Wu Chong-shi a0a1a2a3 ( a4) a5a6a7a8a9a10a11a12a13 §14.1 a14a15a16a17a18a19a20a16a17 star a21a22a23 Newtona24a25a26a27a28a29a30a31a32a33a34a35a36a37a38a39a40a41a42a26a43a44a30a45a31a46 star a47a48a49a42a26a40a50a43a44a30a45a31a51a52a53a54a55a34a35a56a57a51a58a59a60a61a62a63a64a65a46a66a67a25a68a69a54a55a34a35 a30a70a71a72a73a61a74a50a75a76a69a77a51a78a79a71a37a80a39a40a42a26a30a46 star a21a61a81a54a55a34a35a51a82a37a38a39a40a41a83a42a26a41a84a85a86a40a50a87a88a30a89a90a91a35a46 star a25a68a81a54a55a34a35a30a70a71a51a73a61a74a50a75a76a92a77a46a93a94a51a81a54a55a34a35 ?2u(x,y) ?x2 = 0 a30a70a71a95a80 u(x,y) = c1(y) +xc2(y), a96 a72c1(y)a97c2(y)a80ya30a75a76a92a77a46 a98 a61a34a35a51a79a71a36a37a39a40a46a99a89a90a100a101a102a51a82a80a103 a104 a30a51a78a105 star a106a107a108a34a35a109a51a21a110a111a53a112a43a30a113a114a51a36a115a61a110a111a112a43a70a91a116a117a97a57a118a30a119a120a121a122a46a78a123a51 a124a125a126 a101a51a127a128a129a130a131a132a133a134a135a136a46 star a94a137a138a139a140a109a141a61a142a30a143a51a106a107a108a34a35a109a82a36a115a61a110a111a112a43a30a144a145a146a147a46a94a137a148a149a150a151a152a153 a154 a109a30a155a44a51a156a157a51a95a158a159 a126 a51a127a128a160a129a130a131a132t> 0a161a162a163a164a165a46 a105a53a48a49a84a85a40a50a87a61a42a26a71a30a89a90a138a139a51a106a77a33a100a95a80a47a166a167a40a50a26a71a138a139a51a168a52a53a54a55 a34a35a56a57a51a58a59a60a61a169a118a64a65a97a62a63a64a65a46 a170a171a172a173 a170a171a172a173a174a175a176a177a178a179a170a171 a164a165 (t = 0a164)a133a134a135a136a180a181a182a183a162a184a163a185a161a186a187a46 star a188a189a190a31a34a35a101 a126 a51a95a80a191a192a193a29a62a63a109a194a30a195a196a97a197a198 (a94a137a80a32a33a138a139a30a143) a51 uvextendsinglevextendsinglet=0 = φ(x,y,z), ?u?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = ψ(x,y,z), (x,y,z) ∈V. star a188a189a199a200a108a34a35a51a201a189a34a35a72a21a29a202a203a204a92a77u(x,y,z,t)a188ta30a40a68a81a54a205a51a206a159a21a207a193a29 Wu Chong-shi §14.1 a208a209a210a211a212a213a214a210a211 a2152a216 a62a63a109a194a30a217a198 uvextendsinglevextendsinglet=0 = φ(x,y,z), (x,y,z) ∈V. a181a182 a172a173 a30a218a219a220a221a222a223a224a51a47a201a87a88a138a139a72a84a225a30a87a88a146a147a226a26a46a227a161a228a229a230a231a181a182 a172a173 a174a175a176a177a178a179 a181a182a183a232a185a233a162a163a164a165 (t≥ 0)a161a186a187a46 a234 a161a235a236a237 a94a137a238a30 a239 a240a241a242a243a244 a51a67a169a118a64a65a95a80 uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l = 0, t≥ 0. a245 a161a246a236a237 a247x = 0 a241a242a243 a51a67x = 0a248a30a169a118a64a65a249a80 uvextendsinglevextendsinglex=0 = 0. a247a250a163 a241 (x = l) a251xa127a252a183a161a253a254a255a131a51a0a195a117a1a100a30a32 a80 F(t) a46a2a3a107a108a34a35a30a4a5a51a106a248a44 x = la6a7a153a40a8a9 a112a43a51 a10 a198a105 εa46a22a23Newtona24a25a26a27a158a204a51a168a40a8a11a112a43 a206a12a30a13a32 (a57a32a14a113a191a32) a51a191a192a15a189a112a43a30a43a16a17a159a112a43 a72a86a40a44a30a14a197a198a51 ρεS ? 