76
Lesson 18 Linear partial differential equations 1

Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

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Page 1: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

Lesson 18Linear partial differential equations

1

Page 2: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

2

� ;I�LEZI�WIIR�XLEX�3()W�GER�FI�VIHYGIH�XS EPQSWX�FERHIH MR½RMXI�HMQIRWMSREPIUYEXMSRW�YWMRK�'LIF]WLIZ�ERH�YPXVEWTLIVMGEP�TSP]RSQMEPW

� 8LIWI�MR½RMXI�HMQIRWMSREP�IUYEXMSRW�GER�FI WSPZIH�EHETXMZIP] YWMRK�+MZIRW�VSXE�XMSRW�

8LI�I\EGX IVVSV�MR�VIWMHYEP MW�GEPGYPEXIH� [LMGL�GER�FI�YWIH�XS�HIGMHI�GSR�ZIVKIRGI

� 8LMW�PIGXYVI�[I�WII�LS[�XLIWI�MHIEW�GER�FI�YWIH�JSV�PMRIEV�4()W

� 8LI�VIWYPXMRK�QIXLSH�[SVOW�JSV�EVFMXVEV]�PMRIEV�4()W�SR�E�VIGXERKPI

� 8LI�½VWX�WXIT�MW�XS�HS JYRGXMSR�ETTVS\MQEXMSR�SR�E�WUYEVI

Page 3: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

3

2D Function Approximation

Page 4: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

4

� -R��(��[I�ETTVS\MQEXIH�JYRGXMSRW�YWMRK 'LIF]WLIZ�WIVMIW�

f(x) �n�1�

k=0

fkTk(x) = (T0(x), . . . , Tn�1(x))

��f0...

fn�1

��

*YRGXMSRW�EVI�MHIRXM½IH�[MXL ZIGXSVW

� -R��(��[I�YWI�E XIRWSV�TVSHYGX�SJ�'LIF]WLIZ�WIVMIW�

f(x, y) �n�1�

k=0

m�1�

j=0

fkjTk(x)Tj(y)

= (T0(x), . . . , Tn�1(x))

��f00 · · · f0(m�1)...

. . ....

f(n�1)0 · · · f(n�1)(m�1)

��

��T0(y)

...Tm�1(y)

��

*YRGXMSRW�EVI�MHIRXM½IH�[MXL QEXVMGIW

Page 5: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

5

� -R��(��[I�ETTVS\MQEXIH�JYRGXMSRW�YWMRK 'LIF]WLIZ�WIVMIW�

f(x) �n�1�

k=0

fkTk(x) = (T0(x), . . . , Tn�1(x))

��f0...

fn�1

��

*YRGXMSRW�EVI�MHIRXM½IH�[MXL ZIGXSVW

� -R��(��[I�YWI�E XIRWSV�TVSHYGX�SJ�'LIF]WLIZ�WIVMIW�

f(x, y) �n�1�

k=0

m�1�

j=0

fkjTk(x)Tj(y)

= (T0(x), . . . , Tn�1(x))

��f00 · · · f0(m�1)...

. . ....

f(n�1)0 · · · f(n�1)(m�1)

��

��T0(y)

...Tm�1(y)

��

*YRGXMSRW�EVI�MHIRXM½IH�[MXL QEXVMGIW

Page 6: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

6

–1 1

� -R��(��[I�GEPGYPEXIH�XLI�'LIF]WLIZ�WIVMIW�F]�IZEPYEXMRK f(x) EX�XLI 'LIF]WLIZTSMRXW xn�

Page 7: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

6

–1 1

� -R��(��[I�GEPGYPEXIH�XLI�'LIF]WLIZ�WIVMIW�F]�IZEPYEXMRK f(x) EX�XLI 'LIF]WLIZTSMRXW xn�

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

y

x

� -R��(��[I�IZEPYEXI f(x, y) EX�E TVSHYGX�SJ�'LIF]WLIZ�TSMRXW xn,m := xn � x�m�

Page 8: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

7

� -R��(��[I�LEZI

Cnf(xn) �

��f0...

fn�1

��

[LIVI Cn GER�MW�ZMI[IH�EW�ER n � n QEXVM\

� ;I�[MPP�WLS[�XLEX�MR��( [I�LEZI

Cnf(xn,m)C�m �

��f00 · · · f0(m�1)...

