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2,ü69é
?
9F
3á ?Zü6
9Fg' 9F ?? ? 4ú/
y ?á′D'áa ?? ?-W￥1"
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q
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'á K(t) ?? ? L(t)
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0
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L
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Q
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<
L
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ty
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1
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λ
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dQ/dt > 0
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Discussions
3,?^D3 ^
?BQ
W?v^ Lanchester4
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^?s ??^?3 ^?^?
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n
Za ?
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Discussions
4,0t8
=￥s?D
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=?= )()(
~
11
??l￥ ¨Kl=eE? B
β
)(
~
,
1 ii
tct
?
9
0,,
21
→∞→ cct
é ???
i￥0t ??
"
+=
βα
BA
VkD
1130
∫
∞
=
0
11130
)( dttcVkD
BA
V
D
c +==
1
0
1
)0(
AB
BA
k
βα
αβ
+
+
=
)(
13
=
++=+
1321
132112
kk
kkk
αβ
βα
13
21
k
k
αβ
=
211312
kkk+= βα
Discussions
5,P?V
rI￥T¨
V
rI￥T¨D
￥?
éμ
I
11"
ù5
|8
?￥yt
D
'ty
íμ1 ?
'ty
í?Yv
'ty
í?Ylb
?
s
s
?
Hyté ? |8￥V?y ?
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D
?b
!XB?o  |p??ìa/
?
?Z
T??ìa a1
^?M￥b
1 l
1
~?¤é l
2
~V
rIé l = l
1
+ l
2

yt
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s?
áw
0
=M/l
1
?
L
!
2? ?)yt
?üé ? b P?
,?￥
1
^ a′:a,a′+a=1
3?? ?￥?¤V
rI
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yt￥ (?ê
HW )
l
qsY
^ b β
4?üP?,?

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vP? ?


^è
u v >>u
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Q ~ 
B|?yté ? |89

,
1
↓?↑↑ Qlb?↑↓?↑ Qu↓?↓↓↓↑↑ QvaMl,,,,
2
β
v
x
lxlxq
lxxbq
xxqxq
=?
≤≤?
≤≤?
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τβ
τ
,,)(
,0,)(
)()(
1
1
≤≤?
≤≤?
=
lxlxq
v
lxxq
v
b
dx
dq
1
1
),(
0),(
β
ulTdttlqQ
T
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0
1∫
==

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y
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xx?+
)(xq
)( xxq?+
x
v
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x
1
l
l
t=0,x=0? ?P?
q(x,t) ~ yt
@

w(x,t) ~ yt
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00
0
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uwH
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lxeaH
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bl
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lxutetaH
txq
v
lx
v
utlb
v
utxb
1
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1
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,)(
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11
β
v
l
v
utlb
eetutauwtlq
21
)(
),(),(
β
=
t
H YP? ?à x=ut
1) p q(x,0)=q(x)
2) p q(l,t)
3) p w(ut,t)
t
v
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),(),(
=
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t
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l
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β
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a
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0
1),(
4) 9
Q
′
=

v
abut
v
but
v
l
v
bl
aeeee
a
auw
tlq
21
0
),(
β
′
==

∫
v
bla
v
l
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ba
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dttlqQ
12
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β
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e
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1
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2
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v
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2
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l
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r
e
r
v
bla
r
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1
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1
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s
1 QD a,M??1 aM
^yt"? x=l )￥
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2) ~V
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2
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T¨
v
l
e
2
β
v
l
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2
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1
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6B|?{V
rI￥P?1? w
0
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a,v,l (M]
à x=l
1
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v
bla
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l
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ba
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bla
v
bl
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v
lb
e
Q
Q
2
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2
1
=
β
21
QQb <?>β
4ú β-b DFé l
2
rTM]
Discussions
6, | g?? e?
M
-s?? | g??￥×1?
o I
n1 ?
3D
μ§?9 %M
￥ | g
M
- | gs?f
rtrF <(~),(
| g
?Z
Z?
| g9
~)(tN
| g
áf
~),( trp
Kú
M
-~)( ∞→
m
r
)(),(,0),0( tNtrFtF
m
==
r
F
trp
=),(
| g?ZZ?
μ§
q~),( trμ
drtrp ),(
|
M
-
]
,[,
dr
rrt
+
μ§ |
=),( dttt +
|
M
-
]
,[,
1
1
drdrr
drrdtt
++
++
1
drdt =
drdttrptr ),(),(μ=drdttdrrp ),(
1
++?
1
1
,),(),(
)],(),([)],(),([
drdtdttrptr
trpdttrpdttrpdttdrrp
=?=
+++?++
μ
),(),( trptr
t
p
r
p
μ?=
+
B¨
ê±sZ?
≥=
≥=
=
+
0),(),0(
0),()0,(
),(),(
0
ttftp
rrprp
trptr
t
p
r
p
μ
| g?ZZ?
~X?f

