Wu Chong-shi a0a1a2a3a4 a5a6a7a8 a9a10a11a12a13a14a15a16a17a18a12a19a20a21a22a23a24a25a26a27a28a29a30a15a31a32a33a34a35a36a11a12a13a14a15a16 a17a18a12a19a20a37 a38a34a21a30a15a31a32a39 Laplacea31a32a40 Fouriera31a32a41a24a37 §27.1 a42a43 Laplace a44a45a46a47a48a49a50a51a52a53a47a54a55 Laplace a31a32a56a34a35a11a12a57a58a59a21a13a14a15a16a17a18a12a19a20a37a31a32a60a27a61a31a62a21a63a64a65a66a67a68a69 a23a63a37a70a71a27a66a67a72xa40ta41a63a61a31a62a21a13a14a15a16a17a18a12a19a20a27a31a32a60a73a74a75a11a12a38a14a15a16a17 (a61 a31a62a76x)a21a18a12a19a20a37a23a77a78a67a27a60a79a80a65a81a82a83a11a12a37a84a85a11a86a21a72a66a87a21a18a12a19a20a21a12a21a88 a89a64a27a90a91a92a93a94a27a95a96a86a97a66a87a19a20a21a12a37 a98 27.1 a11a99a100a101a21a102a103a104a19a20 ?u ?t ?κ ?2u ?x2 = f(x,t), ?∞<x<∞, t> 0; uvextendsinglevextendsinglet=0 = 0, ?∞<x<∞ a21a12a37 a105 a106a107a108a109a110a111a112a113a114a115a116a117a118a27a119a119a120a121a122a123a124a125a126a110a127a128a37a129a130a131a27a109a110a111a112a27a132a133 a134a135a136a137 a131a113a138a139a27a140a132a133a141a122a106a142a143a144a113a145a146 ( a147 a112a27a148a146a27···) a149 a27a150a151a113a152a153a154a155a156a157a37a158 a159 a27a160a161a162a163a164a165a124a125a114a115a116a117a113a166a27a167a168a169a170a171a126a110a127a128 uvextendsinglevextendsinglex→±∞ → 0. a172Laplacea31a32a37a173 u(x,t) equaldotleftrightU(x,p) = integraldisplay ∞ 0 u(x,t)e?ptdt, a174a34a175a87a176a177a27a39 ?u ?t equaldotleftrightpU(x,p). a178a31a32a60a21a88a89a64a74a179a180a72xa21a89a64a27pa72a181a64a27a182a183 ?2u ?x2 equaldotleftright d2U(x,p) dx2 , a14a184a185a186a73a72a23a187a89a64a21a14a184a37a188a189a23a190a173 f(x,t) equaldotleftrightF(x,p), a84a85a27a36a191a192 Laplacea31a32a60a27a18a12a19a20a73a31a180 pU(x,p)?κd 2U(x,p) dx2 = F(x,p). a36Uvextendsinglevextendsingle x→±∞ → 0 a21a176a177a193a56a183a86a97 U(x,p) = 12 1√κp integraldisplay ∞ ?∞ F(xprime,p)exp braceleftbigg ? radicalbiggp κ|x?x prime| bracerightbigg dxprime. Wu Chong-shi §27.1 a194a195 Laplacea196a197a198a199a200a201a202a203a204a205a199a206a207 a2082a209 a188a210a211 Laplacea31a32a21a93a94a212a213 1√ pe ?α√p equaldotrightleft 1√ pit exp braceleftbigg ?α 2 4t bracerightbigg a183a214a215a30a18a216a27a73a96a217a218a60a86a97 u(x,t) = 12√κpi integraldisplay ∞ ?∞ dxprime integraldisplay t 0 exp braceleftbigg ?(x?x prime)2 4κ(t?τ) bracerightbiggf(xprime,τ) √t?τ dτ. a34Laplacea31a32a11a12a13a14a15a16a17a18a12a19a20a27a219a220a56a183a68a69a61a31a62a21a64a221a183a222a27a223a224a225 a226a89a64a21a88a89a64 (a70a71a16a17a21a227a228a229a230a27a231a21a232a213a56a96a233a234a235) a236a237a238a239 a91a240a241a11a242a27 a36a11a93a94a58a74a75a33a34a215a30a18a216a28a56a37 a98 27.2 a34Laplacea31a32a11a12a99a100a243a21a244a245a19a20 ?2u ?t2 ?a 2? 2u ?x2 = 0, ?