Wu Chong-shi a0a1a2a3 a4a5a6a7a8 ( a9) §17.1 a10a11a12a13a14a15a16a17a18a19 (a20) a21a22a23a24a25 a26a27a28a29a30a31a32a33a34a35a36a37 a38u(x,t) = v(x,t) + w(x,t) a39 a40 ?2u ?t2 ?a 2? 2u ?x2 = f(x,t) uvextendsinglevextendsinglex=0 = 0 uvextendsinglevextendsinglex=l = 0 uvextendsinglevextendsinglet=0 = 0 ?u?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 0 = ?2v ?t2 ?a 2 ? 2v ?x2 = f(x,t) vvextendsinglevextendsinglex=0 = 0 vvextendsinglevextendsinglex=l = 0 + ?2w ?t2 ?a 2? 2w ?x2 = 0 wvextendsinglevextendsinglex=0 = 0 wvextendsinglevextendsinglex=l = 0 wvextendsinglevextendsinglet=0 = ?vvextendsinglevextendsinglet=0 ?w?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = ??v?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 a41a42a43a44a45a46a47a48 f(x,t) a49a50a51a52a53a54a55a39a56a57a58a59 a45a46a47a43a44 a49a60a61a39a62a63a57a64a65 a66a67 a49a61a68a69 a21a22a23a24a70 a71a72a73a74a75a76a77a78a79a80a81a82a83a84a85{X n(x), n = 1,2,3,···}a39 a86a87a88a89a22a90a91a92a93a94a95a96 a39 a97 a98 a39a99a100a101a102a103 u(x,t)a104a105a106a107a108a109a110a105a106a107a111 f(x,t)a112a113 a82a83a84a85a114a115 u(x,t) = ∞summationdisplay n=1 Tn(t)Xn(x), f(x,t) = ∞summationdisplay n=1 gn(t)Xn(x), a116a117a118a76a77a119a120T n(t)a121a100a69a122a123Tn(t) a75a80a124a84a85 a39 a125a126a127 a110 a75a128a129a130 a108a109 (a81)a39 a131 a100a132a133 a119 a103 a134a129a130 a108a109a135a136a137a138a69 a82a83a84a85a81{X n(x)}a110a139a140a141 a142 a137a138a110a143 a77a75 a139a144{Xn(x)}a145a146a147a106a107a148a103a149a150a110 a82a83a84a85 a39 a121 a126a127 a122a106a107 a134a129a130 a108a109a151a106a107a152a153a154a155 ?2u ?t2 ?a 2?2u ?x2 = 0, 0 < x < l, t > 0, uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l = 0, t ≥ 0 a130a156a157a158a159 a136 a79 a110 a82a83a160 a149a150 Xprimeprimen(x) + λnXn(x) = 0, Xn(0) = 0, Xn(l) = 0. Wu Chong-shi §17.1 a161a162a163a164a165a166a167a168a169a170 (a171) a1722a173 a174u(x,t) a151f(x,t)a110 a114a115a175 a151a176a177 a134a129a130 a108a109a39a178a179a111 a129a180 a39 ∞summationdisplay n=1 Tprimeprimen(t)Xn(x) ?a2 ∞summationdisplay n=1 Tn(t)Xprimeprimen(x) = ∞summationdisplay n=1 gn(t)Xn(x). a181a182X n(x) a183 a126a127 a110 a128a129a130 a108a109a39a184a185a186 ∞summationdisplay n=1 Tprimeprimen(t)Xn(x) + a2 ∞summationdisplay n=1 λnTn(t)Xn(x) = ∞summationdisplay n=1 gn(t)Xn(x). a118a187a188a82a83a84a85 a110a189a190a191a39a99a136 a79 T n(t)a183 a126a127 a110 a128a129a130 a108a109 Tprimeprimen(t) + λna2Tn(t) = gn(t). a192a193 a39a102u(x,t)a110 a114a115a175 a176a177a194a195a154a155a39a196a100a136 a79 ∞summationdisplay n=1 Tn(0)Xn(x) = 0, ∞summationdisplay n=1 Tprimen(0)Xn(0) = 0. a187a188a82a83a84a85 a110a189a190a191a39a121a132a197 a120 Tn(0) = 0, Tprimen(0) = 0. a182 a103a105a106a107 a128a129a130 a108a109a110 a128a85a157a198a77 a39a199a200 a182 Laplacea157a201 a39a99a100a101 a119a120 Tn(t) = lnpia integraldisplay t 0 gn(τ)sin npil a(t?