Wu Chong-shi a0a1a2a3 a4a5a6a7a8 ( a9) §16.1 a10a11a12a13a14a15a16a17a18a19a20a21 a22a23a24a25 ?u ?t ?κ parenleftBig?2u ?x2 + ?2u ?y2 parenrightBig = 0, 0 <x<a, 0 <y<b,t> 0, ?u ?x vextendsinglevextendsingle vextendsingle x=0 = 0, ?u?x vextendsinglevextendsingle vextendsingle x=a = 0, 0 ≤y ≤b,t≥ 0, ?u ?y vextendsinglevextendsingle vextendsingle y=0 = 0, ?u?y vextendsinglevextendsingle vextendsingle y=b = 0, 0 ≤x≤a,t≥ 0, uvextendsinglevextendsinglet=0 = φ(x,y), 0 ≤x≤a, 0 ≤y ≤b. a26u(x,y,t) = v(x,y)T(t) a27a28a29a30a31a27a32a33a34a35a27 v(x,y)Tprime(t) ?κ bracketleftbigg?2v ?x2 + ?2v ?y2 bracketrightbigg T(t) = 0 =? ?2v ?x2 + ?2v ?y2 v(x,y) = 1 κ Tprime(t) T(t) = ?λ a36 ?2v ?x2 + ?2v ?y2 +λv(x,y) = 0, Tprime(t) +λκT(t) = 0, a37a38λ a39a32a33a34a35a40a41a42a43a44a45a46a47a48a49a40a50a51a52a53a54a27a55a56a57 a36 ?v ?x vextendsinglevextendsingle vextendsingle x=0 = 0 ?v?x vextendsinglevextendsingle vextendsingle x=a = 0 ?v ?y vextendsinglevextendsingle vextendsingle y=0 = 0 ?v?y vextendsinglevextendsingle vextendsingle y=b = 0 a58a59v(x,y) = X(x)Y(y) a27a42a60a61a32a33a34a35a27 Xprimeprime(x)Y(y) +X(x)Yprimeprime(y) +λX(x)Y(y) = 0 =? X primeprime(x) X(x) +λ = ? Yprimeprime(y) Y(y) = ν a36 Xprimeprime(x) +μX(x) = 0 Yprimeprime(y) +νY(y) = 0 a62a63a64 a41a42a65a46a47μa27a66μ,νa67λ a38a68a69a70a71a39a72a73a43a27a74a75a76a77a78a79μ+ν = λa48 a58a80 a51a52a53a54a32a33a34a35a27a81a56 a36a82 Xprime(0) = 0, Xprime(a) = 0 a67 Yprime(0) = 0, Yprime(b) = 0. a83a23a84a85X(x) a86a87a88a89 a24a25 Xprimeprime(x) +μX(x) = 0 Wu Chong-shi §16.1 a90a91a92a93a94 a95a96a97a98a99a100a101 a1022 a103 Xprime(0) = 0, Xprime(a) = 0 star a104μ = 0a40a27a46a105a32a30a31a43a106a107a39 X(x) = A0x+B0. a28a29(a108a109) a51a52a53a54a27 a36 A0 = 0, B0 a110a111 . a62a112a113λ = 0 a39a60 a71a114a115a116 a27 a114a115a117 a47a56a118a119 X(x) = 1. a120a121a122a123a124a125a126a126a127a128a129a126μ = 0a127a130a131a132 a27 a128a127a133a134a135μ = 0 a136a27 a130a131a132a137a138a139a140a141 a142X(x) = B 0,B0 a127a143a144a145a146 a48 star a104μnegationslash= 0a40a27a46a105a32a30a31a43a106a107a39 X(x) = Asin√μx+Bcos√μx. a28a29(a108a109) a51a52a53a54a27a81 a36a82 A = 0, √μsin√μa = 0. a147a148 a27 √μa = npi a27a149 a114a115a116 μ n = parenleftBignpi a parenrightBig2 , n = 1,2,3,··· a114a115a117 a47 Xn(x) = cos npi a x. a150μ = 0 a67μ> 0a43a151a152a153a154a155a156a27a157a56 a148a158 a60a159a160 a114a115a116 μ n = parenleftBignpi a parenrightBig2 , n = 0,1,2,3,···, a114a115a117 a47 Xn(x) = cos npia x. a49a161a56 a148 a107 a36a162a163Y(y) a43 a114a115a116a164a165 Yprimeprime(y) +νY(y) = 0 Yprime(0) = 0, Yprime(b) = 0 a43a107a119 a114a115a116 ν m = parenleftBigmpi b parenrightBig2 , m = 0,1,2,3,···, a114a115a117 a47 Ym(x) = cos mpib y. Wu Chong-shi a166a167a168a169 a170a171a95a96a172 ( a173) a1023a103 a174a163a175 a45a43na67ma27 a58 a42a60a61a176a177 T00(t) = A00, n = m = 0, Tnm(t) = Anm e?