Chapter 7 INTEGRAL EQUATIONS
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1 hapr 7 INTERAL EQUATIONS
2 hapr 7 INTERAL EUATIONS hapr 7 Igral Eqaios 7. Normd Vcor Spacs. Eclidia vcor spac. Vcor spac o coios cios ( ). Vcor Spac L ( ) 4. ach-baowsi iqali 5. iowsi iqali 7. Liar Opraors - coios opraors - odd opraors 7. Igral Opraor 6.4 Igral qaios - Frdholm igral qaios - Volrra igral qaios - igro-dirial qaios - solio o igral qaio 7.5 Solio hods or Igral Eqaios. hod o sccssiv approimaios or Frdholm IE (Nma sris). hod o sccssiv ssiios or Frdholm IE (Rsolv mhod). hod o sccssiv approimaios or Volrra IE 7.6 ocio w igral qaios ad iiial ad odar val prolms 7.7 Erciss. Rdcio o IVP o h Volrra IE. Rdcio o h Volrra IE o IVP. Rdcio o BVP o h Frdholm IE Fr Topics: 7.7 Fid poi horm (s also [Hochsad Igral qaios, p.5]) Elmar isc horms 7.8 Pracical applicaios (s also [Jrri Irodcio o Igral Eqaios wih Applicaios ]) 7.9 Ivrs prolms (s also [ Jrri, p.7])
3 hapr 7 INTERAL EUATIONS 7. Normd Vcor Spacs W will sar wih som diiios ad rsls rom h hor o ormd vcor spacs which will dd i his chapr.. Eclidia vcor spac Th -dimsioal Eclidia vcor spac cosiss o all pois { (,,..., ) } or which h ollowig opraios ar did: Scalar prodc (,)..., Norm (,)... Disac ρ (,) ovrgc lim i lim. Vcor spac ( ) Vcor spac ( ) cosiss o all ral vald coios cios did o h closd domai : : D coios { } ( ) ( ) Norm ma ( ) ovrgc lim i lim. Vcor spac L ( ) Th spac o cios igral accordig o Lsg (s Scio.) 4. ach-baovs Iqali Ir prodc ( ) ( ) ( ) Norm L ( ) ( ) ( ) : d<,g g d ( ) ( ), d Th ollowig propr ollows rom h diiio o h Lsg igral ( ) d ( )d (,g) g or all,g L ( ) Proo: I,g L ( ), h cios, g ad a comiaio α β g ar also igral ad hror log o L ( ). osidr ( ) λ g L, λ R or which w hav
4 hapr 7 INTERAL EUATIONS ( λ g ) d d λ g d λ g d Th righ had sid is a qadraic cio o λ. Bcas his cio is o-gaiv, is discrima is o-posiiv 4 gd 4 d g d g d d g d (,g) g rom which h claimd iqali ilds (,g) g cas (, g) gd g d g d. 5. iowsi Iqali ( rd propr o h orm) g g or all,g L ( ) Proo: osidr g ( g, g) (, ) (, g) ( g, ) ( g, g) (, g) ( g, ) g g g rom -B iqali ( g ) Th racio o h sqar roo ilds h claimd rsl. No ha h iowsi iqali rdcs o qali ol i cios ad g ar qal p o h scalar mlipl, αg, α R (wh?).