∞<x<∞, t> 0; uvextendsinglevextendsinglet=0 = φ(x), ?u?t vextendsinglevextendsingle vextendsingle t=0 = ψ(x), ?∞<x<∞. a105 a246a36Laplacea31a32a247a193a27 u(x,t) equaldotleftrightU(x,p), a35a72a27a66a67a21a18a12a19a20a73a248a76 p2U(x,p)?a2d 2U(x,p) dx2 = pφ(x) +ψ(x). a56a183a11a86a249a16a17a21a12 ( a250a251a252 a90a253a254a220U(x,p)a36x→ ±∞a21a255a76) U(x,p) = 12ap integraldisplay ∞ ?∞ bracketleftBig pφ(xprime) +ψ(xprime) bracketrightBig exp braceleftBig ?pa vextendsinglevextendsingle vextendsinglex?xprime vextendsinglevextendsingle vextendsingle bracerightBig dxprime = 12a integraldisplay ∞ ?∞ bracketleftBig φ(xprime) + ψ(x prime) p bracketrightBig exp braceleftBig ?pa vextendsinglevextendsingle vextendsinglex?xprime vextendsinglevextendsingle vextendsingle bracerightBig dxprime. a0a76 e?αp equaldotrightleftδ(t?α), 1pe?αp equaldotrightleftη(t?α), a182a183 u(x,t) = 12a integraldisplay ∞ ?∞ φ(xprime)δ parenleftBig t? |x?x prime| a parenrightBig dxprime + 12a integraldisplay ∞ ?∞ ψ(xprime)η parenleftBig t? |x?x prime| a parenrightBig dxprime = 12 integraldisplay ∞ ?∞ φ(xprime)δ(at?|x?xprime|)dxprime + 12a integraldisplay ∞ ?∞ ψ(xprime)η(at?|x?xprime|)dxprime, a1a2a97 δ(at?|x?xprime|) = braceleftBigg 0, |x?xprime| negationslash= at, ∞, |x?xprime| = at, η(at?|x?x prime|) = braceleftBigg 0, |x?xprime|>at, 1, |x?xprime|<at, a73a56a183a11a242 u(x,t) = 12 integraldisplay ∞ ?∞ φ(xprime)δ parenleftBig t? |x?x prime| a parenrightBig dxprime + 12a integraldisplay x+at x?at ψ(xprime)dxprime Wu Chong-shi a3a4a5a6a7 a8 a202a196a197 a2083a209 = 12 bracketleftBig φ(x?at) +φ(x+at) bracketrightBig + 12a integraldisplay x+at x?at ψ(xprime)dxprime. a34 Laplace a31a32a11a12a13a14a15a16a17a18a12a19a20a90a39a23a63a9a10a27a84a73a72 a239 a91a29a227a228a229a21a11 a100a176a177a228a229a248a27a0a76a84a58a66a39a21a13a14a15a16a17a18a12a19a20a21a227a228a229a11a100a176a177a29a12a248a76a38a14 a15a16a17a21a227a228a229a11a100a176a177a27a84a13 a239a14a15 a67a66a16a21a17a18a37 Wu Chong-shi §27.2 Fouriera196a197 a2084a209 §27.2 Fourier a44a45 Fouriera31a32a56a19a20a59a31a62a189a255a37a210a211a20a59a31a62a21a31a248a21a59a27a56a183a22a34 ? Fouriera31a32 a19a35a23a24a25a26(?∞, ∞)a252 a21a89a64f(x)a27a71a27a36a28a2a39a29a21a59 a252 a74a39a39a29a63a30 a31a30a32a40a39a29a63a33a23a34a59a35a10a27a36 a30a15 integraldisplay ∞ ?∞ f(x)dxa37 a19a38a39a27a16a231a21 Fouriera31a32a40a36a27 F(k) = a41[f(x)] ≡ 1√2pi integraldisplay ∞ ?∞ f(x)e?ikxdx, a42a43a31a32 (a93a94) a72 f(x) = a41 ?1[F(k)] ≡ 1√ 2pi integraldisplay ∞ ?