hapr INTERAL EQUATIONS
hapr INTERAL EUATIONS Dcmbr 4, 8 hapr Igral Eqaios. Normd Vcor Spacs. Eclidia vcor spac. Vcor spac o coios cios ( ). Vcor Spac L ( ) 4. achy-byaowsi iqaliy 5. iowsi iqaliy. Liar Opraors - coios opraors - bodd opraors. Igral Opraor.4 Igral qaios - Frdholm igral qaios - Volrra igral qaios - igro-dirial qaios - solio o igral qaio.5 Solio hods or Igral Eqaios. hod o sccssiv approimaios or Frdholm IE (Nma sris). hod o sccssiv sbsiios or Frdholm IE (Rsolv mhod). hod o sccssiv approimaios or Volrra IE.6 ocio bw igral qaios ad iiial ad bodary val problms. Rdcio o IVP o h Volrra IE. Rdcio o h Volrra IE o IVP. Rdcio o BVP o h Frdholm IE.7 Erciss Fr Topics:.7 Fid poi horm (s also [Hochsad Igral qaios, p.5]) Elmary isc horms.8 Pracical applicaios (s also [Jrri Irodcio o Igral Eqaios wih Applicaios ]).9 Ivrs problms (s also [ Jrri, p.7])
hapr INTERAL EUATIONS Dcmbr 4, 8. Normd Vcor Spacs W will sar wih som diiios ad rsls rom h hory o ormd vcor spacs which will b 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 (, y) y y... y,y,... Norm ( ) Disac ρ (,y) y 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 igrabl accordig o Lbsg (s Scio.) 4. achy-byaovsy Iqaliy Ir prodc ( ) ( ) ( ) Norm L ( ) ( ) ( ) : d<,g g d ( ) ( ), d Th ollowig propry ollows rom h diiio o h Lbsg igral ( ) d ( )d (,g) g or all,g L ( ) Proo: I,g L ( ), h cios, g ad ay combiaio α β g ar also igrabl ad hror blog o L ( ). osidr ( ) λ g L, λ R or which w hav
hapr INTERAL EUATIONS Dcmbr 4, 8 ( 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 g d 4 d g d g d d g d (,g) g 5. iowsi Iqaliy ( rd propry o h orm) rom which h claimd iqaliy yilds (,g) g bcas (, g) gd g d g d. g g or all,g L ( ) Proo: osidr g ( g, g) (, ) (, g) ( g, ) ( g, g) (, g) ( g, ) g g g rom -B iqaliy ( g ) Th racio o h sqar roo yilds h claimd rsl. No ha h iowsi iqaliy rdcs o qaliy oly i cios ad g ar qal p o h scalar mlipl, αg, α R (why?).
hapr INTERAL EUATIONS Dcmbr 4, 8. Liar Opraors L ad N b wo liar ormd vcors spacs wih orms ad, N corrspodigly. 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 bodd i hr iss c > sch ha L c or all N Th orm o opraor o sch cosa c L : N ca b did as h gras lowr bod L L sp N Thorm 7. I opraor L : N is bodd h i is coios Proo: L opraor L : N b bodd, 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 ay ε > hr iss N sch ha < ε or all. To prov h horm, show ow ha lim L L sch ha N. W hav o show ha or ay > L L < Ε or all Ε. N L L i N or ha Ε hr iss N Ε hoos ε, h c Ε L L L( ) c < c Ε or all. N N c Ε