2u ?t2 vextendsinglevextendsingle vextendsinglevextendsingle x=l?αε = F(t)S?P(l?ε,t)S, a1814.1 a19a20a21a22a23a24a25a26a24a27a28 a96 a720 ≤α≤ 1a46a29ε→ 0a51a36a30a31 P = E?u?x, a67a61 ?u ?x vextendsinglevextendsingle vextendsinglevextendsingle x=l = 1EF(t). ? a94a137a57a32a105 0a51a32x = la241a230a33a34a161a51a67 ?u ?x vextendsinglevextendsingle vextendsinglevextendsingle x=l = 0. ? a94a137a57a32F(t)a37a80a40a50a42a26a30a35a204a92a77a51a79a80a201a36a37a38a39a30a36a40a32a51a67 F(t) = ?kbracketleftbigu(l,t)?u0bracketrightbig, ka80a36a37a30a41a198a42a77a51a189a80a51 bracketleftbigg ?u ?x + k Eu bracketrightbigg x=l = kEu0. a43a44a45a46a47 a69a48a30a169a118a64a65a61a49a28a50a51a52a53a231 star a181a182a183a232a185a161a54a55a56a57a51 uvextendsinglevextendsingleΣ = φ(Σ,t). Wu Chong-shi a58a59a60a61 ( a62) a63a64a65a66a67a68a69a70a71 a2153a216 Σa116a72a169a118a100a30a73a44a51a74a109a82a116a72a168a75a44a30a76a77a46 star a78a79a164a80a135a83a81a82a78a79a83a84a161a181a182a83a85a86a161 a43a87 a56a57a46 a106a169a118a113 a88 a7a153a40a8a89a90a112a43a51a40a50a91a117a106a112a43a30a116a117a51 a92 a40a50a91a117a106a112a43a113a114a46 a93 a88a30a74a91a117a1a119a15a51 a94 a198a95a189 0a46 a22a23a38a16a96a97a26a27a158a204a51a98a99a100a101a102a103a104a105a106a104a107a108a109a110a111 a112a113 a51 a114a115a116a117a118a119a120 a98a99a121a122a123a124a125a126a127a109a110a111a46 a128 a80a51a151 a112a43a30 a94 a198a95a1890a109a51a70a91 a88 a117a129a31a30a199a16a191a192a95a189 0 (a78a105 a88 a117a1a95a189 0) a51a112a43a30a199a130a16a95a189 0 (a78a105a112a43a30a43a16a95a189 0)a51a78a123a51a70a91a112a43a116a117a129a31a30a199a16a51a191a151a49a114a70a91a89a90a30 a92 a40a91a117a129a131a112a43a113a114a46a189a80a51a158a159a85a29a169a118a64a65 ?u ?n vextendsinglevextendsingle vextendsinglevextendsingle Σ = 1kψ(Σ,t), a1814.2 a132a133a134a135a136a137a138a139a140 a96 a72 ? ?n a141 a105a5a131a54a205a51a142a80a143a198a144a16a106a57a5a145a34a131na100a30a146a147a51 ? ?n = n·? a32 ?u ?n = n·(?u). ? a181a182a148a43a51a67ψ ≡ 0a51 ?u ?n vextendsinglevextendsingle vextendsingle Σ = 0. star a133a134a81a82a181a182a149Newtona150a151 a243a152a153a43 a231a78a79a164a80a81a82a78a79a83a84a154a83a155a253a182a156a157a161 a43a87a158 a133 a134a154a83a54a55u|Σ a155a253a182a54a55u0 a159a160a161a162a163 a46a164a220a93a42a77a105H a51 ?k ?u?n vextendsinglevextendsingle vextendsinglevextendsingle Σ = HparenleftbiguvextendsinglevextendsingleΣ ?u0parenrightbig, a165a166 a85a167 bracketleftbigg ?u ?n +hu bracketrightbigg Σ = hu0. a100a117a29a202a30a169a118a64a65a61a40a50a167a74a30a168a44a231a95a203a204a92a77a79a169a51a142a149a170a80a145a40a30a46 a171a172 a40a173a174a55a51 a58a158a159a55a105a175a52a231 