. . ....

f(n�1)0 · · · f(n�1)(m�1)

��

Page 9: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

8

� -R��(��[I�LEZI

Cnf(xn) �

��f0...

fn�1

��

[LIVI Cn GER�MW�ZMI[IH�EW�ER n � n QEXVM\

� ;I�[MPP�WLS[�XLEX�MR��( [I�LEZI

Cnf(xn,m)C�m �

��f00 · · · f0(m�1)...

. . ....

f(n�1)0 · · · f(n�1)(m�1)

��

Page 10: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

9

� *SV�ER]�½\IH y� [I�GER�I\TERH

f(x, y) =��

k=0

fk(y)Tk(x)

[LIVIfk(y) =

�2

� 1

�1

f(x, y)Tk(x)�1 � x2

x

� -X�JSPPS[W�XLEX�XLI�WQSSXLRIWW�SJ fk(y) MW�MRLIVMXIH�JVSQ f

� ;I�GER�XLYW�I\TERH

fk(y) =��

j=0

fkjTj(y)

Page 11: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

10

� *SV�ER]�½\IH y� [I�GER�I\TERH

f(x, y) =��

k=0

fk(y)Tk(x)

[LIVIfk(y) =

�2

� 1

�1

f(x, y)Tk(x)�1 � x2

x

� -X�JSPPS[W�XLEX�XLI�WQSSXLRIWW�SJ fk(y) MW�MRLIVMXIH�JVSQ f

� ;I�GER�XLYW�I\TERH

fk(y) =��

j=0

fkjTj(y)

Page 12: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

11

� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]

Cnf(xn, y) =:

��fn0 (y)...

fnn�1(y)

��

� ;I�GER�XLYW�I\TERH

Cmfnk (xm) =:

��fnm

k0...

fnmk(m�1)

��

� 8LMW�KMZIW�YW

Cnf(xn, xm)�C�m =

��fn0 (x�

m)...

fnn�1(x

�m)

��C�m

= Cm

�fn0 (xm), . . . , fn

n�1(xm)�

=

��

fnm00 · · · fnm

0(m�1)...

. . ....

fnm(n�1)0 · · · fnm

(n�1)(m�1)

��

Page 13: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

12

� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]

Cnf(xn, y) =:

��fn0 (y)...

fnn�1(y)

��

� ;I�GER�XLYW�I\TERH

Cmfnk (xm) =:

��fnm

k0...

fnmk(m�1)

��

� 8LMW�KMZIW�YW

Cnf(xn, xm)�C�m =

��fn0 (x�

m)...

fnn�1(x

�m)

��C�m

= Cm

�fn0 (xm), . . . , fn

n�1(xm)�

=

��

fnm00 · · · fnm

0(m�1)...

. . ....

fnm(n�1)0 · · · fnm

(n�1)(m�1)

��

Page 14: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

13

� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]

Cnf(xn, y) =:

��fn0 (y)...

fnn�1(y)

��

� ;I�GER�XLYW�I\TERH

Cmfnk (xm) =:

��fnm

k0...

fnmk(m�1)

��

� 8LMW�KMZIW�YW

Cnf(xn, xm)�C�m =

��fn0 (x�

m)...

fnn�1(x

�m)

��C�m

= Cm

�fn0 (xm), . . . , fn

n�1(xm)�

=

��

fnm00 · · · fnm

0(m�1)...

. . ....

fnm(n�1)0 · · · fnm

(n�1)(m�1)

��

Page 15: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

14

� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]

Cnf(xn, y) =:

��fn0 (y)...

fnn�1(y)

��

� ;I�GER�XLYW�I\TERH

Cmfnk (xm) =:

��fnm

k0...

fnmk(m�1)

��

� 8LMW�KMZIW�YW

Cnf(xn, xm)�C�m =

��fn0 (x�

m)...

fnn�1(x

�m)

��C�m

= Cm

�fn0 (xm), . . . , fn

n�1(xm)�

=

��

fnm00 · · · fnm

0(m�1)...

. . ....

fnm(n�1)0 · · · fnm

(n�1)(m�1)

��

Page 16: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

15

Partial derivatives

Page 17: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

16

Page 18: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

17

Page 19: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

18

Page 20: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

19

Page 21: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

20

Page 22: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

21

Page 23: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

22

Page 24: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

23

Page 25: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

24

Page 26: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

25

Page 27: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

26

Page 28: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

27

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.F.

