Chapter 7 INTEGRAL EQUATIONS

Transcription

1 hpr 7 INTERAL EQUATIONS

2 hpr 7 INTERAL EQUATIONS hpr 7 Igrl Eqios 7. Normd Vcor Spcs. Eclidi vcor spc. Vcor spc o coios cios ( ) 3. Vcor Spc L ( ) 4. chy-byowsi iqliy 5. iowsi iqliy 7. Lir Oprors - coios oprors - odd oprors - Lipschiz codiio - corcio opror - sccssiv pproimios - Bch id poi horm 7.3 Igrl Opror 7.4 Igrl qios - Frdholm igrl qios - Volrr igrl qios - igro-diril qios - solio o igrl qio 7.5 Solio hods or Igrl Eqios. hod o sccssiv pproimios or Frdholm IE (Nm sris). hod o sccssiv ssiios or Frdholm IE (Rsolv mhod) 3. hod o sccssiv pproimios or Volrr IE 7.6 ocio w igrl qios d iiil d odry vl prolms 7.7 Erciss. Rdcio o IVP o h Volrr IE. Rdcio o h Volrr IE o IVP 3. Rdcio o BVP o h Frdholm IE Fr Topics: 7.7 Fid poi horm (s [Hochsd Igrl qios, p.5]) (ddd i 7.) Elmry isc horms 7.8 Prcicl pplicios (s [Jrri Irodcio o Igrl Eqios wih Applicios ]) 7.9 Ivrs prolms (s [ Jrri, p.7]) 7. Frdholm s lrivs

3 hpr 7 INTERAL EQUATIONS 7. Normd Vcor Spcs W will sr wih som diiios d rsls rom h hory o ormd vcor spcs which will dd i his chpr (s mor dils i hpr ).. Eclidi vcor spc Th -dimsiol Eclidi vcor spc cosiss o ll pois { (,,..., ) } or which h ollowig oprios r did: Sclr prodc (,y) y y... y,y Norm (,)... Disc ρ (,y) y ovrgc lim i lim is compl vcor spc (Bch spc) rliv o did orm.. Vcor spc ( ) Vcor spc ( ) cosiss o ll rl vld coios cios did o h closd domi : : D coios { } Norm m ovrgc lim i lim is compl vcor spc (Bch spc) rliv o did orm. 3. Vcor spc L ( ) Th spc o cios igrl ccordig o Lsg (s Scio 3.) Ir prodc Norm L : d<,g g d, d Th ollowig propry ollows rom h diiio o h Lsg igrl d d L is compl ormd vcors spcs (Bch spcs) rliv o. 4. chy-byovsy-schwrz Iqliy (s lso Thorm., p.57) (,g) g or ll,g L Proo:,g L, h cios, g d y comiio α β g r I lso igrl d hror log o L. osidr λ g L, λ R or which w hv

4 hpr 7 INTERAL EQUATIONS ( λ g ) d d λ g d λ g d Th righ hd sid is qdric cio o λ. Bcs his cio is ogiv, is discrim ( D 4c) is o-posiiv 4 gd 4 d g d g d d g d d cs (, g) gd g d (,g) g g d, rom which h climd iqliy yilds (,g) g 5. iowsi Iqliy (3 rd propry o h orm Trigl Iqliy ), (s Empl.7 o p.57) g g or ll,g L Proo: osidr g ( g, g) (, ) (, g) ( g, ) ( g, g) (, g) ( g, ) g g g rom -B iqliy ( g ) Th rcio o h sqr roo yilds h climd rsl. No h h iowsi iqliy rdcs o qliy oly i cios d g r ql p o h sclr mlipl, αg, α R (why?).

5 hpr 7 INTERAL EQUATIONS 7. Lir Oprors L d N wo compl ormd vcors spcs (Bch spcs, s h.) wih orms d, corrspodigly. W di opror L s N mp (cio) rom h vcor spc o h vcor spc N : L : N Irodc h ollowig diiios cocrig h oprors i h vcor spcs: Opror L : N is lir i L( α βg) αl βlg or ll, g d ll α, β R Opror L : N is coios i rom i ollows L L i N (h img o h covrg sqc i is covrg sqc i N ) Opror L : N is odd i hr iss c > sch h L c or ll N Th orm o opror o sch cos c L : N c did s h grs lowr od L L sp N Thorm 7. I lir opror L : N is odd h i is coios Proo: L opror L : N odd, h ccordig o h diiio hr iss c > sch h L c. L N i. Th ms h lim. From h diiio o h limi i ollows h or y ε > hr iss N sch h < ε or ll ε. To prov h horm, show ow h lim L L N ε L L i N or h. W hv o show h or y Ε > hr iss K N sch h L L < Ε or ll K N Ε. Ε hoos ε, h c Ε L L L( ) c < c Ε or ll N N Ε c KΕ. c Rmr: I is lso r h i lir opror is coios h i is odd (prov s rcis). Thror, or lir oprors, propris coios d odd r qivl. Ε

