l 0 bracketleftbigg G(x,?t;xprimeprime,?tprimeprime)? 2G(x,t;xprime,tprime) ?x2 ?G(x,t;x prime,tprime)?2G(x,?t;xprimeprime,?tprimeprime) ?x2 bracketrightbigg dx = integraldisplay l 0 bracketleftbigg G(x,?t;xprimeprime,?tprimeprime)?G(x,t;x prime,tprime) ?t ?G(x,t;x prime,tprime)?G(x,?t;xprimeprime,?tprimeprime) ?t bracketrightbigg∞ 0 dx ? integraldisplay ∞ 0 bracketleftbigg G(x,?t;xprimeprime,?tprimeprime)?G(x,t;x prime,tprime) ?x ?G(x,t;x prime,tprime)?G(x,?t;xprimeprime,?tprimeprime) ?x bracketrightbiggl 0 dt, a146a147a26a135a29a148a27a60a61a89a65a66a60a61a23a38a10a149a78a23a150a151a29 a143 a137a150 a23a80a10a36a152a78a18 Green a43a44a50a55a56 a142a29a95a96a97a153a47 a56 a142a29a154a155a97a23 G(xprimeprime,tprimeprime;xprime,tprime) = G(xprime,?tprime;xprimeprime,?tprimeprime), a156a157 a105xprimeprime a89tprimeprime a158a159a160xa89ta23 G(x,t;xprime,tprime) = G(xprime,?tprime;x,?t). a50 a63a94a135a161a162a163a23 a105ta89tprime a95a164a165a166a47a78a79a29a167a168a23a169a170a85a171a18a47a56 a29a172a173 a59a174a175a176 a23a177a178a36a179 a26a180 a51 a75a181a182a29 a92 a109a17 Wu Chong-shi a183a184a185a186 Green a112a113(a184) a1143a115 a187 Green a116a117a188a189a190a191a192 f(x,t), μ(t), ν(t) a193φ(x), ψ(x) a194a195a196a197a198 a118 a196 u(x,t) a199a200a201a202 a15 a81 a23 a105 a20a35a30a31a163a29a203 a176a204 a158a159a160xprime a89tprime a23 ?2u(xprime,tprime) ?tprime2 ?a 2? 2u(xprime,tprime) ?xprime2 = f(x prime,tprime), 0 <xprime <l, tprime > 0, u(xprime,tprime)vextendsinglevextendsinglexprime=0 = μ(tprime), u(xprime,tprime)vextendsinglevextendsinglexprime=l = ν(tprime), tprime > 0, u(xprime,tprime)vextendsinglevextendsingletprime=0 = φ(xprime), ?u(x prime,tprime) ?tprime vextendsinglevextendsingle vextendsingle tprime=0 = ψ(xprime), 0 <xprime <l. a133 a159 a78 Green a43a44 a29a20a35a30a31 bracketleftBigg ?2 ?(?tprime)2 ?a 2 ?2 ?xprime2 bracketrightBigg G(xprime,?tprime;x,?t) = δ(x?xprime)δ(t?tprime), 0 <x,xprime <l, t,tprime > 0, G(xprime,?tprime;x,?t)vextendsinglevextendsinglexprime=0 = 0, G(xprime,?tprime;x,?t)vextendsinglevextendsinglexprime=l = 0, t,tprime > 0, G(xprime,?tprime;x,?t)vextendsinglevextendsingle?tprime<?t = 0, ?G(x prime,?tprime;x,?t) ?t vextendsinglevextendsingle vextendsingle ?tprime<?t = 0, 0 <x,xprime <l. a205 a101Green a43a44 a29a95a96a97a153a154a155a97a135a161a23a206a38a10 a158a159a160bracketleftbigg ?2 ?tprime2 ?a 2 ?2 ?xprime2 bracketrightbigg G(x,t;xprime,tprime) = δ(x?xprime)δ(t?tprime), 0 <x,xprime <l, t,tprime > 0, G(x,t;xprime,tprime)vextendsinglevextendsinglexprime=0 = 0, G(x,t;xprime,tprime)vextendsinglevextendsinglexprime=l = 0, t,tprime > 0, G(x,t;xprime,tprime)vextendsinglevextendsingletprime>t = 0, ?G(x,t;x prime,tprime) ?t vextendsinglevextendsingle vextendsingle tprime>t = 0, 0 <x,xprime <l. a105a136 a94a13a14a137a138a139a10G(x,t;xprime,tprime)a89u(xprime,tprime)a23a41a140a23a133 a143 a137a23 integraldisplay l 0 dxprime integraldisplay ∞ 0 G(x,t;xprime,tprime)f(xprime,tprime)dtprime ?u(x,t) = integraldisplay l 0 dxprime integraldisplay ∞ 0 bracketleftbigg G(x,t;xprime,tprime)? 2u(xprime,tprime) ?tprime2 ?u(x prime,tprime)?2G(x,t;xprime,tprime) ?tprime2 bracketrightbigg dtprime ?a2 integraldisplay ∞ 0 dtprime integraldisplay l 0 bracketleftbigg G(x,t;xprime,tprime)? 2u(xprime,tprime) ?xprime2 ?u(x prime,tprime)?2G(x,t;xprime,tprime) ?xprime2 bracketrightbigg dxprime. a146a147a148a27a60a61a89a65a66a60a61a23a36a38a10a207a208a15 u(x,t) = integraldisplay l 0 dxprime integraldisplay ∞ 0 G(x,t;xprime,tprime)f(xprime,tprime)dtprime ? integraldisplay l 0 bracketleftbigg G(x,t;xprime,tprime)?u(x prime,tprime) ?tprime ?u(x prime,tprime)?G(x,t;xprime,tprime) ?tprime bracketrightbigg∞ 0 dxprime + a2 integraldisplay ∞ 0 bracketleftbigg G(x,t;xprime,tprime)?u(x prime,tprime) ?xprime ?u(x prime,tprime)?G(x,t;xprime,tprime) ?xprime bracketrightbiggl 0 dtprime = integraldisplay l 0 dxprime integraldisplay t 0 G(x,t;xprime,tprime)f(xprime,tprime)dtprime ? integraldisplay l 0 bracketleftbigg G(x,t;xprime,0)ψ(xprime)?φ(xprime) ?G(x,t;x prime,tprime) ?tprime vextendsinglevextendsingle vextendsinglevextendsingle tprime=0 bracketrightbigg dxprime ? a2 integraldisplay t 0 bracketleftbigg ν(tprime) ?G(x,t;x prime,tprime) ?xprime vextendsinglevextendsingle vextendsinglevextendsingle xprime=l ?μ(tprime) ?G(x,t;x prime,tprime) ?xprime vextendsinglevextendsingle vextendsinglevextendsingle xprime=0 bracketrightbigg dtprime. Wu Chong-shi §30.1 a110a111 Greena112a113 a1144a115 a209 a201Green a116a117a118a210a211a212a213 bracketleftBig ?2 ?t2 ?a 2 ?2 ?x2 bracketrightBig G(x,t; xprime,tprime) = δ(x?xprime)δ(t?tprime), 0 <x,xprime <l, t,tprime > 0, G(x,t;xprime,tprime)vextendsinglevextendsinglex=0 = 0, G(x,t;xprime,tprime)vextendsinglevextendsinglex=l = 0, t,tprime > 0, G(x,t;xprime,tprime)vextendsinglevextendsinglet<tprime = 0, ?G(x,t;x prime,tprime) ?t vextendsinglevextendsingle vextendsingle t<tprime = 0, 0 <x,xprime <l a214a41a42 a58a59 a30a31a29a215a216 a43a44a217a218 a23 G(x,t;xprime,tprime) = ∞summationdisplay n=1 Tn(t)sin npil x, a219a47a23 a105δa43a44 a206a214a45a220a215a216 a43a44a217a218 