THE NUMERICAL SOLVING OF A NON LINEAR INTEGRAL EQUATION OF HAMMERSTEIN TYPE

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1 U.P.B. S. Bll. Seres A Vol. 68 No. 4 6 THE NUMERIAL SOLVING OF A NON LINEAR INTEGRAL EQUATION OF HAMMERSTEIN TYPE Mara IUGA * Aes arol îş prope să realzeze o reere sară î revsă a ora dre ele a des îâle eode ere de rezolvare a eaţlor egrale. Apo de aseeea se vor a ele dre eleeele de bază ale eore sal. Parea orgală a aroll o repreză abordarea e egrale de p Haerse e poae îâlă î adrl eore sal ş despre are se va arăa ă se poae rezolva pr eode ere. Ths arle res o aheve a sar o oe o he os ell o eral ehods or solvg egral eqaos. I he sae e soe elees abo dead aer-heor ll be red. The orgal par o hs arle s represeed b he solvg o a Haerse eqao hh a be od he dead-aer heor ad ll be deosraed ha a be solved sg eral ehods. Ke ords: dead aer Haerse egrals. Irodo Aog he oders o he lear egral eqao s heor e ll eobesde Volerra ad Fredhol also Davd Hlber (86-943) ad Erhar Shd ( ). I s also pora o reeber he Roaa aheaa Traa Lales ho hs doorae hess eled Sr l éqao de Volerra ad ssaed Pars 98 sed or he rs e he sessve approao ehod or he egrao o a Volerra eqao. He also roe he rs boo ro he ere orld abo egral eqaos pblshed Bhares Roaa lagage 9 ad aer ha also pblshed Pars sg Freh lagage oe ear laer respevel 9. Nolear egral eqaos are a d o eqaos hh he o o a be od der he sg o he egral soe oplaed a. For eaple: l () s g( s ) [ () d h() s * Lerer Dep. o Maheas Valaha Uvers ROMANIA

2 36 Mara Iga. Haerse-pe egrals A. Haerse sded olear egral eqaos loog le: ( ) K( ) [ ( ) d (.) The a also be eeded o o -desoal spaes b hs does o volve daeal derees. We ll se he ollog pora hpohess - Fredhol heore s re or he lear egral eqao havg he erel K - he erel K s seral : K ( ) K(. ) - he erel K s posve hh eas ha all s ege vales are o he posve d I hese odos are llled e a sa ha he egral eqao reall s o he Haerse pe. Haerse sed he a ha aordg o relao (.): ( ) K( ) g( ) d h g( ) [ ( ) es also g( ) L he ( ) seres havg he or: a be represeed le a or overge ( ) ( ) (.) sg ( ) ( ) le he orooralsed ege vales or he erel K ( ) orrespodg o he ege vales ad beg o osas. The bease: ( ) ( ) d ( ) d K( ) [ ( ) [ ( ) d K( ) ( ) d [ ( ) ( )d d he proble o solvg he gve eqao s eqvale h he oe o solvg a e sse o eqaos havg a e ber o os: ( ) ( ) hh d h 3 (.3)

3 The eral solvg o a olear egral eqao o haerse pe 37 I s oral o o osder he approae solo: ( ) ( ) (.4) h he osas 3 havg o ver he sse h eqaos ad os: hh ( ) ( ) d 3 (.5) h We ll as abo he esee o he solo or hs sse ess or o. Haerse shoed ver el ha he sses o he d (.5) have a leas oe solo b deosrag ha he o ( ) s a oos oe ad veres a odo o he ollog pe: ( ) (.6) h ad are o posve osas ad s less he he rs ege vale o he posve erel K ( ). Eva he relao (.6) a be relaed Haerse deosraed ha he odo < a o be geerall elarged. For shog hs Haerse sed he oos o: H ( ) F hh ( ) d (.7) h h F( ) ( v) dv s a o havg paral dervaos losel relaed h he ( ) H solos or he sse (.5) bease: H hh ( ) ( )d. (.8) h I s eas o observe ha he o H has a loer l. Usg he relaos (.6) ad (.7) e ll ge ha: F( ) (.9) ad s saller ha a arbrar osa ad also sg he eqal: a a b b > (.) 4b