1 0 0 0 …

0 1

40 0 …

- 23

0 1

60 …

0 - 38

0 1

8…

1

60 - 4

150 …

0 1

80 - 5

24…

Å Å Å Å Å

+

1 0 - 23

0 1

60 0 0 …

0 1

40 - 3

80 1

80 0 …

0 0 1

60 - 4

150 1

100 …

0 0 0 1

80 - 5

240 1

12…

Å Å Å Å Å Å Å Å Å

.F.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

=

Page 29: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

28

Boundary conditions

Page 30: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

29

� ;I�[ERX�XS�WSPZI�4()W�[MXL�FSYRHEV]�GSRHMXMSRW

� *SV�I\EQTPI� 4SMWWSR�[MXL�(MVMGLPIX XEOIW�XLI�JSVQ

u(�1, y) = g1(y), u(1, y) = g2(y)

u(x, �1) = h1(x), u(x, 1) = h2(x)

�u(x, y) = f(x, y)

� ;I�GER�VITLVEWI�XLMW�MR�XIVQW�SJ�XLI�GSIJ½GMIRXW�SJ u

Page 31: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

30

� )\TERH

u(x, y) = (T0(x), T1(x), . . .)U

��T0(y)T1(y)

...

��

� 8LIR�[I�LEZI

�u(�1, y)u(1, y)

�=

�T0(�1) T1(�1) . . .T0(1) T1(1) . . .

�U

��T0(y)T1(y)

...

��

=

�1 �1 . . .1 1 . . .

�U

��T0(y)T1(y)

...

��

=: BU

��T0(y)T1(y)

...

��

Page 32: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

31

� )\TERH

u(x, y) = (T0(x), T1(x), . . .)U

��T0(y)T1(y)

...

��

� 8LIR�[I�LEZI

�u(�1, y)u(1, y)

�=

�T0(�1) T1(�1) . . .T0(1) T1(1) . . .

�U

��T0(y)T1(y)

...

��

=

�1 �1 . . .1 1 . . .

�U

��T0(y)T1(y)

...

��

=: BU

��T0(y)T1(y)

...

��

Page 33: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

32

� 8LYW�XLI�GSRHMXMSR �u(�1, y)u(1, y)

�=

�g1(y)g2(y)

FIGSQIWBU = G�

[LIVI

G = (g1, g2)

=

��g10 g20

g11 g21...

...

��

Page 34: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

33

� 7MQMPEVP]� XLI�GSRHMXMSR �u(x, �1)u(x, 1)

�=

�h1(x)h2(x)

FIGSQIWUB� = H

[LIVI

H =�h1, h2

=

��h10 h20

h11 h21...

...

��

Page 35: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

34

� -R�SXLIV�[SVHW� 4SMWWSR�[MXL�(MVMGLPIX

u(�1, y) = g1(y), u(1, y) = g2(y)

u(x, �1) = h1(x), u(x, 1) = h2(x)

�2u

�x2+

�2u

�y2= f(x, y)

FIGSQIW

BU = G�,

UB� = H,

D2US�0�2 + S0�2UD�

2 = F

Page 36: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

35

� %RSXLIV�I\EQTPI� ,IPQLSPX^�[MXL�(MVMGLPIX

u(�1, y) = g1(y), u(1, y) = g2(y)

u(x, �1) = h1(x), u(x, 1) = h2(x)

�2u

�x2+

�2u

�y2+ k2u = f(x, y)

FIGSQIW

BU = G�,

UB� = H,�D2 + k2S�

0�2

�US�

0�2 + S0�2UD�2 = F

Page 37: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

Sylvester matrix equations

36

Page 38: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

37

� 'SRWMHIV�E 7]PZIWXIV�W�QEXVM\�IUYEXMSR

AX + XB = F

[LIVI A, B, F EVI n � n QEXVMGIW

� %R]�IUYEXMSRAXC + DXB = F

GER�FI�VIHYGIH�XS�XLMW�JSVQ�EWWYQMRK C ERH D EVI�MRZIVXMFPI�F]�MRZIVXMRK�SR�IEGLWMHI

D�1AX + XBC�1 = D�1FC�1

-J D ERH C EVI EPQSWX�FERHIH �EW�MW�XLI�GEWI�JSV�YW � XLIR�ETTP]MRK�XLI�MRZIVWIXS�E�ZIGXSV�MW�ER O(n) EPKSVMXLQ� WS�ETTP]MRK�XS�IEGL�GSPYQR�SJ A� B ERH FMW O

