Wu Chong-shi a0a1a2a3 a4a5a6a7a8 ( a9) §15.1 a10a11a12a13a14a15a16a17a18a19 (a20) a21a22a23a24 a25a26a27a28l a29a30a31a32a33a34a35a34a36a37a38a39a40a41a42a43a33a44a45a46 a28 ?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. a47a48a49a50a51a52a53a54 star a55a56 a57a58a59a60a61a62a34a63a64a44u(x,t) = X(x)T(t) star a65a66 a67a68X(x)a69a70a34a71a72a57a41a42a73a74a75a45a46a76a43T(t)a69a70a34a71a72a57a41a42 star a77a78 a79a72a57a41a42a73a74a75a45a46a80a81a82a83a34 a84a85a86a84a87a88a89X(x)a87a90a91a92a93a94a95a96a97a98 a40 a99a100a101a50a91a92a93a94a102a103a104a105a95a90a89λ a40 a95a96a106a107a101a108a109a110a111a112a113a106a107a114a115a116a87a95a96a97a98a117a118a119a90a91a92a93a94a87a120a121a97a98a114 a122a123a109a119a124a125λa121 a40a126 a104a127a128a129a110a111a90a91a92a93a94 a29a130 a128a129a110a111a112a113a106a107a87a123a131a96a114 a132a104a133λ a134a135a136 a99a95a121a137 a40a138 a104a127a128a129a110a111a90a91a92a93a94 a29a130 a128a129a110a111a112a113a106a107a87a123a131 a96X(x)a114 λa87a115a136 a99a95a121a139a140a141a142a143 a40 a144a145a87a123a131a96a139a140a141a142a146a147a114 a88a89X(x)a87a90a91a92a93a94a95a96a97a98 a40 a139a140a141a142a143a23a24a114 a47a148a49a50a149a22 a141a142a143a23a24 a150a151a152 λ n = parenleftBignpi l parenrightBig2 , n = 1,2,3,··· a150a151 a67a68 Xn(x) = sin npil x. a153a154a155a156 a34 a150a151a152a157a158a159a160a161 a40a162a163a164a76a165a166a167a68na168a169a40a170a171a40a172a173a174a34a175a176a177a40a178 a150a151a152 a73 a179a180 a34 a150a151 a67a68a80a169 a28λ n a73Xn(x) a114 Wu Chong-shi §15.1 a181a182a183a184a185a186a187a188 a189a190 ( a191) a1922a193 a47a194a49a50a149a195a22 a40a196a197a198a199a200a201 a22 a172 a155 a44a202 a150a151a152a203a204a205 a40a206a207a208 a48a161a150a151a152λ n a40a37a41a42 Tprimeprime(t) +λa2T(t) = 0 a164a76 a155a209a179a180 a34Tn(t)a40 Tn(t) = Cn sin npil at+Dn cos npil at. a170a171a40a210a211 a156a212 a202a69a70a79a72a57a41a42a73a74a75a45a46a34a213a44 un(x,t) = parenleftBig Cn sin npil at+Dn cos npil at parenrightBig sin npil x (n = 1,2,3,···). star a153a154a34a213a44a157a158a159a160a161 star a208 a48a161 a213a44a80a69a70a82a83a79a72a57a41a42a73a82a83a74a75a45a46 star a48a214a215a216a40 a217a218a219a220a48a161 a213a44a221a164a222a210a223a224a69a70a33a44 a203a204 a177a34a225a226a45a46a40 a227a48a214a158a228a229a212 a71 a68Cn a73Dn a40a69a70 Dn sin npil x = φ(x), Cnnpial sin npil x = ψ(x). star a79a72a57a41a42a73a74a75a45a46a80a81a82a83a34a40a178a162a163a34 (a230a231a232a233a234)a213a44a235a236a237a216a40a238a239a81a69a70a82a83 a41a42a73a82a83a74a75a45a46a34a44 a114 