数学物理方程教案

发布于:2021-09-19 09:25:04

1??

???§

8?
1 ?§ 1.1 1.2 1.3 1.4 2 ?J 2.1 2.2 2.3 2.4 2.5 u u ?! ?)^? ?o? ???§ . . . . . . . . . . . . . . . . . . . . . . . . . ? . . . . . . . . . . . . . . . . . . . . . . . . . ??§ 2 2 4 6 6 7 7 7 9 9 12 12 12 13 15 15 17 17 17 19 19 21
第1页

?)^? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?)?K·?5Vg . . . . . . . . . . . . . . . . . . . . . . . ú?! ? U\ ??§ D? ?J ){ . . . . . . . . . . . . . . . . . . .

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

D?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?6?m! ?????K??? . . . . . . . . . . . . . . . . . àgz n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?lC?{

3 ?>??K 3.1 3.2 3.3 3.4 3.5

Fourier?ê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?lC?{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ) ?n?? . . . . . . . . . . . . . . . . . . . . . . . . . . . ?/ . . . . . . . . . . . . . . . . . . . . . . . . . ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …??K ? . . . . . . . . . . . . . . . . . . . . . . . . . J{ . . . . . . . . . . . . . . . . . . . . . . . . . . ?àg?§

4 p‘???§ 4.1 4.2 4.3 4.4 ?)^?

??§

???{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ü‘{ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5 5 ? 5.1 5.2 5.3

?àg???§…??K ) . . . . . . . . . . . . . . . . . . D??P~ ?6??! ?????K??? . . . . . . . . . . . . . . . . . ¨?d n(Huygens)! ? ‘? . . . . . . . . . . . . . . . . P~ . . . . . . . . . . . . . . . . . . . . . . . . . ???§)

22 22 22 23 23 25 25 27 30

6 U?? 6.1 6.2 7

?! ???§) ??5?-?5 ??5?-?5 . . . . . . . . . . . . . . . . . ??5?-?5 . . . . . . . . . . . . . . . . . .

?>??K) …??K)

??S??

1
1.1 ?o?
? ? 1.1 ?



?! ?)^?

???§
? ? ? §: ?k???ê9? êê?m 'X); ? ?Y ); ê ê. ê, …ò§“\ ê ?§(??gC?!

???ê9? ? PDE

) : X?3?ê?k?§¤I?

?§?U??§¤?? ?, K?T?ê??§ ? PDE (order): PDE¤?k ???ê?p ê??5

? XJ?§'u???ê9??

, K?d?§?? 5

? § (linear), ??????5?§ (nonlinear); ? X??5?§é???ê ¤k?p ê o N 5`??5 ,K

?§?[ ? 5 ? § (quasilinear); ? X??5?§??§é???ê ? ? ? 5 ? § (fully nonlinear). ê ‘, d?§??? à g ? ?p ê???1 , K?§

? XJ?§??k?????ê9?

§ (nonhomogeneous), ????àg?§ (homogeneous); F ( x, y, u( x, y), u x ( x, y), uy ( x, y)) = F ( x, y, u, u x , uy ) = 0
第2页

F ( x, y, u, u x , uy , u xx , u xy , uyy ) = 0 1. 2. 3. 4. 5. 6. 7. u x + uy = 0 u x + uuy = 0 utt ? u xx + u3 = 0 ut + uu x + u xxx = 0 u xx + uyy = 0 utt + u xxxx = 0 ut ? i u xx = 0 transport shock wave wave with interaction dispersive wave Laplace’s equation vibrating bar quantumn mechanics

8. (cos xy2 ) u x ? y2 uy = tan( x2 + y2 ) ~ 1.1 (1) u xy = 0; ): (2) XJ?\?? n = (a, b) = ai + bj, K au x + buy = gradu · n = u( x, y) ÷ ? ?n 7 ? ~ ê. ? ?u = f (bx ? ay). ,): ?C?“? ξ = ax + by ??E??êó?? {K, ux = uy = ?u ?ξ ?u ?η + = auξ + buη , ?ξ ? x ?η ? x ?u ?ξ ?u ?η + = buξ ? auη . ?ξ ?y ?η ?y )?u = f (η) = f (bx ? ay). η = bx ? ay. ?u = 0. ?n u( x, y) = ? 6 ubx ? ay, = ? § (2) au x + buy = 0; (3) u x + yuy = 0.

)

dau x + buy = (a2 + b2 )uξ = 0 ?uξ = 0. ? ?§ ?S? (1) 4u x ? 3uy = 0, u(0, y) = y3 ; (2)

u x + 2 xy2 uy = 0.

第3页

(1) u( x, y) = (3 x + 4y)3 /64;

1 (2) u( x, y) = f ( x2 + ). y

1.2

u
?

??§

?
? .; ?!R^ u, ???l, 3 $?5?. ?, ?d????? ? ?, ?u??: ???'?±

??{: ‰???üà ?NC?? :u

??^e3?? ? u??!

????u

-?, ??—?ρ ?~ê; ? u3???S?? ? ?: =u ???3???NC, ?; ?????u ? u??

:?3????SR??T?? ???? ? u?R^ ?', u , §3/C??-| u -? ?±? . u T ( x + ? x, t ) α2

-. u??:m

????—,

?/C???'X?lHooke ??. ,

, ???

α1 T ( x, t ) x O ????1 x + ?x x ? ?/. X??á ?

??, ?k????

??^?u

?IX, u( x, t) L?u??:3t ??÷R?u x ?? u?( x, x + ? x), ?l?? ?s =
x x+? x

?. 3u??

1+

?u dx ≈ ?x

2

x+? x x

d x = ? x.

(1.1) ?

dHooke? ? ?, u ? z ? : ¤ ? ? ? ? ? m ? ', ? ò x : ? ?T ( x, t) P ?T ( x), ? ? ? o ? ÷ X u 3 x : ? ?????", = T ( x + ? x) cos α2 ? T ( x) cos α1 = 0, ? ? ? ?.

u?Y?

(1.2)
第4页

cos α1 = 1 cos α2 = 1 ?

1 + [u x ( x, t)]2 ≈ 1, 1 + [u x ( x + ? x, t)]2 ≈ 1,

(1.3) (1.4) (1.5) u?? O\, k

T ( x + ? x) ? T ( x) = 0.