∞ F(k)eikxdk. a84a44a21 Fourier a31a32a40a43a31a32a21a232a213a56a96a40a45a79a46a47a21a232a213a48a39 a239a49 a37a232a213a50a51a19 a52a27a50a53a54a76a55a216a56a57a182a58a34a37 ? a59 a243a31a32a40a60a243a31a32 a71a27f(x)a72a18a61a36a62a23a24a25a26[0, ∞) a252 a27a16a56a210a211x = 0 a63 a11a100a176a177 a21 a239a49 a34a64a27a22a34 a59 a243a31a32 F(k) = radicalbigg2 pi integraldisplay ∞ 0 f(x)sinkxdx, f(x) = radicalbigg2 pi integraldisplay ∞ 0 F(k)sinkxdk, a65a60a243a31a32 F(k) = radicalbigg2 pi integraldisplay ∞ 0 f(x)coskxdx, f(x) = radicalbigg2 pi integraldisplay ∞ 0 F(k)coskxdk.. ? a39a29a59 a243a66a60a243a31a32 a71a27f(x)a72a18a61a36a39a100a21a59 a252 a27a16a33a58a34a39a29 a59 a243a65a60a243a31a32a37 a67a68a9a10a99a100a20a59 a252 a21 Fouriera31a32a37 a34Fouriera31a32a67a11a12 a252 a23a68a21a70 27.1a40a7027.2a37 Wu Chong-shi a3a4a5a6a7 a8 a202a196a197 a2085a209 star a19a35a70 27.1a27a28a99a100a101a21a102a103a104a19a20 ?u ?t ?κ ?2u ?x2 = f(x,t), ?∞<x<∞, t> 0; uvextendsinglevextendsinglet=0 = 0, ?∞<x<∞. a69a246u(x,t)a21Fouriera31a32a40a36a27 U(k,t) = 1√2pi integraldisplay ∞ ?∞ u(x,t)e?ikxdx, a13a246 F(k,t) = 1√2pi integraldisplay ∞ ?∞ f(x,t)e?ikxdx, a84a85a27a36a172 Fouriera31a32a60a27a18a12a19a20a73a31a76 dU(k,t) dt +κk 2U(k,t) = F(k,t), U(k,t)vextendsinglevextendsinglet=0 = 0, a34a38a64a31a83a26a11a12a84a63a23a70a38a14a15a16a17a21a175a71a19a20a27a73a86a97 U(k,t) = e?κk2t integraldisplay t 0 F(k,τ)eκk2τdτ. a188a11a93a94a27 u(x,t) = 1√2pi integraldisplay ∞ ?∞ U(k,t)eikxdk = integraldisplay t 0 bracketleftbigg 1 √2pi integraldisplay ∞ ?∞ F(k,τ)e?κk2(t?τ)eikxdk bracketrightbigg dτ. a174a34 integraldisplay ∞ 0 e?t2 cos2xtdt = 12√pie?x2, a56a183a186a242 1√ 2pi integraldisplay ∞ ?∞ e?κk2(t?τ)eikxdk = 1√2pi integraldisplay ∞ ?∞ e?κk2(t?τ) coskxdk = 1radicalbig2κ(t?τ) exp bracketleftbigg ? x 2 4κ(t?τ) bracketrightbigg , a188a174a34 f(x,t) = 1√2pi integraldisplay ∞ ?∞ F(k,t)eikxdk, a210a211Fouriera31a32a21a215a30a212a213a27 a41[f1(x)]a41[f2(x)] = a41 bracketleftbigg 1 √2pi integraldisplay ∞ ?∞ f1(ξ)f2(x?ξ)dξ bracketrightbigg , a73a96a218a60a86a97 u(x,t) = integraldisplay t 0 braceleftBigg 1√ 2pi integraldisplay ∞ ?∞ f(ξ,τ)radicalbig 2κ(t?τ) exp bracketleftbigg ? (x?ξ) 2 4κ(t?τ) bracketrightbigg dξ bracerightBigg dτ = 12√κpi integraldisplay t 0 braceleftbiggintegraldisplay ∞ ?∞ f(ξ,τ)exp bracketleftbigg ? (x?ξ) 2 4κ(t?τ) bracketrightbigg dξ bracerightbigg dτ √t?τ. a40 a252 a23a68a72a86a97a21a12a213a73a74a23a85a37 a75a12a26 a252 a179a27 Fouriera31a32a21a93a94a19a20a76a77a78a65Laplacea31a32a79a80a23a224a27a81a81 a239 a75a78 a34a82a64a18a216a67a83a186a93a94a72a242a84a21a18a30a15a37a73a67a70a42a85a27a41a24a16a26 a238 a78a34a97a215a30a212a213a37 Wu Chong-shi §27.2 Fouriera196a197 a2086a209 star a188a67a12a70 27.2a27a99a100a243a252 a21a61a86a87a245a19a20a27 ?2u ?t2 ?a 2? 2u ?x2 = 0, ?∞<x<∞, t> 0; uvextendsinglevextendsinglet=0 = φ(x), ?u?t vextendsinglevextendsingle vextendsingle t=0 = ψ(x), ?∞<x<∞. a88a246u(x,t)a21Fouriera31a32a40a36a27 U(k,t) = 1√2pi integraldisplay ∞ ?∞ u(x,t)e?ikxdx, a13a246 Φ(k) = 1√2pi integraldisplay ∞ ?∞ φ(x)e?ikxdx, Ψ(k) = 1√2pi integraldisplay ∞ ?