| g?°
~
3a
q
e? | g
m

0
t
r
)(
0
rp
rt =
)(tf
rt <
rt >
)(),( rtr μμ =
>?
≤≤?
=
∫
∫
rtertf
rtetrp
trp
r
r
tr
dss
dss
,)(
0,)(
),(
0
)(
)(
0
μ
μ
∫
=
r
dstsptrF
0
),(),(
∫
=
m
r
dstsptN
0
),()(
3a
q￥s3
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o? )(~),( trk
3a
o? )(~),( trb
a
- uW~],[
21
rr
∫
=
2
1
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r
r
drtrptrktrbtf
)(),( rhtrh =
0 1
r
2
r
r
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∫
=
2
1
1),(
r
r
drtrh
h~
3a

T
∫
=
2
1
),()(
r
r
drtrbtβ
β~9
3a
q
∫
=
2
1
),(),(),()()(
r
r
drtrptrktrhttf β
| g?ZZ?
3a
q
∫
=
2
1
),(),(),()()(
r
r
drtrptrktrhttf β
)(tβ
~9
3a
q —— e?
3a￥

),( trh
~
3a

T —— e?
3a￥*?

á
>?
≤≤?
=
∫
∫
rtertf
rtetrp
trp
r
r
tr
dss
dss
,)(
0,)(
),(
0
)(
)(
0
μ
μ
),(),( trptr
t
p
r
p
μ?=
+
)(tf
)(
0
rp
),( trp
)(tβ
?Q ?"d
ìaT¨v
| g·
∫
=
m
r
drtrptN
0
),()(
1 | g9
∫
=
m
r
drtrrp
tN
tR
0
),(
)(
1
)(
∫
∞
∫
=
t
drtr
detS
t
τ
τ
μ
0
),(
)(
2
ü (
M
-
3
ü (
p

t
H Y
3￥ |
μ§
q? μ(r,t) 9
￥
ü (i
HW
)(/)()( tStRt =ω
4 o
-·
e? N(t)?Vv
e?
3a
q
e? ω(t)?Vú
Discussions
7,?ü￥ ? ?Dh
>
? b?I?ü_
1? ? ?????
?i; u×b
C`

ù5
?i; u×? ?v ?a u×H??v
ü
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êlKa?üh
>b
y ?
?
í
?ü ? ?h
>V?sh
>
HWDòy
í￥1"b
ù5
s
í k bW?
§
H?÷?á￥ ? ?V?¨=¨
ê
±sZ?
í
?ü
i￥Mb
43￥?üh
>D?ü;L￥
l[#N

üà￥
2
ù?μ1b
1?üí k bW ? ??
sv1?
￥?Y ? ??V ￡.??
pb
?
L
!
2;L,V?ü
H; <￥h
D?ü
i
??1 í?ü￥v ??Y; <b
3,V?üé ?N ￥;Loμ
üà-
s
üà?K?N 
2
ù %?b
?
y ?
1?ü
i￥M
p
),,,( tzyxC
gradCkq=
G
￡.??
p?ê
HWYV?êE
_
￥
@
D
i0??1
21
QQ =
∫∫∫
+?=
V
dVttzyxCtzyxCQ )],,,(),,,([
2
∫∫∫
+
=
tt
t
s
dtdnqQ σ
GG
1
V
S
n
G
1
Q
q
G
@
YV+ ],[ ttt
=?ü?M
?
∫∫ ∫∫∫
=?
sV
dVqdivdnq
GGG
σ
w
s￥í
f
T
gradCkq=
G
1?ü
i￥M
p
),,,( tzyxC
+
+
==
2
2
2
2
2
2
)]([
z
C
y
C
x
C
kgradCdivk
t
C
1?ü
i￥M
p
),,,( tzyxC
0,,,,
2
2
2
2
2
2
>∞<<∞?
+
+
=
tzyx
z
C
y
C
x
C
k
t
C
),,()0,,,( zyxQzyxC δ=