T ???~ê. ??3(t, t + ?t) ?m?S ??A
t+?t t t+?t

(T sin α2 ? T sin α1 )dt =
t

T
x+? x x

?u( x + ? x, t) ?u( x, t) ? dt ?x ?x ?u( x, t + ?t) ?u( x, t) ? dx ?t ?t

=
t+?t x+? x

ρ

?
t x

T

?2 u( x, t) ?2 u( x, t) d xdt = 0 ?ρ 2 ?x ?t2

d? x, ?t

??5? utt ? a2 uxx = 0, (a2 = T/ρ). ?x
x+? x

(1.11)

í

'…? ?s T ( x) 1 + u2 x T ux 1 + u2 x
x+? x



=0
x x+? x x

=
x

ρutt d x ? ?^, ü ??¤?

í21: XJu3 ?? F ( x , t ) , k

?L§???

??

f ( x, t) L?ü í22:

2 ?2 u 2? u ? a = F ( x, t)/ρ = f ( x, t). ?t2 ? x2 ??3 x :?¤? ?.

(1.14)

‘???§(X

?) (1.15)

2 ?2 u ?2 u 2 ? u = a + + f ( x, y, t). ?t2 ? x2 ?y2

í23: n‘???§(X>^?! (? D?)
2 ?2 u ?2 u ?2 u 2 ? u = a + + + f ( x, y, z, t). ?t2 ? x2 ?y2 ?z2

(1.16)
第5页

1.3

?)^?
??^?(Initial Condition): u3????t = 0 u( x, 0) = ?( x), ?u( x, 0) = ψ( x) ?t ???? (1.18)

(0 ≤ x ≤ l).

>.^?(Boundary Condition): 1. 1?a>.^?(Dirichlet >.^?): u(0, t) = 0, 2. 1 a>.^?(Neuman >.^?): ?u ?x
x =0

u(l, t) = 0.

(1.17)

= 0 or

?u ?x

x=0

= ?(t).

3. 1na>.^?(Robin >.^?): ?u -σ1 u ?x ??σ1 , σ2 ??? = ?(t) or
x=0

?u +σ2 u ?x

= ν(t).
x=l

ê. ? ? §(1.14) ? ? ) ^

> . ^ ? ? ? ? ^ ? o ? ? ? ) ^ ?. u

?(1.17)! (1.18) (??5, ?)?K, ???>??K, ?·??K: ? ? ? utt ? a2 u xx = f ( x, t), ? ? ? ? ? ? ? ? t = 0 : u = ?( x), ut = ψ( x), ? ? ? ? x = 0 : u = 0, ? ? ? ? ? ? x = l : u = 0.

1.4 ?)?K·?5Vg
? ? 1.2 ?5; ? XJ???)?K ·? )??3 , ?? , …?-? , ?d?K? ? ?)?K ? 3 5! ? ? 5! - ? 5 ? ? ? ? ) ? K ·

(well-posed), ?K??·?

(ill-posed).

第6页

2 ?J
2.1 U\
J

ú?! ?

D?

n
? n?¤ ) ) J uù ) ?? .d ?ü? ) ??^3???N?¤ \???\ \???±^ü? n??U\

?n?, A???

\\. ~XA?

??gü??^3T?N?¤ ?

n(Superposition Principle). U\ ?ny–5`, ??¤á . ~X:

néu^?5?§??5?)^??

utt ? a2 u xx = f1 ( x, t) utt ? a2 u xx = f2 ( x, t) utt ? a2 u xx = C1 f1 ( x, t) + C2 f2 ( x, t) ?

? ?

u1 ( x, t) u2 ( x, t)

(2.1) (2.2)

u( x, t) = C1 u1 ( x, t) + C2 u2 ( x, t) (2.4)

2.2

u

??§

?J

){

???K?…?(Cauchy)?K: utt ? a2 u xx = f ( x, t) (t > 0, ?∞ < x < ∞), t = 0 : u = ?( x), ut = ψ( x) (?∞ < x < ∞), ??U\ 0.5 utt ? a2 u xx = 0, t = 0 : u = ?( x), ut = ψ( x) 0.5 utt ? a2 u xx = f ( x, t), t = 0 : u = 0, ut = 0 e?? gd (2.9) (2.10) (2.7) (2.8) n, ò??…??K?)?Xeü????K (2.5) (2.6)

??? ???K(2.7)-(2.8), ?\Xe/?C?“? ξ = x + at, η = x ? at, (2.11)

第7页

·?k ?u ?u ?ξ ?u ?η ?u ?u = + = + , ? x ?ξ ? x ?η ? x ?ξ ?η ?2 u ?2 u ?2 u ?2 u = + 2 + , ? x2 ?ξ2 ?ξ?η ?η2 ò??“\(2.7), {z? ?u ?u ?ξ ?u ?η ?u ?u = + =a ?a , ?t ?ξ ?t ?η ?t ?ξ ?η 2 2 2 2 ?u ?u ?u ?u = a2 ?2 + , ?t2 ?ξ2 ?ξ?η ?η2

?2 u = 0. ?ξ?η

(2.12)

é?§(2.12)?? u( x, t) = F (ξ) + G(η) = F ( x ? at) + G( x + at), ? ?F ?G ? ? ? ü ? ? ? ? r(2.14) ‘\??^?(2.8), u|t=0 = F ( x) + G( x) = ?( x), ut |t=0 = a(?F ( x) + G ( x)) = ψ( x), é(2.16) ?ü>??
x

(2.14) ? ).

ü C ? ? ê. d = ?PDE (2.7)

(2.15) (2.16)

a(?F ( x) + G( x)) + C =
x0

ψ(α)dα,

(2.17)

?? x0 ????:,

C ???~ê. d(2.15) ?(2.17) ?)?F ?G: F ( x) = G ( x) = 1 1 ?( x ) ? 2 2a 1 1 ?( x ) + 2 2a
x

ψ(α)dα +
x0 x

C , 2a C . 2a

ψ(α)dα ?
x0

ò§?“\(2.14)

???K(2.7)-(2.8)

)?
x+at x?at

u( x, t) =

1 1 ?( x + at) + ?( x ? at) + 2 2a

ψ(α)dα

(2.19)

???d’Alembert ú?. ? n 2.1 ?( x) ∈ C 2 (R), ψ( x) ∈ C 1 (R), @ o ? ? ? K(2.7)-(2.8) ? 3 ? ?

) u( x, t), § d d’Alembert ú?(2.19) ‰?.
第8页

2.3

D??
l(2.14) ???, d’Alembert ú??k?w u( x, t) = F ( x ? at) ?n??. X?

(a > 0), ? ?N3

w,§??§(2.7) ??? ?A ?.

). ‰t ±?? ?, ??±w??‘gd u t=0 O x1 x1 + at0 at0 x2 x2 + at0 x t = t0

?d, ·??/XF ( x ? at)

)?mD??, /XG( x + at)

)??D??.