a81a240a164a222a69a70a225a226a45a46a241 star a178a242a243 a158a159a160a161 a213a44a235a236a237 a216 u(x,t) = ∞summationdisplay n=1 parenleftBig Cn sin npil at+Dn cos npil at parenrightBig sin npil x, a244a245a246 a68a247 a157 a70a248a224a34a249a250a251 (a252a253a40a164a76a254a255 a155a148a0 a79a72a1) a40a2a3a40 a153a154a156a212 a34u(x,t)a210 a238a239 a81a82a83a79a72a57a41a42a172a82a83a74a75a45a46a4a34a44 a114 a153a5 a61a62a34a44a6 a28 a200a201 a22a114 a162a221a7a207a79a72a57a41a42a34a8a44a40a170 a28 a48a214 a44a221 a244 a81a69a70a79a72a57a41a42a40a9a10a69a70a82a83a74a75a45a46 a11a125a12a13a108a14a96a102a87a15a16a17a89C n a18 Dn a241 ∞summationdisplay n=1 Dn sin npil x = φ(x), (maltesecross) ∞summationdisplay n=1 Cnnpial sin npil x = ψ(x) (dividemultiply) a47a19a49a50a20a21 a141a142a146a147a22a23a24a25a21 a197a198a26 a147 a27a28a29a30 a141a142a146a147a22a23a24a25 integraldisplay l 0 Xn(x)Xm(x)dx = 0, nnegationslash= m. Wu Chong-shi a31a32a33a34 a35a36a37a38a39 ( a40) a1923a193 a172(maltesecross)a62a30a31a7a41a76 sin mpil xa40a254a255a42a57a40a211 a156a212 integraldisplay l 0 φ(x)sin mpil xdx = integraldisplay l 0 ∞summationdisplay n=1 Dn sin npil xsin mpil xdx = ∞summationdisplay n=1 Dn integraldisplay l 0 sin npil xsin mpil xdx = Dm · l2. a43 a76 Dn = 2l integraldisplay l 0 φ(x)sin npil xdx. a7 a154 a40a37(dividemultiply)a62a40a164a76 a156a212 Cn = 2npia integraldisplay l 0 ψ(x)sin npil xdx. a153a154 a211 a155a156 a202a167 a161 a33a44 a203a204 a34a44 a114 star a141a142a146a147a23a24a25a22a44a45 a46 X n(x) = sin npi l x a73Xm(x) = sin mpi l x a81a57a47a206 a180 a207 a150a151a152 λ n a73λm a34a30 a161a150a151 a67a68a40 λn negationslash= λm(a227nnegationslash= m)a114a162a163a57a47a69a70 Xprimeprimen(x) +λnXn(x) = 0, Xn(0) = 0, Xn(l) = 0, a73 Xprimeprimem(x) +λmXm(x) = 0, Xm(0) = 0, Xm(l) = 0. a165Xm(x) a41a76Xn(x)a34a41a42a40a165Xn(x) a41a76Xm(x)a34a41a42a40 a179a48 a40 (Xm(x)Xprimeprimen(x) ?Xn(x)Xprimeprimem(x)) + (λn ?λm)Xm(x)Xn(x) = 0, a172a49a50 [0,l]a173a42a57a40 a227a156 (λn ?λm) integraldisplay l 0 Xn(x)Xm(x)dx = integraldisplay l 0 [Xn(x)Xprimeprimem(x) ?Xm(x)Xprimeprimen(x)]dx = [Xn(x)Xprimem(x)?Xm(x)Xprimen(x)] vextendsinglevextendsingle vextendsingle l 0 = 0. a173a174a165 a212 a202Xn(x)a73Xm(x) a69a70a34a74a75a45a46 a114a25a26 a212 λ n negationslash= λm a40a211a51 a156 a141a142a146a147a22a23a24a25 integraldisplay l 0 Xn(x)Xm(x)dx = 0, nnegationslash= m. square triangle a172a173a174a34a51a52a177 a244 a165 a212 a202 a50 1. a150a151a67a68a69a70a34a72a57a41a42 2. a150a151a67a68a69a70a34a74a75a45a46 a53a157 a165 a212a150a151 a67a68a34a247a54a67a68a61a62 triangle a170a171a40a55a56 a141a142a146a147a57a58a22a59 a51a60a61a62 Xprimeprime(x) +λX(x) = 0, a63 a175a176 (λn ?