2010/10/25 MSV-Lab. 1

PDE Homework 3PDE Homework 3

ReporterReporter ：： Y. FanY. Fan

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

2010/10/25 MSV-Lab. 20