�n2

�STIVEXMSRW

Page 39: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

38

� 'SRWMHIV�E 7]PZIWXIV�W�QEXVM\�IUYEXMSR

AX + XB = F

[LIVI A, B, F EVI n � n QEXVMGIW

� %R]�IUYEXMSRAXC + DXB = F

GER�FI�VIHYGIH�XS�XLMW�JSVQ�EWWYQMRK C ERH D EVI�MRZIVXMFPI�F]�MRZIVXMRK�SR�IEGLWMHI

D�1AX + XBC�1 = D�1FC�1

-J D ERH C EVI EPQSWX�FERHIH �EW�MW�XLI�GEWI�JSV�YW � XLIR�ETTP]MRK�XLI�MRZIVWIXS�E�ZIGXSV�MW�ER O(n) EPKSVMXLQ� WS�ETTP]MRK�XS�IEGL�GSPYQR�SJ A� B ERH FMW O

�n2

�STIVEXMSRW

Page 40: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

39

� 'SRWMHIV�E 7]PZIWXIV�W�QEXVM\�IUYEXMSR

AX + XB = F

[LIVI A, B, F EVI n � n QEXVMGIW

� %R]�IUYEXMSRAXC + DXB = F

GER�FI�VIHYGIH�XS�XLMW�JSVQ�EWWYQMRK C ERH D EVI�MRZIVXMFPI�F]�MRZIVXMRK�SR�IEGLWMHI

D�1AX + XBC�1 = D�1FC�1

-J D ERH C EVI EPQSWX�FERHIH �EW�MW�XLI�GEWI�JSV�YW � XLIR�ETTP]MRK�XLI�MRZIVWIXS�E�ZIGXSV�MW�ER O(n) EPKSVMXLQ� WS�ETTP]MRK�XS�IEGL�GSPYQR�SJ A� B ERH FMW O

�n2

�STIVEXMSRW

Page 41: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

40

� *SV�XLI 7]PZIWXIV�W�QEXVM\�IUYEXMSR

AX + XB = F

½VWX HMEKSREPM^I A = V �V �1 ERH B = W�W�1

� 2S[�PIXXMRK Y = V �1XW [I�LEZI

V �V �1X + XW�W�1 = V �Y W�1 + V Y �W�1 = F

� -R�SXLIV�[SVHW��Y + Y � = V �1FW

Page 42: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

41

� *SV�XLI 7]PZIWXIV�W�QEXVM\�IUYEXMSR

AX + XB = F

½VWX HMEKSREPM^I A = V �V �1 ERH B = W�W�1

� 2S[�PIXXMRK Y = V �1XW [I�LEZI

V �V �1X + XW�W�1 = V �Y W�1 + V Y �W�1 = F

� -R�SXLIV�[SVHW��Y + Y � = V �1FW

Page 43: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

42

� *SV�XLI 7]PZIWXIV�W�QEXVM\�IUYEXMSR

AX + XB = F

½VWX HMEKSREPM^I A = V �V �1 ERH B = W�W�1

� 2S[�PIXXMRK Y = V �1XW [I�LEZI

V �V �1X + XW�W�1 = V �Y W�1 + V Y �W�1 = F

� -R�SXLIV�[SVHW��Y + Y � = V �1FW

Page 44: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

43

� &YX�[I�LEZI

�Y + Y � =

���1y00 · · · �1y0(n�1)

.... . .

...�ny(n�1)0 · · · �ny(n�1)(n�1)

�� +

���1y00 · · · �ny0(n�1)

.... . .

...�1y(n�1)0 · · · �ny(n�1)(n�1)

��

=

��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)

.... . .

...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)

��

� 8LYW �Y + Y � = P MW�WSPZIH�F]

yij =pij

�i + �j

� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1

Page 45: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

44

� &YX�[I�LEZI

�Y + Y � =

���1y00 · · · �1y0(n�1)

.... . .

...�ny(n�1)0 · · · �ny(n�1)(n�1)

�� +

���1y00 · · · �ny0(n�1)

.... . .

...�1y(n�1)0 · · · �ny(n�1)(n�1)

��

=

��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)

.... . .

...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)

��

� 8LYW �Y + Y � = P MW�WSPZIH�F]

yij =pij

�i + �j

� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1

Page 46: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

45

� &YX�[I�LEZI

�Y + Y � =

���1y00 · · · �1y0(n�1)

.... . .