2.4

?6?m! ?????K???
dd’Alembert ú ? ?, ? ? ? K(2.7)-(2.8) ) 3 ? ? ? ?t ≥ 0 ? ?m[ x ? at, x + at] ? ?u( x, t) d??] ?( x) 9ψ( x) 3 x ?

:( x , t ) ? ?6?m.

?¤??(?,

??( x) 9ψ( x) 3T?m

??', ?d?m?:( x, t)

t ( x, t )

O é???t = 0 ? L: x2 ?

x ? at ?6?m x + at

x ?? x = x1 + at, ¤n /??,

???m[ x1 , x2 ], L: x1 ?

??1/a

???1/a

?? x = x2 ? at, §???m[ x1 , x2 ] ???? (domain of dependence).

d?????m[ x1 , x2 ]

第9页

t ( x, t ) x = x1 + at x = x2 ? at

???? O X 3 ? ?? ?t = 0, ? ? ] 6?). @o, ?L?mt x1 ?( x) ?ψ( x) x2 x

? 3 ? m[ x1 , x2 ] ? k C ?(? ? ‰??d? (t > 0) k G ? (2.21) . ?(2.21) ¤L? ?

, T6?¤D x1 ? at ≤ x ≤ x2 + at

¤??,

3d‰?

K??K?, E?u

????m[ x1 , x2 ]

K??? (domain of in?uence). t

K??? x = x1 ? at O ?? x = x0 ± at ?????§ x1 A x2 x = x2 + at x )L?

?. ù?r?)?K(2.7)-(2.8)

?mD????D???U\ ?{, ??D??{(1?{). ~ 2.1
2 ?2 u 2? u ? a = 0 (t > 0, 0 < x < ∞), ?t2 ? x2 ?u t = 0 : u = ?( x), = ψ( x) (0 ≤ x < ∞), ?t x = 0 : u = 0.

(2.22) (2.23) (2.24)

) : ?§(2.22)

)?u( x, t) = F ( x ? at) + G( x + at), “\??^?k F ( x) + G( x) = ?( x), ?aF ( x) + aG ( x) = ψ( x).
第10页

) F ( x) = G ( x) = ? x + at ≥ 0, G( x + at) = 5? x ? at ????, F ( x ? at) = 1 1 ?( x + at) + 2 2a x ? at ≥ 0 ? 1 1 ?( x ? at) ? 2 2a
x?at x+at 0

1 ?( x ) ? 2 1 ?( x ) + 2

1 2a 1 2a

C , 2 0 x C ψ(ξ)dξ + . 2 0 ψ(ξ)dξ ? C . 2

x

ψ(ξ)dξ +

ψ(ξ)dξ ?
0

C . 2

x ? at < 0 ?, r>.^?(2.24) “\u( x, t) = F ( x ? at) + G( x + at) k F (?at) + G(at) = 0 ? F (? x) = ?G( x),
at? x

1 1 F ( x ? at) = F (?(at ? x)) = ?G(at ? x) = ? ?(at ? x) ? 2 2a n? d?)?K )? ? ? 1 ? ? ? [?( x + at) + ?( x ? at)] + ? ? ? 2 ? ? ? u( x, t) = ? ? ? ? ? ? 1 ? ? ? ? [?( x + at) ? ?(at ? x)] + 2 1 2a 1 2a
x+at x?at x+at at? x

ψ(ξ)dξ ?
0

C . 2

ψ(ξ)dξ

( x ≥ at),

ψ(ξ)dξ

(0 ≤ x < at). (2.28)

~ 2.2

2 ?2 u 2? u ? a = 0 (t > 0, 0 < x < ∞), ?t2 ? x2 ?u t = 0 : u = ?( x), = ψ( x) (0 ≤ x < ∞), ?t x = 0 : u x = 0.

u( x, t) =

1 1 [?( x + at) + ?( x ? at)] + 2 2a
x+at 0

x+at x?at

ψ(ξ)dξ

( x ≥ at),

u( x, t) =

1 1 [?( x+at)+?(at ? x)]+ 2 2a

at? x

ψ(ξ)dξ +
0

ψ(ξ)dξ

(0 ≤ x < at).
第11页

2.5

àgz

n
n ) eW ( x, t; τ) ????K ? 2 ? ? ? Wtt ? a W xx = 0 (t > τ), ? ? ? t = τ : W = 0, Wt = f ( x, τ)

? n 2.2 (Duhamel

(2.33)

) (? ? τ ? ? ê ), K
t

u( x, t) =
0

W ( x, t; τ)dτ

(2.34)

????K utt ? a2 u xx = f ( x, t), t = 0 : u = 0, ut = 0 ). (2.29) (2.30)

3 ?>??K
3.1 Fourier?ê
Fourier u?ê


?lC?{

φ( x) =
n=1

An sin nπ x d x, l

nπ x l n = 1, 2, . . .

An = Fourier{u?ê

2 l

l

φ( x) sin
0

φ( x) = 2 l
l

A0 + 2



An cos
n=1

nπ x l

An =

φ( x) cos
0

nπ x d x, l

n = 0, 1, 2, . . .

第12页

3.2

?lC?{

utt ? a2 u xx = f ( x, t), u( x, 0) = ?( x), u(0, t) = 0, ut ( x, 0) = ψ( x), u(l, t) = 0.

(3.1) (3.2) (3.3)

? ? ? ? ? ? ? ? ? ? (I) ? ? ? ? ? ? ? ? ?

?2 u1 ?2 u1 ? a2 2 = 0, 2 ?t ?x ?u1 t = 0 : u1 = ?( x), = ψ( x), ?t u1 (0, t) = 0, u1 (l, t) = 0; (3.1)-(3.3)

? ? ? ? ? ? ? ? ? ? (II) ? ? ? ? ? ? ? ? ?

?2 u2 ?2 u2 ? a2 2 = f ( x, t), 2 ?t ?x ?u2 t = 0 : u2 = 0, = 0, ?t u2 (0, t) = 0, u2 (l, t) = 0;

)?u( x, t) = u1 ( x, t) + u2 ( x, t). ?lC?{(Method

of Separation of Variables) utt ? a2 u xx = 0, u(0, t) = 0, u(l, t) = 0, ut ( x, 0) = ψ( x). ±e/? ??…) (3.7) u( x, 0) = ?( x), (3.4) (3.6) (3.5)

?ké÷v?§(3.4) ?>.^?(3.6)

u( x, t) = X ( x)T (t), ò(3.7) “\?§(3.4) ?(3.6) ? T (t) + λa2 T (t) = 0, X ( x) + λX ( x) = 0, X (0) = 0, (3.10)-(3.11) A ?ê?A ?? Xk ( x) = Ck sin λ = λk = kπ x, l (k = 1, 2, . . . ), (k = 1, 2, . . . ). X (l) = 0.