λnmκt, a37a178a179a180, a55a56 a148 a159a160 a158 a60a43 a180a181 Tnm(t) = Anm e?λnmκt, n = 0,1,2,3,···, m = 0,1,2,3,···, λnm = μn +νm = parenleftBignpi a parenrightBig2 + parenleftBigmpi b parenrightBig2 . a182a183 a27a157a176 a36a184a71 a45a107 a164a165 a43a185a107 unm(x,y,t) = Xn(x)Ym(y)Tnm(t) = Anm cos npia xcos mpib ye?λnmκt a67a60a186a107 u(x,y,t) = ∞summationdisplay n=0 ∞summationdisplay m=0 unm(x,y,t) = ∞summationdisplay n=0 ∞summationdisplay m=0 Anm cos npia xcos mpib ye?λnmκt = ∞summationdisplay n=0 ∞summationdisplay m=0 Anm cos npia xcos mpib yexp braceleftbigg ? bracketleftbiggparenleftBignpi a parenrightBig2 + parenleftBigmpi b parenrightBig2bracketrightbigg κt bracerightbigg . a28a29a187a188a53a54a27 a69 u(x,y,t)vextendsinglevextendsinglet=0 = ∞summationdisplay n=0 ∞summationdisplay m=0 Anm cos npia xcos mpib y = φ(x,y). a189 a60a61a157a190a104a191a192 a114a115a117 a47a43a193a194a195a45a177a196a197a198a47a48a199a200a201a202a203 a82 {X n(x), n = 0,1,2,···}a43a193a194 a195a27a81a202a203 a82{Y m(y),m = 0,1,2,···}a43a193a194a195a27a204a60a205a56a48 a37 a109a27a206a207 a82 a74a75a43a193a194a208a60a195integraldisplay a 0 Xn(x)Xnprime(x)dx = a2 (1 +δn0)δnnprime, integraldisplay b 0 Ym(y)Ymprime(y)dy = b2 (1 +δm0)δmmprime. a209a210a38a211a212 a202a213a214a215a32n = 0a216nnegationslash= 0a67m = 0 a216mnegationslash= 0a43 a179a180 a48 a209a210 a43a151a152a39 Anm = 4ab 1(1 +δ n0)(1 +δm0) integraldisplay a 0 integraldisplay b 0 φ(x,y)cos npia xcos mpib ydxdy. Wu Chong-shi §16.2 a217a218a219 a220a221a222a223a224a225a226 a2274 a228 §16.2 a12a229a230a18a231a17a232a233a234a235 a236a237a238a239a240a241a242a120a236a237a243a244a245a246a247a240a248a249a250a251a252a253a254a255a0a1a2a3a133a134a241a242a120a243a244a245a246 a127a236a237a126 a27 a240a248a249a250a4a5a6a7a8 a48 a9a10a11a142a137a138 a252a126a241a242a120a243a244a245a246a124a127a236a237a126 a27a12 a139a13a139a14a15a16a2a240a248a249a250a251a17 a22a23a24a25 ?2u ?t2 ?a 2?2u ?x2 = f(x,t), 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. a119a65a18a177 a174a163 a30a31a19a108a109a20a43a21a22a27 a62a63a23a24a25a26 a50a27a28a41a155a43 a70a29a30 a45a31a43a32a33a34a35a27a31a43a187 a36a37 a67a187a38a39a40a119 0a48 a41 a87 a23a42a43 a44a45a46a47a48a49a50a51a52a53a54a55 u(x,t) = v(x,t) +w(x,t), a200 a80 a19a108a109a30a31a108a109a56a43a49a40a27a76a77a57a58a59 a69 a43a108a109a51a52a53a54a205a34a48 a107a60a43 a162a61 a157a200 a163 a176 a36 a185a107v(x,t)a48 a62 a203 a163f(x,t)a180a181a63a64a65a66 a43 a179a180 a48 a23 a67a68 a176a107a19a108a109a30a31a43a60a69a70a60a27a71a176 a36 a19a108a109a30a31a43a60 a71 a185a107 v(x,t) a27 ?2v ?t2 ?a 2?2v ?x2 = f(x,t). a128a72 a27 a9a10a73u(x,t) = v(x,t) +w(x,t) a27 a74 ?