τ)dτ. a202a203 a103 a77 a39a204a145a113a146a147a106a107a149a150a110 a82a83a84a85a114a115a77 a69 a118a182a202a203 a108 a77a119 a103a205 16.2 a71 a110a148a103a149a150 ?2u ?t2 ?a 2?2u ?x2 = A0 sinωt, 0 < x < l, t > 0, uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l = 0, t ≥ 0, uvextendsinglevextendsinglet=0 = 0, ?u?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 0, 0 ≤ x ≤ l. a23 a146a147a106a107a149a150a110 a82a83a84a85a206a207 15.1 a208 a71a209a120 a39a210a211a100 a76 u(x,t) = ∞summationdisplay n=1 Tn(t)sin npil x, a102a105a106a107a111 A0 sinωta196a113 a202a80a81a82a83a84a85a114a115 a39 A0 sinωt = 2A0pi ∞summationdisplay n=1 1 ? (?1)n n sin npi l xsinωt, a176a177a108a109a151a194a195a154a155a39a99a136 a79 Tprimeprime(t) + parenleftBignpi l a parenrightBig2 Tn(t) = 2A0pi 1 ? (?1) n n sinωt, T(0) = 0, Tprime(0) = 0. a103a212a121a136 Tn(t) = 2A0l 2 pi 1 ? (?1)n n 1 (npia)2 ? (ωl)2 sinωt Wu Chong-shi a213a214a215a216 a217a218a219a220a221 ( a222) a1723a173 ?2A0ωl 3 pi2a 1 ? (?1)n n2 1 (npia)2 ? (ωl)2 sin npi l at. a210a211a184a100a101 a119a120 a205 16.3a110a223 a80a203a224a175 a110a103 u(x,t) = 4A0l 2 pi ∞summationdisplay n=0 1 2n + 1 1 [(2n + 1)pia]2 ? (ωl)2 sin 2n + 1 l pix sinωt ? 4A0ωl 3 pi2a ∞summationdisplay n=0 bracketleftbigg 1 (2n + 1)2 1 [(2n + 1)pia]2 ? (ωl)2 sin 2n + 1 l pix sin 2n + 1 l piat bracketrightbigg . a225 a123a226a148a149a150a39a205a227a39 Poissona108a109a110a228 a80a229 a152 a160 a149a150 ?2u ?x2 + ?2u ?xy2 = f(x,y), 0 < x < a, 0 < y < b, uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=a = 0, 0 ≤ y ≤ b, uvextendsinglevextendsingley=0 = 0, uvextendsinglevextendsingley=b = 0, 0 ≤ x ≤ a, a230a116 a196a100 a182 a113a146a147a106a107a149a150 a82a83a84a85a114a115 a110a231 a77a119 a103a69a205a227a39a100 a76 u(x,y) = ∞summationdisplay n=1 Yn(y)sin npia x, f(x,y) = ∞summationdisplay n=1 gn(y)sin npia x. a176a177a108a109a151a152a153a154a155a39a100a136 Y primeprimen (y) ? parenleftBignpi a parenrightBig2 Yn(y) = gn(y), Yn(0) = 0, Yn(b) = 0, a119a120Y n(y)a39a196a99 a209a120a232 a103 u(x,y)a69a233a234a235a236a237a39a196a100a101 a76 u(x,y) = ∞summationdisplay m=1 Xm(x)sin mpib y, f(x,y) = ∞summationdisplay m=1 hm(x)sin mpib y, a159a117 a197 a120X m(x) a126a127 a110a105a106a107 a128a129a130 a108a109a152 a160 a149a150 Xprimeprimem(x) ? parenleftBigmpi b parenrightBig2 Xm(x) = hm(x), Xm(0) = 0, Xm(a) = 0, a119a120X m(x) a121a100a69 a238a239a240a241 a68a242a243a244a245a246a247a69a248a249a49a250a251a252 a45a46a47a48 g n(y) a253 hm(x) a49a254a255a50a51a63a0a250 a251a39a1a2a3a4a5 Yn(y)a253Xm(x) a49 a45a46a47a239a6a7a8a9a43a44a10 a243a11 a6a12a13 a5a58a61a69 Wu Chong-shi §17.1 