a23 δ(x?xprime) = 2l ∞summationdisplay n=1 sin npil xprime sin npil x, a51 a37a23T n(t) a36a221a222a223a224a137a13a14a29a65a225a30a31 Tprimeprime(t) + parenleftBignpia l parenrightBig2 Tn(t) = 2l sin npil xprimeδ(t?tprime), Tn(t<tprime) = 0, Tprimen(t<tprime) = 0. a35a226a144a145 Tn(t) = 2npia sin npil xprime sin npil a(t?tprime)η(t?tprime). a80a10a23 Green a43a44G(x,t;xprime,tprime)a36a37 G(x,t;xprime,tprime) = 2pia ∞summationdisplay n=1 1 n sin npi l x prime sin npi l x sin npi l a(t?t prime)η(t?tprime). Wu Chong-shi a183a184a185a186 Green a112a113(a184) a1145a115 a133a24a25a33a94a93a227 a55a56 a29a16a228a17a63a47a29 Green a43a44G(r,t;rprime,tprime)a221a222a20a35a30a31bracketleftBig ?2 ?t2 ?a 2?2 bracketrightBig G(r,t;rprime,tprime) = δ(r ?rprime)δ(t?tprime), t,tprime > 0, G(r,t;rprime,tprime)vextendsinglevextendsinglet<tprime = 0, ?G(r,t;r prime,tprime) ?t vextendsinglevextendsingle vextendsinglevextendsingle t<tprime = 0. a229Fourier a176 a164 g(r,ω;rprime,tprime) = 1√2pi integraldisplay ∞ ?∞ G(r,t;rprime,tprime)eiωt dt, a51 a37a36 a105 a20a35a30a31a207a15 bracketleftBig (?iω)2 ?a2?2 bracketrightBig g(r,ω;rprime,tprime) = 1√2pi eiωtprime δ(r ?rprime) a144 bracketleftBig ?2 + parenleftBigω a parenrightBig2 bracketrightBig g(r,ω;rprime,tprime) = ? 1√2pia2 eiωtprime δ(r ?rprime), a230a23127.2 a232 a29a233a181a23a38a10a145a234 g(r,ω;rprime,tprime) = 1√2pia2 eiωtprime 14pi|r ?rprime|ei(ω/a)|r?rprime|. a229a235 a176 a164a23a36a26 G(r,t;rprime,tprime) = 1√2pi integraldisplay ∞ ?∞ g(r,ω;rprime,tprime)e?iωtdω = 14pia2 1|r ?rprime| 12pi integraldisplay ∞ ?∞ e?iω(t?tprime) · ei(ω/a)|r?rprime|dω = 14pia2 1|r ?rprime|δ parenleftbigg|r ?rprime| a ?(t?t prime) parenrightbigg = 14pia 1|r ?rprime|δ(|r ?rprime|?a(t?tprime)). a67a68a69a70a71a236a19a74 tprime a47a53 a50rprime a237a238a239a29a240a168a23ta47a53a33a20a234a241a242 rprime a54 a15a(t?tprime)a29a243a244a142a17 a230a231a63a94 Green a43a44 a23a83a84a36a38a10a145a234a93a227a245a27 a55a56 a163a11a12a13a14a29a65a225a30a31 ?2u(r,t) ?t2 ?a 2?2u(r,t) = f(r,t), t> 0, u(r,t)vextendsinglevextendsinglet=0 = φ(r), ?u(r,t)?t vextendsinglevextendsingle vextendsingle t=0 = ψ(r) a29a35a23 u(r,t) = 14pia2 integraldisplayintegraldisplayintegraldisplay |rprime?r|<at f(rprime,t?|rprime ?r|/a) |rprime ?r| dr prime + 14pia bracketleftBiggintegraldisplayintegraldisplay Σprime ψ(rprime) |rprime ?r|dΣ prime + ? ?t integraldisplayintegraldisplay Σprime φ(rprime) |rprime ?r|dΣ prime bracketrightBigg , a246a163Σprime a37a10r a54 a15a243a247a248ata15a249a250a29a243a244 |rprime ?r| = ata17