star a176 a119a177a178a179a180a181 a231a182a183 a178a179a184a185a186 a109a187a188a189 star a176a190 a177a178a179a180a181 a231a182a183 a178a179a184a185a186 a187a188a109a191a192a193a194a189 star a176a195 a177a178a179a180a181 a231a182a183 a178a179a184a185a186 a109a187a188a189a196a191a192a193a194a189 a112a197 a109a198a199a200a201 Wu Chong-shi §14.1 a208a209a210a211a212a213a214a210a211 a2154a216 a202a203 a102 a178a179 a104 a184 a51 a185a186 a109 a178a179a180a181a204a205a119a206a207a208a209a119 a109a210a211a212a51a213 a205a214a215a216a217a117a119a218 a177a219 a46a220a221 a184 a104a222a223a109a224a199a225a109 a178a179a180a181 a51a226a227a228a229a46 star a230a182a231a80a161 a46a47 a168a109a30a169a118a64a65a95a191a151a232a233a234a57a235a236a233a230a237a238a239a161a240a241a242a243a51a93a94 a92a77a244a245a142a30a108a77a106a246a247a248a6a61a118a46 a202a208a179a249 a197 a109a250a251a252a51 a208a253 a213a127a183a254 a208a179a180a181 a46 a255 a228a51a0a1a2a3a4a5a6a7a201a83 a8 a6a7a201 a9a10 a6a7a201 a253 a51 a11 a193a194?u/?r a202 a6a7a12 a186a13a14a15a16 a46a17a18a126a127a19a20a21a22a23a24a25 a26a27a28 a12 a29a30a31a32a33a34a35a36a37a38a39a35a36a40 Wu Chong-shi a41a42a43a44 ( a45) a46a47a48a49a50a51a52a53a54 a555a56 §14.2 a57a58a59a60a61a62a57a63 a64a65a66a67a68a69 a25a70a71a72a73a74a71a75a76 a64 a74a25a77a78a74a25a79a80a75a81a70a74a82 star a71a74a76 a64a83 a70a71a72a73a84a71 a40 a85a86a87a88a89a90a91a92 a25 a93a94a95a96 a25 a97a87a88a98a99a100a88a40a101a85 a25 a85a86a102a103a104a105a106a107a108a109a110a111a87 a25 a112a102 a103a104a113a105a106a114a108a110a115a116a117a118a87a108a119a120a121a122a40a123a109a124a105a106a125a126a93a94a95a96a108a40 star a71a74a77a78 a83 a70a71a72a73a74a71a75a77a78a74 a40 a85a86a87a88a89a90a127a128 a25 a87a88a98a99a108a88a125a127a126a129a102a108a40 a130a131 a25 a132a133a134a135a136a137a138a135a139a26a140a141a142a143 a25 a144a145a132a133a134a135a136a137a146a147a148 a149a150a151a152 a25 a134a135a35a36 a132a153a154a153a155 a25a156a157a158a159 a40 star a71a74a81a70 a83 a160a161 a70a71a72a73a162a74a163a164 a67a68( a165 a160a166a167a168 a70 a71 a67a68 a162a74a163a164a169a170)a84a171 a172a173a174a175 a25a71a176a177a84a171 a172 a74 a173a174a40 a178a179a180a87a88a98a99a181 a25 a127a182a183a184a185a186a105a121a187a188a189a190a191a40a192a193 a25a194a195 a178a196a87a197a198a199a182a108a200a201a202a198 a121a108a187a188a189a190a191a203a126 a195a204a205 a108a40 a206a207 a70a71a72a73a71a74a76 a64a83a208 a77a78 a83a209 a81a70 a83 a25a210a211a212a70 a83a40 a177a213a214a215a216a217a218a72a73a74a219a220a75a221a218a74a25 a222a223a67a68 a74a224a75a225a226a227 a208 a224a70a227a228a229a230 a222a223a175a231( a232a233 a234a235t = 0) a236a237a238a239a240a241a242a243a244a245a246a247a78a248a74a249a250a25a242a243 a67a68 a74a224a75a225a226a251a80a224a70a227a228a229a230a242a243 a244a245a246a247a78a248 