a161a162a163a164a165a166a167a168a169a170 (a171) a1724a173 a14 a100a101a15a16a17a18 a80a19 a110a143 a77 a39a121a102 u(x,y) a151 f(x,y) a192a20a21a113 a82a83a84a85 {X n(x)} a22a184a113 a82a83a84a85 {Ym(y)}a114a115 (a145a23a24a25 a85) u(x,y) = ∞summationdisplay n=1 ∞summationdisplay m=1 cnm sin npia xsin mpib y, f(x,y) = ∞summationdisplay n=1 ∞summationdisplay m=1 dnm sin npia xsin mpib y, a114a115a26a85c nm a27 a119 a69a210a145f(x,y)a75a206a28a84a85a39a183a101 cnm a196 a75a206a28 a110a69 a207a29 a23a24a25 a85a114a115a20 a39 a230a116 a206a30 a15a16 a232 a152a153a154a155a69a102a31a32a110 a114a115a175 a176a177a108a109a39a121a136 ? ∞summationdisplay n=1 ∞summationdisplay m=1 cnm bracketleftbiggparenleftBignpi a parenrightBig2 + parenleftBigmpi b parenrightBig2bracketrightbigg sin npia xsin mpib y = ∞summationdisplay n=1 ∞summationdisplay m=1 dnm sin npia xsin mpib y. a187a188a82a83a84a85 a110a189a190a191a39a133a33 a26a85 a39a121a136 ?cnm bracketleftbiggparenleftBignpi a parenrightBig2 + parenleftBigmpi b parenrightBig2bracketrightbigg = dnm. a123 a75 cnm = ? dnmparenleftBignpi a parenrightBig2 + parenleftBigmpi b parenrightBig2. a142a117 a39 a119 a136a103 u(x,y) = ? ∞summationdisplay n=1 ∞summationdisplay m=1 dnmparenleftBig npi a parenrightBig2 + parenleftBigmpi b parenrightBig2 sin npia xsin mpib y. a202a203 a143 a77 a110a34a35 a75 a233a234a36a37 a232a119 a103a105a106a107 a128a129a130 a108a109a69 Wu Chong-shi a213a214a215a216 a217a218a219a220a221 ( a222) a1725a173 §17.2 a38a39a40a41a42a43a44a15a39a40a45 a46a47a48a49a50 a39a51a52a3a53a54a55a56 a10a57 a249a243a11a58 a9a59a60a61a62 a65a5a54a63a64a65a255a22a1a2a66a67a252 a45a46a47 a49a57a68a39a69a70a71a252a249a58 a59a60a61a62 a252 a46a47 a49a69 a49a72a73a59a60a61a62a74a75 a252 a46a47 a49a76 ? a45a46a47a59a60a61a62a250a0a9a77a78a79 ? a80a243a81a82 a46a47a43a44 a253 a46a47a59a60a61a62 a49a60a61a63a64a83a84a85a86a0a81a82 a46a47a43a44 a253 a46a47a59a60a61a62 ? a87a88a89a90a49a91a1a92a93 a46 a90a94a254a255a49a95a96a97 a45a46a47a59a60a61a62a41a98a99a100 a76 a101 a101a102a103a108a109a110a148a103a149a150a145a205a69 a145 a232a104a120 a105a106a107a152a153a154a155a110a35a105a39a106a148a108a109a151a194a195a154a155a107 a75 a106a107a110a69 ?2u ?t2 ?a 2?2u ?x2 = 0, 0 < x < l, t > 0, uvextendsinglevextendsinglex=0 = μ(t), uvextendsinglevextendsinglex=l = ν(t), t ≥ 0, uvextendsinglevextendsinglet=0 = 0, ?u?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 0, 0 ≤ x ≤ l. a145 a232 a147 a182a130a156a157a158a77 a39a108a109a139a144a39 a86a131a110 a102a105a106a107a152a153a154a155a106a107a185a39a121a111 u(x,t) = v(x,t) + w(x,t), a112a230 a139a144v(x,t) a39a113a212 a126a127 v(x,t)vextendsinglevextendsinglex=0 = μ(t), v(x,t)vextendsinglevextendsinglex=l = ν(t). a202a193 a39w(x,t) a230a116 a99 a80 a148 a126a127 a106a107a152a153a154a155 w(x,t)vextendsinglevextendsinglex=0 = 0, w(x,t)vextendsinglevextendsinglex=l = 0. a80a114a115 a135a39 w(x,t) a183 a126a127 a110a108a109a151a194a195a154a155a107a102 a75 a105a106a107a110a39 ?2w ?t2 ?a 2?2w ?x2 = ? parenleftbigg?2v ?t2 ?a 2 ?2v ?x2 parenrightbigg , wvextendsinglevextendsinglet=0 = ?vvextendsinglevextendsinglet=0, ?w?