D
?
Mathematical Modeling
? [ o
=
> ?D
ligangliu@zju.edu.cn
http://www.math.zju.edu.cn/ligangliu/Courses
/MathematicalModeling_2005-2006
D
??
D$
null
Dμ
àμ¨$
null
D?
^
àμ¨7
^?¨
null Cμ￥
Dy ?
3 %
μ
L=ù5
null á <ù????
Dù57
sE
null 8
1¨$
null 3 %
L=ù5
null
D
D
?D
Dy
null
D
?
Mathematical Model
^¨
D?| a
D
T0 a?? am??
L= [5
'é
?￥`7?e±￥ Y
3
d
t Z4
C`
??? ?￥?Z
p
1 e?
BC`
￥?Z4
?il/￥Ka|
{?z|
{ b
null
Dy
Mathematical Modeling
?¨?
MV
L= [5?` a4 ?
D
?￥V? b
D
?*ü?
null á
ìVlü¤(V
D
?
null ?¨5
null,JY
1M 750 ú/VJ?Y
¨
￡t?3
30l
HVY?J
I
￡t?3 50l
Hùt
￡
??$,
null t8
null,V
üáμ
￥l/
ˉB v
Fhà
￡?μ
^
h
^/?$,
null
Dé ×
null …
D
?í
?
null °è
3
null g'
null %|
null ò?ò<
null ü6
null á μ
null <ù?
5×
null t ?
null 9
ù?
è 1,
mè 5￥ê4
null X? ??è 5
sò 0.4í
25
í?á
*ü? 5
sò 0.6í?¨
?á
null ù ê4
'? 51?
8 +$
è 2.u
￡ù5
null
??
5:00 à 5:30 -W 7
￡\￥?
?XA ?
B? |?
'μ?(?
ê
ê
??
HμBt | B? |]z+
?
Ch?¤? ? |
n?
b
?
| ??Kz￥÷E? ?
^ ?′Q
¨?￥ -4/D V
*1¤(
Ch b
Y
L
vE9'
^?"S￥ b ú
?
H
ù?] ?
Ch￥ |ü
?%??
y￥ 139%?
";￥ 13 b
null L
!C μ 2?
￡
Ch 10 ? | ?u
￡
? |
)"
?ˉ
uBˉ1 1sò?
^B?èn￥ f ? b
null ZE A üZE b?"?μ
|??
H
? |ò¨B?
C
h1| 10? |u
9
￥??
HW
^
2*(2+4+6+8+10)=60 sò
null ZE B
Qs
￥
￡
Ch
H?a5
@K -
￥ | b?"?μ
|??
H?B? |5¨
?
Ch?
eu?
?=? |
¨ b??ZE/9￥??
HW
^
1+2+3+4+5+6+7+8+9+10=55 sò
null 2TaB?ZE$￡
ü
^÷μr
q￥ b9ü
^
a?? A ?μ
t1
2￥Z???èè$á
ì 13￥Z?
Y
L
^B?÷
?￥Z? b
null M] ?
￥i??
è 3.?ù5
null ??Sy
?i | +
? | ?
a?
%?
^BQé y￥ü ? bá
ü (
Q ?
| +?à
1
??l
H￥
HW?
\
Pá I
n
^?μ÷E Cμ3 g￥ f
?/4ú??"d￥r
q b
null ?] ?
￥1??
è 4.￡μ? ??
p￥?C
null 7
à ?v?
p
null ??E?
^B???
t ê?N??￥B???
b
null ?? ?ê
HW
= áV￥
?M b
null ????
ù￥
üZ?1???é?à￥ ?QZ1
è"
?
??7?M
'è
b
null
d? 7
à ??
p
d?=?
p ?¨±
sZEw?
d? ??
p'￡μ? ??
p b
Proof
Dy
￥B???
null
3ù5￥
L= ?
ü ?y
"￥
l"g?A1
￥
'
b
null
ü ?y
"￥g?A1'
￥$
YV'
￥s9
s ?1T¨￥y
íüA1￥ú
? ae4 ??? Z4
L=￥L
! b
null
TL
!￥$
?¨
a?￥
Dy ? Yò
M
-W￥1"y ?M?￥
D2 ——'y ?