...�ny(n�1)0 · · · �ny(n�1)(n�1)

�� +

���1y00 · · · �ny0(n�1)

.... . .

...�1y(n�1)0 · · · �ny(n�1)(n�1)

��

=

��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)

.... . .

...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)

��

� 8LYW �Y + Y � = P MW�WSPZIH�F]

yij =pij

�i + �j

� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1

Page 47: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

46

� &YX�[I�LEZI

�Y + Y � =

���1y00 · · · �1y0(n�1)

.... . .

...�ny(n�1)0 · · · �ny(n�1)(n�1)

�� +

���1y00 · · · �ny0(n�1)

.... . .

...�1y(n�1)0 · · · �ny(n�1)(n�1)

��

=

��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)

.... . .

...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)

��

� 8LYW �Y + Y � = P MW�WSPZIH�F]

yij =pij

�i + �j

� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1

Page 48: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

47

Complexity

� A = V �V �1 ERH B = W�W�1

� V �1FW

� 7SPZMRK �Y + Y � = V �1FW

� X = V Y W�1

O�n3

�O

�n3

O�n3

O�n3

O�n3

Page 49: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

Reducing PDEs to Sylvester equations

48

Page 50: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

1 0 - 23

0 1

6…

0 1

40 - 3

80 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

= F

K 1 -1 1 -1 …1 1 1 1 …

O.U =

U.

1 1-1 11 1

-1 1Å Å

=

GT

H

Page 51: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

1 0 - 23

0 1

6…

0 1

40 - 3

80 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

= F

K 1 -1 1 -1 …1 1 1 1 …

O.U =

U.

1 1-1 11 1

-1 1Å Å

=

GT

H

Row reduction

1

2

�1 1

�1 1

�1

2

�1 1

�1 1

1

2

�1 �11 1

�1

2

�1 �11 1

Page 52: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

1 0 - 23

0 1

6…

0 1

40 - 3

80 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

= F

=

=

PT

R

K 1 0 1 0 …0 1 0 1 …

O.U

U.

1 00 11 00 1Å Å

Page 53: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

1 0 - 23

0 1

6…

0 1

40 - 3

80 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

= F

= PTK 1 0 1 0 …0 1 0 1 …

O.U

= RU.

1 00 11 00 1Å Å

�1

14

� �1

14

�D�

2 D�2

Page 54: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

1 0 - 23

0 1

6…

0 1

40 - 3

80 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

= F

PTK 1 0 1 0 …0 1 0 1 …

O.U

= RU.

1 00 11 00 1Å Å

�1

14

� �1

14

�D�

2 D�2– –

Page 55: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

=

= RU.

1 00 11 00 1Å Å

F

0 0 - 53

0 - 56

0 0 0 - 58

0 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

Page 56: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

=

= RU.

1 00 11 00 1Å Å

F

0 0 - 53

0 - 56

0 0 0 - 58

0 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

1 0 0 0 Å

0 1

40 0 Å

- 23

0 1

60 Å

0 - 38

0 1

… … … … Å

Page 57: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

= F

0 0 - 53

0 - 56

0 0 0 - 58

0 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å

- 53

0 1

60 Å

0 - 58

0 1

… … … … Å

Page 58: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

= F

0 0 - 53

0 - 56

0 0 0 - 58

0 …

0 0 1

60 - 4

15…

0 0 0 1

80 …

Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å

.U.

0 0 0 0 Å0 0 0 0 Å

- 53

0 1

60 Å

0 - 58

0 1

… … … … Å

U =

�U11 U12

U21 U22

�2 � 2 2 � �

� � 2 � � �

Page 59: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

= F

U =

�U11 U12

U21 U22

�2 � 2 2 � �

� � 2 � � �

- 53

0 - 56

0 - 58

0 …1

60 - 4

15…

0 1

80 …

Å Å Å Å

.U22.

4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

4 0 0 0 …0 6 0 0 …0 0 8 0 …0 0 0 10 …Å Å Å Å Å

.U22.

- 53

0 1

60 Å

0 - 58

0 1

… … … … Å

Page 60: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

= F

U =

�U11 U12

U21 U22

�2 � 2 2 � �

� � 2 � � �

- 53

0 - 56

0 - 58

0 …1

60 - 4

15…

0 1

80 …

Å Å Å Å

.U22.

4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å

+

4 0 0 0 …0 6 0 0 …0 0 8 0 …0 0 0 10 …Å Å Å Å Å

.U22.