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 a75a76a77 a5a78a79a142v(x,t) a27a80 a14a6 a77a81w(x,t) a126 a75a82 a142 w(x,t) = ∞summationdisplay n=1 parenleftBig Cn sin npil at+Dn cos npil at parenrightBig sin npil x, Wu Chong-shi a166a167a168a169 a170a171a95a96a172 ( a173) a1025a103 a83a6 u(x,t) = v(x,t) + ∞summationdisplay n=1 parenleftBig Cn sin npil at+Dn cos npil at parenrightBig sin npil x, a84a85a86a87a245a246 a27 ∞summationdisplay n=1 Dn sin npil x = ?v(x,t)vextendsinglevextendsinglet=0, ∞summationdisplay n=1 Cnnpial sin npil x = ??v(x,t)?t vextendsinglevextendsingle vextendsingle t=0 , a88a2a130a131a89a146a126a90a91a92 a75a93 a27 a11 a81a94a95a96 a146 Cn = ? 2npia integraldisplay l 0 ?v(x,t) ?t vextendsinglevextendsingle vextendsingle t=0 sin npil xdx, Dn = ? 2l integraldisplay l 0 v(x,0)sin npil xdx. ? a128a97a142a251a98a134a241a242a120a243a244a245a246a126a125a136a236a237a99a48 ? a247a100a140a236a237a241a242a236a237a99a126a125a136a27a101a102a103a104a105a139a126a236a237a243a244a245a246a124a249a48 ? a142a251a126a255a0a80a247a106a77a5a79a142v(x,t)a48a107 a2a106f(x,t) a108a109a110a111a112a113 a126a114 a108a48 ? a236a237a86a87a245a246a126a115a116a14a6a117a118a48 a119 16.1 a176a107a45a107 a164a165 ?2u ?t2 ?a 2?2u ?x2 = f(x), 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, a37a38f(x) a119a120a121 a117 a47a48 a23 a68a175 a177a107 a165 a43a122a202a123a124a48 a50 a163 a30a31a43a19a108a109a20 a68 a39xa43 a117 a47a27a157a56 a148a150 a108a109a56 a117 a47a55a118a119 a68 a39xa43 a117 a47a27a149 a26 u(x,t) = v(x) +w(x,t), a37a38v(x) a78a79a46a105a32a30a31a43a51 a116a164a165 vprimeprime(x) = ? 1a2f(x), v(0) = 0, v(l) = 0; a125w(x,t) a126a78a79a45a107 a164a165 ?2w ?t2 ?a 2?2w ?x2 = 0, 0 <x<l, t> 0, wvextendsinglevextendsinglex=0 = 0, wvextendsinglevextendsinglex=l = 0, t≥ 0, wvextendsinglevextendsinglet=0 = ?v(x), ?w?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 0, 0 ≤x≤l. Wu Chong-shi §16.2 a92a127a128 a98a129a97a130a131a132a133 a1026 a103 a119 16.2 a176a107a45a107 a164a165 ?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, a37a38a, A 0 a134 ωa40a119a120a121a46a47a48 a23 a26 u(x,t) = v(x,t) +w(x,t), a206a207 a82 a19a108a109a20a43a135a136 a180a181 a27a56 a80 a108a109a56 a117 a47v(x,t)a118a119 v(x,t) = f(x)sinωt. a137a36v(x,t) a78a79a19a108a109a30a31 a134 a108a109a51a52a53a54a27 ?2v ?t2 ?a 2?2v ?x2 = A0 sinωt, 0 <x<l, t> 0, vvextendsinglevextendsinglex=0 = 0, vvextendsinglevextendsinglex=l = 0, t≥ 0, a55a157a39a138a139f(x)a27 a137a36 ?