λm) integraldisplay l 0 Xn(x)Xm(x)dx = [Xn(x)Xprimem(x) ?Xm(x)Xprimen(x)] vextendsinglevextendsingle vextendsingle l 0 a238a239a64a65a114 Wu Chong-shi §15.1 a181a182a183a184a185a186a187a188 a189a190 ( a191) a1924a193 triangle a253a176a66 a150a151 a67a68a69a70a34a74a75a45a46a67 a28 α1X(0) +β1Xprime(0) = 0, α2X(l) +β2Xprime(l) = 0, a68 a177α1 a73β1 a29α2 a73β2 a69 a221a7a70 a28 0 a40 a63a157 α1Xn(0) +β1Xprimen(0) = 0, α1Xm(0) +β1Xprimem(0) = 0 a73 α2Xn(l) +β2Xprimen(l) = 0, α2Xm(l) +β2Xprimem(l) = 0. a170 a28α 1 a73β1 a221a7a70 a28 0 a40 a43 a76 vextendsingle vextendsinglevextendsingle vextendsinglevextendsingleXn(0) X primen(0) Xm(0) Xprimem(0) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = 0. a71 a170 a28α 2 a73β2 a221a7a70 a28 0 a40 a43 a76 a71a157 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle Xn(l) Xprimen(l) Xm(l) Xprimem(l) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle = 0. star a175a72 a50 a73a74 a141a142a143a23a24 Xprimeprime(x) +λX(x) = 0, α1X(0) +β1Xprime(0) = 0, α2X(l) +β2Xprime(l) = 0 a141a142a146a147a22a23a24a25 integraldisplay l 0 Xn(x)Xm(x)dx = 0, nnegationslash= m a75a76a77a78a114 triangle a173a174a34a74a75a45a46a79a80a202 a48 a29 a148 a29 a194a81a194a5a81a82 a34a74a75a45a46 a114 star a141a142a146a147a83a60a84 bardblXnbardbl2 ≡ integraldisplay l 0 X2n(x)dx = l2. a84 bardblXnbardbl a85a86a87a88a89a90a91a92a93a94a95a96a97a98a99a100a101a102a103a90 1 bardblXnbardbl2 integraldisplay l 0 X2n(x)dx = 1 a104a105a106a107 a87 Xn(x)/bardblXnbardbla85a108a90 1a100a109a110a111a112a113a114a115a116a117a118integraldisplay l 0 Xn(x)Xm(x)dx = l2δnm. a89a90a91a92a93a94a95a119a120a96a97a121a100 Wu Chong-shi a31a32a33a34 a35a36a37a38a39 ( a40) a1925a193 star a122a123a124a125a126a127 a21a128a129a22a130a131a132a61 a28 a202a133 a217 a237a134a40 a238 a76 a217a135 a37a225a136a137a138a237a34a139a39 a28 a252 a114 a140 t > 0 a70a40a225a136a137a210a141a172 a158 a75a35a173a57a47a142a143a144a145a146a40a221a7a147a148a81 a212a149 a31a150 x = 0 a151 x = l a70a40 a152a153a154a155a156a216 a40a196a157 a157a158a159 a34 a179 a136a160a161 pi(a227a172a31a150 x = 0a73x = la152a153a162a163a164a165a40 a153 a81a37a30a31 a32a33 a153a154 a34a74a75a45a46a166a33a34) a114 a211a35a173 a219a167a48 a150a172 a219a167a48a161 a70a168a34a136a137a9a169a40a162a211a81a225a136a137a172a30 a161 a31a150a50 a160 a83 a154a170a154a155 a9a235a236 a209 a34a175a176 a114 a206a207a225a171a172a173a174a34a139a39a40 a140a239 a210a164a76 a81a175a176a177 a72 a114 