(3.9) (3.10) (3.11)

(3.13) (3.12)
第13页

k2 π2 , l2

òA

?λk “\?§(3.9) ?, ? ?)? T k (t) = Ak cos kπa kπa t + Bk sin t l l (k = 1, 2, . . . ) ?ê (k=1,2,. . . ) (3.14)

ù

?

?

/X(3.7) ?

÷v?§(3.4) ?>.^?(3.6) kπa kπa kπ t + Bk sin t sin x l l l )

Uk ( x, t) = Xk T k = Ak cos òù )U\

?>??K(I)


u( x, t) =
k=1

Ak cos

kπa kπa kπ t + Bk sin t sin x. l l l

(3.15)

d??^?(3.5) ? ? ? 2 l kπ ? ? ? A = ?(ξ) sin ξdξ, ? ? ? k l 0 l ? l ? ? 2 kπ ? ? ? ψ(ξ) sin ξdξ. ? Bk = kπa l 0 ψ(l) = 0, X ê Ak 9 Bk d (3.16) ?(?. @o?ê
∞ ∞

(3.16)

? n 3.1 e ?( x) ∈ C 3 , ψ( x) ∈ C 2 , ? …?(0) = ?(l) = ? (0) = ? (l) = ψ(0) =

k2 |Ak |,
k =1 k =1

k2 | Bk |

??. ? n 3.1 e ? ê ?( x) ∈ C 3 , ψ( x) ∈ C 2 , ?… ?(0) = ?(l) = ? (0) = ? (l) = ψ(0) = ψ(l) = 0, Ku ??§ ? ) ? K(I) )??3 (3.17)

, § ? ± ^ ? ê(3.15) ‰ ?, ?

? Ak 9 Bk d (3.16) ? ( ? . l?n?w, ?( x) ?ψ( x) ?OL?u ??÷v?n
n

??

??????, §??Y
n

?

^?. d?, ?r?( x) ?ψ( x) ?Ow¤?ê Ak sin
k=1

?n ( x) =

kπ x, l

ψn ( x) =
k=1

Bk kπa kπ sin x l l

????4?. ???K utt = a2 u xx , t = 0 : u = ?n ( x), ut = ψn ( x)
第14页

)?
n

un ( x , t ) =
k=1

Ak cos

kπa kπa kπ t + Bk sin t sin x, l l l /?).

(3.18)

n → ∞ ?, §?????(3.15) ?¤‰ §?f)?2?).

§???;), ·??

3.3 )

?n??
kπa kπ kπa t + Bk sin t sin x l l l kπ = Nk cos(ωk t + θk ) sin x l Ak cos kπa , l cos θk = Ak
2 A2 k + Bk

uk ( x, t) =

Nk =

2 A2 k + Bk ,

ωk =

,

sin θk =

? Bk
2 A2 k + Bk

? Nk , ~ 3.1 ? . ? h, u

?? ωk , ?? üà ?3 x ?

θk , !:, 7?, 7?{, ??, ?? x = 0 9 x = l ?, 3 : x = c (0 < c < l) ? ?

?m?gd

?, ??$?5?.

3.4

?àg?§

?/

? n 3.2 (à g z

n (Duhamel n)) ? ? Wtt ? a2 W xx = 0 (t > τ), ? ? ? ? ? t = τ : W = 0, Wt = f ( x, τ), ? ? ? ? ? ? x = 0? x = l : W = 0 )(??τ ≥ 0 ??ê), K
t

(3.26)

e W ( x, t; τ) ? ± ? ? > ? ?K

u( x , t ) =
0

W ( x, t; τ)dτ

(3.27)

?±e?>??K

) utt ? a2 u xx = f ( x, t), u( x, 0) = 0, u(0, t) = 0, ut ( x, 0) = 0, u(l, t) = 0. (3.23) (3.24) (3.25)
第15页

?>??K(3.26)

)?


W ( x, t; τ) =
k =1

Bk (τ) sin 2 kπa
l

kπa kπ (t ? τ) sin x, l l kπ ξdξ. l

(3.29)

Bk (τ) = dàgz n,

f (ξ, τ) sin
0

(3.30)

?)?K(3.23)-(3.25)
t ∞ t

)? Bk (τ) sin kπa kπ (t ? τ)dτ · sin x. l l (1)

u( x, t) =
0

W ( x, t; τ)dτ =
k =1 0

2 ?2 u 2? u ? a = f ( x, t), ?t2 ? x2 ?u t = 0 : u = ?( x), = ψ( x), ?t u(0, t) = ?1 (t), u(l, t) = ?2 (t).

(3.32) (3.33) (3.34)

?2 u3 ?2 u3 ? a2 2 = 0, 2 ?t ?x ?u3 t = 0 : u3 = 0, = 0, ?t u3 (0, t) = ?1 (t), u3 (l, t) = ?2 (t). ?)(3.35)-(3.37) ?{?>.^?àgz?{. ~X, ex U ( x, t) = ?1 (t) + (?2 (t) ? ?1 (t)). l

(3.35) (3.36) (3.37)

(3.38)

?C?V ( x, t) = u3 ( x, t) ? U ( x, t), ?\???êV ( x, t), ?§÷v?àg?§
2 x ?2 V 2? V ? a = ??1 (t) ? (?2 (t) ? ?1 (t)) 2 2 ?t ?x l

??àg??^? x V ( x, 0) = u3 ( x, 0) ? U ( x, 0) = ??1 (0) ? (?2 (0) ? ?1 (0)), l ?V ?u3 ?U x ( x, 0) = ( x, 0) ? ( x, 0) = ??1 (0) ? (?2 (0) ? ?1 (0)). ?t ?t ?t l

第16页

3.5

(
~^ ??K: ? λn = n2 π2 nπ , Xn ( x) = Cn sin x, l2 l (n = 1, 2, . . . ) n2 π2 nπ , Xn ( x) = Cn cos x, 2 l l (n = 0, 1, 2, . . . )

? ? ? ? X ( x ) + λ X ( x ) = 0, ? ? ? X (0) = X (l) = 0.

? ? ? ? X ( x) + λX ( x) = 0, ? ? ? X (0) = X (l) = 0.

?

λn =

? ? ? ? X ( x) + λX ( x) = 0, ? ? ? X (0) = X (l) = 0. ? ? ? ? X ( x) + λX ( x) = 0, ? ? ? X (0) = X (l) = 0. ? ? ? ? Φ (?) + λΦ(?) = 0, ? ? ? Φ(? + 2π) = Φ(?).