a64t≥ 0 a74a249a250a25 a252a66 a25 a253a254a255a0 a74a70a71a72a73a1a78a70a75a212a70a74a25a176a1a75a2a25a71a78a70a75 a76 a64 a74 a208 a77a78a74a25a79a80a75a81a70a74 a40 a3a4a94a5a108a98a99a126 a25 a222a223a67a68a209 a242a243 a67a68 a162a6a7a74a163a164a169a170a8a9a10a11a78a70a74a12a13 a83 a213a14 a40 a15a16a17a18a98a99a19a101a40a85a86a20a21a89a90a126 u(x,y,z,t)vextendsinglevextendsingleΣ = f(Σ,t), a22a23a24a89a90a126 u(x,y,z,t)vextendsinglevextendsinglet=0 = φ(x,y,z), a25a26 a25 a125a27a28 a195 f(Σ,t)vextendsinglevextendsinglet=0 = φ(x,y,z)vextendsinglevextendsingleΣ. a195a29 a87a88a98a99a127a102a87a30a128a123a124a105a106a40a182a15a31a32 a25 a33a23a24a34a201a35a36a19φ(x,y,z)a108a102a37a38a39 a40a117a102a124a41a34a42a43(a101a85a34a201a41a19u 0) a44a25 a45a22a46a38a39a47a104a108a34a201a48a49a50a51a117a41a34u 0 a25 a85a86a105a106a108a52a201a199a182 a25 a38a39a47a104a53a54a55a56a34a91a57a108a58a59a182a15a60a61 a25 a25a26 a25 a125a182a15a187a62a185 a63a20a21a89a90a64a65 u(x,y,z,t)vextendsinglevextendsingleΣ = u0. Wu Chong-shi §14.2 a51a52a53a54a66a67a51a68 a556a56 a123a69a70a108a71a86 a25 a72a73a189a52a118a108a20a21a89a90a74 a195a75a76a25 a77 a194 a105a123a78 a75a76 a128a79a80 a25 a25a26 a25 a88a108 a196a87a197a125a81a82a83a84 a25a85 a4a86a87a108a88a108 a75a88 a48a126a128a79a80a108a40a28a193 a25 a85a86a83a84a125a126a105a89a90a123 a78a53a54a55a56a34a91a57a108a58a59 a25 a123a78a190a191a125a126a127a182a91a108a40 Wu Chong-shi a41a42a43a44 ( a92) a48a93a94a95a96 a557a56 a97a98a99a100 ( a101) a102 a103 a104 a105 a106 a107a108a109a110a111a134a135a136a137a138a112a113a114a135a115 a25 a109a116a117a118a115a40 a135a113a108a109a110a111a134a135a136a137a119 a25 a120a113a121a145a122a133a123a108a109a110a111a138a124a135 a25 a125a126a127a128a129a138a124a135a130a131a123 a120a135 a25a132a133 a114a134a135a35a36 ( a134a135a136 a35a36) a134a123a130a131a137a138a40 a143a139a126a127a107a108a109a110a111a138a133a135a136a137 a25a140a141 a138a110a115a142a145a143a144a145a143a139a126a127a113a108a109a110a111a146a138a133 a135a136a137a40 a147a148a149a139a131a150a151a152a139a138a107a108a109a110a111a134a135a136a137 a25a23a24 a33a153a153a154a155a156a157a158a131 a122a133a123 a107a108a109a110 a111a138a120a135 a25 a125a148a120a135a159a160a33a161a134a162a138 a25 a143a163a164a165 a25 a166a131a167a168a169a170a134a135a35a36a134a123a40a145a171a133 a135a107a108a109a110a111a138a134a135a136a137 a25 a144a172a173a174a133a135a175a176a131a131a177a178a138a179a180 a40 §14.3 a181a182a183a57a184a61a185a186a187a188 a70a71a72a73 a189a190a191 a19la208a109a110a111a87a108a107a108a192 a85a193a194a25 a103a57a195a87a88a89a90a19 ?2u ?t2 ?a 2?2u ?x2 = 0, 0 <x<l, t> 0, uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=l = 0, t≥ 0, uvextendsinglevextendsinglet=0 = φ(x), ?u?t vextendsinglevextendsingle vextendsingle t=0 = ψ(x), 0 ≤x≤l. a103a57a189a20a21a89a90a196a126a197a198a108 a25 a22a23a24a89a90a126a199a197a198a108a40 a200a201a202a203a133a148a138a124a135a204a33a109a116a117a118a138a205a206 a25 a156 u(x,t) = X(x)T(t). star a63u(x,t)a207a208 a103a57 a25a209a210 X(x)Tprimeprime(t) = a2Xprimeprime(x)T(t). a211a212a109a110a213a15X(x)T(t) a25 1 a2 Tprimeprime(t) T(t) = Xprimeprime(x) X(x) . a178a123a124a211a212 a44a25 a214 a110 a194 a126ta108a215a216 ( a217a218a219a220a25 a3xa100a5) a221a110 a194 a126xa108a215a216 ( a217a218a219a220a25 a3ta100a5) a222a4 a25 a214 a110a189a221a110a94a211 a25 a125a223a224a225a226a211a227a102a124a228a3xa100a5a208a113a3ta100a5a108a229a216a40a31a19?λ a25 a230a104a108 a71a86a125a182a15a188a65 Tprimeprime(t) +λa2T(t) = 0, Xprimeprime(x) +λX(x) = 0. Wu Chong-shi §14.3 a231a232a233a234a235a236a237a233a234 a558a56 star a63u(x,t)a207a208 a20a21a89a90 a25a210 X(0)T(t) = 0, X(l)T(t) = 0. a123a181a223a224 a195 X(0) = 0, X(l) = 0. a123a69a125a238a65a239a122a35a240a241a242a243a106a88a244a245a35a103a57a87a88a98a99a108 a246a102a247a155a248a249a174a250 star a251a252 a155a35a240a241a242a253a212a108a199a254a88u(x,t) = X(x)T(t) star a255a161a155a215a216X(x)a30a128a108a229a245a35a103a57a189a20a21a89a90a15a195T(t)a30a128a108a229a245a35a103a57 star a67a68a155a244a245a35a103a57a189a20a21a89a90a196a126a197a198a108 a0a178a1a0a108a215a216X(x)a108a229a245a35a103a57a87a88a98a99 a25 a2a115a126a155a245a35a103a57 a44a3a195a4 a87a229a216λ a25 a87a88a89a90a126a102a5a197a198a20a21a89a90a40a123a69a108a87a88a98a99a127a226a227a229a245a35a103a57a108a23a6a98a99a40 a7a199a5a227a8a9λa6 a25 a196 a195 a228a30a128a197a198a229a245a35a103a57a208a113a30a128a197a198a20a21a89a90a108a199a254a88a40 a194a195 a28λa91a10 a29 a2a87a6a181 a25 a203 a195 a228a30a128a197a198a229a245a35a103a57a208a113a30a128a197a198a20a21a89a90a108a199a254 a88X(x)a40 λa108a123a29a2a87a6a11a19a12a13a14a15 a94a27a108a199a254a88a11a19a12a13 a169a170 a40 a215a216X(x)a108a229a245a35a103a57a87a88a98a99a15a11a19a12a13a14 a72a73 a40 a246a16a247a155 a14a71 a12a13a14 a72a73 star a17λ = 0a15a245a35a103a57a108a18a88a126 X(x) = A0x+B0. a207a208 a20a21a89a90 X(0) = 0, X(l) = 0, a125a182a15a87a1 A0 = 0, B0 = 0. a123 a220a19λ = 0a181a245a35a103a57a194a195 a254a88a40 a217a218a219a220 a15λ = 0a127a126a20a21a6a40 star a28λnegationslash= 0a181a15a229a245a35a103a57 Xprimeprime(x) +λX(x) = 0 Wu Chong-shi a41a42a43a44 ( a92) a48a93a94a95a96 a559a56 a108a18a88a126 X(x) = Asin√λx+Bcos√λx, a207a208 a20a21a89a90a15a125 a195 B = 0, Asin √ λl = 0. a222a19Anegationslash= 0a15a22a223 a195 √λl = npia15 a209 a20a21a6 λ n = parenleftBignpi l parenrightBig2 , n = 1,2,3,···. a94a27a108a20a21a215a216a125a126 Xn(x) = sin npil x. a123a69a106 a210 a108a20a21a6 a195 a100a23a24a124a15a25a84a182a15a122a26a27a216n a28a29 a15a222a30a15a31a230a32a33a34a35a36a15a37a20a21a6a38 a39a40a33a20a21a215a216a196 a29a41λn a38Xn(x)a42