4 38 Mara Iga deg h a h ad b h ( ) e ge: 3 h 3. ( ) Bease: e ge: bease: The: H H < < F ( ) 3. (.) h h 3 d h ( ) ( ) Usg ha loer l 3. h 3 ( ) (.) h < he rgh sde s s o egave. So H has a Thereore ll es a leas oe se o vales ( ) ( ( or he al so ha he oos o H reah s absole vale d. We a be sre b hoosg or eaple ( ). Mlplg h he posve ber 4 b he relao (.) s eqvale h he saee ha: a 4ab 4b ( a b). The sse (.5) ll be vered bease or: ( ) (.3) e ll oba: H H H. 3

5 The eral solvg o a olear egral eqao o haerse pe 39 No s pora o s ha h hs a o hoosg he qaes he s S (.4) has a pper l depede b. I a h havg he vales ro (.3) ad sg he relao (.) e ge ha: ( ) d ad he ( ) eag ha d 3 S. O he oher had as ar as H ( ) H ( ) 3 d 3 e ge ha d d ( ) Wh d de oba: d 3 S D. (.5) Ths eas ha e are able o re: D ( ) d. (.6) No e have o s ha or he al o os realze a approao or he gve eqao. For he begg e ll deosrae ha he o ( ) ( ) K( ) [ ( ) χ d (.7) goes orl o or.

6 Mara Iga 4 I a sg Hlber-Shd heore e ge: ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) [ ( ) d d K d ξ ξ ξ χ Usg he relao (.5) e have ha: ( ) [ ( ) d ad ( ) ( ) ( ) [ ( ) d χ. Fall e a sa ha: ( ) ( ) ( ) [ ( ) d χ. (.8) ( ) [ ( ) ( ) ( ) [ ( ) [ d d d d. ( ) ( ). So < ad sg (.) h b a e ge ha: ( ) 4. Togeher h (.6) o e have: ( ) [ ( ) [ 4 4 D D d d. (.9) The he eqal (.8) beoes: ( ) ( ) D χ.

7 The eral solvg o a olear egral eqao o haerse pe 4 I s ell o ha he e seres ( ) K ( ). The χ ( ) goes orl o zero or. So e deosraed ha he seqee ( ) ( ) o ( ) goes orl o goes o a l ad he Lebesge daeal heore a be appled or he evalao o he l he sao ha or he egral: K ( ) [ ( ) d he he l o ( ) o o sd he seqee ( ) ( ) s a solo or he al eqao (.). We have overge. We a easl s ha hh s orl gog o a l ad ha ll be eve oos. For deosrao e ll se: e a hoose a sbseqee ( ) ( ) o ( ) ω ( ) χ ( ) ( ) K( ) [ ( ) The seqee { ω } relao(.9) e ge ha: d s eqall boded bease as a oseqee o he ( ) K ( ) d [ ( ) d D K ( ) ω d. [ ω ( ) ω ( ) [ K( ) K( ) [ ( ) D D D [ K( ) K( ) d [ ( ) [ K ( ) K( ) K( ) K ( ) [ K ( ) K ( ) K ( ) [ K ( ) K ( ) D [ K ( ) K ( ) d As ar as l χ ( ) e a pass ro { } So e sed he ollog: d ω o{ }. d

