Differential Operators - 'LIIHUHQWLDO (TXDWLRQV 6HFRQG...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 'LIIHUHQWLDO (TXDWLRQV 6HFRQG 4XDUWHU 6<  /HVVRQ  +LJKHU±2UGHU )LUVW 'HJUHH 'LIIHUHQWLDO (TXDWLRQV /LQHDU 'LIIHUHQWLDO (TXDWLRQV ZLWK &RQVWDQW &RHIILFLHQWV 6SHFLILF 2EMHFWLYHV $W WKH HQG RI WKH OHVVRQ WKH VWXGHQWV DUH H[SHFWHG WR x GHILQH GLIIHUHQWLDO RSHUDWRU ' x GLIIHUHQWLDWH KRPRJHQHRXV DQG QRQKRPRJHQHRXV KLJKHU RUGHU ILUVW GHJUHH '( x SHUIRUP WKH RSHUDWLRQV RQ GLIIHUHQWLDO RSHUDWRUV 6WDQGDUG )RUP § GQ\ · § G Q \ · § G Q  \ · G\  DQ \ D ¨ Q ¸  D ¨ Q  ¸  D ¨ Q   ¸    DQ  G[ © G[ ¹ © G[ ¹ © G[ ¹ 5 [ ZKHUH D D D   DQ± DUH FRQVWDQWV DQG 5 [ LV D IXQFWLRQ RI [ 3UHOLPLQDU\ 7KHRU\  ,QLWLDO±9DOXH DQG %RXQGDU\ 9DOXH 3UREOHPV )RU D OLQHDU GLIIHUHQWLDO HTXDWLRQ DQ QWK±RUGHU LQLWLDO±YDOXH SUREOHP LV § GQ\ · § G Q \ · G\ 6ROYH D ¨ Q ¸  D ¨ Q  ¸    DQ   DQ \ G[ © G[ ¹ © G[ ¹ 6XEMHFW WR \ \   \ \  \ Q  \ Q  5 [ $QRWKHU W\SH RI SUREOHP LQ ZKLFK WKH GHSHQGHQW YDULDEOH \ RU LWV GHULYDWLYHV DUH VSHFLILHG DW GLIIHUHQW SRLQWV LV FDOOHG D ERXQGDU\±YDOXH SUREOHP G \ G\  D  D \ 5 [  G[ G[ 6XEMHFW WR \ D \   \ E \ 6ROYH D 7KH SUHVFULEHG YDOXHV \ D \  DQG \ E \ DUH FDOOHG ERXQGDU\ FRQGLWLRQV BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 6LOYD $OYLVR DQG /ODFXQD ± 0DWKHPDWLFV 'HSDUWPHQW 0,7  'LIIHUHQWLDO (TXDWLRQV 6HFRQG 4XDUWHU 6<   +RPRJHQHRXV (TXDWLRQV 6WDQGDUG )RUP § G Q\ · § G Q \ · § G Q  \ · G\ D ¨ Q ¸  D ¨ Q  ¸  D ¨ Q   ¸    DQ   DQ \ G[ © G[ ¹ © G[ ¹ © G[ ¹ ZKHUH D D D    DQ ±  DUH FRQVWDQWV DQG 5 [  ±±±    LV D KRPRJHQHRXV OLQHDU GLIIHUHQWLDO HTXDWLRQ VLQFH WKH GHJUHH RI WKH GHULYDWLYHV RI HDFK WHUP LV RQH  1RQ±+RPRJHQHRXV (TXDWLRQV 6WDQGDUG )RUP § GQ\ · § G Q \ · § G Q  \ · G\ D ¨ Q ¸  D ¨ Q  ¸  D ¨ Q   ¸    DQ   DQ \ G[ © G[ ¹ © G[ ¹ © G[ ¹ 5 [ ZKHUH D D D   DQ± DUH FRQVWDQWV DQG 5 [ LV D IXQFWLRQ RI [  'LIIHUHQWLDO 2SHUDWRUV '±2SHUDWRUV '±2SHUDWRU $Q RSHUDWRU LV D V\PERO LQGLFDWLQJ DQ RSHUDWLRQ WR EH SHUIRUPHG )RU '±RSHUDWRU LW PHDQV WDNLQJ WKH GHULYDWLYH RI D IXQFWLRQ ZLWK UHVSHFW WR [ 7KXV G\ '\ G[ G \ \ ' \  G[ G \ \ ' \ G[  GQ\ Q \ 'Q \ Q G[ \ BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 6LOYD $OYLVR DQG /ODFXQD ± 0DWKHPDWLFV 'HSDUWPHQW 0,7  