D
? b
null
? p3 b
null
?￥sD_ b
L8?
(
)
L
!
y
p3
￡
?¨
D
?￥s ?
s ?S
s ?S
8 ?Y
8 ?Y
?
L=ù5
3￥
' ??
aQ
? aQ
? aQ
?
??M
￥+
?
???
? a ? ??
? ???
? a
?
??
y
?
¨￥
DZE
?
? a±sZ?
? aμsZ?
? aa
??
ù? [5￥
L=
S
| g
? a
3
"d
? a?Y
@
? aü 6
? a y
??
?￥
!
null
?
￥ ?
null43
? as
? aB
,
?
) ?
?
null D V
￥
HW
= °?iD áX?¨￥?
M￥'
5
null Google
null m
6
null 7?￥
?
Bte?
L è
? Bte?
L è
è 1
|
ü
H/?9
^???
HW?r
) ?
?a
e
y0 7ú¤
eíE bμB?
e1
ü
H4*
?
Esò?r?)?
^N |ü"
y0 ?¤
e
￥Z_??í ?i o???
y0?B?
e
1
ü
H4 -
Esò?EùN |
??
é
H
W$
Hq?
w b b
DB?XEù5ü? ?7
3
bL ?
e￥
y0??
ea ˉ
"
e 7a1?
*
1?B
?
eü?4 -íE
b4 -￥
Esò
HWV7 ?$
A ?
^???
8
VM???
??V?RíM???B
^￥?##?M????3
7 5sò b7N |4 -
?
Esò?
r?#M?
H
eX??
=
E
?sò b
h
± IB/'53s??c
'tL
!
$
è
|?B?? "1 ? #1?=??
#1e
^Rí " 1 bù
I
1Hq
/ V[ ￡o?à
i B1N |
??￥]B
HW?r?1 b
s '5
'5
μ?`
μ?`
D?
D?
3￥i
3￥i
?Hq
?Hq
#￡
ü?
#￡
ü?
?
?
? ú￥ f ?1e?¤ b
? ú￥ f ?1e?¤ b
L ?á
ìDB?XEü?=?￥Rí?M?
6B | ]
B?? # ? "ù5ü1
I
1Hq/
|à
o?
M?BQ?"2
ü ?^¤
o1 ?B |￥?
r
HW??
6B |￥?
HW
|A o?M? b
h1- Nó?ì￡
ü
è ?Y§
s§D?￡§
Hμ
B?V
i i
B
HW￥
§ b hs§??
ò b
!XB/§￥T¨
^
I
1?
4 A
§ ￥
^Tá￥T¨i
±
^
1￡§
L ?
F
Tt h
?'Tú bTú
^31
HW￥ ?
HW
=ú
ˉ|_ -?
RB
? L b?ü
^
a ?§ g ?1 L
)i "BHTúL
D5
à$
1
nm 1-4 b?
*t§
HXVL￥ú
5?? ￡
ì ˉ
,V
^ b
^￥ zD
^ ?^?¤ ￥ù5￥1o ? L
￥ ?? b1 ?? L??| Ls1
L1
L2 ? L1
^
3 ?C§
#
??
?ú￥Q?
HW
=
RV￥
^? L21 ?ú??
aú
RV￥
^? b L1? ?^9
?Y?
ê
3￥
ü (Q?
HW t1*μ?
Q?
HWV
é| I?Pv7N§?￥?
R
v 9
^?5?
ê*X?z￥
"￥
^
P?Y
@
K
v V
6y
?ù?V7 L1=v*t1 b ?ú ?
L2; V¨ wL
EZE¤9 V ?¨
d?=
?
p9
?
=T 5 b
§?è??
òC XüM¤ b#
b?
B?59
L? v?
P An§￥
3
T¤tú b?=?§
￥
HW?? ?XVL
￥ú
¨ ?,V
^' T à
??r?
L+D /v b
D
L
è ?6
??1!v
￥
01
Z
L
?6
^?" b!
0￥ 5¨ ?
￡Y
Y!B/ bé ￡
￡y!′
￡??
t
ú?5
m?9?
t??5??