?

λn =

1 (n + 1 )2 π2 n+ 2 2 , X ( x ) = C sin π x, n n l2 l (n = 0, 1, 2, . . . )

?

1 (n + 1 )2 π2 n+ 2 2 λn = , Xn ( x) = Cn cos π x, l2 l (n = 0, 1, 2, . . . )

?

λn = n2 , Φn (?) = An cos n? + Bn sin n?, (n = 0, 1, 2, . . . )

4
4.1 ??§
?( ? ? 5 N

p‘???§
?
?) ? b :

…??K

??é ?? ?,

, ?????-?. ? 3 ? ? ? ? S,

??!



?—?ρ ?~ê; ?????

??:3R?ù???

¤? , §é

???T??R?; -C/?? )??-|?. 3( x, y) ?3??t b , K3 ??
第17页

? ò ?:

?R^ ??

??u??Oxy ?, ^u( x, y, t) P : éu??;

?. N?IIy?

, ?÷vX??

??T ?~?.

3

??

?

??, § 3Oxy ? ? ?

? K ??. e ? O ? 3 ? m ? ??Cz.

?(t, t + ?t) S?^u ?? ??±9T??mSù? u u( x, y, t) -?{? ν ? λ τ T -?Γ

ν = (?u x , ?uy , 1) ???? τ×

s = (cos( x, s), cos(y, s), 0) O ? x ν ??? Γ ?(α1 , α2 , α3 ), ?? ?u uy , ?s α2 = ? cos( x, s) ? ?u ux , ?s y s n -?λ ???? ?u ?s

τ = cos( x, s), cos(y, s),

α1 = cos(y, s) +

α3 = u x cos(y, s) ? uy cos( x, s) = u x cos( x, n) + uy cos(y, n) = dd?, ??T 3R??? ??? α3
2 2 α2 1 + α2 + α3

?u . ?n

Tu = T

≈T ???

?u . ?n

3?m?(t, t + ?t) S?^u?
t+?t

T
t Γ

?u ds + ?n

F ( x, y, t)d xdy dt.
?

(4.1)

qù??m?S

?? ρ
?

??Cz? ?u ?u ( x, y, t + ?t) ? ρ ( x, y, t) d xdy. ?t ?t (4.2)

b

u 'u x, y
t+?t

ê??Y, |^Green ú??(4.1)=(4.2) ? T
?

t

?2 u ?2 u ?2 u + + F ( x , y , t ) ? ρ d xdydt = 0 ? x2 ?y2 ?t2 ??5, =
2 2

5?

?m???m??? ρ
2

??§

?u ?u ?u =T + + F ( x, y, t). 2 ?t ? x2 ?y2

第18页

2 ?2 u ?2 u 2 ? u = a + + f ( x, y, t). ?t2 ? x2 ?y2

T F = a2 , = f ρ ρ

(4.3)

Green ú?:
?D

[F cos(n, x) + G cos(n, y)]d s =
D

?F ?G + d xdy ?x ?y

4.2

?)^?
??^?:

J{

u( x, y, 0) = ?( x, y), >.^?: 1. 1?a>.^? u( x, y, t)|Γ = 0 2. 1 a>.^? ?u ?n

?u ( x, y, 0) = ψ( x, y). ?t

(4.5)

u( x, y, t)|Γ = ?( x, y, t) ?u ?n

(4.6)

Γ

=0

Γ

= ?( x, y, t)

(4.7)

3. 1na>.^? ?u + σu ?n =0
Γ

?u + σu ?n

= ?( x, y, t)
Γ

(4.8)

4.3 ???{
n‘???§…??K
2 ?2 u ?2 u ?2 u 2 ? u = a + + , ?t2 ? x2 ?y2 ?z2

(4.14) (4.15) x2 + y2 + z2

u|t=0 = ?( x, y, z),

?u ?t

t =0

= ψ( x, y, z).

???( x, y, z), ψ( x, y, z) ?k?é?5? k', ??é??6ut ?r

?/, =?, ψ =?r = ?

)u(r, t). n‘???§? ?2 u ?2 u 2 ?u = a2 + . 2 ?t ?r2 r ?r

第19页

2-v = ru, ???

ù`??^?J ?

2 ?2 v 2? v = a , ?t2 ?r2 ú?? (4.14) ?k?é?/?

¤

). ??

? ? ? {: ? \ ? ? ' uu( x, y, z, t) 3 ? k ? ? ? %! ? ? ? ? ????ê Mu , ?á Mu ¤÷v ?L Mu u L??. ??m??Y…k? ?Y ??S r ? ê ??? h( x1 , x2 , x3 ) ? 3 ), ,

PDE ??A…??K, d?KN?? ???

ê, ???êh 3±( x1 , x2 , x3 ) ?%! ±r ??? Mh ( x1 , x2 , x3 , r) = ??dr σ L???S r ? ? n 4.1 3 ?2 ) ? xi2 i=1 ??? . 1 4πr2

h( x, y, z)dr σ,
Sr

(4.16)

h ∈ C 2 , K ? ? ? ? ê Mh ( x1 , x2 , x3 , r)

g?Y?

, … ÷ v(? =

?2 2 ? + Mh ( x1 , x2 , x3 , r) = Mh ( x1 , x2 , x3 , r) ?r2 r ?r Mh |r=0 = h( x1 , x2 , x3 ), ? Mh ?r
r=0

(4.17) (4.18)

= 0.

u( x1 , x2 , x3 , t) ?…??K(4.14)! (4.15) Mu ( x1 , x2 , x3 , r, t) = ? n 4.2 1 4π

), §'u x1 , x2 , x3 ?????ê (4.24)

u( x1 + rα1 , x2 + rα2 , x3 + rα3 , t)dω
S1

u( x1 , x2 , x3 , t) ? … ? ? K(4.14)! (4.15) ?ê, ÷v
2 ?2 Mu 2 ? 2 ? ? a + Mu = 0 2 2 ?t ?r r ?r

), K d(4.24) ? ? ?

Mu ? ? r, t

(4.25) (4.26) (4.27)

Mu |t=0 = M? ( x1 , x2 , x3 , r), ? Mu ?t
t =0

= Mψ ( x1 , x2 , x3 , r).

ò Mu ( x1 , x2 , x3 , r, t) (?A/ M? 9 Mψ ) rMu ( x1 , x2 , x3 , r, t) ÷v

r < 0 ???óò?, K§3?∞ < r <

∞, t ≥ 0 ?E ÷ v(4.25)-(4.27). u ?, ? x1 , x2 , x3 ? ? ê, v( x1 , x2 , x3 , r, t) = vtt ? a2 vrr = 0, (4.28)
第20页

v|t=0 = rM? ( x1 , x2 , x3 , r, t), vt |t=0 = rMψ ( x1 , x2 , x3 , r, t), u?v ?d?J ? n 4.1 3?? ú?)?. l Mu = v/r, 2-r → 0 ?? u.