- 53

0 1

60 Å

0 - 58

0 1

… … … … Å

Truncate to get Sylvester’s equation!

Page 61: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

60

Recover solution

Page 62: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

61

� ;I�LEZI�XLYW�GEPGYPEXIH U22� ERH�[I�NYWX�RIIH�XS�VIGSZIV U11, U12 ERH U21

� ;I�YWI�XLI FSYRHEV]�GSRHMXMSRW�

P� =

�1 0 1 0 · · ·0 1 0 1 · · ·

��U11 U12

U21 U22

=

���U11 +

�1 0 1 0 · · ·0 1 0 1 · · ·

�U21

U12 +

�1 0 1 0 · · ·0 1 0 1 · · ·

�U22

���

[VMXMRK P� = (P1, P2) [LIVI P1 MW 2 � 2 ERH P2 MW 2 � �� [I�KIX

U12 = P2 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U22

Page 63: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

62

� ;I�LEZI�XLYW�GEPGYPEXIH U22� ERH�[I�NYWX�RIIH�XS�VIGSZIV U11, U12 ERH U21

� ;I�YWI�XLI FSYRHEV]�GSRHMXMSRW�

P� =

�1 0 1 0 · · ·0 1 0 1 · · ·

��U11 U12

U21 U22

=

���U11 +

�1 0 1 0 · · ·0 1 0 1 · · ·

�U21

U12 +

�1 0 1 0 · · ·0 1 0 1 · · ·

�U22

���

[VMXMRK P� = (P1, P2) [LIVI P1 MW 2 � 2 ERH P2 MW 2 � �� [I�KIX

U12 = P2 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U22

Page 64: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

63

� ;I�LEZI�XLYW�GEPGYPEXIH U22� ERH�[I�NYWX�RIIH�XS�VIGSZIV U11, U12 ERH U21

� ;I�YWI�XLI FSYRHEV]�GSRHMXMSRW�

P� =

�1 0 1 0 · · ·0 1 0 1 · · ·

��U11 U12

U21 U22

=

���U11 +

�1 0 1 0 · · ·0 1 0 1 · · ·

�U21

U12 +

�1 0 1 0 · · ·0 1 0 1 · · ·

�U22

���

[VMXMRK P� = (P1, P2) [LIVI P1 MW 2 � 2 ERH P2 MW 2 � �� [I�KIX

U12 = P2 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U22

Page 65: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

64

� 7MQMPEVP]� [I�KIX

R =

�U11 U12

U21 U22

��

����

1 00 11 0...

...

����

=

�������������

U11 + U12

����

1 00 11 0...

...

����

U21 + U22

����

1 00 11 0...

...

����

�������������

� 8LYW�[I�KIX�JSV R =

�R1

R2

U21 = R2 � U22

����

1 00 11 0...

...

����

Page 66: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

65

� 7MQMPEVP]� [I�KIX

R =

�U11 U12

U21 U22

��

����

1 00 11 0...

...

����

=

�������������

U11 + U12

����

1 00 11 0...

...

����

U21 + U22

����

1 00 11 0...

...

����

�������������

� 8LYW�[I�KIX�JSV R =

�R1

R2

U21 = R2 � U22

����

1 00 11 0...

...

����

Page 67: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

66

� 7MQMPEVP]� [I�KIX

R =

�U11 U12

U21 U22

��

����

1 00 11 0...

...

����

=

�������������

U11 + U12

����

1 00 11 0...

...

����

U21 + U22

����

1 00 11 0...

...

����

�������������

� 8LYW�[I�KIX�JSV R =

�R1

R2

U21 = R2 � U22

����

1 00 11 0...

...

����

Page 68: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

67

� *MREPP]� [I�LEZI

U11 = P1 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U21

= R1 � U12

����

1 00 11 0...

...

����

� *SV�XLIWI�XS�QEXGL� [I�RIIH

P1 � BU21 = P1 � B�R2 � U22B�

= R1 ��P2 � BU22

�B� = R1 � U12B�

� 8LYW�[I�[ERXBR = P�B�

� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW

BH = G�B�

Page 69: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

68

� *MREPP]� [I�LEZI

U11 = P1 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U21

= R1 � U12

����

1 00 11 0...

...