ê b??X?
8 7|?6 oeX
3By
￡
￡?? V[!
0 h
Fú
ey
s
B/?Bù5 b
0μvl
^
I
1"￥
0$
0
^8"!￥ l l l
?^ L
! á
ì
3?
0vl
M] (1Lé?
!′
H5
|B?
0?
￡
￡? ?a
B b! b
?
4 A
^
￡ ￥? % ?
!
0￥
b
0
^5¨ ?
￡
!V￥ a V
¨ b
￡
!÷D ￡
￡idy1
￡
t#
7
^y1
￡? ￡
b
*
1 ￡
￡1
I
1M ?;$L ?
FXyB??ú%￥
?
F?
???ü
￡y a b ?
￡￥y
í? I
né ???6 oe￥e
io
^X
3B/By ￡
￡
ü (
v? V[!
0 ê
ò¨
d?$
?^ V[4[/ eL
!
1
￡y a b
￡?9o I
n
0
￡
0￥vl a?
M]
2
0
S?D ?M]!
?a￥?D
￡?M]
3
￡y?￥
￡
1è
7
S?
1 T1K?D
￡
H￥?1 T2
4
?
0￥!′
HW ?T
^B
?è
b
?BL
!
.à V[ ??
?1
?
eL
! ?¨ ￡
o
?
p?6 oe￥ù5ü ?
^ís
? ?
F???°
B/By
￡￥é
^
°B
/L
￥
￡"
é
? b
Vn L
!Hq ￥4? ?
F ù￥
ù5 μ1
F! ?¨
't?
M a!y ?
I
1"￥
?[#
F
!ù?￥
' ??μ1'
F4L
!
H
Fy
￥ OXü'qz
b
è |?é
M]￥ vBB_·a?
?bàD V
1ü
%?ùZùKù V
[ü
% v ? b
! v
^ (é￥éD×
( 1
×? ?? é)C¨B
,E w? b
Z
n
(n1)
n
(n1)
?? n v
s?￥
? ?￥ ? M?μ
1/2-Z
n
= (n1) Z
n
# Z
n
=1/(2n)V7
n v_·w￥
9 ?1
∑
=
n
k
k
1
2
1
+∞=→+∞→
∑∑
∞
== 11
2
1
2
1
,
H
n
n
k
nk
n
# v_· V?à ?iù ?B2T
μ? |i
b
è
|t
?L?í??L
^
1 " a #
1W?
? sò " a #
1ò?B?úN |è ?EKí￥ $
??ú
e?C
B?
7
e?? /￥C
` 'v
f ?/5?_￥9
^?
# ? "￥ú
4?? # ? "￥úQ t
$ h
Fúù
esB/ey
" #?úQA ?
^B" ￥
?5B)￥ú
| b
?? ??]
! "? $?
R s
ò #? $1?
R sò I3B
?
HWé1 sò￥ uW è
? V[V "Z_ ?￥ú
R ? $_
H 7
S a￥ sò
=?r￥
e Zn?5 ?￥ú (1 # 7a "
￥?μKa sò?r￥e Z?
n? ? " ?￥ú5? b?N Vn
?TN | ? $_?ú￥
HW
^
￥5
e5?
#Z_ ?￥ú￥à
q1 b
è f
>
Y
H(01?u 7?
?
L b1
T b
>
Yey |
ìA?
D ysí(0 b ??(0￥ê?A? ?
?
￥Z_ ?
k
!9Btùs
(0￥ZE b??1 ??
??
à
1
¨N _?
Q"d
F
Y5??(0?
L￥ < b
ZEB
?;÷?￥
L ò?)￥vD ??;÷￥ ?
￥
üZ?Q1'
2
k/dI =
(0
Z_ ?^ ??1o ? ?? ? b
!
]BZ_?]ê?_?
Q?¤￥vsY1 I
1
I
2
?
?W￥ ?1 a5μ
2
22
12
)(
//
+
=
+
=
d
ad
ad
k
d
k
II
= 1
1
2
/
I
I
ad
ZE=
ZEB ?