∞ ψ(x)e?ikxdx, a35a72a27a36a172 Fouriera31a32a60a27a18a12a19a20a73a31a76 d2U(k,t) dt2 +k 2a2U(k,t) = 0, U(k,t)vextendsinglevextendsinglet=0 = Φ(k), U(k,t)vextendsinglevextendsinglet=0 = Ψ(k). a84a72a23a63a89a70a38a14a15a16a17a21a175a71a19a20a27a12a247a28a86 U(k,t) = Φ(k)coskat+Ψ(k)sinkatka . a210a211Fouriera31a32a21a93a94a212a213a27a73a56a183a11a242 u(x,t) = 1√2pi integraldisplay ∞ ?∞ bracketleftbigg Φ(k)coskat+Ψ(k)sinkatka bracketrightbigg eikxdk. a1a2 1√ 2pi integraldisplay ∞ ?∞ Φ(k)coskateikxdk = 1√2pi 12 integraldisplay ∞ ?∞ Φ(k) bracketleftBig eik(x+at) + eik(x?at) bracketrightBig dk = 12bracketleftbigφ(x+at) +φ(x?at)bracketrightbig, a34a76a54a27a90a39 1√ 2pi integraldisplay ∞ ?∞ Ψ(k)sinkatka eikxdk = 1√2pi integraldisplay ∞ ?∞ Ψ(k) bracketleftbiggintegraldisplay t 0 coskaτ dτ bracketrightbigg eikxdk = integraldisplay t 0 bracketleftbigg 1 √2pi integraldisplay ∞ ?∞ Ψ(k)coskaτ eikxdk bracketrightbigg dτ = 12 integraldisplay t 0 bracketleftBig ψ(x+aτ) +ψ(x?aτ) bracketrightBig dτ = 12a integraldisplay x+at x?at ψ(ξ)dξ, a90a91 a252a92 a21a93a27a27a218a60a73a86a97 u(x,t) = 12 bracketleftBig φ(x+at) +φ(x?at) bracketrightBig + 12a integraldisplay x+at x?at ψ(ξ)dξ. a84a94a95a40a34 Laplacea31a32a86a97a21a232a213a73a74a23a85a37 Wu Chong-shi a3a4a5a6a7 a8 a202a196a197 a2087a209 a98 27.3 a11a12a96a97a99a100a20a59a244a245a16a17a21a18a12a19a20a27 ?2u ?t2 ?c 2?2u = f(r,t),t> 0, uvextendsinglevextendsinglet=0 = φ(r), ?u?t vextendsinglevextendsingle vextendsingle t=0 = ψ(r). a105 a98a99a172 Fouriera31a32a37a173 U(k,t) = 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay u(r,t)exp{?ik·r}dr, Φ(k) = 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay φ(r)exp{?ik·r}dr, F(k,t) = 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay f(r,t)exp{?ik·r}dr, Ψ(k) = 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay ψ(r)exp{?ik·r}dr, a16a18a12a19a20a248a76a38a14a15a16a17a175a71a19a20 d2U dt2 +k 2c2U(k,t) = F(k,t), Uvextendsinglevextendsinglet=0 = Φ(k), dUdt vextendsinglevextendsingle vextendsingle t=0 = Ψ(k). a188a172Laplacea31a32 U(k,t) equaldotleftrightU(k,p), F(k,t) equaldotleftrightF(k,p). a35a72a27a18a12a19a20a189a23a190a31a180a90a64a16a17 p2U(k,p)?pΦ(k)?Ψ(k) +k2c2U(k,p) = F(k,p). a12a247a28a86 U(k,p) = 1p2 +k2c2bracketleftbigF(k,p) +pΦ(k) +Ψ(k)bracketrightbig. a11a93a94a37a99a172 Laplacea31a32a21a93a94a27a39 U(k,t) = 1kcΨ(k)sinkct+ Φ(k)coskct+ 1kc integraldisplay t 0 sinkcτF(k,t?τ)dτ. a188a172Fouriera31a32a21a93a94a27 u(r,t) = 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay U(k,t) exp{ik·r}dk = 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay Ψ(k) sinkctkc exp{ik·r}dk + 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay Φ(k) coskctexp{ik·r}dk + 1(2pi)3/2 integraldisplayintegraldisplayintegraldisplay bracketleftbigg 1 kc integraldisplay t 0 sinkcτF(k,t?