����

� *SV�XLIWI�XS�QEXGL� [I�RIIH

P1 � BU21 = P1 � B�R2 � U22B�

= R1 ��P2 � BU22

�B� = R1 � U12B�

� 8LYW�[I�[ERXBR = P�B�

� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW

BH = G�B�

Page 70: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

69

� *MREPP]� [I�LEZI

U11 = P1 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U21

= R1 � U12

����

1 00 11 0...

...

����

� *SV�XLIWI�XS�QEXGL� [I�RIIH

P1 � BU21 = P1 � B�R2 � U22B�

= R1 ��P2 � BU22

�B� = R1 � U12B�

� 8LYW�[I�[ERXBR = P�B�

� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW

BH = G�B�

Page 71: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

70

� *MREPP]� [I�LEZI

U11 = P1 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U21

= R1 � U12

����

1 00 11 0...

...

����

� *SV�XLIWI�XS�QEXGL� [I�RIIH

P1 � BU21 = P1 � B�R2 � U22B�

= R1 ��P2 � BU22

�B� = R1 � U12B�

� 8LYW�[I�[ERXBR = P�B�

� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW

BH = G�B�

Page 72: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

71

� *MREPP]� [I�LEZI

U11 = P1 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U21

= R1 � U12

����

1 00 11 0...

...

����

� *SV�XLIWI�XS�QEXGL� [I�RIIH

P1 � BU21 = P1 � B�R2 � U22B�

= R1 ��P2 � BU22

�B� = R1 � U12B�

� 8LYW�[I�[ERXBR = P�B�

� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW

BH = G�B�

Page 73: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

72

� *MREPP]� [I�LEZI

U11 = P1 ��

1 0 1 0 · · ·0 1 0 1 · · ·

�U21

= R1 � U12

����

1 00 11 0...

...

����

� *SV�XLIWI�XS�QEXGL� [I�RIIH

P1 � BU21 = P1 � B�R2 � U22B�

= R1 ��P2 � BU22

�B� = R1 � U12B�

� 8LYW�[I�[ERXBR = P�B�

� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW

BH = G�B�

Page 74: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

73

� )\EQTPI� (MVMGLPIX

� ;I�LEZIB(h1, h2) =

�h1(�1) h2(�1)h1(1) h2(1)

ERH �g1

g2

�B� =

�g1(�1) g1(1)g2(�1) g2(1)

� 8LYW�GSQTEXMFMPMX]�MW�IUYMZEPIRX�XS GSRXMRYMX]�

h1(�1) =x��1

u(x, �1) =y��1

u(�1, y) = g1(�1)

h2(�1) =x��1

u(x, 1) =y�1

u(�1, y) = g1(1)

h1(1) =x�1

u(x, �1) =y��1

u(1, y) = g2(�1)

h2(1) =x�1

u(x, 1) =y�1

u(1, y) = g2(1)

Page 75: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

74

� )\EQTPI� (MVMGLPIX

� ;I�LEZIB(h1, h2) =

�h1(�1) h2(�1)h1(1) h2(1)

ERH �g1

g2

�B� =

�g1(�1) g1(1)g2(�1) g2(1)

� 8LYW�GSQTEXMFMPMX]�MW�IUYMZEPIRX�XS GSRXMRYMX]�

h1(�1) =x��1

u(x, �1) =y��1

u(�1, y) = g1(�1)

h2(�1) =x��1

u(x, 1) =y�1

u(�1, y) = g1(1)

h1(1) =x�1

u(x, �1) =y��1

u(1, y) = g2(�1)

h2(1) =x�1

u(x, 1) =y�1

u(1, y) = g2(1)

Page 76: Lesson 18 Linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew mw xli gewi jsv yw xlir ettp]mrk xli mrzivwi xs e zigxsv mw ero(n) epksvmxlq ws ettp]mrk

75

� )\EQTPI� (MVMGLPIX

� ;I�LEZIB(h1, h2) =

�h1(�1) h2(�1)h1(1) h2(1)

ERH �g1

g2

�B� =

�g1(�1) g1(1)g2(�1) g2(1)

� 8LYW�GSQTEXMFMPMX]�MW�IUYMZEPIRX�XS GSRXMRYMX]�

h1(�1) =x��1

u(x, �1) =y��1

u(�1, y) = g1(�1)

h2(�1) =x��1

u(x, 1) =y�1

u(�1, y) = g1(1)

h1(1) =x�1

u(x, �1) =y��1

u(1, y) = g2(�1)

h2(1) =x�1

u(x, 1) =y�1

u(1, y) = g2(1)