lesson 18 linear partial differential equations · 2013. 10. 14. · -jd erhc eviepqswx ferhih ew...
TRANSCRIPT
![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/1.jpg)
Lesson 18Linear partial differential equations
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/2.jpg)
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� 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/3.jpg)
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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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/4.jpg)
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k=0
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/5.jpg)
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� -R��(��[I�ETTVS\MQEXIH�JYRGXMSRW�YWMRK 'LIF]WLIZ�WIVMIW�
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/6.jpg)
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–1 1
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8
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/10.jpg)
9
� *SV�ER]�½\IH y� [I�GER�I\TERH
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10
� *SV�ER]�½\IH y� [I�GER�I\TERH
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/12.jpg)
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� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
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�
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/13.jpg)
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� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
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fnm00 · · · fnm
0(m�1)...
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fnm(n�1)0 · · · fnm
(n�1)(m�1)
�
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![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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/14.jpg)
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� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
Cnf(xn, y) =:
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fnn�1(y)
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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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/15.jpg)
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� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
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fnn�1(y)
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Cnf(xn, xm)�C�m =
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�m)
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= Cm
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n�1(xm)�
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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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/16.jpg)
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Partial derivatives
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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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/29.jpg)
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Boundary conditions
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� ;I�[ERX�XS�WSPZI�4()W�[MXL�FSYRHEV]�GSRHMXMSRW
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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
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� )\TERH
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� )\TERH
u(x, y) = (T0(x), T1(x), . . .)U
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��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)
...
�
��
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32
� 8LYW�XLI�GSRHMXMSR �u(�1, y)u(1, y)
�=
�g1(y)g2(y)
�
FIGSQIWBU = G�
[LIVI
G = (g1, g2)
=
�
��g10 g20
g11 g21...
...
�
��
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33
� 7MQMPEVP]� XLI�GSRHMXMSR �u(x, �1)u(x, 1)
�=
�h1(x)h2(x)
�
FIGSQIWUB� = H
[LIVI
H =�h1, h2
�
=
�
��h10 h20
h11 h21...
...
�
��
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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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/36.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/37.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/38.jpg)
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
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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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/40.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/41.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/42.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/43.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/44.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/45.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/46.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/47.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/48.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/49.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/50.jpg)
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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/51.jpg)
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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/52.jpg)
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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/53.jpg)
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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/54.jpg)
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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/55.jpg)
=
= 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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/56.jpg)
=
= 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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/57.jpg)
= 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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/58.jpg)
= 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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/59.jpg)
= 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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/60.jpg)
= 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
8Å
… … … … 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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/61.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/62.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/63.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/64.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/65.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/66.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/67.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/68.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/69.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/70.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/71.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/72.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/73.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/74.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/75.jpg)
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](https://reader035.vdocuments.net/reader035/viewer/2022071512/61327b8adfd10f4dd73a7a7a/html5/thumbnails/76.jpg)
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)