Welcome

Our website is made possible by displaying online advertisements to our visitors.
Please disable your ad blocker to continue.

Mathematical Engineering - Matematica Numerica

ODE

Divided by topic

6RPPDULR'HULYD]LRQH 'LIIHUHQ]DLQDYDQWL (UURUH 'LIIHUHQ]DDOO LQGLHWUR (UURUH 'LIIHUHQ]DFHQWUDOH (UURUH 0HWRGRJHQHUDOH 'LIIHUHQ]DFHQWUDOHGHOVHFRQGRRUGLQH (UURUH 2'( 3UREOHPDGL&DXFK\ 7HRUHPDGLHVLVWHQ]DHXQLFLWjORFDOH 7HRUHPDGLHVLVWHQ]DHXQLFLWjJOREDOH 6WDELOLWjGHO3G& 6WDELOLWjGL/LDSXQRY 0HWRGLRQHVWHS 2QHVWHS96PXOWLVWHS 0HWRGRLPSOLFLWR96PHWRGRHVSOLFLWR 0HWRGRGL(XOHURHVSOLFLWR 0HWRGRGL(XOHURLPSOLFLWR 0HWRGRGL&UDQN1LFROVRQ 0HWRGRGL+HXQ 0HWRGR $QDOLVLGHLPHWRGLRQHVWHS 0HWRGRFRQYHUJHQWH (UURUHORFDOHGLWURQFDPHQWR(7/ (UURUHJOREDOHGLWURQFDPHQWR*7/ &RQYHUJHQ]DGL(( &RQYHUJHQ]DGL(, &RQYHUJHQ]DGL&1H+HXQ $QDOLVLGHOPHWRGRGL(( )RUPDJHQHUDOHGLPHWRGLRQHVWHSHVSOLFLWL 5HVLGXR &RQVLVWHQ]DGLXQPHWRGRQXPHULFRSHUO DSSURVVLPD]LRQHGHO3G& =HURVWDELOLWj 'HILQL]LRQH &6SHUODVWDELOLWj]HUR 6WDELOLWjDVVROXWD 3UREOHPDWHVW 'HILQL]LRQHGLVWDELOLWjDVVROXWD 5HJLRQHGLVWDELOLWjDVVROXWD &DVRJHQHUDOH $VWDELOLWj 5LDVVXQWR 0HWRGLGL5XQJH.XWWD &RQVLVWHQ]D 0HWRGRSHUODGHWHUPLQD]LRQHGHLFRHIILFLHQWL 0HWRGRHVSOLFLWR 0HWRGRVHPLHVSOLFLWR (UURUHGLWURQFDPHQWR &RQVLVWHQ]D 1XPHURPLQLPRGLSDVVL 0HWRGLPXOWLVWHS )RUPXOD]LRQHHTXLYDOHQWH 0HWRGLGL$GDPV $GDPV%DVKIRUWKHVSOLFLWR $GDPV0RXOWRQLPSOLFLWR 0HWRGL%')LPSOLFLWR $QDOLVL &RQYHUJHQ]D (UURUHORFDOHGLWURQFDPHQWR &RQVLVWHQ]D =HURVWDELOLWj $VVROXWDVWDELOLWj 7KHWDVWDELOLWj 6LVWHPLGLRGHGHOƒRUGLQH $VVROXWDVWDELOLWj $6SHUVLVWHPL 2VVHUYD]LRQH 2'(GLRUGLQHVXSHULRUH 0HWRGRGL/HDSIURJ 'HULYD]LRQH 'LIIHUHQ]DLQDYDQWL (UURUH 6LD $FFXUDWRDOSULPRRUGLQH 'LIIHUHQ]DDOO LQGLHWUR (UURUH 6LD $FFXUDWRDOSULPRRUGLQH 'LIIHUHQ]DFHQWUDOH (UURUH 6LD $FFXUDWRDOVHFRQGRRUGLQH 