5 hapr 7 INTERAL EUATIONS 7. Liar Opraors L ad N wo liar ormd vcors spacs wih orms ad, N corrspodigl. W di a opraor L as a map (cio) rom h vcor spac o h vcor spac N : L : N Irodc h ollowig diiios cocrig h opraors i h vcor spacs: Opraor L : N is liar i L( α βg) αl βlg or all, g ad all α, β R Opraor L : N is coios i rom i ollows L L i N (h imag o h covrg sqc i is a covrg sqc i N ) Opraor L : N is odd i hr iss c > sch ha L c or all N Th orm o opraor o sch cosa c L : N ca did as h gras lowr od L L sp N Thorm 7. I opraor L : N is odd h i is coios Proo: L opraor L : N odd, h accordig o h diiio hr iss c > sch ha L c. L N i. Tha mas ha lim. From h diiio o h limi i ollows ha or a ε > hr iss N sch ha < ε or all. To prov h horm, show ow ha lim L sch ha L N L L i N or ha. W hav o show ha or a Ε > hr iss N L L < Ε or all Ε. N Ε hoos ε, h c Ε L L L( ) c < c Ε or all. N N c Ε
6 hapr 7 INTERAL EUATIONS 7. Igral Opraor osidr a opraor calld a igral opraor giv h qaio (, ) ( ) R Oviosl, ha igral opraor is liar. Fcio (, ) rls (, ) L ( ), hror (, ) d < is calld a rl o h igral opraor. W will cosidr I a cas o R, h domai ( a,), whr a, ca ii or iii. Thorm 7. L h igral opraor wih a rl (, ) coios i [ a,] [ a,]. Th opraor is odd, ad, hror, coios. orovr: ) : L ( a,) [ a,] a or L ( a,) ) : L ( a,) L ( a,) ( a) or L ( a,) ) : [ a,] [ a,] ( a) or [ a,] Proo: Sic (, ) is coios i h closd domai [,] [ a,] > sch ha ma (, )., [ a, ] ) L L ( a,). Th cas cio (, ) [ a,] [ a,], h cio ( )( ) is coios i [ a,] : L ( a,) [ a,]. osidr ma ( )( ) [ ] a, ma [ a,] a (, ) ( ) a, hr iss is coios i ma [ a,], ad, hror ( (, ), ( ) ) ma (rom ach-baowsi iqali) [ a,] ma [ a,] a (, ) ma [ a, ] a a
7 hapr 7 INTERAL EUATIONS ) (( )( ),( )( ) ) ( )( ) a d a a (, ) ( ) d a d a a (, ) d a a d a a ( a) d ) Ercis
8 hapr 7 INTERAL EUATIONS 7.4 Igral Eqaios Igral qaios ar qaios i which h ow cio is dr h, igral sig. Th pical igral qaios or ow cio ( ) R (i his chapr, w cosidr ( a,) R i h orm o igral opraor wih h rl (, ) ) icld igral rm (, ) ( ) Th mai ps o igral qaios ar h ollowig: I Frdholm igral qaio ) Frdholm s igral qaio o h s id: (, ) ( ) ( ) o-homogos q (, ) ( ) homogos q ) Frdholm s igral qaio o h d id: λ is a paramr ( ) (, ) ( ) ( ) λ λ o-homogos q ( ) (, ) ( ) λ λ homogos q II Volrra igral qaio L (,a) R. (, ) is calld a Volrra rl i (, ) a or < < < a ) Volrra s igral qaio o h s id: (, ) ( ) ( ) ) Volrra s igral qaio o h d id: ( ) λ (, ) ( ) ( ) a III Igro-Dirial Eqaio iclds a ow cio dr h igral sig ad also a drivaiv o h ow cio. For ampl: d ( ) (, ) ( ) ( ) d A impora rprsaio o h igro-dirial qaio is a Radiaiv Trasr Eqaio dscriig rg raspor i h asorig, miig ad scarig mdia (aalogos qaios appar i h hor o ro raspor). Som ohr ps o igral qaios will cosidrd i h Scio 8..4.
9 hapr 7 INTERAL EUATIONS Solio o igral qaio is a cio ( ) saisig his qaio: λ o-homogos qaio λ homogos qaio Th val o h paramr λ or which h homogos igral qaio has a o-rivial solio L which is calld a igval o h rl (, ), ad h corrspodig solio is calld a igcio o his rl. Eigval prolm W will disigish igval prolms or h igral rl (igral qaio): λ ad or h igral opraor λ Th igvals o h igral opraor ar rcipical o igvals o h igral rl, ad igcios ar h sam i oh cass.