SHq
Q~
?
db￥?ü9
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kt
zyx
e
kt
Q
tzyxC
4
2
3
222
)4(
),,,(
++
=
π
 ?i t,C￥?′
^ o
x
2
+y
2
+z
2
=R
2; R↑→C↓
?? t→∞, ?i? (x,y,z),C→0
2,V?ü; <￥M
p
Z_￥?ü
i
Z_￥; <
llC
llI
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; <￥h
D?
ü
i??1
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dl
dI
α?=
00
)( IlI =
00
)( Ill ￥; <1?é ??ü ≤
∫
=
l
l
dssC
eIlI
0
)(
0
)(
α
3N 
2
ùD?ü
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∫
=
l
l
dssC
eIlI
0
)(
0
)(
α
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i ??M
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>￥V?
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üà-
s
üà?K?N 
2
ù %?b
μ
α
=
∫
∞
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1
),,,( dztzyxC
e
4?2T1àN 
2
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0
μμ?<II
N 
z
-∞
∞
!;÷ z=-∞,N  z=∞,54??￥
üà?K1
~?i; u×H?
4?i; u×H?￥M
p
tk
Q
kttr
μπ
α
4
ln4)( =
α
μ
π
=
+
kt
yx
e
kt
Q
4
22
4
∫
∞
∞?
= adxe
a
x
π
2
μ
α
=
∫
∞
∞?
1
),,,( dztzyxC
e
kt
zyx
e
kt
Q
tzyxC
4
2
3
222
)4(
),,,(
++
=
π
lμ
α
μ
(≈
μα?
=
∫
∞
∞?
1
1
ln
1
),,,( dztzyxC
kt
yx
e
kt
Q
dztzyxC
4
22
4
),,,(
+
∞
∞?
∫
=
π
222
ryx =+
 ?i t,?i; u×H?
^??
?i; u×
H???
)(
4
1
Kv′
e
Q
rr
ek
Q
tt
m
πμ
α
μπ
α
====
0,
4
2
=== r
k
Q
tt
μπ
α
r(t)
r
m
0
t
1
t
2
t
tk
Q
kttr
μπ
α
4
ln4)( =2Ts
112
7.2 tett =?=

4???i; u×H?r?Kv￥
H Y t
1
 V[??üh
>￥
H Y t
2
↑↑↓?↑↑
m
rtQ,,,
1
μα
↑↓?
1
tk
Discussions
8,￡μ? ??
p￥?C
tZ<?Z ?ó4?ú ?
“1?
a,?(
?
aa
D, °?
a,
? ú
{
8?
7
à??? ??
p M
?￥9
ZE
dB M?μ ?Dey
d? ??
p
dù?M
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ü±s
@
E
7
à ??
p
￡μ? ??
p
d??=?
p
u1 ? SD-
De ?v (1687)
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!
t (0,0)US" (r,θ)
1,??E?
??ê?_? ))(),(()( ttrtr θ
G
O (
t )
θ
P (?? )
r
r
G
a~é?à,b~
?à,e~ ??
q
)1(,,
cos1
222
2
eab
a
b
p
e
p
r?==
+
=
θ
2,?ê
HW áV
1è
A
r
G
Ar =2/
2
θ

32
aT λ=
3,????
ù T λ ~ 'è
rmf

G
G
=
4,???
s ?
f
G
m ~ ??é

?y ?
O (
t )
θ
P (?? )
r
r
G
_? ￥_

r
G
jiu
jiu
r
GG
G
GG
G
θθ
θθ
θ
cossin
sincos
+?=
+=
r
u
G
θ
u
G
r
urr
GG
=
r
r
uu
uu
G

G
G

G
θ
θ
θ
θ
=
=
θ
θ
θθθ
θ
urrurrr
ururr
r
r
G



G



G
G

G


G
)2()(
2
++?=
+=
Ar =2/
2
θ

32
4
,
2
r
rA
r
A


== θθ
02 =+ θθ



rr
r
urrr
G



G
)(
2
θ?=
θcos1 e
p
r
+
=
3
2
)(4
,
sin2
pr
rpA
r
p
Ae
r
==

θ
r
u
pr
A
r
G

G
2
2
4
=
rmf

G
G
=
r
r
rr
pr
mA
f
G
GG
G
=?=
00
2
2
,
4
r
urr
GG
=
?y ?
￡μ? ??
p
r
r
rr
pr
mA
f
G
GG
G
=?=
00
2
2
,
4
0
2
r
r
kMm
f
G
G
=
3￡
ü 4A
2
/p =kM

D
'B R??í1
A~?ê
HW áV
r
G
32
aT λ=
abTA π=
O (
t )
θ
P (?? )
r
r
G
)1(,,
cos1
222
2
eab
a
b
p
e
p
r?==
+
=
θ
λπ
//
22
=
pA kM=λπ /4
2
(5 )
Discussions