hapr INTERAL EUATIONS Dcmbr 4, 8. Igral Opraor osidr a opraor calld a igral opraor giv by h qaio (, y) ( y) R Obviosly, ha igral opraor is liar. Fcio (, y) rls (, y) L ( ), hror (, y) d < is calld a rl o h igral opraor. W will cosidr I a cas o R, h domai ( a,b), whr a, b ca b ii or iii. Thorm 7. L b h igral opraor wih a rl (, y) coios i [ a,b] [ a,b]. Th opraor is bodd, ad, hror, coios. orovr: ) : L ( a,b) [ a,b] b a or L ( a,b) ) : L ( a,b) L ( a,b) ( b a) or L ( a,b) ) : [ a,b] [ a,b] ( b a) or [ a,b] Proo: Sic (, y) is coios i h closd domai [,b] [ a,b] > sch ha ma (, y).,y [ a,b ] ) L L ( a,b). Th bcas cio (, y) [ a,b] [ a,b], h cio ( )( ) is coios i [ a,b] : L ( a,b) [ a,b]. osidr ma a, b ( )( ) [ ] ma [ a,b ] b a (, y) ( y) a, hr iss is coios i, ad, hror ma [ a,b ] ( (, y), ( y) ) ma (rom achy-byaowsi iqaliy) [ a,b ] ma [ a,b ] b a (, y) b ma [ a,b ] a b a
hapr INTERAL EUATIONS Dcmbr 4, 8 ) (( )( ),( )( ) ) ( )( ) b a d b b a a (, y) ( y) d b a d b b a a (, y) d b b a a d b b a a ( b a) d ) Ercis
hapr INTERAL EUATIONS Dcmbr 4, 8.4 Igral Eqaios Igral qaios ar qaios i which h ow cio is dr h, igral sig. Th ypical igral qaios or ow cio ( ) R (i his chapr, w cosidr ( a,b) R i h orm o igral opraor wih h rl (, y) ) icld igral rm (, y) ( y) Th mai yps o igral qaios ar h ollowig: I Frdholm igral qaio ) Frdholm s igral qaio o h s id: (, y) ( y) ( ) o-homogos q (, y) ( y) homogos q ) Frdholm s igral qaio o h d id: λ is a paramr ( ) (, y) ( y) ( ) λ λ o-homogos q ( ) (, y) ( y) λ λ homogos q II Volrra igral qaio L (,a) R (, y) is calld a Volrra rl i (, y). a y or < < y < a ) Volrra s igral qaio o h s id: (, y) ( y) ( ) ) Volrra s igral qaio o h d id: ( ) λ (, y) ( y) ( ) a III Igro-Dirial Eqaio iclds a ow cio dr h igral sig ad also ay drivaiv o h ow cio. For ampl: d ( ) (, y) ( y) ( ) d A impora rprsaio o h igro-dirial qaio is a Radiaiv Trasr Eqaio dscribig rgy raspor i h absorbig, miig ad scarig mdia (aalogos qaios appar i h hory o ro raspor). Som ohr yps o igral qaios will b cosidrd i h Scio 8..4.
hapr INTERAL EUATIONS Dcmbr 4, 8 Solio o igral qaio is ay cio ( ) saisyig 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 (, y), ad h corrspodig solio is calld a igcio o his rl. Eigval problm W will disigish igval problms 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 boh cass.
hapr INTERAL EUATIONS Dcmbr 4, 8.5 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 by: ( ) ( ) ( ) λ,,... Lmma 7. ( ) λ whr ( ( )) ims Proo by mahmaical idcio (assm ha h ormla or is r): ( ) λ coirmd ( ) ( ) λ by diiio λ λ by assmpio λ liariy 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 b h Nma Sris Esimaio o iraios ( ) ( b a) Thorm 7. () ( b a) ( b a)
hapr INTERAL EUATIONS Dcmbr 4, 8 λ λ ( b a) [ ( b a) ] λ gomric sris λ ( b a) covrgs i λ < ( b a) Thror, h Nma sris λ covrgs or Do h sm o h Nma sris as a cio ( ) : λ <. ( b a) ( ) λ Show ha his cio saisis igral h qaio iraios ( ) λ h lim ( ) ( ) λ lim b ( ) (, y) lim ( y) λ a b λ a (, y) ( y) λ. osidr Ad, rcallig simaio, ( ) λ ( b a) Show ha his solio is iq. For ha, i is ogh o show ha h homogos qaio λ has oly a rivial solio. Idd, i λ, h [ a,b] ad, accordig o Thorm 6. ), λ b a, h ( ) ( b a) λ [ ] [ ] > Bcas λ <, ( b a) ( b a) yilds, ha ( ) or all [ a,b] homogos qaio. λ ad, hror,. Tha. So, oly h rivial solio iss or h Th o-homogos qaio λ ca b rwri i h orm ( I λ ) whr I is a idiy opraor Th solio o his qaio ca b rad as a ivrsio o h opraor ( I λ ) Thror, i λ <, h hr iss a ivrs opraor ( I λ ). b a ( ) Th abovmiod rsls ca b ormlad i h ollowig horm:
hapr INTERAL EUATIONS Dcmbr 4, 8 Thorm 7. Frdholm s igral qaio λ wih λ < ad coios rl (, y) ( b a) iq solio ( ) [ a,b] or ay ( ) [ a,b]. This solio is giv by a covrg Nma sris has a ad saisis I λ < ( I ) λ. ( b a) ( ) λ ( ) λ ( b a), h hr iss a ivrs opraor. odiios o Thorm 7. ar oly js sici codiios; i hs codiios ar o saisid, solio o h igral qaio sill ca iss ad h Nma sris ca b covrg. Eampl 7. Fid h solio o h igral qaio ( ) ( y) by h mhod o sccssiv approimaios ad i h orm o h Nma sris. Idiy: (, y) ( ) b a λ hc codiio: λ < < < ( b a) ) iraios: ( ) ( ) y ( y) ( ) ( ) ( ) ( y) [ ] Th solio o h igral qaio is a limi o iraios ( ) lim ( ) lim This rsl ca b validad by a dirc sbsiio io h giv igral qaio.
hapr INTERAL EUATIONS Dcmbr 4, 8 ) Nma sris: ( ) λ ( ) λ λ ( ) ( ) Th h Nma sris is ( ) ( ) ( ) ( ) ( ) ( ) ( ) So, h Nma sris approach prodcs h sam solio.
hapr INTERAL EUATIONS Dcmbr 4, 8. Th hod o Sccssiv Sbsiios or Frdholm s Igral Eqaio (h Rsolv hod) Irad rl L igral opraor has a coios rl (, y) Rpad opraor ( ) ( ), h di:,,... I has a has a rl (, y) (, y ) ( y, y) Idd, ( )( ) (, y) ( y) ( )( ) [ ( )]( ) (, y ) ( y, y) ( y) (, y ) ( y, y) ( y) rl (, y) (, y ) ( y, y) is calld a irad rl. rls (, y) ( a,b), h (, y) ( b a) (, y ) ( y, y) ar coios, ad i domai Rsolv Fcio did by h iii sris is calld a rsolv. R (, y, λ ) λ (, y) Thorm 6.4 Solio o igral qaio λ wih coios rl (, y) is iq i [ a,b] or λ <, ad or b a ay [ a,b] is giv by b ( ) ( ) λ R(, y, λ) ( y) a i.. hr iss ivrs opraor ( I λ ) I λr, λ < ( b a) ( )
hapr INTERAL EUATIONS Dcmbr 4, 8 Eampl 6. Fid solio o igral qaio ( ) y( y) 6 8 by h rsolv mhod. Idiy: (, y) y ( ) b a 6 λ 8 hc codiio: Irad rls: (, y) y λ < < < 8 ( b a) y y y y y (, y) (, y ) ( y, y) y y y (, y) (, y ) ( y, y) y y y y (, y) y Rsolv: R (, y,λ) λ (, y) Solio: y y y y y...... 8 8 8 8 y...... 8 8 8 8 y 4 4 y ( ) ( ) λ R(, y, λ) ( y) 6 6 6 4 b 8 a 4 y y 6 y y
hapr INTERAL EUATIONS Dcmbr 4, 8. 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 b rdcd o h d id by dirrio) ( ) λ ( ) ( ) ( ), y y whr (, y) is a coios Volrra rl. Th mhod o sccssiv approimaio is did by h ollowig iraios: ( ) ( ) ( ) λ λ Thorm 6.5 Th Volrra igral qaio o h d id ( ) λ ( ) ( ) ( ), y y wih coios Volrra rl (, y) ad wih ay λ R has a iq solio ( ) [,a] or ay ( ) [,a]. This solio is giv by a iormly covrg Nma sris ( ) λ ( )( ) ad is orm saisis λ a ( ) Eampl 6. Fid solio o igral qaio ( ) ( y) by h mhod o sccssiv approimaios. Idiy: (, y) ( ) λ ( ) (, y)( )( y) [ y ] (, y)( )( y) y y y y (, y)( )( y)! Solio: ( ) ( )( ) λ!