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = ? ?v?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 . a116a182 a228 16.2a208a199 a82a117 a228 1a208a110a231 a77 a39a99a100a101 a119a120 w(x,t) a39a196a99 a209a120a232 a103 u(x,t) a69 star a118a119a120a121 a35a36a37a91a92 v(x,t) a76 a210a145a122 a87a119 v(x,t) a126a127 a152a153a154a155 v(x,t)vextendsinglevextendsinglex=0 = μ(t), v(x,t)vextendsinglevextendsinglex=l = ν(t), a183a101 a131 a146 a230a123 a110a139a144a124a237a69 Wu Chong-shi §17.2 a125a126a127a128a129a130a131a166a126a127a132 a1726a173 a227a133 a174t a134a186 a75a135a85 a39 a202 a99 a86a87a119a207(x,y) a136a32a31a110a137a138y = v(x,t) a139a140 a209 a148a110a141a142(0,μ(t)) a151(l,ν(t))a121a100a69 a205a227a39a100a140a143a138 v(x,t) = A(t)x + B(t), a176a177a152a153a154a155a39a121a100a148 a120 B(t) = μ(t), A(t) = 1lbracketleftbigν(t) ?μ(t)bracketrightbig. a196a100a140a144a145a138 v(x,t) = A(t)x2 + B(t), A(t) = 1l2bracketleftbigν(t) ?μ(t)bracketrightbig, B(t) = μ(t), a199 v(x,t) = A(t)(l ?x)2 + B(t)x2, A(t) = 1l2μ(t), B(t) = 1l2ν(t). a146 17.1 a119 a103a148a103a149a150 ?u ?t ?κ ?2u ?x2 = 0, 0 < x < l, t > 0, uvextendsinglevextendsinglex=0 = Asinωt, uvextendsinglevextendsinglex=l = 0, t ≥ 0, uvextendsinglevextendsinglet=0 = 0, 0 ≤ x ≤ l. a23 a15a16 a79 a105a106a107a152a153a154a155a110a147a148 a224a175 a39a100 a76 a106a107a185 a84a85 a145 v(x,t) = A parenleftBig 1? xl parenrightBig sinωt. a123 a75 a111 u(x,t) = A parenleftBig 1 ? xl parenrightBig sinωt + w(x,t), a149w(x,t) a126a127 a148a103a149a150 ?w ?t ?κ ?2w ?x2 = ?Aω parenleftBig 1 ? xl parenrightBig cosωt, 0 < x < l, t > 0, wvextendsinglevextendsinglex=0 = 0, wvextendsinglevextendsinglex=l = 0, t ≥ 0, wvextendsinglevextendsinglet=0 = 0, 0 ≤ x ≤ l. a102w(x,t) a151a108a109a110a105a106a107a111 1 ?x/la107a113a146a147a106a107a149a150a110 a82a83a84a85a114a115 a39 a131 w(x,t) = ∞summationdisplay n=1 Tn(t)sin npil x, 1 ? xl = ∞summationdisplay n=1 2 npi sin npi l x. a187a188T n(t)a183a147a150 a126a127 a105a106a107 a80a151a128a129a130 a108a109 Tprimen(t) + κ parenleftBignpi l parenrightBig2 Tn(t) = ?2Aωnpi cosωt a194a195a154a155 Tn(0) = 0, a152a198a119a120 Tn(t) = 2Aωl 2 npi 1 κ2(npi)4 + ω2l4 braceleftbigg κ(npi)2 exp bracketleftBig ? parenleftBignpi l parenrightBig2 κt bracketrightBig ?κ(npi)2 cosωt?