τ)dτ bracketrightbigg exp{ik·r}dk. a174a34Fouriera31a32a21a100a30a212a213a27a73a56a183a11a242 a252a101a102 a230a21a93a94a27a75a42a73a218a103a11a242a18a12a19a20a21a12a37 a58a34ka20a59a21a104a105a106a27a56a183a186a242a107 star a33a23a230 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay sinkct kc exp{ik·r}dk = 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay sinkct kc e ikrcosθk2 sinθdkdθdφ = 1√2pic integraldisplay ∞ 0 k sinkctdk integraldisplay pi 0 eikrcosθ sinθdθ = 1√2pic integraldisplay ∞ 0 sinkct 1?ir eikrcosθ vextendsinglevextendsingle vextendsingle pi 0 dk Wu Chong-shi §27.2 Fouriera196a197 a2088a209 = 1√2pic2r integraldisplay ∞ 0 sinkct sinkrdk = radicalbiggpi 2 1 crδ(r?ct). a182a183a27a210a211 Fouriera31a32a21a100a30a212a213a27a73a39 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay Ψ(k) sinkctkc exp{ik·r}dk = 1(2pi)3/2 radicalbiggpi 2 1 c integraldisplayintegraldisplayintegraldisplay 1 |r ?rprime|δ(|r ?r prime|?ct)ψ(rprime)drprime = 14pic integraldisplayintegraldisplay Σprime 1 |r ?rprime|ψ(r prime)dΣprime, a108a72Σprime a72a183r a76a104a109a66cta76a110a111a21a104 a92 |r ?rprime| = cta37 star a33a89a230 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay Φ(k) coskct exp{ik·r}dk = 1(2pi)3/2 ??t integraldisplayintegraldisplayintegraldisplay Φ(k) sinkctkc exp{ik·r}dk = 14pic ??t integraldisplayintegraldisplay Σprime 1 |r ?rprime|φ(r prime)dΣprime. star a33a96a230 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay bracketleftbigg 1 kc integraldisplay t 0 sinkcτF(k,t?τ)dτ bracketrightbigg exp{ik·r}dk = integraldisplay t 0 braceleftbigg 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay bracketleftbiggsinkcτ kc F(k,t?τ) bracketrightbigg exp{ik·r}dk bracerightbigg dτ = integraldisplay t 0 braceleftbigg 1 4pic integraldisplayintegraldisplayintegraldisplay 1 |r ?rprime|δ(|r ?r prime|?cτ)f(rprime,t?τ)drprime bracerightbigg dτ = 14pic integraldisplayintegraldisplayintegraldisplay 1 |r ?rprime| bracketleftbiggintegraldisplay t 0 δ(|r ?rprime|?cτ)f(rprime,t?τ)dτ bracketrightbigg drprime, a112a95a27 integraldisplay t 0 δ(|r ?rprime|?cτ)f(rprime,t?τ)dτ = ?? ? 1 cf(r prime,t?|r ?rprime|/c), |r ?rprime|<ct; 0, |r ?rprime|>ct. a182a183a27 1 (2pi)3/2 integraldisplayintegraldisplayintegraldisplay bracketleftbigg 1 kc integraldisplay t 0 sinkcτF(k,t?τ)dτ bracketrightbigg exp{ik·r}dk = 14pic2 integraldisplayintegraldisplayintegraldisplay |r?rprime|<ct 1 |r ?rprime|f(r prime,t?|r ?rprime|/c)drprime. a178 a252a92 a21a93a27a113a72a114a67a27a73a218a60a11a86 u(r,t) = 14pic ? ? integraldisplayintegraldisplay Σprime 1 |r ?rprime|ψ(r prime)dΣprime + ? ?t integraldisplayintegraldisplay Σprime 1 |r ?rprime|φ(r prime)dΣprime ? ? + 14pic2 integraldisplayintegraldisplayintegraldisplay |r?rprime|<ct 1 |r ?rprime|f(r prime,t?|r ?rprime|/c)drprime.