star a128a22a178a179a54 a172 a219a48 a70a168ta40a35a34a39a222a73a136a222a57a47a81 1 2 integraldisplay l 0 ρ parenleftbigg?u ?t parenrightbigg2 dx a73 1 2 integraldisplay l 0 T parenleftbigg?u ?x parenrightbigg2 dx, a180 a222a60 a28 E(t) = 12 integraldisplay l 0 ρ parenleftbigg?u ?t parenrightbigg2 dx+ 12 integraldisplay l 0 T parenleftbigg?u ?x parenrightbigg2 dx. a66a44a62a181a182a40a183a165 a150a151 a67a68a34a166a184a185 a48 a251a40a211a186a187 a155a156 E(t) = mpi 2a2 4l2 ∞summationdisplay n=1 n2bracketleftbig|Cn|2 +|Dn|2bracketrightbig. a188 a62a144a31a189 a239 a81a71a68a40a190 ta158a191a40 a227 a35a34 a180 a222a60a192a193 a84a114 star a22a22a194a200 a25 a253a176a171a33a44 a203a204a157 a30 a161 a44a40u1(x,t)a73u2(x,t)a40a2a3a40v(x,t) ≡u1(x,t)?u2(x,t)a211 a48 a33a69a70 a33a44 a203a204 ?2v ?t2 ?a 2?2v ?x2 = 0, 0 <x<l,t> 0, vvextendsinglevextendsinglex=0 = 0, vvextendsinglevextendsinglex=l = 0, t≥ 0, vvextendsinglevextendsinglet=0 = 0, ?v?t vextendsinglevextendsingle vextendsingle t=0 = 0, 0 ≤x≤l. a244a245 a222a248a51a52v(x,t) = 0 a227 a164 a114 a195a196a197 a173a164a76a198a199a40 a153a200 a33a81a166a201a34 a114 a195 a222a60a192a193a34 a245a155a216a202 a40 a140 t = 0 a70a35a34 a180 a222a60 a28 0 a40a170a171a76 a205 a34 a219a48 a70a168 ta40E(t) a69 a280a114a153a167a203a204a48 a33 a157 ?v ?x = 0, ?v ?t = 0, a227v(x,t) a28 a71a68 a114 a37a225a226a45a46a151a74a75a45a46a40a80a222a33 a209 a171a71a68 a28 0a114 a84 a205a206a207 a85a208a209a102a210a211 13.6a212a85a213a209a111a214a215a216a217 dE/dt = 0a111a218a219a220a221a222a223a224a85a225a226a227a209 (a228a229a111a230a231a232a233a209) a100 Wu Chong-shi §15.1 a181a182a183a184a185a186a187a188 a189a190 ( a191) a1926a193 star a20a21a51a52a53a54a234a149a22a235a59a51a60a61a21a22a23a24a22a236a141a237a238 1. a47a48a49a40a57a58a59a60a114 a115a108a239a240a241a242a243a244a245a84 a40a246a247 a106a107a101a248a91a92a93a94 a18 a112a113a106a107 a126 a101a110a111a87a114a249a92a250a251a252 a87a253a254 a40 a101a255a0a1(a108a2a3a4a2) a103a104a105a95a90a89a87a110a111a90a91a92a93a94 a18 a110a111a112a113a106a107 a40a5 (a108a2a3a4a2) a6a7 a121a97a98a114 2. a47a148a49a40a155a44a150a151a152a203a204a114 3. a47a194a49a40 a155a209 a242a243a34a213a44a40a196a8 a48a49 a235a236 a209a48a214 a44 a114 a9a10a11 a246a12 a104a124a125a13a14a15a16a17a102a87a124a125a108a2a99a96a114 4. a47a19a49a40a183a165 a150a151 a67a68a34a166a184a251a33a235a236a18a68 a114 a19a20a215a216 a40a173a174 a156a212 a34a21 a244 a81a61a62a44 a114 a206a207a247a54 a203a204 a40a21 a152a153a22 a51 a50 1. a115a116a255a0a87u(x,t)a101a23a128a129a248a91a92a93a94a40a24a25a26a27a40a28a89a96a101a23a29a242a30a31a32a33a34a248a91a35a36 