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