0HWRGRJHQHUDOH6LGLYLGHO LQWHUYDOORFRQ QRGL GRYH H FRQ 'LIIHUHQ]DFHQWUDOHGHOVHFRQGRRUGLQH (UURUH 6LD 2'( 3UREOHPDGL&DXFK\ 6LD 6LGHILQLVFHVROX]LRQHGHO3G&XQDIXQ]LRQH\FKHVRGGLVIDLOVLVWHPD 6HIqFRQWLQXDULVSHWWRDWDOORUDHVLVWHODVROX]LRQHLQWHJUDOH 7HRUHPDGLHVLVWHQ]DHXQLFLWjORFDOH 6HIqORFDOPHQWH/LSVFKLW]LDQDLQ ULVSHWWRD\DOORUDLO3G&DPPHWWHXQDHXQDVRODVROX]LRQH QHOO LQWRUQRGL /DFRQGL]LRQHqVRGGLVIDWWDVHIKDGHULYDWDFRQWLQXDULVSHWWRD\ 7HRUHPDGLHVLVWHQ]DHXQLFLWjJOREDOH 6LD SHUXQGDWR76LD HIXQLIRUPHPHQWH/LSVFKLW]LDQDULVSHWWRD\WDOHFKH GRYH/qODFRVWDQWHGL/LSVFKLW]$OORUDLO3G&DPPHWWHXQDHXQDVRODVROX]LRQHLQ, 6WDELOLWjGHO3G& 3G&SHUWXUEDWR 6WDELOLWjGL/LDSXQRY 6LD,XQLQVLHPHOLPLWDWR,O3G&qVWDELOHQHOVHQVRGL/LDSXQRYVX,VHSHURJQLSHUWXUED]LRQH FKH VRGGLVID FRQ VXIILFLHQWHPHQWHSLFFRORGDJDUDQWLUHO HVLVWHQ]DGHOODVROX]LRQHGHO3G&SHUWXUEDWRDOORUD LQGLSHQGHQWHGD WDOHFKH 0HWRGLRQHVWHS /HFRQGL]LRQLSHUO HVLVWHQ]DHO XQLFLWjGHOODVROX]LRQHVRQRVRGGLVIDWWH 6LD HVLD O LQWHUYDOORGLLQWHJUD]LRQH6RQRSHUWDQWR LQRGLGL GLVFUHWL]]D]LRQHFRQ GDFXL ‡ ‡ 2QHVWHS96PXOWLVWHS 8QPHWRGRQXPHULFRSHUO DSSURVVLPD]LRQHGHO3G&qGHWWR ‡2QHVWHSVH GLSHQGHVRORGD ‡ 0XOWLVWHSDOWULPHQWL 0HWRGRLPSOLFLWR96PHWRGRHVSOLFLWR8QPHWRGRqGHWWR ‡(VSOLFLWRVH SXzHVVHUHFDOFRODWRGLUHWWDPHQWHDSDUWLUHGDYDORULSUHFHGHQWL ‡,PSOLFLWRVH GLSHQGHLPSOLFLWDPHQWHGDI 0HWRGRGL(XOHURHVSOLFLWR8VRGHOODGLIIHUHQ]DLQDYDQWL IXQFWLRQ>WKXK@ HXOHURBH[SIWBWBPD[\K WK WBKWBPD[ 1 OHQJWKWK XK \ IRULW 1 XKLW XKLWK IWKLWXKLWHQG HQG 0HWRGRGL(XOHURLPSOLFLWR'LIIHUHQ]DDOO LQGLHWUR IXQFWLRQ>WKXK@ HXOHURBLPSDEWBWBPD[\K WK WBKWBPD[ 1 OHQJWKWK XK \ IRULW 1 XKLW XKLWK EWKLWK DWKLW HQG HQG 0HWRGRGL&UDQN1LFROVRQ6RPPDGL(,H(( ‡ 0HWRGRLPSOLFLWR ‡ 3XzHVVHUHULFDYDWRXWLOL]]DQGRODIRUPXODGHLWUDSH]LSHUDSSURVVLPDUHODIRUPDLQWHJUDOHGHO3G& IXQFWLRQ>WKXKLWHUBSI@ &1IWBWBPD[\K WK WBKWBPD[ 1 OHQJWKWK