hapr INTERAL EUATIONS Dcmbr 4, 8.6 ocio bw igral qaios ad iiial ad bodary val problms. 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 (, y). Rdcio o h Volrra igral qaio o IVP Rcall h Libiz rl or diriaio o prssios wih igrals: d b( ) b( ) g g(, y) (, y ) d a( ) I pariclarly, a( ) g [,b( ) ] ( ) db d g [,a( ) ] ( ) d da d d g ( y) g( ) d g g(, y) (, y ) d g (, ) Rdcio o h Volrra igral qaio o IVP is prormd by cosciv diriaio o h igral qaio wih rspc o variabl ad sbsiio or sig o h iiial codiios. Eampl 7.5 Rdc h Volrra igral qaio ( ) ( y) ( y) iiial val problm. sbsi o g iiial codiio ( ) ( y) ( y) ( ) ( y) ( y) ( )
hapr INTERAL EUATIONS Dcmbr 4, 8 ( ) ( y) ( y) ( ) ( y) ( y) ( ) ( ) ( y) ( ) ( y) ( ) ( ) 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 bodary val problm y y, ( ) ( ) ( ) y ( ) y( ) o h Frdholm igral qaio. S y ( ) ( ) igra y ( ) d ( ) y d ( ) y ( ) ( ) igra [ y ( ) y ( ) ] d ( ) d d Us h irs bodary codiio I his prssio, ( ) y y y d ( ) y( ) y ( ) ( ) d d ( ) y( ) y ( ) ( ) ( )d rpad igraio ( ) y ( ) ( ) ( )d y is o ow. Sbsi ad apply h scod bodary codiio
hapr INTERAL EUATIONS Dcmbr 4, 8 ( ) ( ) ( ) ( )d y y ( ) ( ) ( )d y Solv or ( ) y ( ) ( ) ( )d y Th ( ) y ( ) ( ) ( ) ( )d d ( ) ( ) ( ) ( )d d Now sbsi his prssio or ( ) y ad ( ) ( ) y 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 yilds a Frdholm igral qaio ( ) ( ), d wih a rl ( ) ( ) ( ),
Erciss hapr INTERAL EUATIONS Dcmbr 4, 8. Prov par ) o h Thorm 6... lassiy ach o h ollowig igral qaios as Frdholm or Volrra igral qaio, liar or o-liar, homogos or o-homogos, idiy h paramr λ ad h rl (,y ) : y( y ) a) ( ) b) ( ) ( ) c) ( ) y ( y ) y ( y ) d) ( ) ( ) ) ( ) y ( y ) 4 y ( y ). Rdc h ollowig igral qaio o a iiial val problm ( ) ( ) ( ) y y 4. Fid h qival Volrra igral qaio o h ollowig iiial val problm y ( ) y ( ) cos ( ) y y ( ) 5. Driv h qival Frdholm igral qaio or h ollowig bodary val problm (,) y ( ) ( ) y y y 6. Solv h ollowig igral qaios by sig h sccssiv approimaio mhod ad h rsolv mhod: a) ( ) λ ( ) y y b) ( ) ( ) cos y 4 7. Solv h ollowig igral qaio by sig h sccssiv approimaio mhod ( ) ( ) ( ) y y 8. Solv h ollowig igral qaios: a) ( ) ( ) ( ) si s si s ds as b) ( ) ( s) ds ( )