More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 More oundr-vlue Prolems nd genvlue Prolems n Os Lrr retto Menl ngneerng 5 Semnr n ngneerng nlss ovemer 9, 7 Outlne Revew oundr-vlue prolems Soot nd tr nte fferenes Toms lgortm Oter oundr vlues Grdent oundr vlues Med oundr vlues genvlue prolems Soure of su prolems Soluton metods Revew Lst Leture In oundr-vlue prolem, we ve ondtons set t two dfferent lotons seond-order O d /d = g(,, ), needs two oundr ondtons Smplest re () = nd (L) = n lso ve d/d+ = t =, L Two soluton pproes: Soot-nd-tr nte dfferenes Soot-nd-Tr ppro Tke n ntl guess of dervtve oundr ondtons t = nd use n ntl-vlue routne to get (omp) (L) t te oter oundr ompre te vlue of (omp) (L) found from te prevous step to te oundr ondton on (L) Use te dfferene etween (omp) (L) nd (L) to terte te ntl vlue of z = d/d = nd ontnue untl (omp) (L) (L) 4 Soot-nd-Tr mple Solve d /d +6sn( ) = wt = t = nd = L = Must fnd pr of frst order equtons Set d/d = z s one O Orgnl O eomes dz/d = 6sn( ) We know () =, ut we need z() guess z () () = [(L) ()]/L = ( )/ = Ts z () () gves () =.684 (RK4, =.) Tr z () () = [(L) () (L) ()]/L = [ (.684) ] =.76 () = () (L) (L) =.684 = -.684 5 6 M 5 Semnr n ngneerng nlss Pge
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 Soot-nd-Tr Lner Os or lner O te soluton n e found on te trd terton omplete two soot-nd-tr solutons, () () nd () () nd for two ntl guesses, z () ( ) nd z () ( ) nte fferene ppro efne unform or non-unform grd; unform s eser nd s ger order trunton error; note: = ( )/ ----- -------- ------------- ~ ~ ------- --- - - t e node wrte fnte-dfferene equvlent to dfferentl equton Hndle oundr ondtons t nd (smplest f = () nd = (L) gven, ut n ve grdent oundres) 7 8 nte-fferene mple Solve d T/d + T = nte dfferene equton t node [d T/d + T] = (T + + T - T )/ + T + O( ) = Ignore trunton error Ignore trunton error nd get fntedfferene equton sstem T + + T - T + T = Hve + nodes numered from to wt oundr ondtons t nd 9 Trdgonl Mtr qutons nte-dfferene equtons n mtr form wt = ve trdgonl form solved Toms lgortm used wt u splne T T T T T T T rror nd rror Order Get overll mesure of error (lke norm of vetor) Tpll use mmum error (n solute vlue) or root-men-squred (RMS) error = s m =.4 - nd RMS =.8 -. or =, m =.4-5 nd RMS =.7-5. Seond-order error n soluton RMS ( T et T ) numerl oundr Grdents Use seond-order dervtve epressons dt T 4T T q k k d dt T 4T T q k k d -q et /k -q/k rror.995..57.68.995..999.6 -.95. -.9.786 -.95. -.955. M 5 Semnr n ngneerng nlss Pge
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 M 5 Semnr n ngneerng nlss Pge Toms lgortm Generl formt for trdgonl equtons 4 Toms lgortm II Guss elmnton upper trngulr form Hve to fnd nd 5 Toms lgortm III orwrd omputtons Intl: = / = / ppl equtons elow for =, -: t fnl pont k susttute: = + + 6 Oter oundr ondtons Generl ondton dt/d + T = =, = for eumnn (grdent gven) =, = for rlet (vlue gven) Wrte grdent usng seond order forwrd ( = ) or kwrd dfferene ( = ) omne wt equton for frst node n from te oundr to elmnte term wt seond node from oundr Result onforms to trdgonl sstem 7 Generl oundr mple Look t = oundr; results for = follow smlr dervton 4 d d 4 ) ( dd tese two equtons elmntng ) ( 8 Generl oundr mple II quton just derved s seen to gve orret rlet result for =, = ) ( Smlr dervton t = gves ) ( qutons sown ere wll work for = or =, ut t lest one must e nonzero
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 onlner Prolems Soot-nd-tr requres no spel proedures for nonlner prolems or fnte dfferene or fnte elements, solve lnerzed equton mple s pendulum equton d /dt = (-g/l) sn (usull solved wt sn ) Tlor seres: sn = sn + [d(sn )/d] ( ) = sn + os ( ) Reple sn lner result to terte 9 onlner mple Strt wt d /dt = (-g/l) sn Reple sn lnerzed seres Wrte n tertve form wt (m+) s new terton nd use (m) n nonlner terms d (m+) /dt = (-g/l) [sn (m) + os (m) ( (m+) (m) ) efne = g/l nd rerrnge d (m+) /dt + (m+) os (m) = [sn (m) (m) os (m) ] = r onlner mple II onvert d (m+) /dt + (m+) os (m) = r to (lner) fnte-dfferene form n (m+) ( m) ( m) r ( m) ( m) os Hve trdgonl sstem ( m) ( m) sn ( m) r ( m) ( m) r ( m) onlner mple III Mke ntl guesses for () Lner profle () (t) = () + [(L) ()]t/t nd ll nodl vlues for () usng () to ompute te nonlner terms Repet te proess untl te dfferenes etween tertons s good enoug ompute resduls to test onvergene R ( m) ( m) ( m) r ( m) Hndlng oundr ondtons We ve used n emple prolem wt fed (rlet) oundr ondtons We lso mentoned grdent (eumnn) nd med oundr ondtons. If te dependent vrle s u nd te ntl nd fnl nodes n fntedfferene grd re nd te fed oundr ondtons re u = t = nd u = t = Grdent nd Med oundres We use dretonl dervtve to represent oundr grdents 4 4 ouple ts oundr ondton equton wt frst (lst) possle fnte dfferene equton t = nd = 4 M 5 Semnr n ngneerng nlss Pge 4
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 Grdent oundr t = nte-dfferene equton for d /d = s (u + u + u - )/ + u = Wrte ts s u + + u + u - = were = rst ( = ) equton s u + u + u = Grdent t = : g = -u + 4u u omne tese to get g = -u + 4u u = -u + 4u + (u + u ) = (4 + )u u or TM soluton omned equton [-u + (4 + )u = g ] s frst; prevous frst equton,u + u + u =, s seond5 Grdent oundr t = Strt wt sme fnte-dfferene equton u + + u + u - = were = Lst ( = ) equton s u + u - + u - = Grdent t = : g = u 4u - + u - omne tese to get g = u 4u - + u - = u 4u - (u - + u ) = (4 + )u - + u or TM soluton omned equton [u (4 + )u - = g ] s lst; prevous lst equton,u - +u - + u =, s seond-to-lst 6 Med oundr ondton Med oundr ondton reltes oundr grdent to oundr vlue of dependent vrle ommon emple s onveton et trnsfer oundr ondton: t = : k(dt/d) = (T T) t = : k(dt/d) = (T T ) Use forwrd/kwrd epresson for grdents 7 Med oundr ondton II or = : k(dt/d) = (T T) wt forwrd dfferene eomes or = : k(dt/d) = (T T ) wt kwrd dfferene eomes omne wt fnte dfferene equtons: T + T + T = / T - + T - + T = 8 Med oundr t = nte-dfferene: T + T + T = oundr: 4 4 4 Ts eomes te frst equton n te TM lgortm nd te frst fnte-dfferene equton ove eomes te seond 9 Med oundr t = nte-dfferene: T - + T - + T = oundr: 4 4 4 Ts eomes te lst equton n te TM lgortm nd te frst fnte-dfferene equton ove eomes te seond-to-lst M 5 Semnr n ngneerng nlss Pge 5
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 M 5 Semnr n ngneerng nlss Pge 6 genvlue Prolems umerl egenvlue prolems our wen te numer of oundr ondtons s greter tn te order of te dfferentl equton mple of ts s soluton for urnng velot of lmnr flme s ppro s to use fntedfferenes nd trnsform prolem nto numerl mtr egenvlue prolem genvlue Prolems II Look t smple prolem wt known soluton s n emple d /d + = wt () =, () = nd d = Hve tree oundr ondtons nd onl seond order equton ontrvl soluton: = sn wt = n Use seond order fnte dfferenes ( + + - )/ + = genvlue Prolem III nte-dfferene equtons n mtr form wt = ; wt s soluton? 4 genvlue Prolem IV Hve mtr egenvlue prolem wt = - s te egenvlue 5 genvlue Prolems V Solve numerl tenques for fndng mtr egenvlues Te ur of te egenvlues depends on te grd Often need onl one (lowest or gest) n onl fnd s mn egenvlues s tere re grd nodes (not ountng oundr nodes) 6 mple genvlue Prolem O: d /d + k = wt ()=()= k s n unknown prmeter (egenvlue) Soluton s = sn(k) were k = n Solve wt =. nte dfferene equton s Δ Wrte s mtr equton for =.
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 mple genvlue Prolem II.4k.4k.4k.4k Ts s egenvlue equton ( I) = Here =.4k et( I) = s (.4k ) 4 (.4k ) + = umerl solutons for k ompred to et vlues on net slde 7 mple genvlue Prolem III genvlues Perent umerl t error.9.4.66% 5.878 6.8 6.45% 8.9 9.45 4.6% 9.5.566 4.% ote lrger errors for ger egenvlues Joe. Hoffmn, umerl metods for ngneers nd Sentsts, ( nd ed), Mrel ekker (), p. 48. 8 MTL O genvlues MTL s two solvers vp4 nd vp5 for solvng oundr-vlue Os MTL doumentton sows te use of vp4 for omputng te egenvlue of n O ttps://www.mtworks.om/elp/mtl/re f/vp4.tml mple sows te omputton of sngle egenvlue s unknown prmeter n te soluton sed on ntl guess of egenvlue 9 M 5 Semnr n ngneerng nlss Pge 7