(4.29)

? ∈ C 3 , ψ ∈ C 2§@on‘???§ ) u( x, y, z, t) = ? 1 ?t 4πa2 t
M S at

… ? ? K(4.14)! (4.15) ? 1 4πa2 t

?dS +

ψdS ,
M S at

(4.30) ???

M L ? ± : M ( x, y, z) ? ? %! at ? ? ? ? ? S at

? ?, dS ? ? ?

. (4.30) ? ? ? ? t (Poisson) ú?.

4.4 ü‘{
‘???§Cauchy ?K utt = a2 (u xx + uyy ), u|t=0 = ?( x, y), ut |t=0 = ψ( x, y).
2

(4.36) (4.37)

???{?U? ‘???§). ?n‘???§

? 1 ? + , ?Uz¤? 2 ?r r ?r E?|^n‘???§Cauchy ?K Poisson ú??). ? A^(Why? ?(4.17) ??àC? utt = a2 (u xx + uyy + uzz ), u|t=0 = ?( x, y), ut |t=0 = ψ( x, y) (4.38) (4.39) ?$‘???§)

)u( x, y, z, t) ??z ?' ?ê, K§?? ‘???§Cauchy ?K(4.36)! (4.37) ), ù?|^p‘???§Cauchy ?K ? ? 1 ? ? ? ? u( x, y, z, t) = ? ? 2πa ?t + ) ?{??ü‘{. ?(ξ, η)dξdη
M Σat

M Σat

? (ξ ? x)2 ? (η ? y)2 ? ? ψ(ξ, η)dξdη ? ? ? ? ?. 2 2 2 (at) ? (ξ ? x) ? (η ? y) (at)2
2π 0

? 1 ? ? ? ? ? = ? 2πa ? ?t
at

at 0 2π 0

?( x + r cos θ, y + r sin θ) (at)2 ? r2 (at)2 ? r2

rdθdr (4.40)

+
0 M Σat :

ψ( x + r cos θ, y + r sin θ)

? ? ? ? rdθdr? ? ?

(ξ ? x)2 + (η ? y)2 ≤ a2 t2
第21页

4.5

?àg???§…??K

)

utt = a2 (u xx + uyy + uzz ) + f ( x, y, z, t), u|t=0 = ?, ut |t=0 = ψ. e??(4.41) ÷vàg?^??K ) u|t=0 = 0, ut |t=0 = 0. ??àgz n, w( x, y, z, t; τ) ?Xe?K wtt = a2 (w xx + wyy + wzz ), w|t=τ = 0, wt |t=τ = f ( x, y, z, τ) ), K(4.41)! (4.43) )?
t

(4.41) (4.42)

(4.43)

(4.44) (4.45)

u( x, y, z, t) =
0

w( x, y, z, t; τ)dτ. f (ξ, η, ζ, τ) r dS ,
r=a(t?τ)

(4.46)

w( x, y, z, t; τ) =

1 4πa

M Sa (t?τ)

u( x, y, z, t) = = = 3??t! ?

1 4πa 1 4πa2 1 4πa2

t 0 at 0
M Sa (t?τ)

f (ξ, η, ζ, τ) r

dS dτ
r=a(t?τ) r ) a

f (ξ, η, ζ, t ? dS dτ M r Sr r f (ξ, η, ζ, t ? a ) dV r r≤at

τ=t?

r a (4.47)

u M ( x, y, z) ?)u

ê?d f 3??τ = t ?

r a

?

?3d?

N???L?, ?ù

???í??.

5
5.1
ù:

?

D??P~

?6??! ?????K???
? ? :( x0 , y0 , t0 ). ? ?Poisson ú ?(4.40), ) 3 S ?
第22页

? ? 5.1 3 ( x, y, t) ? m S ,

ê ? d ? ? ? ? t = 0 ? ±( x0 , y0 ) ? ? %! at0 ? ? ?

? ^ ? ?( x, y) 9 ψ( x, y) ?t = 0 ?

? ? ¤ L ?,

??6u

? ?ψ

?. ? d ? (5.1) ??

2 ( x ? x0 )2 + (y ? y0 )2 ≤ a2 t0

? ? ? : ( x0 , y0 , t0 ) ]

? 6 ? ?. ? ?, ? ? ? ?t = 0 ? ? ?(5.1) ? ±( x0 , y0 , t0 ) ??:! ±T???. (t ≤ t0 ) (5.1)

? ?ψ ??/??

IN?? (5.2) ? ? ? ?.

( x ? x0 )2 + (y ? y0 )2 ≤ a2 (t ? t0 )2 ? ). ? d,

I N (5.2) ? ? ? ? ?t = 0 ?

( x0 , y0 , 0) ? ? ? ? ? ? :, ÷v^? ( x ? x0 )2 + (y ? y0 )2 ≤ a2 t2 (t > 0), : ( x, y, t) ? 6 ? ? ? ? :( x0 , y0 , 0). K(5.3) 3( x, y, t) ? m S I N, ? 1 ? ?t ? A I. ?arctan a.
2 2

(5.3) ¤??

± ( x0 , y0 , 0) ? ? : ?? ‘???§

I N(5.3) ?

? ? ? ? ? ? : ( x0 , y0 , 0)

K ? ? ?. I ?( x ? x0 ) + (y ? y0 ) = a2 (t ? t0 )2 ,

5.2

¨?d
én = 3

n(Huygens)! ?
? ?,

‘?
? ? 6 ?, 3 ? ?t ? é 6?, 3D?L§?, ? ?. ù?y–??¨?d ? ? 6 ?, 3 ? ?t ? é ? > .± 6?, ???aq. ù D?. : ) K ?, ù ? ? ? Q ? c

t = 0 ?, :( x0 , y0 , z0 ) ? ??? , ?k?w

±( x0 , y0 , z0 ) ? ? %! at ? ? ? ?, ? ? k?w c ?. ?, 3???? ? ?,

t = 0 ?, u)3k.??S

n(Huygens’ principle). ~X(? D?. én = 2 ±( x0 , y0 ) ? ?kc ?y–??? t = 0 ?, :( x0 , y0 ) ? ?S %! at ? ? ? : ? kK ?, ù ?

?. d?vk?(

?. ék.??S

‘?(dispersion of wave). ~XY?

5.3

???§)
b ??]