'LIIHUHQWLDO (TXDWLRQV 6HFRQG 4XDUWHU 6<  $OJHEUDLF 3URSHUWLHV RI '±2SHUDWRU  &RPPXWDWLYH ZLWK UHVSHFW WR $GGLWLRQ0XOWLSOLFDWLRQ 'P  'Q \ 'Q  'P \ ' P 'Q \ ' ' \ Q P IRU GLIIHUHQWLDO RSHUDWRV ZLWK FRQVWDQW FRHIILFLHQWV  $VVRFLDWLYH ZLWK UHVSHFW WR $GGLWLRQ 0XOWLSOLFDWLRQ ª ' P  ' Q  ' S º \ ª' P  ' Q  ' S º \ ¬ ¼ ¬ ¼ ª ' P ' Q ' S º \ ¬ ¼ ª' P ' Q ' S º \ ¬ ¼  'LVWULEXWLYH ZLWK UHVSHFW WR $GGLWLRQ ª' P ' Q  ' S º \ 'P' Q  ' P' S \ ¬ ¼  /LQHDULW\ 4 ' > \  \     \ Q @ 4 ' \  4 ' \     4 ' \ Q ZKHUH 4 ' LV D SRO\QRPLDO LQ ' ([DPSOH 3HUIRUP WKH LQGLFDWHG RSHUDWLRQV  '   [  62/87,21 '   [  '   [ '   [   ' [  [ [    [  [     [  '   [H  [  62/87,21 '    [H  [ '    '   [H  [ %XW ' [H [  H  H  RU' [H  [H  H ' [H >[  H  H @   H RU' [H [H  H [H  H    [H  H  [H 6XEVWLWXWLQJ '   [H '    [H  [ [  [ '  [H  [   ' [H  [  [H  [ [ [ [  '    [H  [ [ [ [ [ [  [ [ [ [ [ [ [ [ [H  [  H  [   [H  [  H  [  [H  [ BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 6LOYD $OYLVR DQG /ODFXQD ± 0DWKHPDWLFV 'HSDUWPHQW 0,7  'LIIHUHQWLDO (TXDWLRQV 6HFRQG 4XDUWHU 6<  '    [H  [   '    VLQ  [ 62/87,21 '  '  VLQ  [   VLQ  [   VLQ  [ ' VLQ  [  FRV  [ '  VLQ  [    VLQ  [  VLQ  [ 6XEVWLWXWLQJ '    VLQ  [ '     VLQ  [   VLQ  [  VLQ  [  '  '  P H P[ 62/87,21 '  '  P H P[  P P H '  H P[  P'  H P[ '  '  P H P[ P  H P[ '  '  P H P[ P  H P[  P  H P[ '  '  P H P[  PH H P PH H P PH VLQFH ' H P[ ' ' P[ P[ P[  P[ P[  P[ ,Q JHQHUDO ' Q H P[ P  H P[ P  H P[ P Q H P[ 6HDWZRUN 3HUIRUP WKH LQGLFDWHG RSHUDWLRQV  ' [ DQV [  ' ± '   [   FRV[ DQV [ ± [    VLQ[   FRV [ +RPHZRUN 3HUIRUP WKH LQGLFDWHG RSHUDWLRQV  ' [ OQ[ DQV   OQ[  ' [  VLQ [ ± OQ[ DQV [   FRV [ ± [ BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 6LOYD $OYLVR DQG /ODFXQD ± 0DWKHPDWLFV 'HSDUWPHQW 0,7  'LIIHUHQWLDO (TXDWLRQV 6HFRQG 4XDUWHU 6<   ' ± '   [ FRV[ ±  DQV [VLQ[  [FRV[ ±VLQ[ ±FRV[    '  ' ±  H[VLQ[  H[ DQV H[ FRV[  VLQ[  H[ BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 6LOYD $OYLVR DQG /ODFXQD ± 0DWKHPDWLFV 'HSDUWPHQW 0,7  ...
View Full Document

This note was uploaded on 01/17/2012 for the course MATH 24 taught by Professor Dantesilva during the Spring '11 term at Mapúa Institute of Technology.

Ask a homework question - tutors are online