8 4 Mara Iga Esee Theore: I he erel K sases () () ş () ad he oos o ( ) s verg he odo(.6) he he olear egral eqao(.) has a leas oe solo (oos oe).. Dead aer heor The ed o he XIX-h er ad he begg o he XX-h er represeed or he los ehas perods o eree ese vesgaos. These geeraed pora ors hs eld o av. Aog he s also he av heor ho org s H. Helholz (868) ad G. Krhho (869) ors a heor elaboraed h he prpose o epla he D Aleber parado. D Aleber parado represes he orado beee he heoreal resl sag ha drg a sragh ad or ovg o a bod hrogh a deal ld ll be o ressae og ro he ld ad he epereal observao ha hs ressae es. Helholz reaed a ahea odel ad so he sared a heor hh beae a pora oe sall reerred as dead aer heor. I hs heor here are soe olear egral eqaos. Oe o hose o Haerse pe s solved hs paper sg soe eral ehods. 3. Neral resls Or prpose s o solve he ollog eqao: σ s T ( σ ) T () e l ( sσ ) r( σ ) sσdσ σ s The o hh s be egraed has a logarh sglar ad hs oe s a ee sglar. We ll rere he egral eqao as: σ ( σ ) s T ( σ ) T () e l ( sσ ) r( σ ) sσdσ σ s e T ( σ ) T () [ e ( sσ ) r( σ ) sσ e ( s ) r( ) s T () ( s ) r( ) s l σ dσ l σ dσ

9 The eral solvg o a olear egral eqao o haerse pe 43 The rs ad he seod egral a be allaed sg rapeza ehod ad he hrd oe ll be aalall allaed. We ll osder [ he ods { } h. Usg rapeza ehod ( σ ) dσ ( ) ( ) ( ) ad beases s : T ( ) e T e ( ) T ( ) ( ) l( s )( s ) r( ) T ( ) ( s ) r( ) s l T ( s ) r( ) s e T ( [ ) e ( s ) r( ) s e ( ) ( s ) r( ) s [( ) l( ) l l s s Thereore or T ( ) ( ) ( s ) r( ) e T s s ( ) l ( s ) r( ) s T e s l So all e have o solve he algebr sse: ( s ) l l( ) l s T T e (*)

10 44 Mara Iga h T T( ) ad Respevel s l s ( s ) r( ) s ( ) ( ) s r s l( s ) l l( ) l We ed o esae he derees: e T e S e T e S T S Bease o he deo o e sa ha: s l s σ σ ( sσ ) r( σ ) sσdσ σ s l σ σ s s s s σ sσdσ ( sσ ) sσd ( sσ os σ ) s dσ s σdσ

11 The eral solvg o a olear egral eqao o haerse pe 45 s s os σ ( ( ) ) s s ( L) ( )( ) ( 3 ) So e a sa ha he sse (*) has ol oe solo or < < ad hs solo a be od sg he sessve approao 3 ehod. olsos Soe elds o aves or eaple he oe regardg he los sdes a reae aer beg odeled a aheaal a soe olear egrals ad her solvg s o eqal eres or aheaas ad also or he oes org ore praal aspes. R E F E R E N E S. E. Kedall Aso A srve o eral ehods or solvg olear egral eqaos Joral o egral eqaos ad applaos Vol. 4 Nr. 99 p V.I. Arold Meode aeae ale ea lase Edra Şţă ş elopedă Breş Fel E. Broder Nolear oal aalss ad olear egral eqaos o Haerse ad Ursoh pe. orbos o olear oal aalss (Pro. Spos. Mah. Res. eer Uv. Wsos Madso Ws. 97) Aade Press Ne Yor V. Brîzăes; O. Săăşlă Maea speale Edra All Breş E. araol V.N. osaes Daa ldelor opresble Edra Aadee Roâe E. araol V.N. osaes Daa ldelor opresble Edra Aadee Roâe D.V. Ioes Eaţ dereţale ş egrale Edra Ddaă ş Pedagogă Breş 964.

12 46 Mara Iga 8.. Mor Approao ehods or solvg he ah proble. zehoslova Mah J.Aad. S. zeh Repb Vol.55 o N. Marov; A. arabea Eaţ egrale elare î sdl şărlor lde l lbere Mah. ReporsNo. 3 Breş S. Popp Modele aeae î eora avaţe Edra Tehă Breş 985.