P~
. 1w5^?, …?k;| 5?. ê?t?1 ;
第23页

??t → ∞ ?, ???§Cauchy ?K) ì?5 ? 9ψ ÷vPoisson ú?¤?? O\ P~(attenuation) 8, ??6?‘?m 1.

t → ∞ ?, n‘???§Cauchy ?K )?—?u"

2.

t → ∞ ?,

‘???§Cauchy ?K )?—?u"

ê?t? 2 ;

1

3. ? ‘ ? ? ? § kP~5.

Cauchy ? K, dd’Alembert ú ? ?, ) 3t → ∞ ? v

e?±n‘???§?~?1??: éPoisson ú?(4.30) ?1U 1 4πa2 t ??dω ?ü ?dS = 1 4π
|α|=1

M S at

t?( x1 + atα1 , x2 + atα2 , x3 + atα3 )dω,

??

??? . ét ? ?g

= =

1 ? ?dS M ?t 4πa2 t S at ? 1 ? ? ? ? ?( x + atα)dω + ? 4π ? |α|=1 1 4πa2 t2
M S at

3

at
|α|=1 i=1

? ? ? ? ? xi ( x + atα) · αi dω? ? ?

?( M ) + ??( M ) · MM dS M , C ? :, dS M ? ? ? ? . u ?Poisson ú

? ?M ? ? ? ? ? ? ?(4.30)? ¤ 1 4πa2 t2

u( M, t) = ??

M S at

tψ( M ) + ?( M ) + ??( M ) · MM dS M .

(5.7)

??]

?, ψ ? k ; | 8, K ? 3 ? ? ~ êρ > 0, ?? 9ψ 3 ± ? BO ρ ? u", ?? ≤ C1 ? xi 3? BO ρ S¤á (i = 1, 2, 3),

:??%! ρ ???

|ψ|, |?|, ??C1 ???

M ~ê. q3S at ∩ BO ρ ?

|tψ( M ) + ?( M ) + ??( M ) · MM | ≤ C2 t + C3 ,
M (S at ∩ BO ?? ≤ 4πρ2 . ρ)

(5.9) (5.10)

u?,

t≥1? |u( M, t)| ≤ 1 |tψ( M ) + ?( M ) + ??( M ) · MM |dS M M ∩ BO 4πa2 t2 S at ρ 1 ≤ (C2 t + C3 ) · 4πρ2 ≤ Ct?1 , (5.11) 4πa2 t2
第24页

??C ??~ê.

6 U??
6.1 ?>??K)

?! ???§)
??5?-?5

??5?-?5

utt = a2 (u xx + uyy ) + f ( x, y), ( x, y) ∈ ? u|t=0 = ?( x, y), ut |t=0 = ψ( x, y),

(6.8) (6.9) (6.10)

u|Γ = ?( x, y, t). 3vk ??^ ??e( f = 0), E (t) = …oU?AT??, =k ? oU?? ¤

?

2 2 2 [u2 t + a (u x + uy )]d xdy,

(6.13)

d E (t) = 0. (6.14) dt e?y?éu÷vàg???§utt = a2 (u xx + uyy ) 9àg>.^?u|Γ = 0 ???êu( x, y, t) ¤á(6.14) ?. d E (t ) =2 dt =2 =2 =2 =0 ? E (t) = E (0) = Green ú?:
?D 2 [ψ2 + a2 (?2 x + ?y )]d xdy.

?

[ut utt + a2 (u x u xt + uy uyt )]d xdy ut utt ? a2 (ut u xx + ut uyy ) + a2 ? ? (ut u x ) + (ut uy ) d xdy ?x ?y
Γ

?

?

ut [utt ? a2 (u xx + uyy )]d xdy + 2a2 ut [utt ? a2 (u xx + uyy )]d xdy

ut (?uy d x + u x dy) (6.15)

?

?

Pd x + Qdy =
D

?Q ?P ? d xdy ?x ?y {, § ? ? ? ?

? n 6.1 ? ? ? § ? > ? ? K(6.8)-(6.10) ? .

)XJ?3

第25页

y:

u1 , u2 ??)?K(6.8)-(6.10)

ü?), K?

u = u1 ? u2 ÷v?A

àg?§9àg?>?^?, ?d3????kE (0) = 0, E (t) =
? 2 2 2 [u2 t + a (u x + uy )]d xdy = 0,

? ut = u x = uy = 0 ? u( x, y, t) = const. qdu3????u = 0, u( x, y, t) ≡ 0. e?^U??? ?{???>??K utt = a2 (u xx + uyy ) + f , u|t=0 = ?( x, y), ut |t=0 = ψ( x, y), (6.16) (6.17) (6.18)

u|Γ = 0. )'u??^???§mà?àg‘ ?Y?65. l(6.15) ? d E (t ) =2 dt
?t

?

ut f d xdy ≤

?

u2 t d xdy + t ??, ?

?

f 2 d xdy ≤ E (t) +

f 2 d xdy.
?

(6.19)

^e ????mü>, ?l0 d ?t (e E (t)) ≤ e?t dt f 2 d xdy,
?

t

E (t) ≤ et E (0) +
0

e?τ
?

f 2 d xdydτ .

u?é0 ≤ t ≤ T , ?k(C0 ???=?T k'
T

~ê) f 2 d xdydτ . (6.20)

E (t) ≤ C0 E (0) +
0

?

??????êu( x, y, t) ?? E0 (t) = d E 0 (t ) =2 dt , uut d xdy ≤

O. P u2 ( x, y, t)d xdy.
?

(6.21)

?

?

u2 d xdy +

?

u2 t d xdy ≤ E 0 (t) + E (t).

re?t ???üà?l0 d ?t (e E0 (t)) ≤ e?t E (t), dt

t ??
t

E0 (t) ≤ et E0 (0) + et
0

e?τ E (τ)dτ.

(6.22)
第26页

(?(6.20) ?, ?

é0 ≤ t ≤ T ¤á
T

E (t) + E0 (t) ≤ C E (0) + E0 (0) +
0

f 2 d xdydt ,
?

(6.23) ?, ?U? O?

??C ?????T k' O?. d ???k O??3b O?.

~ê. (6.23) ?(6.20) ??U?? )?3 cJe

, ?kù?A:

? n 6.2 ? ? ? § ? ? ? K(6.16)-(6.18) ? (?, ψ) ? ? § =?6uε ?T mà‘ f ?-? η > 0, ? ? ?1 ? ?2 f1 ? f2
L2 (?)

)u( x, y, t) 3 e ? ? ? e ' u ? ε > 0, ? ? ? ± é

: éu??‰?