6 hpr 7 INTERAL EQUATIONS Diiio Lir opror L: N sisis h Lipschiz codiio wih cos i L Lg g or ll,g Oviosly h i lir opror sisis h Lipschiz codiio (i is clld Lipschiz opror) h i is odd ( vcor g ) d, hror, i is coios. Diiio Lir opror L: N is corcio i i sisis h Lipschiz codiio wih cos <. disc w imgs coms smllr L S closd ss o Bch spc, S, d l L:S S opror. Diiio Solio o opror qio L is clld id poi o opror L. Diiio Sccssiv pproimios is sqc {,,,... } cosrcd i h ollowig wy: S is srig poi L L L Schmic vislizio o sccssiv pproimios:

7 hpr 7 INTERAL EQUATIONS Sccssiv pproimios c sd or solio o opror qio L For mpl, i his cs, h sccssiv pproimios covrg o h id poi: B hy do o lwys covrg o h id poi o opror qio. This mpl shows h v h choic o h srig poi clos o h id poi yilds h divrg sqc o sccssiv pproimios (pprly hy r o vry sccssiv ): Th ollowig horm slishs h sici codiio or covrgc o sccssiv pproimios o h id poi o opror qio.

8 hpr 7 INTERAL EQUATIONS Thorm (Bch Fid Poi Thorm, 9) Proo: L S o-mpy closd ss o Bch spc, S, S. Ad l L:S S corcio opror wih cos <. Th h sqc o sccssiv pproimios L, S covrgs o h iq id poi { } S, L or y srig poi S d h ollowig sim is vlid Usig mhmicl idcio, show h ( ) Vriy or Assm or : Show or : r Idd, Show h { } osidr L L diiio o s.. Lipschiz codiio ssmpio is chy sqc, i.. lim m,m m m Lm Lm L L (dd d src) Apply iowsi iqliy wic: m m Lm Lm L L m Lm m L Lipschiz codiio m Lm L m m m diiio o s.. qio ( ) m wh m,

9 hpr 7 INTERAL EQUATIONS Bcs vcor spc is compl, chy sqc { } covrgs o som pois), S. Ad cs S d s S is closd (iclds ll limiig. Thror i limi, qio o sccssiv pproimios L lim lim L lim covrgs o L Ad hror, S is id poi. L lim (Uiqss) L,g S wo id pois o opror L : L g Lg Th rom g L Lg g yilds ( ) g Bcs > g Th is possil oly i g Thror, g So h id poi is iq. Hgo Sihs, h collg d rid o S Bch, ormld h id poi horm i h ollowig wy: hdghog co comd

10 hpr 7 INTERAL EQUATIONS 7.3 Igrl Opror osidr opror clld igrl opror giv y h qio K K (, y) ( y) dy R Oviosly, h igrl opror is lir. Fcio (, y) rls K(, y) L ( ), hror K(, y) ddy < K is clld rl o h igrl opror. W will cosidr I cs o R, h domi (,), whr, c ii or iii. Thorm 7. L K h igrl opror wih rl K (, y) coios i [,] [,]. Th opror K is odd, d, hror, coios. orovr: ) K : L (,) [,] K or L (,) ) K : L (,) L (,) K ( ) or L (,) 3) K : [,] [,] K ( ) or [,] Proo: Sic K(, y) is coios i h closd domi [,] [,] > sch h m K(, y).,y [, ] ) L L (,). Th cs cio (, y) [,] [,], h cio ( K ) is coios i [,] K : L (,) [,]. osidr, hr iss K is coios i, d, hror K diiio o orm i [, ] m K [,] m K, y y dy [,] diiio o igrl opror ( ) ir prodc i L (, ) m K, y, y [,] m K chy-byowsi iqliy [,] m K (, y) dy [, ] diiio o orm i L (, ) m dy [, ] rplc y m K,y [, ] (, y) clclig dii igrl