XK ]HURV1 XK \ SDUDPHWULSHUOHLWHUD]LRQLGLSXQWRILVVR 1PD[ WROO H LWHUBSI ]HURV1 IRULW 1 SKL #XXKLWK IWKLWXKLWIWKLWXM HUU WROO Z XKLW ZKLOHHUU!WROO M1PD[ Z SKLZ HUU DEVZZM MZ Z HQG LWHUBSILW M XKLW ZHQG HQG 0HWRGRGL+HXQ8VRGL((SHUUHQGHUH&1HVSOLFLWR 0HWRGR 6RPPDSDU]LDOHGL((H(, FRQ ‡ ((XQLFRHVSOLFLWR ‡ (, ‡ &1 ‡3HU qLPSOLFLWR ‡3HU KDRUGLQH ‡3HU KDRUGLQH $QDOLVLGHLPHWRGLRQHVWHS 0HWRGRFRQYHUJHQWH 8QPHWRGRQXPHULFRSHUO DSSURVVLPD]LRQHGHO3G&qGHWWRFRQYHUJHQWHVH SHUTXDOFKH FRQ 3qGHWWRRUGLQHGHOPHWRGR ‡ËULVSHWWDWDODUHOD]LRQH (UURUHORFDOHGLWURQFDPHQWR(7/ (UURUHJOREDOHGLWURQFDPHQWR*7/ &RQYHUJHQ]DGL(( 6LD ODVROX]LRQHGHO3G&HVLD $OORUDODVROX]LRQHHVSOLFLWDGL((FRQYHUJHFRQRUGLQH LH &RQYHUJHQ]DGL(,6LD ODVROX]LRQHGHO3G&HVLD $OORUDODVROX]LRQHHVSOLFLWDGL(,FRQYHUJHFRQRUGLQH LH /HGXHFRVWDQWLGLSHQGRQRGD 3HU7JUDQGHSRVVRQRHVVHUHPROWRJURVVH &RQYHUJHQ]DGL&1H+HXQ (QWUDPELFRQYHUJRQRFRQRUGLQH $QDOLVLGHOPHWRGRGL(( )RUPDJHQHUDOHGLPHWRGLRQHVWHSHVSOLFLWL FRQ IXQ]LRQHLQFUHPHQWR /DVROX]LRQHHVDWWDSXzHVVHUHVFULWWDFRPH 5HVLGXR &RQVLVWHQ]DGLXQPHWRGRQXPHULFRSHUO DSSURVVLPD]LRQHGHO3G& =HURVWDELOLWj &RPSRUWDPHQWRULVSHWWRDOOHSHUWXUED]LRQL 'HILQL]LRQH &6SHUODVWDELOLWj]HUR 5HJRODDXUHD 6H q/LSVFKLW]LDQDULVSHWWRDOƒDUJRPHQWRTXLQGLLOPHWRGRq]HURVWDELOHDOORUD / HUURUHWHQGHD]HURFRPH 6WDELOLWjDVVROXWD 3UREOHPDWHVW ,OQRVWURFDVRG LQWHUHVVHqVROR SHUFXLODVROX]LRQHFRQYHUJH 'HILQL]LRQHGLVWDELOLWjDVVROXWD 8QPHWRGRQXPHULFRSHUO DSSURVVLPD]LRQHGHOSUREOHPDWHVWqDVVROXWDPHQWHVWDELOHVH SHU 5HJLRQHGLVWDELOLWjDVVROXWD 6LDKXQLQWHUYDOORGLGLVFUHWL]]D]LRQH/DVROX]LRQHQXPHULFD GHOSUREOHPDWHVWGLSHQGHGD HGDK6L GHILQLVFHUHJLRQHGLVWDELOLWjDVVROXWD &DVRJHQHUDOH 3HUSDVVDUHGDOSUREOHPDWHVWDO3G&VLWURYDQRGXHFRVWDQWL H HVLSRQH $VWDELOLWj 8QPHWRGRq$VWDELOHLQFRQGL]LRQDWDPHQWHDVVROXWDPHQWHVWDELOHVH FLRqVHSHU OD