≤ η, ≤ η, ≤ η,

?1 x ? ?2 x ψ1 ? ψ2

L2 (?)

≤ η, (6.24)

?1y ? ?2y

L2 (?)

L2 (?)

≤ η,

L2 ((0,T )×?)

@ o ± (?1 , ψ1 ) ? ? ?! f1 ? m à ‘ ‘ ) u2 ? 30 ≤ t ≤ T ?÷v u1 ? u2
L2 (?)

)u1 ? ±(?2 , ψ2 ) ? ? ?! f2 ? m à

≤ ε, ≤ ε,

u1 x ? u2 x u1t ? u2t

L2 (?) L2 (?)

≤ ε, ≤ ε.

u1y ? u2y

(6.25)

L2 (?)

y : Pv( x, y, t) = u1 ( x, y, t) ? u2 ( x, y, t), Kv( x, y, t) ÷v vtt = a2 (v xx + vyy ) + f1 ? f2 , v|t=0 = ?1 ( x, y) ? ?2 ( x, y), vt |t=0 = ψ1 ( x, y) ? ψ2 ( x, y), (6.26) (6.27) (6.28)

v|Γ = 0. l |^U?? ?(6.23) = ¤I?(?.

6.2 …??K)
e?|^U? 5. ?? X?O?

??5?-?5
'X?5?????§Cauchy ?K) ???t =~ê? U? E1 (t) =
R2 2 2 2 [u2 t + a (u x + uy )]d xdy

??5?-?

(6.29)

第27页

?U?u? ?

, ?d?UO?3( x, y) ??

,?k???? ?

U?. ·? IK : (6.30)

??‘?mO\

???t , §3( x, y, t) ?m ( x ? x0 )2 + (y ? y0 )2 ≤ (R ? at)2 .

¤??A

INK 3??t = 0 ? ?0 : INK ??.??0

?=?.???? ( x ? x0 )2 + (y ? y0 )2 ≤ R2 , (6.31)

????. t ( x0 , y0 , t0 ) K ?t

O ( x0 , y0 ) x 3??t ?, ?? R ?0

y

?t : ( x ? x0 )2 + (y ? y0 )2 ≤ (R ? at)2 ??:?) ê?d (6.31) ? ??^?¤ ? ?, 3 (6.31)

(6.32) U?

3??t ???D? eu( x, y, t) 3A

?t ?. ?d3???t ? IK S÷vàg???§ utt ? a2 (u xx + uyy ) = 0,

oU????L?0 ?

oU?, =E1 (?t ) ≤ E1 (?0 ).

(6.8’)

K3K S??

??t ?¤áU?? ? E1 (?t ) = ≤
?0 ?t 2 2 2 [u2 t + a (u x + uy )]d xdy 2 2 2 [u2 t + a (u x + uy )]d xdy = E 1 (?0 ).

(6.33)
第28页

?Iy?E1 (?t ) 30 ≤ t ≤ R/a ??t

üN~ ?ê, = (6.34) >.

d E1 (?t ) ≤ 0. dt y : Pr = dE1 (?t ) d = dt dt =2
0 R?at 0 2πr

( x ? x0 )2 + (y ? y0 )2 , d s L? l l?rdθ, Γt ????t
2 2 2 [u2 t + a (u x + uy )]d xdy = 2πr

?t R?at

d dt

R?at 0 0

2πr 2 2 2 [u2 t + a (u x + uy )]d sdr 2 2 2 [u2 t + a (u x + uy )]d s

[ut utt + a2 (u x u xt + uy uyt )]d sdr ? a ut [utt ? a2 (u xx + uyy )]d sdr
0 0

Γt

=2 +2 ≤ 0.

Γt

a 2 a2 [u x ut cos(n, x) + uy ut cos(n, y)] ? [u2 + a2 (u2 x + uy )] d s 2 t

D

?F ?G d xdy = + ?x ?y

?D

F dy ? Gd x =

?D

[F cos(n, x) + G cos(n, y)]d s

? n 6.3 ? ? ? § ? > ? ?K(6.8) u|t=0 = ?( x, y), Cauchy ? K )??? .

??^? ut |t=0 = ψ( x, y) (6.35)

y : ? I y ? §(6.8’) ? k " ? ? ^ ? ?(6.33) ? , XJt = 0 ?, u( x, y, 0) = ?kE1 (?0 ) = 0, l E1 (?t ) ??U

) 7 ? " ). ? ? U ? ?

?u ( x, y, 0) = 0, ?t u", ?dut = u x = yy = 0, =u =~ê.

2????^?u 3INS ?Y5 ?u = 0. ? n 6.4 ? ? ? § (6.8’) e'u????-? ?T ψ1 ? ψ2 η > 0, ? ? ?1 ? ?2
L2 (?0 )

? ? ^ ?(6.35) : éu??‰?
L2 (?0 )

Cauchy ? K
L2 (?0 )

)3e??? = ? 6 uε
L2 (?0 )

ε > 0, ? ? ? ± é

≤ η, ?1 x ? ?2 x

≤ η, ?1y ? ?2y

≤ η,

≤ η, K é A u ? ? ?(?1 , ψ1 ) ≤ ε, ≤ ε,

)u1 ? é A u ? ? ?(?2 , ψ2 ) ≤ ε, ≤ ε,

) u2 ?

3 0 ≤ t ≤ R/a ?¤á u1 ? u2
L2 (?t )

u1 x ? u2 x u1t ? u2t

L2 (?t ) L2 (?t )

u1y ? u2y

(6.36)
第29页

L2 (?t )

q3IN K ?¤á u1 ? u2 =
K

L2 ( K )

(u1 ? u2 )2 d xdydz ≤ ε. ?êu( x, y, t), ???? u2 ( x, y, t)d xdy.
?t

(6.37)

y : éu??÷vàg?§(6.8’) E0 (?t ) = ò§'ut ? , uut d xdy ? a

(6.38)

dE0 (?t ) =2 dt

u2 d s ≤
Γt ?t

?t

u2 d xdy +

?t

u2 t d xdy

≤ E0 (?t ) + E1 (?t ), u??(6.22) ??, ?
t

E0 (?t ) ≤ et E0 (?0 ) +
0

et?τ E1 (?τ )dτ.

(6.39)

(?(6.33) ?

éu0 ≤ t ≤ R/a ¤á E1 (?t ) + E0 (?t ) ≤ C (E1 (?0 ) + E0 (?0 )), (6.40)

l

¤á(6.36), 2'ut ??= (6.37).

7

??S??
?;

1. ?????§9???)^? J{??n 2. ???)?)?K 3. ??U??? 4. ??A ??A O; I.

?{(1?{??lC?{);

第30页


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