11 hpr 7 INTERAL EQUATIONS ) K (( K ),( K ) ) diiio o orm i L (, ) ( K ) d ir prodc i L (, ) K(, y) ( y) dy d diiio o igrl opror K d chy-byowsi iqliy K(, y) dy d corig dy d rplc y m K,y [, ] (, y) dy d clclig dii igrl ( ) 3) K diiio o orm i [, ] m K [,] m K, y y dy [,] diiio o igrl opror m K, y y dy [ ], m y dy [ ] rplc y, m K,y [, ] (, y) y dy dos o dpd o m y dy y [,] dy diiio o orm i [, ] dy ( ) clclig dii igrl

12 hpr 7 INTERAL EQUATIONS 7.4 Igrl Eqios Igrl qios r qios i which h ow cio is dr h, igrl sig. Th ypicl igrl qios or ow cio (i his chpr, w cosidr (,) i h orm o igrl opror wih h rl K (, y) K K,y y dy ) icld igrl rm Th mi yps o igrl qios r h ollowig: I Frdholm igrl qio ) Frdholm s igrl qio o h s id: K,y y dy K o-homogos q K,y y dy K homogos q ) Frdholm s igrl qio o h d id: λ is prmr λ K,y y dy λ K o-homogos q λ K,y y dy λk homogos q II Volrr igrl qio ) Volrr s igrl qio o h s id: K (, y) ( y) dy ) Volrr s igrl qio o h d id: λ K(, y) ( y) dy No h Volrr s qios c viwd s spcil cs o Frdholm s qios wih K (, y) or < < y < (i is clld Volrr rl). y

13 hpr 7 INTERAL EQUATIONS III Igro-Diril Eqio iclds ow cio dr h igrl sig d lso y driviv o h ow cio. For mpl: d K(, y) ( y) dy d A impor rprsio o h igro-diril qio is Rdiiv Trsr Eqio dscriig rgy rspor i h sorig, miig d scrig mdi (logos qios ppr i h hory o ro rspor). Solio o igrl qio is y cio sisyig his qio: λ K o-homogos qio λk homogos qio Th vl o h prmr λ or which h homogos igrl qio hs o-rivil solio L which is clld igvl o h rl K (, y), d h corrspodig solio is clld igcio o his rl. Eigvl prolm W will disigish igvl prolms or h igrl rl (igrl qio): λk d or h igrl opror K λ Th igvls o h igrl opror r rcipicl o igvls o h igrl rl, d igcios r h sm i oh css. Eisc o h solio o Frdholm s igrl qio osidr Frdholm s igrl qio o h d id: λ K ( ) wih odd igrl opror K which lso sisis h Lipschiz codiio: K K, Rwri igrl qio i h orm T ( ) whr opror T is did y T λk No h opror T is o lir. Oviosly, i is id poi o opror qio ( ), h is solio o igrl qio ( ). osidr T T λk λk λk λk ( ) λ K λ I λ <, h opror T is corcio d ccordig o Bch id poi horm, hr is iq id poi o qio ( ). This iq id poi is lso solio o Frdholm s qio ( ). Thror, h ollowig coclsio c md: Frdholm s igrl qio o h d id wih odd rl hs iq solio or sicily smll λ, i c λ <.

14 hpr 7 INTERAL EQUATIONS 7.5 Solio hods or Igrl Eqios. Th hod o Sccssiv Approimios or Frdholm s Igrl Eqio For h igrl qio λ K h ollowig irios o h mhod o sccssiv pproimios r s y: K λ,,... Lmm 7. λ K whr K K( K( K )) ims Proo y mhmicl idcio (ssm h h orml or is r): K λ coirmd K λ y diiio K λ λ K y ssmpio λ K liriy p p λ K chg o id p p λ K p λ K p p p p λ K p λ K chg o id p Nm Sris K λ is clld o h Nm Sris Esimio o irios K K K ( ) K Thorm 7. (3) ( ) K ( )

15 hpr 7 INTERAL EQUATIONS λ K λ K chy-byovsy iqliy ( ) λ Thorm 7. 3) [ ( ) ] λ gomric sris λ ( ) covrgs i λ < ( ) Thror, h Nm sris λ K covrgs or Do h sm o h Nm sris s cio : λ <. ( ) λ K Show h his cio sisis h igrl qio irios λ K h lim λ K lim (, y) lim ( y) dy λ K λ K (, y) ( y) dy Ad, rcllig simio, λ ( ) λ K. osidr show h his solio is iq. For h, i is ogh o show h h homogos qio λk hs oly rivil solio. Idd, i λ, d, ccordig o Thorm 6. 3),, h [ ] λ ( ) ( ) K [ λ ], h [ ] > Bcs λ <, ( ) ( ) yilds, h or ll [,] homogos qio. λ d, hror,. Th. So, oly h rivil solio iss or h Th o-homogos qio λk c rwri i h orm ( I λ K ) whr I is idiy opror Th solio o his qio c rd s ivrsio o h opror I λ K Thror, i λ <, h hr iss ivrs opror ( I K ) ( ) λ. Th ov miod rsls c ormld i h ollowig horm:

16 hpr 7 INTERAL EQUATIONS Thorm 7.3 Frdholm s igrl qio λ K wih λ < d coios rl (, y) ( ) iq solio [,] or y [,]. This solio is giv y covrg Nm sris d sisis I λ < ( I K ) λ. ( ) λ K λ ( ). K hs, h hr iss ivrs opror odiios o Thorm 7.3 r oly js sici codiios; i hs codiios r o sisid, solio o h igrl qio sill c iss d h Nm sris c covrg. Empl 7. Fid h solio o h igrl qio ( y) dy y h mhod o sccssiv pproimios d i h orm o h Nm sris. Idiy: K (, y) λ hc codiio: λ < < < ( ) ) irios: dy ( y) ( y) dy [ ] dy dy dy ( y) Th solio o h igrl qio is limi o irios lim lim This rsl c vlidd y dirc ssiio io h giv igrl qio.

17 hpr 7 INTERAL EQUATIONS ) Nm sris: λ K λ K λ K K dy K ( ) dy K Th h Nm sris is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) So, h Nm sris pproch prodcs h sm solio.

18 hpr 7 INTERAL EQUATIONS. Th hod o Sccssiv Ssiios or Frdholm s Igrl Eqio (h Rsolv hod) Ird rl L igrl opror K hs coios rl K (, y) Rpd opror K K( K ) ( K )K, h di:,3,... I hs rl K (, y) K(, y ) K ( y, y) dy Idd, ( K )( ) K,y K (,y ) ( K ) [ K( K )] y dy K (, y ) K( y, y) ( y) dy dy K (,y ) K ( y,y) dy ( y) dy K (,y) Krl K (, y) K(, y ) K ( y, y) dy K is clld ird rl. Krls (, y) (,), h K (, y) ( ) (, y ) K( y, y) dy K r coios, d i domi Rsolv Fcio did y h iii sris is clld rsolv. R (, y, λ ) λ K (, y) Thorm 7.4 Solio o igrl qio λk wih coios rl K (, y) is iq i [,] y [,] is giv y or λ R(, y, λ) ( y)dy i.. hr iss ivrs opror ( I λ K ) I λr, λ < ( ) λ < ( ), d or

19 hpr 7 INTERAL EQUATIONS Empl 7. Fid solio o igrl qio 3 y( y) dy 6 8 y h rsolv mhod. Idiy: K (, y) y 3 6 λ 8 hc codiio: Ird rls: (, y) K y λ < < < 8 ( ) 3 y y,y dy y y ydy y 3 K (, y) K (,y ) K y K,y K y,y dy y ydy 3 K 3 (, y) y 3 3 y 3 y 3 y 3 K (, y) y 3 Rsolv: R (, y,λ) λ K (, y) Solio: y y y y y y y 4 4 y 3 λ R(, y, λ) ( y)dy y 3 y dy y 3 ydy

20 hpr 7 INTERAL EQUATIONS 3. Th hod o Sccssiv Approimios or h Volrr Igrl Eqio o h d id osidr h Volrr igrl qio o h d id λ K (,y) ( y) dy whr K (, y) is coios rl, K (,y) ( [,] [,] ). Th mhod o sccssiv pproimio is did y h ollowig irios: λ K λ K Thorm 7.5 Th Volrr igrl qio o h d id λ K,y y dy wih coios rl (, y) K d wih y λ R hs iq solio [,] or y [,]. This solio is giv y iormly covrg Nm sris λ ( K ) d is orm sisis λ Empl 7.3 Fid solio o igrl qio ( y) dy y h mhod o sccssiv pproimios. Idiy: K (, y) λ K K K(, y)( K )( y)dy dy [ y ] K K(, y)( K )( y)dy ydy y K 3 y K(, y)( K )( y)dy dy K! Solio: ( K ) 3 y λ 3! 3 3