_??D(0
ê?B°L
?B?1? ?^
S??1 ??
^2Tv?
￥ú1 p?ú
￥μ/?2T￥
vM'
ù?? <C4
6BZE "
??¤(0Z_a ? #? ?Z_ " #
?1 a ? # " $ % ? " # $ & ?¨??? ?¤
d a T J O% T J O % &
b31·￥
^?
(0ê??ù)7 % ??l
H % & V
d
è¤í 4
? " $ #¤í? 7 T J O
% & ?
zê?s
y72T￥ú ???Y9
v1
P2T?z?
P a9M?v b
B
A
C
a
%
&
Course Goals
null ?]D
ì??
null 4ú?Cù53 %ù5￥
?
null ¨?
Mùs?
M￥
?
null Dμ
¨9 <? ??
null ZE
null
± Is
null
Ll
?!/
null
D?
M
null s}
+à
d9a …
null èq
P¨
null Microsoft Word,Visio,LaTeX
null Matlab,Mathematica,Maple,Lindo,
Lingo…
null I?
null C/C++
null GUI Programming
Grading Policies
null General homework (20%)
null Every class/week
null Large projects (40%)
null 4-5
null Final exams (40%)
Grading Policies
5+
ZE?? F
ódèz
5
?y ? p3 ?
z
4
?y ? p3 ?
S
3
?y ? p3' ??
B?
2
?y ? p3μù5
B?
1
?y ??? ?
&Y
μù5
0
μù5Lm null
F￥T<??￥Q ?Ríó
F b
Requirements(1)
null
?á
null ?S
null ó3mV b
null
a
ü
null }
4?
null ó?
a
ü ??
?T?
null ??
? IrYV
null a
êu
null ?"?A1￥óq ? \debug
"
c
Requirement(2)
null Assignment submissions
null No late assignments will be accepted
null Late assignment follow the following rules
null 25% deduction for 1-day late
null 50% deduction for 2-day late
null Not accepted after being 2-day late
null Feedbacks on your codes will be returned
to you after judging
Requirement(3)
null ? ???
null Moúù
null vT
null '?
?ò
HW
null
?=/? 910?3:30-5:30
null ,á???\ 1
L 2|\
null
sP
My FTP
null ftp://10.13.61.167:21
null Username,MM05
null Password,MM05
null Hand in your homework
null Upload by FTP
null Send via E-mail
Q&A
?mμ??Z?,
θcos1 e
p
r
=
θdrdA
2
2
1
=
O?
áV￥
A￥±s1,
? 7
à?=?
p,
== wr
dt
dA
2
2
1
è
?'¤,
+== wrwrrwr
dt
d
22
2)(0
',
02 =+
wrwr
??
wTrdt
dt
dA
ab
T
2
0
2
1
==
∫
π
?N¤ ==
T
ab
wr
π2
2
è
e?w? ?/
??
r
θ
t
á
ì3
??￥F
1N31y ?
?
?]￥USO b?B?
^%?￥[
t 1USe?
éàZ_￥?ê_
,1 i,
àZ_￥?ê_
:
1 j?
^
jir θθ sinrcosr +=
é7μ F
·cossin)(2(sin)(cos(
)sinr(
dt
d
)cosr(
dt
d
2
2
2
2
2
jiji
jira
θθθθ
θθ
+?+++?=
+==
wrwrrwr
[??1USe?y ??OS
???￥?ê_
sY
^
jiejie θθθθ cossinsincos +?=+=
θr
yN¤
r
rwr ea )(
2
=
?? 02 =+
wrwr
9ü
^
a??￥F
1
r
rT
a
ea
22
32
14
=
π
? 7
à? ??
p?
23
/Ta
1è
b ?:
2
32
4
MT
a
G
π
=
*
1ü?÷
￥
￡μ? ??
p
|??Z?
)cos1( θerp?=
H±s
Q¤
0)(
1
)(
22
3
2
=+?
wr
rr
p
rwr
| -
¤?￥2T
??
} ?'¤
22
32
2
rT
a
rwr
14
=?
π
T
ab
wr
π2
2
=
a
b
p
2
=
r
r
Mm
GF e
2
=
