Post on 20-Dec-2015
2010/10/25 MSV-Lab. 2
First order Quasi-linear PDEFirst order Quasi-linear PDE
u)y,c(x,u)uy,b(x,u)uy,a(x,
PDElinear -Quasi of formula General
yx :
(s)us) u(0 dt
duc
(s)ys) y(0 dt
dyb
(s)xs) x(0 dt
dx a
0
0
0
,,
,,
,,
dataCaucthy called is and curve space a is
(s)u
(s)y
(s)x
0
0
0
Failure
s)u(t
s)y(t
s)x(t
,,,
Success
--------
--->
----------->
--------
->
--------
->
--------->
--------->
Construct surface
Can not Construct surface Unlucky
Unlucky
Lucky
Lucky
Get explicit solution
Can not Get explicit solution
Can not Get explicit solution
Get explicit solution
--->
--->
--->
--->
X
But we have already get this graph
y)u(xu ,
y)u(xu ,
--->
<---
The luckiest
The unluckiest
2010/10/25 MSV-Lab. 3
2
yx
x0) u(xsatisfy 2p
pu yu xu
Question
,,
:
dt
dy
y
u
dt
dx
x
u
dt
du From
2t3
t2
t1
ecu2udt
duc
ecyydt
dyb
ecxxdt
dxa
Sol :
sc
0c
sc
0When t
23
2
1
2t2
t
esu
0y
sex
solutionParameter
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
2010/10/25 MSV-Lab. 4
2t2
t
esu
0y
sex 2xy)u(x, +
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
-2
0
2
-4
-2
0
2
-5
0
5
10
-2
0
2
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
2010/10/25 MSV-Lab. 5
Classification
2t2
t
esu
0y
sexFailure
Success
--------
--->
----------->
--------
->
--------->
Construct surface
Can not Construct surface Unlucky
Lucky
Get explicit solution
Can not Get explicit solution
--->2xy)u(x,
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
-2
0
2
-4
-2
0
2
-5
0
5
10
-2
0
2
2010/10/25 MSV-Lab. 6
2
yx
yy) u(0satisfy 2p
pu yu xu
Question
,,
:
dt
dy
y
u
dt
dx
x
u
dt
du From
2t3
t2
t1
ecu2udt
duc
ecyydt
dyb
ecxxdt
dxa
Sol :
sc
sc
0c
0When t
23
2
1
2t2
t
esu
sey
0x
solutionParameter
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
2010/10/25 MSV-Lab. 7
2t2
t
esu
sey
0x
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
+ 2yy)u(x, -2
0200.5
11.522.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
-2
0
2
-2
0
2
-5
0
5
10
-2
0
2
2010/10/25 MSV-Lab. 8
Classification
Failure
Success
--------
--->
----------->
--------
->
--------->
Construct surface
Can not Construct surface Unlucky
Lucky
Get explicit solution
Can not Get explicit solution
--->
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
-2
0
2
-2
0
2
-5
0
5
10
-2
0
2
2t2
t
esu
sey
0x
2yy)u(x,
2010/10/25 MSV-Lab. 9
22
yx
yxy) u(x satisfy 2p
pu yu xu
Question
,,
:
2t3
t2
t1
ecu2udt
du
ecyydt
dy
ecxxdt
dx
Sol :
1c
Sin(s)c
Cos(s)c
Sin(s)y
Cos(s)xcircle ofsolution Parametr
0When t
3
2
1
2t
t
t
eu
Sin(s)ey
Cos(s)ex
solutionParameter
dt
dy
y
u
dt
dx
x
u
dt
du From
-20
2
-2-1
01
2
-5
0
5
10-2-1
01
2
2010/10/25 MSV-Lab. 10
2t
t
t
eu
Sin(s)ey
Cos(s)ex + 22 yxy)u(x,
-20
2
-2-1
01
2
-5
0
5
10-2-1
01
2
-2
0
2-4
-2
0
2
4
0
10
20
-2
0
2
-20
2
-2-1
01
2
-5
0
5
10-2-1
01
2
2010/10/25 MSV-Lab. 11
Classification
2t
t
t
eu
Sin(s)ey
Cos(s)exFailure
Success
--------
--->
-----------> --------
->
--------->
Construct surface
Can not Construct surface
Unlucky
LuckyGet explicit solution --->
Can not Get explicit solution
-20
2
-2-1
01
2
-5
0
5
10-2-1
01
2
-2
0
2-4
-2
0
2
4
0
10
20
-2
0
2
22 yxy)u(x,
The Luckiest!!
2010/10/25 MSV-Lab. 12
ss) u(ssatisfy 2p
pu yu xu
Question
yx
,,
:
2t3
t2
t1
ecu2udt
du
ecyydt
dy
ecxxdt
dx
sc
sc
sc
0When t
3
2
1
2t
t
t
seu
sey
sex
solutionParameter
dt
dy
y
u
dt
dx
x
u
dt
du From
Sol :
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
2010/10/25 MSV-Lab. 13
2t
t
t
seu
sey
sex
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
Bad news, it’s the unluckiest situation
2010/10/25 MSV-Lab. 14
Classification
Failure
Success
--------
--->
----------->
--------
->
--------->
Construct surface
Can not Construct surface Unlucky
Lucky
Get explicit solution
Can not Get explicit solution
---> X
2t
t
t
seu
sey
sex
-20
200.511.5
22.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
Oh~Shit!!
The unluckiest!!
2010/10/25 MSV-Lab. 15
cxy
eeee
clnxlny
x
dx
y
dy
dy
dx
dt
dydt
dx
y
x
11 clnxclnxlny
1
2p pu yu xu
equation thisinto Looking
yx ,:
Characteristic line
2010/10/25 MSV-Lab. 16
22
yx
yxy) u(x satisfy 2p
pu yu xu
Question
,,
:
-20
2
-2-1
01
2
-5
0
5
10-2-1
01
2
2010/10/25 MSV-Lab. 17
2
yx
x0) u(xsatisfy 2p
pu yu xu
Question
,,
:-2
0200.5
11.522.5
-5
0
5
1000.5
11.522.5
-5
0
5
10
2010/10/25 MSV-Lab. 18
判別式 牽絲原則判別式 牽絲原則
0ba
(s)y(s)x '0
'0
無法牽絲,,若 b)(a // ) (s)y' (s) x'( 特徵線帶出的訊息
bdt
dy
adt
dx
整理出來,千變萬化的解答
點之斜率於SdataCauthy (s)y'
(s)x'
2010/10/25 MSV-Lab. 19
2uyuxu yx
無法牽絲
,
0-xyx
10
ba
(s)y'(s)x'
yy) u(0 2
可以牽絲
,
0-1
(s))cos(s)-(sinxcos(s)--ysin(s)
yx
cos(s)sin(s)-
ba
(s)y'(s)x'
sin(s)y
cos(s)x yxy) u(x
22
22
無法牽絲
,
0x-y
yx
11
ba
(s)y'(s)x'
ss) u(s
無法牽絲
,
0yyx
01
ba
(s)y'(s)x'
x0) u(x 2
Conclusion