2. a115a116a255a0a87u(x,t)a101a23a128a129a112a113a106a107a40a24a25a26a27a40 a28a89a96a87 a18 a88a89a101a23a37a38a36 3. a85a95a15a16a17a89a137a40 a30a31a39a92a101a23a40a41a42 a191a43a44a45a46a47a48a49a50a51a52a53a54a55a56a57a58a59a60a42a61a43a62a55C n a63 Dn a64 a61φ(x) a63 ψ(x) a65a66 a57a49 a67a68φ(x) a63 ψ(x) a57a60a69a70a65a66a71a72 a44a45a46a47a48a57a73a74a42 a75a76a77a78a79a49a80a81a82a83a84a57a85a86a49a87a88 a65 a43a89a90a91a46a92a93a94 1. a95a96a97a98a99a100a101a102 2. a103a101a98a99a104a101a105a103a106a107a108a109a95a96a110a111a112a113 a49a114a115a116a117a49 a95a96a110a111a104a118a119a120a121a122a104a102 3. a95a96a110a111a105a103a123a100a124a125a126 a42 a127a128a129a130a131a132a133a134a73a74a44a91a46a47a48a42 Wu Chong-shi a135a136a137a138 a139a140a141a142a143 ( a144) a1457a146 a147a148a149a150a151a152 a153a154a155a56 un(x,t) = parenleftBig Cn sin npil at+Dn cos npil at parenrightBig sin npil x = An sin(ωnt+δn)sinknx, a156a157 ωn = npil a, kn = npil , An cosδn = Cn, An sinδn = Dn. star un(x,t) a158a159a160a46a161a162 star An sinknxa159a163a164 a78a165a166a57a167a168a80a169 star sinparenleftbigωnt+δnparenrightbiga159a163a170a171 a67a172 star ωn a64 a161a162a57a173a174a175a49a176a177a178a179a180 a66a164 a57a180a181a174a175a182a183a184a174a175a49a185a186a187a92a93a188a189 star kn a176a177a162a55a49 a64a190 a171a191a192 a78a162a57a193a194a55 star δn a64 a186 a170a171 a49a61a186a187a92a93 a65a66 star a130knx = mpia49a195x = mpi/kn = (m/n)l,m = 0,1,2,3,···,na57a165a166a78a49a167a196a57a167a168a197a177 0a49a176 a177a162a198a42 a199a200 a164 a57a178a46a179a166a130a201a49a162a198a166a202a181n+ 1a46a42 star a130knx = (m+1/2)pia49a195x = (2m+1)pi/2kn = (2m+1)l/2n,m = 0,1,2,3,···,n?1a57a165a166a78a49 a167a196a167a168a57a203 a72a204 a197a177a205a206a49a176a177a162a207a42a162a207a166a202a181na46a42 star a208 a46a47a48a57a56a209 a64 a44a210a161a162a57a211a212a42 a124a120a213a214a215a216a217a213 a49 a215a218a101 a41a219a220 a214a221a222a223 a42 a70a178a179a180 a66 a57 a164a224 a79a49a180a181a174a175a157a181 a160 a46a205a225 a204 a49a195 ω1 = pila, a176a177a226a227a49a156a228a180a181a174a175ω n a50 a64a229 a174ω 1 a57 a208 a55a230a49 ωn = nω1, n = 2,3,···, a176a177a231a227a42 star a164 a57 a229 a174a70 a65a66a71a232a233a234a235 a57a236a237a42a130 a164a238a239 a157a49a132 a164 a57a69a240 a160a66 ( a195ρ a160a66) a133a49a241a242a243 a82 a164 a57a244a245a246 a192 (a195a243a82a247a248T a57a206a225) a49a70a249a127a250a198 a229 a174ω 1 a57a206a225a42 star a56a251a157 a229 a174 a63 a230a174a57a211a212a62a55{C