FRQGL]LRQLGLDVVROXWDVWDELOLWjqVRGGLVIDWWDSHUWXWWLLYDORULGLK ‡ 1RQHVLVWRQRPHWRGLHVSOLFLWL$VWDELOL 5LDVVXQWR 0HWRGLGL5XQJH.XWWD GRYH ,FRHIILFLHQWL FDUDWWHUL]]DQRLOPHWRGRHVLUDFFKLXGRQRQHOO DUUD\GL%XWFKHU &RQVLVWHQ]D,OPHWRGR5.qFRQVLVWHQWHVHHVRORVH 0HWRGRSHUODGHWHUPLQD]LRQHGHLFRHIILFLHQWL 0HWRGRHVSOLFLWR ,OPHWRGRqHVSOLFLWRVH 0HWRGRVHPLHVSOLFLWR ,OPHWRGRqVHPLHVSOLFLWRVH (UURUHGLWURQFDPHQWR &RQVLVWHQ]D ,OPHWRGRqFRQVLVWHQWHGLRUGLQHSVH VH FRQ 1%RUGLQHGLFRQVLVWHQ]D RUGLQHGLFRQYHUJHQ]D 1XPHURPLQLPRGLSDVVL 0HWRGLPXOWLVWHS6LXWLOL]]DQROHLQIRUPD]LRQLGHOO XOWLPRSDVVRHGHLSSUHFHGHQWL VWHSV ‡ 6H ORVFKHPDqHVSOLFLWR ‡6H ORVFKHPDqLPSOLFLWR ‡ GHYRQRHVVHUHQRWL )RUPXOD]LRQHHTXLYDOHQWH ‡ ‡ SHU ‡ SHU 0HWRGLGL$GDPV 6LSDUWHGHOODIRUPDLQWHJUDOH 6LDSSURVVLPD FRQLOVXRSROLQRPLRLQWHUSRODQWHVXRSSRUWXQL QRGLGLVWLQWLFRQ HVLLQWHJUD LOSROLQRPLRLQWHUSRODQWH $GDPV%DVKIRUWKHVSOLFLWR 1RGLGLLQWHUSROD]LRQH FRQ ‡ (XOHURHVSOLFLWR ‡ $%DSDVVL ‡ $%DSDVVL $GDPV0RXOWRQLPSOLFLWR 1RGLGLLQWHUSROD]LRQH FRQ ‡ (XOHURLPSOLFLWR ‡ &1 ‡ $0DSDVVL ‡ $0DSDVVL 0HWRGL%')LPSOLFLWR 1RGLGLLQWHUSROD]LRQH FRQ 6LSDUWHGDOODIRUPDGLIIHUHQ]LDOH HVLDSSURVVLPDLOYDORUHGHOODGHULYDWDSULPD LQ WUDPLWHODGHULYDWDSULPDGHOSROLQRPLRLQWHUSRODQWHGL LQRSSRUWXQL QRGL $QDOLVL &RQYHUJHQ]D 8QPHWRGRPXOWLVWHSqFRQYHUJHQWHVH 5HJRODDXUHD 7HRUHPL 8QPHWRGRPXOWLVWHSFRQVLVWHQWHqFRQYHUJHQWHVHHVRORVHq]HURVWDELOHHO·HUURUHVXLGDWLLQL]LDOLqLQILQLWHVLPRSHU ,QROWUHLOPHWRGRFRQYHUJHFRQRUGLQHTVHVLDlKVLDO·HUURUHVXLGDWLLQL]LDOL VRQRLQILQLWHVLPLGLRUGLQHTULVSHWWRDGKSHUK 1RQHVLVWRQRPHWRGLPXOWLVWHSOLQHDUL]HURVWDELOLDTSDVVLFRQRUGLQHPDJJLRUHGL VHTGLVSDULH VHTSDUL (UURUHORFDOHGLWURQFDPHQWR &RQVLVWHQ]D ,OPHWRGRqFRQVLVWHQWHVH SHU ,QROWUHVH SHUTXDOFKH DOORUDLOPHWRGRVLGLUjGLRUGLQHT 7HRUHPLGLFRQYHUJHQ]D,OPHWRGRqFRQVLVWHQWHVHHVRORVH H 6HODVROX]LRQH LOPHWRGRqGLRUGLQHTVHHVRORYDOH ‡ LOWHRUHPD ‡ =HURVWDELOLWj 8QPHWRGRPXOWLVWHSq]HURVWDELOHVHHVRORVHqVRGGLVIDWWDODFRQGL]LRQHGHOOHUDGLFL &RQGL]LRQHGHOOHUDGLFL 6LDLOSROLQRPLR GHILQLWRFRPH /DFRQGL]LRQHqVRGGLVIDWWDVH‡ ‡ SHUWXWWLLMWDOHFKH ,QSUDWLFDODFRQGL]LRQHqVRGGLVIDWWDVHWXWWHOHUDGLFLDSSDUWHQJRQRDOFHUFKLRXQLWDULRHTXHOOHVXOERUGRVRQR VHPSOLFL ‡,PHWRGLGL$GDPVVRQR]HURVWDELOLSHUFKp HTXLQGL H FRQPROWHSOLFLWjS ‡,PHWRGL%')VRQR]HURVWDELOLILQRD DOPHQR $VVROXWDVWDELOLWj 6LDLOSROLQRPLR FRQ ,OPHWRGRqDVVROXWDPHQWHVWDELOHVHH VRORVHODFRQGL]LRQHDVVROXWDGHOOHUDGLFLqVRGGLVIDWWD &RQGL]LRQHDVVROXWDGHOOHUDGLFL /DFRQGL]LRQHqVRGGLVIDWWDVHHVLVWH WDOHFKHWXWWHOHUDGLFL GHOSROLQRPLR VRGGLVILQRODUHOD]LRQH 7HRUHPD 8QPHWRGRPXOWLVWHSOLQHDUHHVSOLFLWRQRQSXzHVVHUH$VWDELOH,QROWUHQRQHVLVWRQRPHWRGLPXOWLVWHSOLQHDUL $VWDELOLGLRUGLQHVXSHULRUHD 7KHWDVWDELOLWj /DFRQYHUJHQ]DD]HURGHOOHVROX]LRQLGLVFUHWHQRQDYYLHQHVXWXWWRLOVHPLSLDQRDSDUWHUHDOHQHJDWLYDPDVROR VXXQVHWWRUHGLDPSLH]]DaFRQWHQHQWHLOVHPLDVVHUHDOHQHJDWLYR 6LVWHPLGLRGHGHOƒRUGLQH 7HRUHPD 6LD FRQ7ILQLWR6LVXSSRQJD FRQWLQXDVX HFKH WF $OORUD VROX]LRQH GHOVLVWHPDFRQWLQXDHGLIIHUHQ]LDELOHLQ 7HRUHPD 7XWWLLPHWRGLYLVWLSHUOH2'(SRVVRQRHVVHUHHVWHVLDOFDVRGHLVLVWHPL $VVROXWDVWDELOLWj3UREOHPDWHVW 6LDQR JOLDXWRYDORULGL$GLVWLQWL$OORUDODVROX]LRQHGHO3Wq GRYH VRQRFRVWDQWLH VRQRJOLDXWRYHWWRUL /DVROX]LRQHFRQYHUJHVH $6SHUVLVWHPL8QPHWRGRQXPHULFRq$6VHDSSOLFDWRDO3WqWDOHFKH 2VVHUYD]LRQH 6LD4ODPDWULFHQRQVLQJRODUHSHUODTXDOH FRQ HVLDLOFDPELRGLYDULDELOL ,O3WqULVFULWWRFRPH ,ULVXOWDWLYLVWLSUHFHGHQWHPHQWHVLHVWHQGRQRDOFDVRYHWWRULDOHDQDOL]]DQGRFLDVFXQDGHOOHQHTXD]LRQL 2'(GLRUGLQHVXSHULRUH8Q RGHGLRUGLQHNSXzHVVHUHULVFULWWDFRPHXQVLVWHPDGLNHTXD]LRQLGHOƒRUGLQHLQWURGXFHQGR YDULDELOL DXVLOLDULH 0HWRGRGL/HDSIURJ6LXWLOL]]DSHUDSSURVVLPD]LRQHGLUHWWDSHU2'(GHOƒRUGLQH ËDFFXUDWDDOƒRUGLQHHVSOLFLWRFRQGL]DVVROXWDPHQWHVWDELOH