# Outline. Numerical Analysis Boundary Value Problems & PDE. Exam. Boundary Value Problems. Boundary Value Problems. Solution to BVProblems

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1 Oulie Numericl Alysis oudry Vlue Prolems & PDE Lecure 5 Jeff Prker oudry Vlue Prolems Sooig Meod Fiie Differece Meod ollocio Fiie Eleme Fll, Pril Differeil Equios Recp of ove Exm You will o e le o rig lpop oudry Vlue Prolems oudry Vlue Prolems come i my forms We will e lookig oe exmple muliple wys No Ml! Will e le o rig ook y y y() Noes y() lculor Soluio o VProlems We will e lookig for wy o evlue y() oudry Vlue Prolems y y y() over e iervl [, ] y y y() y() Tere re soluios of e form y() y c e c e 5 6

2 oudry Vlue Prolems oudry Vlue Prolems We c use e oudry vlues o elp y() c e c e c c y() c e c e c c y() c e c e c e c e y() c e c e c e c e Tis gives us sysem c c e e 7 oudry Vlue Prolems e e c c >> f(.5) wic we c solve o ge e exc swer e e e e e e e e e e 5 y() oudry Vlue Prolems s.6776 >> f(.5) s >> f(.75) s Sooig Meod Sooig Meod y y y y y() y() y() y()

3 Sooig Meod Sooig Meod y y Suer sows ow o wrie is i Ml Use IVP solver, suc s ode5() y() Wrie fucio F ke iiil codiios, compues vlue w edpoi y() F reurs w - y() Fid iiil codiios so F(p) > d F(q) < Feed F io roo fider isec(f, [p, q]) Sooig Meod Fiie Differece Meod Replce derives wi fiie differeces Replce differeil equio wi discree pproximios W does e Sooig Meod give you? Gives you se of iiil codiios Ly dow grid, d evlue sep y sep Turs VP io IVP Use IV o esime soluio sep y sep Fiie Differece Meod Fiie Differece Meod y y We pproxime y () s wi wi wi wi wi wi wi wi wi or We express e sic relio i discree form wi d y () y() s wi wi ( )wi wi

4 Fiie Differece Meod ( )wi wi wi Fiie Differece Meod If we ke we ge e followig ree equios Te firs d ls express e oudry codiios. ( )w w w ( )w w w ( )w Fiie Differece w w w yi wi ( )w w w ( )w w w ( )w Tis gives us sysem w A w Fiie Differece Meod A W does Fiie Differece Meod give? Esimes of y i You c ierpole ollocio ollocio Our ex emp srs wi sis We work roug exmple wi e sis i () j Te soluio will e ve e form y() c c c... c y() d se of pois j < < <... < j () j c j j

5 y y ollocio ollocio y() y() y() Plug is expressio io our differeil equio y() j () j c j j j () c j j j Plug i oudry vlues, y() d y() j () j () j j y j y() y() c j j c j j y c j j c c... c j 5 6 Plug i oudry Vlues Drop Summio Noio Le s ry is wi Le s ry is wi y() c j j c c j y() j () j y() c j j c Plug i oudry vlues, y() d y() j j () j j c c c c c j Soluio is c d c y 7 eyod e oudry Vlues Tis gives us wo equios We ge more evluig e di ereil equio e pois i. Trslig y y o y P j (j )(j ) j y we ge P j j ollocio Le s ry is ou wi d d. We would ve y() c c c c y () c 6c

6 ollocio ollocio y() c c c c c Evlue y () c 6c Rewriig y y we ge c (c c c c ) c 6c c c c ( ) c (6 ) c c c ( d c ( c ) (6 c c c ( 6 c ) (6 c c ( c 6 ) c c c 5 c c 7 c 76 c c c c c c A (c, c, c, c ) (.,.6,.7,.6) c c 76 c c 7 c 5 c c 7 76 c c 7 c 5 7 c 76 c 7 c c A ollocio Solve o ge y() 6 ) c c c c ) 6 c ) (6 5 c c 7 c c c c c c c ollocio c c ) (6 c omie our equios from e oudry vlues d ierior pois c ( ) d ollocio c ) o ge c ) c (6 y() We plo e ew vlues our old pois P i yi wi ci

7 ollocio Ulike prior soluios, we ve polyomil I geerl, ve lyic soluio y c e c e ke derivives We c improve our soluio y usig More pois, iger degree ollocio Ulike prior soluios, we ve polyomil I geerl, ve lyic soluio ke derivives We c improve our soluio y usig More pois, iger degree, Orogol sis 7 ollocio Digressio o sis fucios Ulike prior soluios, we ve polyomil I geerl, ve lyic soluio ke derivives We c improve our soluio y usig W do we look for i sis for fucios? Sps useful suse for e prolem domi Esy o del wi More pois, iger degree, Differeies d iegres esily Orogol sis, Esy o evlue eysev or Legedre pois Exmple: Music Exmple: Music

8 Exmple: Music Fiie Eleme Meod We pproxime y d y wi fucios u use fucios re mosly zero We sy ey ve ouded suppor Fiie Eleme Meod We pproxime y d y wi fucios u use fucios re mosly zero We piece ogeer soluio Fiie Eleme Meod Give our di ereil equio y () f (, y, y ) We express y usig sis y() 5 Fiie Eleme Meod y () f (, y, y ) For ec i (), ke e do produc wi y f (, y, y ) y () i ()d 7 f (, y, y ) i ()d P j i () j () 6 Fiie Eleme Meod We use iegrio y prs o elimie e secod derivives. y () i ()d i ()y () i ()y () i ()y () y () i ()d y () i ()d

9 Fiie Eleme Meod Fiie Eleme Meod We c solve ese for ci i e fuciol form We pu ese ogeer o ge se of equios for ec f (, y, y ) i ()d i ()y () i ()y () i () y () y() i ()d j () j y() j () c j y() j () c j Sice oly d oly () is o-zero () Fiie Eleme Meod is o-zero Fiie Eleme Meod y y for i... we use e equios f (, y, y ) i ()d y () i ()d Usig our expressio of y() s summio, we c rewrie is s f (, y, y ) i ()d i ()d j i () j j j () j i () j ()d o d i () i ()d ( i ()) d d d i () i ()d d ( i ()) d A j () i () d j () i ()d o 5 Ideiies we will eed j 5 Exmple 6 Sep roug for i. Fixed j c c c i () d vry () j ()d () ()d () ()d () ()d j () j () ()d () ()d () ()d () ()d

10 Exmple Work roug oe smll pr: c () ()d Fiie Eleme Meod Sep roug for i. Fixed () ()d j Usig ese ideiies i () i ()d To ge i () i ()d c d d c 6 c c were c c c A c c d vry () j ()d () ()d () ()d () ()d j () j () ()d () ()d () ()d () ()d c PDE We ur ow o Pril Differeil Equios D or D vs D Tere re my pplicios I geerl, we mig kow oudry vlues We will pply similr ides He Equio 5 Forwrd Differece Meod Eslis grid Use discree secod derivive uild soluio ou From kow o ukow Temperure log leg of r s e spreds 5 c c Gives sysem of equios i () 6

11 ckwrd Differece Meod Forwrd differece c e usle Replce explici equio wi Implici Equio ckwrd Differece Meod rk-nicolso Meod omiio of meods Euler d ckwrds Euler erl Differece d Trpezoid Sle 6 6 Fiie Eleme Meod Summry Fucios wi ouded suppor Use Glerki Meod 6 Review of opics we ve see is semeser Sysem of Lier Equios Suile for Ierive Soluios Piecewise lier pproximios Approximio wi sum fucios Orogol sis Les Squres 6

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