n}a63 {Dn}a57a170a72 a206a225 a65a66a71a234a235 a57a174a252a80a169a49a195 a65a66a71a234 a235 a57a236a253a42 Wu Chong-shi §15.1 a254a255a0a1a2a3a4a5 a6a7 ( a8) a1458a146 star a63 a55 ∞summationdisplay n=1 n2bracketleftbig|Cn|2 +|Dn|2bracketrightbig a185 a164 a57a9a10a83a85a11a12a49 a232 a127a70 a65a66a71a234a235 a57a13a14a42 star a15a16a17a18a223 a148a147a19a20 a222 a147a148a21a22 a129a186a187a92a93φ(x) a63 ψ(x) a23a24a25a26 Φ(x) = ? ? ? ?φ(?x), ?l≤x≤ 0, φ(x), 0 ≤x≤l, Ψ(x) = ? ? ? ?ψ(?x), ?l≤x≤ 0, ψ(x), 0 ≤x≤l, a27a128a28 a25a26 a177a193a194a177 2la57a193a194a29a55 ( a30a31 a177Φ(x) a63 Ψ(x)) a42a44a32a25a26 a57a33a34a35a36 a71 a130a179a166 x = l a37 a64 a24a25a26 a42a129Φ(x) a63 Ψ(x) a38a39 a177Fouriera54a55 Φ(x) = ∞summationdisplay n=1 αn sin npil x, Ψ(x) = ∞summationdisplay n=1 βn sin npil x, a156a157 αn = 1l integraldisplay l ?l Φ(x)sin npil xdx = 2l integraldisplay l 0 φ(x)sin npil xdx, βn = 1l integraldisplay l ?l Ψ(x)sin npil xdx = 2l integraldisplay l 0 ψ(x)sin npil xdx. a185a40a41 a66a42 a57C n a63 Dn a170 a12a43a49a70a249a127a154 a42 αn = Dn, βn = npial Cn. a232 a127 u(x,t) = ∞summationdisplay n=1 parenleftBig Cn sin npil at+Dn cos npil at parenrightBig sin npil x = 12 ∞summationdisplay n=1 Dn bracketleftBig sin npil (x?at) + sin npil (x+at) bracketrightBig + 12 ∞summationdisplay n=1 Cn bracketleftBig cos npil (x?at)?cos npil (x+at) bracketrightBig = 12 ∞summationdisplay n=1 αn bracketleftBig sin npil (x?at) + sin npil (x+at) bracketrightBig + 12 ∞summationdisplay n=1 βn npia bracketleftBig cos npil (x?at)?cos npil (x+at) bracketrightBig = 12 [Φ(x?at) +Φ(x+at)] + 12a integraldisplay x+at x?at Ψ(x)dx. a63a44 a162a56a57a45a251a46a47 a160a48 a49a49a50a242a44a51a57Φ(x) a63 Ψ(x) a64 a61a186a187a92a93φ(x) a63 ψ(x)a52a53 a40a41a57a84a209 a25a26 a68a54a57a42 a44a32a54a53a57a56a251u(x,t)a49a132a27a49a131a55a43a56a57 0 ≤x≤l a157a42 Wu Chong-shi a135a136a137a138 a139a140a141a142a143 ( a144) a1459a146 §15.2 a58a59a60a61a62a63a64a65a66a67 a68a69a70a71a72a219a73a74a75a76a77a78a79a80a81a82 a103a98a99 (a83a84 a49 Laplacea79a80) a104a103a101a98a99 a42 a85a181 a66 a56a47a48 ?2u ?x2 + ?2u ?y2 = 0, 0 <x<a, 0 <y<b, uvextendsinglevextendsinglex=0 = 0, ?u?x vextendsinglevextendsingle vextendsingle x=a = 0, 0 ≤y ≤b, uvextendsinglevextendsingley=0 = f(x), ?u?y vextendsinglevextendsingle vextendsingle y=b = 0, 0 ≤x≤a. a30 a55a80a81a82a83a84a86a56a42a87 u(x,y) = X(x)Y(y), star a158a88a89a246a49a80a81a82a83a49a195a54 Xprimeprime(x)Y(y) = ?X(x)Yprimeprime(y). a43 a64 a70a54a53 Xprimeprime(x) X(x) = ? Yprimeprime(y) Y(y) . a130a44a46a90a251a157a49 a91a92a93 a120xa104a110a111(a94ya95a96) a97 a92a93 a120ya104a110a111 (a94xa95a96) a67a98a49 Xprimeprime(x) X(x) = ? Yprimeprime(y) Y(y) = ?