10 hapr 7 INTERAL EUATIONS 7.5 Solio hods or Igral Eqaios. Th hod o Sccssiv Approimaios or Frdholm s Igral Eqaio For h igral qaio λ h ollowig iraios o h mhod o sccssiv approimaios ar s : ( ) ( ) ( ) λ,,... Lmma 7. ( ) λ whr ( ( )) ims Proo mahmaical idcio (assm ha h ormla or is r): ( ) ( ) λ coirmd ( ) λ diiio λ λ assmpio λ liari p p λ chag o id p λ p p λ p p p p λ p λ chag o id p Nma Sris λ is calld o h Nma Sris Esimaio o iraios ( ) ( a) Thorm 7. () ( a) ( a)
11 hapr 7 INTERAL EUATIONS λ λ ( a) [ ( a) ] λ gomric sris λ ( a) covrgs i λ < ( a) Thror, h Nma sris λ covrgs or Do h sm o h Nma sris as a cio ( ) : λ <. ( a) ( ) λ Show ha his cio saisis igral h qaio iraios ( ) λ h lim ( ) ( ) λ lim ( ) (, ) lim ( ) λ a λ a (, ) ( ) λ. osidr Ad, rcallig simaio, ( ) λ ( a) Show ha his solio is iq. For ha, i is ogh o show ha h homogos qaio λ has ol a rivial solio. Idd, i λ a, ad, accordig o Thorm 6. ),, h [ ] λ ( a) ( a) [ λ ], h [ ] > Bcas λ <, ( a) ( a) ilds, ha ( ) or all [ a,] homogos qaio. λ ad, hror,. Tha. So, ol h rivial solio iss or h Th o-homogos qaio λ ca rwri i h orm ( I λ ) whr I is a idi opraor Th solio o his qaio ca rad as a ivrsio o h opraor I λ ( ) Thror, i λ <, h hr iss a ivrs opraor ( I ) ( a) λ. Th aovmiod rsls ca ormlad i h ollowig horm:
12 hapr 7 INTERAL EUATIONS Thorm 7. Frdholm s igral qaio λ wih λ < ad coios rl (, ) ( a) iq solio ( ) [ a,] or a ( ) [ a,]. This solio is giv a covrg Nma sris ad saisis I λ < ( I ) λ. ( a) ( ) λ ( ) λ ( a). has a, h hr iss a ivrs opraor odiios o Thorm 7. ar ol js sici codiios; i hs codiios ar o saisid, solio o h igral qaio sill ca iss ad h Nma sris ca covrg. Eampl 7. Fid h solio o h igral qaio ( ) ( ) h mhod o sccssiv approimaios ad i h orm o h Nma sris. Idi: (, ) ( ) a λ hc codiio: λ < < < ( a) ) iraios: ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) Th solio o h igral qaio is a limi o iraios ( ) lim ( ) lim This rsl ca validad a dirc ssiio io h giv igral qaio.
13 hapr 7 INTERAL EUATIONS ) Nma sris: ( ) λ ( ) λ λ ( ) ( ) Th h Nma sris is ( ) ( ) ( ) ( ) ( ) ( ) ( ) So, h Nma sris approach prodcs h sam solio.
14 hapr 7 INTERAL EUATIONS. Th hod o Sccssiv Ssiios or Frdholm s Igral Eqaio (h Rsolv hod) Irad rl L igral opraor has a coios rl (, ) Rpad opraor ( ) ( ), h di:,,... I has a has a rl (, ) (, ) (, ) Idd, ( )( ) (, ) ( ) ( )( ) [ ( )]( ) (, ) (, ) ( ) (, ) (, ) ( ) rl (, ) (, ) (, ) is calld a irad rl. rls (, ) ( a,), h (, ) ( a) (, ) (, ) ar coios, ad i domai Rsolv Fcio did h iii sris is calld a rsolv. R (,, λ ) λ (, ) Thorm 6.4 Solio o igral qaio λ wih coios rl (, ) is iq i [ a,] a [ a,] is giv ( ) ( ) λ R(,, λ) ( ) a i.. hr iss ivrs opraor ( I λ ) I λr, λ < or ( a) λ < ( a), ad or
15 hapr 7 INTERAL EUATIONS Eampl 6. Fid solio o igral qaio ( ) ( ) 6 8 h rsolv mhod. Idi: (, ) ( ) a 6 λ 8 hc codiio: Irad rls: (, ) λ < < < 8 ( a) (, ) (, ) (, ) (, ) (, ) (, ) (, ) Rsolv: R (,,λ) λ (, ) Solio: ( ) ( ) λ R(,, λ) ( ) a 8 4 6
16 hapr 7 INTERAL EUATIONS. Th hod o Sccssiv Approimaios or h Volrra Igral Eqaio o h d id osidr h Volrra igral qaio o h d id (o ha qaio o h s id ca rdcd o h d id dirrio) ( ) λ (,) ( ) ( ) whr (, ) is a coios Volrra rl. Th mhod o sccssiv approimaio is did h ollowig iraios: ( ) ( ) ( ) λ λ Thorm 6.5 Th Volrra igral qaio o h d id ( ) λ ( ) ( ) ( ), wih coios Volrra rl (, ) ad wih a λ R has a iq solio ( ) [,a] or a ( ) [,a]. This solio is giv a iorml covrg Nma sris ( ) λ ( )( ) ad is orm saisis λ a ( ) Eampl 6. Fid solio o igral qaio ( ) ( ) h mhod o sccssiv approimaios. Idi: (, ) ( ) λ ( ) (, )( )( ) [ ] (, )( )( ) (, )( )( )! Solio: ( ) ( )( ) λ!