ωl2 sinωt bracerightbigg . a202a193 a99 a119 a136 a232 w(x,t) a39 a118 a176a153a154a39a99a136 a79 a148a103a149a150a110a103 u(x,t) a69 Wu Chong-shi a213a214a215a216 a217a218a219a220a221 ( a222) a1727a173 a155a156 a250a251a49 a46a47a157 a254a255v(x,t)a39 a158a159 a49w(x,t)a49a54a61a55a56a160 a161a162 a62a250a251a39a58 a159 a49 w(x,t) a162 a62a250a251a69a87a252a39a54a61a55a56a49a61a49a163a3a164a11a97a39a62a165a166a52a88a167a168 a159 a49u(x,t)a11a54a252a169a251 a49a39a170a171a172a173a51a49a50a51a63a0a243a174a250a251a69 a202a193 a99a100a101a175 a120a80a176 a17a177a110 a87a119 a141a139a144a178 a112 a110a106a107a185 a84a85 v(x,t) a39a113 w(x,t) a183 a126a127 a110a148a103a149a150a179 a100a132a137a138a180a69 a142 a105 a74 a110a181a182a39 a230a116 a99 a75 a141a183a184a185a135 u(x,t) a110a108a109 a75 a183 a75 a106a107a110a39 a142a186 w(x,t) a110a108 a109 a75 a106a107a110a69a99a31a32a110a148a103a149a150 a159a187 a39 a202a188a189a190a87a119 a106a107a185 a84a85 v(x,t) a196 a75 a108a109a110a103a39 ?2v ?t2 ?a 2 ?2v ?x2 = 0. a225 a123a191a180a192a193a110 μ(t) a151ν(t)a39 a75 a100a101a143 a79a202a80 a142a110a69 a183a184a185a135a110a108a109 a75 a183 a75 a106a107a110a39a194a195 a174a202a203 a108 a77 a107a204a145 a102a108a109a151a152a153a154a155 a192a20 a106a107a185a69 a146 17.2 a119 a103a148a103a149a150 ?2u ?t2 ?a 2?2u ?x2 = 0, 0 < x < l, t > 0, uvextendsinglevextendsinglex=0 = 0, ?u?x vextendsinglevextendsingle vextendsinglevextendsingle x=l = Asinωt, t ≥ 0, uvextendsinglevextendsinglet=0 = 0, ?u?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 0, 0 ≤ x ≤ l. a23 a196a207 a99a197a198 a78a79a80a176 a106a107a185 a84a85 a39a102a108a109a151a152a153a154a155 a192a20 a106a107a185a69 a145a211a39 a76u(x,t) = v(x,t) +w(x,t) a39a15a16 a79 a105a106a107a152a153a154a155a110a147a148 a84a85a224a175 a39a100a140a106a107a185 a84a85 v(x,t) a145 v(x,t) = f(x)sinωt, a159f(x)a75a199a200a128a129a130 a108a109a152 a160 a149a150 fprimeprime(x) + parenleftBigω a parenrightBig2 f(x) = 0, f(0) = 0, fprime(l) = A a110a103a39 f(x) = Aaω 1 cos ωla sin ωax. w(x,t) a183 a126a127 a110a148a103a149a150 a75 ?2w ?t2 ?a 2?2w ?x2 = 0, 0 < x < l, t > 0, wvextendsinglevextendsinglex=0 = 0, ?w?x vextendsinglevextendsingle vextendsinglevextendsingle x=l = 0, t ≥ 0, Wu Chong-shi §17.2 a125a126a127a128a129a130a131a166a126a127a132 a1728a173 wvextendsinglevextendsinglet=0 = 0, ?w?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = ? Aa cos ωla sin ωax, 0 ≤ x ≤ l. a201a80a114 a103a145 w(x,t) = ∞summationdisplay n=0 parenleftbigg Cn sin 2n + 12l piat + Dn cos 2n+ 12l piat parenrightbigg sin 2n + 12l pix. a187a188 a194a195a154a155a39a100a101a148 a120 Cn = ? 4A picos ωla 1 2n+ 1 integraldisplay l 0 sin ωaxsin 2n + 12l pixdx = (?)n 4Aω(2n + 1)pia 1parenleftBigω a parenrightBig2 ? parenleftbigg2n + 1 2l pi parenrightbigg2, Dn = 0. a102 a202a193a119 a136a110 v(x,t) a151w(x,t) a146a202a39a99 a142a117a209a120a232 a103 u(x,t)a69