ω2f(x) ?a2fprimeprime(x) = A0, f(0) = 0, f(l) = 0. a62a71 a19a108a109a46a105a32a30a31a43a106a107a119 f(x) = ?A0ω2 +Asin ωax+Bcos ωax. a28a29a108a109a51a52a53a54a56 a148 a45a177 B = A0ω2, A = A0ω2 tan ωl2a. a163 a39 f(x) = ?A0ω2 bracketleftbiggparenleftBig 1?cos ωax parenrightBig ?tan ωl2a sin ωax bracketrightbigg = ?A0ω2 bracketleftbigg 1? cos(ω(x?l/2)/a)cos(ωl/2a) bracketrightbigg . a62 a161a157a140a141a177w(x,t) a147 a78a79a43a45a107 a164a165 a27 ?2w ?t2 ?a 2?2w ?x2 = 0, 0 <x<l,t> 0, w vextendsinglevextendsingle vextendsingle x=0 = 0, wvextendsinglevextendsinglex=l = 0, t≥ 0, wvextendsinglevextendsinglet=0 = 0, ?w?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = ?ωf(x), 0 ≤x≤l. a74a43a60a186a107a119 w(x,t) = ∞summationdisplay n=1 bracketleftBig Cn sin npil at+Dn cos npil at bracketrightBig sin npil x. Wu Chong-shi a166a167a168a169 a170a171a95a96a172 ( a173) a1027a103 a142 a203a143a144a43a187a188a53a54a157a56 a148 a45a177 Dn = 0, Cn = ? 2ωnpia integraldisplay l 0 f(x)sin npil xdx = ?2A0ωl 3 pi2a 1?(?)n n2 1 (npia)2 ?(ωl)2. a68a69n = a145a47a40a27Cn a146 a205a1190 a48 a62 a161a27a147a148a157a176a177a65 w(x,t) = ?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 a67 u(x,t) = ? A0ω2 bracketleftbigg 1? cosω(x?l/2)/acos(ωl/2a) bracketrightbigg 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 . a149a150a151a152a3 a32a33a28a43a153a154a155ωa193a156a39a31a43a157a158 a30a69 a154a155a27 ω = (2k+ 1)pia/l, ka119a157 a71a159 a45a43a19a160 a184 a47 a31a200a32a33a28a43a161a203 a189a162a163a164a165 a34a199a166a48 a119 16.3 a176a107a45a107 a164a165 ?2u ?x2 + ?2u ?y2 = xy, 0 <x<a, 0 <y<b, uvextendsinglevextendsinglex=0 = 0, uvextendsinglevextendsinglex=a = 0, 0 ≤y ≤b, uvextendsinglevextendsingley=0 = φ(x), uvextendsinglevextendsingley=b = ψ(x), 0 ≤x≤a. a23 a167a168 a176a177a30a31a43a106a107 1 6x 3y +f(x+ iy) +g(x?iy). a62 a104a138a139 a117 a47f a67ga27a169a170a27 f(x+ iy) +g(x? iy) = ?a 2 24i bracketleftbig(x+ iy)2 ?(x?iy)2bracketrightbig = ?1 6a 2xy, a137a36a82 a43a107 v(x,y) = 16 parenleftbigx2 ?a2parenrightbigxy a78a79a108a109a51a52a53a54 v(x,y)vextendsinglevextendsinglex=0 = 0, v(x,y)vextendsinglevextendsinglex=a = 0. a59 u(x,y) = v(x,y) +w(x,y), Wu Chong-shi §16.2 a92a127a128 a98a129a97a130a131a132a133 a1028 a103 a157a56 a148 a141a177w(x,t) a147a190a78a79a43a45a107 a164a165 a27 ?2w ?x2 + ?2w ?y2 = 0, 0 <x<a, 0 <y<b, wvextendsinglevextendsinglex=0 = 0, wvextendsinglevextendsinglex=a = 0, 0 ≤y ≤b, wvextendsinglevextendsingley=0 = φ(x), wvextendsinglevextendsingley=b = ψ(x) ? b6 parenleftbigx2 ?a2parenrightbigx, 0 ≤x≤a. a37a38 a43a30a31a67a60 a174 a51a52a53a54a171a39a108a109a43a27 a182a183 a27a172 a167a168 a176a107a48