21 hpr 7 INTERAL EQUATIONS 7.6 ocio w igrl qios d iiil d odry vl prolms. Rdcio o IVP o h Volrr igrl qio Empl 7.4 Rdc IVP 3 o h Volrr igrl qio. Igr h diril qio rom o : y 3y y dy dy 3y dy s h iiil codiio 3 y y dy is Volrr qio wih K (,y) y 3 y y dy. Rdcio o h Volrr igrl qio o IVP Rcll h Liiz rl or diriio o prssios wih igrls: d g(, y) dy d I priclrly, g,y d d dy g, g, d d d d g ( y) dy g d g g(, y) (, y ) dy d dy g (, ) Rdcio o h Volrr igrl qio o IVP is prormd y cosciv diriio o h igrl qio wih rspc o vril d ssiio or sig o h iiil codiios. Empl 7.5 Rdc h Volrr igrl qio 3 ( y) ( y) iiil vl prolm. dy ssi o g iiil codiio 3 3 ( y) ( y) dy ( ) y y dy

22 hpr 7 INTERAL EQUATIONS 3 ( y) ( y) dy 3 ( y) ( y) dy 3 ( y) dy 3 ( y) dy 6 Thror, h igrl qio is rdcd o IVP or 3 rd ordr ODE: 6 3. Rdcio o BVP o h Frdholm igrl qio Rcll rpd igrio orml: d d d d d 3 ( ) ( )! Empl 7.6 Rdc h odry vl prolm y y, y y( ) o h Frdholm igrl qio. S y igr y () d () y d y igr [ y ( ) y ] d ( ) d d Us h irs odry codiio I his prssio, y y y d y y ( ) d d y y ( ) d rpd igrio y ( ) d y is o ow. Ssi d pply h scod odry codiio

23 hpr 7 INTERAL EQUATIONS d y y d y Solv or y d y Th y d d () ()d d Now ssi his prssio or y d y io h origil diril qio d d () () d d () ()d d d d d () () d d d () () ()d d d ()d d () ()d d I yilds Frdholm igrl qio K, d wih rl, K

24 hpr 7 INTERAL EQUATIONS 7.7 Erciss. Prov pr 3) o h Thorm 7... lssiy ch o h ollowig igrl qios s Frdholm or Volrr igrl qio, lir or o-lir, homogos or o-homogos, idiy h prmr λ d h rl K (,y ) : ) y( y )dy ) ( ) y ( y )dy c) y ( y)dy d) ( ) ) y ( y )dy dy 4 y ( y) 3. Rdc h ollowig igrl qio o iiil vl prolm ( ) y y dy 4. Fid h qivl Volrr igrl qio o h ollowig iiil vl prolm y y cos y y 5. Driv h qivl Frdholm igrl qio or h ollowig odry vl prolm (,) y y y y 6. Solv h ollowig igrl qios y sig h sccssiv pproimio mhod d h rsolv mhod: ) λ y y dy ) cos y dy 4 7. Solv h ollowig igrl qio y sig h sccssiv pproimios mhod ( ) y y dy

25 hpr 7 INTERAL EQUATIONS 8. Solv h ollowig igrl qios: ) ( ) si s si s ds s ) ( s) ds 9. Usig mhmicl idcio prov idiy or ird rl (7.5 ): K (, y) K(, y ) K ( y, y) dy. Usig mhmicl idcio vriy h ollowig sim or ird rls (7.5 ): K (, y) ( ). Vriy rsl o Empl 7.4 y solvig oh IVP d drivd igrl qio.

26 hpr 7 INTERAL EQUATIONS S Bch (89-945) Scoish é Lvov Th Scoish cé i Lvov (Uri) ws mig plc or my mhmicis icldig Bch, Sihs, Ulm, zr, Kc, Schdr, Kczmrz d ohrs. Prolms wr wri i oo p y h ldlord. A collcio o hs prolms pprd lr s h Scoish Boo. R D ldi, Th Scoish Boo, hmics rom h Scoish é (98) cois h prolms s wll s som solios d commris. Ivr Frdholm (866 97) Frdholm is s rmmrd or his wor o igrl qios d spcrl hory. Fid o mor : hp://www-hisory.mcs.s-drws.c./hisory/hmicis/frdholm.hml Vio Volrr (86-94) Volrr plishd pprs o pril diril qios, priclrly h qio o cylidricl wvs. His mos mos wor ws do o igrl qios. H plishd my pprs o wh is ow clld ' igrl qio o Volrr yp'. Fid o mor : hp://www-hisory.mcs.s-drws.c./hisory/hmicis/volrr.hml