17 hapr 7 INTERAL EUATIONS 7.6 ocio w igral qaios ad iiial ad odar val prolms. Rdcio o IVP o h Volrra igral qaio Eampl 7.4 Rdc IVP ( ) o h Volrra igral qaio. Igra h dirial qaio rom o : ( ) d ( ) d ( ) d ( ) ( ) d s h iiial codiio ( ) ( ) d is a Volrra qaio wih (, ). Rdcio o h Volrra igral qaio o IVP Rcall h Liiz rl or diriaio o prssios wih igrals: d ( ) ( ) g g(, ) (, ) d a( ) I pariclarl, ( ) a g [,( ) ] ( ) d d g [,a( ) ] ( ) d da d d g ( ) g( ) d g g(, ) (, ) d g (, ) Rdcio o h Volrra igral qaio o IVP is prormd cosciv diriaio o h igral qaio wih rspc o varial ad ssiio or sig o h iiial codiios. Eampl 7.5 Rdc h Volrra igral qaio ( ) ( ) ( ) iiial val prolm. ssi o g iiial codiio ( ) ( ) ( ) ( ) ( ) ( ) ( )
18 hapr 7 INTERAL EUATIONS ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 ( ) Thror, h igral qaio is rdcd o IVP or rd ordr ODE: ( ) 6 ( ) ( ) ( ) ( ). Rdcio o BVP o h Frdholm igral qaio Rcall rpad igraio ormlas: d d d d d ( ) ( ) ( ) ( )! Eampl 7.6 Rdc h odar val prolm, ( ) ( ) ( ) ( ) ( ) o h Frdholm igral qaio. S ( ) ( ) igra () d () d ( ) ( ) ( ) igra [ ( ) ( ) ] d ( ) d d Us h irs odar codiio I his prssio, ( ) d ( ) ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) ( )d rpad igraio ( ) ( ) ( ) ( )d is o ow. Ssi ad appl h scod odar codiio
19 hapr 7 INTERAL EUATIONS ( ) ( ) ( ) ( )d ( ) ( ) ( )d Solv or ( ) ( ) ( ) ( )d Th ( ) ( ) ( ) ( ) ( )d d ( )() ( )()d d Now ssi his prssio or ( ) ad ( ) ( ) io h origial dirial qaio ( ) ( ) ( ) ( ) d d ( )() ( )() d d ( )() ( )()d d ( ) ( ) ( ) ( ) ( ) ( )d d d ( )() ( )() ( ) ( )d d d ( )() ( )() ( ) ()d d d ( ) ( ) ( ) ( ) ()d d ( ) () ( ) ()d d I ilds a Frdholm igral qaio ( ) ( ), d wih a rl ( ) ( ) ( ),
20 hapr 7 INTERAL EUATIONS Erciss. Prov par ) o h Thorm 6... lassi ach o h ollowig igral qaios as Frdholm or Volrra igral qaio, liar or o-liar, homogos or o-homogos, idi h paramr λ ad h rl (, ) : a) ( ) ( ) ) ( ) ( ) ( ) c) ( ) ( ) d) ( ) ( ) ) ( ) ( ) 4 ( ). Rdc h ollowig igral qaio o a iiial val prolm ( ) ( ) ( ) 4. Fid h qival Volrra igral qaio o h ollowig iiial val prolm ( ) ( ) cos ( ) ( ) 5. Driv h qival Frdholm igral qaio or h ollowig odar val prolm (,) ( ) ( ) 6. Solv h ollowig igral qaios sig h sccssiv approimaio mhod ad h rsolv mhod: a) ( ) λ ( ) ) ( ) ( ) cos 4 7. Solv h ollowig igral qaio sig h sccssiv approimaio mhod ( ) ( ) ( ) 8. Solv h ollowig igral qaios: a) ( ) ( ) ( ) si s si s ds as ) ( ) ( s) ds ( )
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