λ =? X primeprime(x) +λX(x) = 0 a63 Yprimeprime(y)?λY(y) = 0. star a158a88 a189a99xa57 a160a72a100a101a102a103 a92a93 X(0)Y(y) = 0, Xprime(a)Y(y) = 0, a37a249a127a80a81a82a83a54 X(0) = 0, Xprime(a) = 0. a44a32a49a104a54a53 a71a160 a46a183a184 a204 a47a48 Xprimeprime(x) +λX(x) = 0, X(0) = 0, Xprime(a) = 0. star a105 a147a106a107a108a109a110 a111λ = 0a49a112a113a80 a89 a246a57a241a56 a64 X(x) = A0x+B0. a158a88(a100a101) a102a103 a92a93a49a54A 0 = 0,B0 = 0a114 a67a98a113a80 a89 a246a49a181a115a56 a114 =? λ = 0a50 a64 a183a184 a204a114 Wu Chong-shi §15.2 a116a117a118a119a120a3a121a1a122a123 a14510a146 a111λnegationslash= 0a49a112a113a80 a89 a246a57a241a56a70 a64 X(x) = Asin √ λx+Bcos √ λx. a158a88(a100a101) a102a103 a92a93a49a54B = 0,Anegationslash= 0, cos√λa = 0 a114 a99 a64 a49a70a86 a42a71 a183a184 a204 λn = parenleftbigg2n+ 1 2a pi parenrightbigg2 , n = 0,1,2,3,··· a183a184a29a55 X n(x) = sin 2n+ 1 2a pix. a170a124a125 a49 Yn(y) = Cn sinh 2n+ 12a piy +Dn cosh 2n+ 12a piy. a99 a64 a49a70a54a53 a71a126a127a128 Laplacea89 a246a129a104 a127a128a100a101a102a103 a92a93a57a155a56 un(x,y) = parenleftbigg Cn sinh 2n+ 12a piy +Dn cosh 2n+ 12a piy parenrightbigg sin 2n+ 12a pix. a129a44a188a130a131a46a155a56a211a212a132 a224 a49a70a54a53 a160a133 a56 u(x,y) = ∞summationdisplay n=0 parenleftbigg Cn sinh 2n+ 12a piy +Dn cosh 2n+ 12a piy parenrightbigg sin 2n+ 12a pix. a158a88 a189a99ya57 a160a72(a134a100a101) a102a103 a92a93a49 uvextendsinglevextendsingley=0 = ∞summationdisplay n=0 Dn sin 2n+ 12a pix = f(x), ?u ?y vextendsinglevextendsingle vextendsinglevextendsingle y=b = ∞summationdisplay n=0 2n+ 1 2a pi parenleftBig Cn cosh 2n+ 12a pib+Dn sinh 2n+ 12a pib parenrightBig sin 2n+ 12a pix = 0, a28 a101a135a136 a183a184a29a55a57a11a137a138 a160 a60a49 integraldisplay a 0 sin 2n+ 12a pixsin 2m+ 12a pixdx = a2δnm, a70a249a127a86a54 Dn = 2a integraldisplay a 0 f(x)sin 2n+ 12a pixdx a63 Cn cosh 2n+ 12a pib+Dn sinh 2n+ 12a pib = 0, a61a98a54 Cn = ?Dn tanh 2n+ 12a pib. a44a32a49a70a205a128a86 a42a71a139 a45a56a140a201Laplace a89 a246 a102a204 a47a48a57a54a55a56 a114a141 a34a142a143 a71f(x) a57a144a145a45a251a49 a70a249a127a146 a160a147 a86 a42 a211a212a62a55C n a63 Dn a57a144a145a45a251a114 star a44a46a47a48 a64a148 a66 a47a48a49a185a133a57ta188a189a49a67a98a50 a42a149 a186a187a92a93 a114 star a55a80a81a82a83a84a86a56a133a49a150a55a100a101a102a103 a92a93a151a85a183a184 a204 a47a48a49a68a55 a134a100a101a102a103 a92a93 a66 a211a212a62a55 a114