Numerical Methods using the Successive Approximations for the Solution of a Fredholm Integral Equation

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1 ece Advce Appled d eorecl ec uercl eod u e Succeve Approo or e Soluo o Fredol Ierl Equo AIA OBIŢOIU epre o ec d opuer Scece Uvery o Peroş Uvery Sree 6 Peroş OAIA rdorou@yoo.co Arc: pper pree wo eod or ppro e oluo o Fredol erl equo u e ucceve pproo eod w e rpezod orul d e recle orul repecvely or pproe clculo o erl pper e er o e ucceve pproo equece. ee pproo eod wll e eled uder e codo o eore o eece d uquee o e oluo o erl equo e pere B ; r ee preeed 7. ey-word: - Fredol erl equo pproo o e oluo ucceve pproo eod rpezod orul recle orul Overvew We coder e Fredol erl equo: d were or J J cloed ervl d. I e pper 7 ee uded e eece d uquee o e oluo o erl equo e pce d e pere B ; r repecvely. u ve ee oed wo eore we pree elow. Fr o ll we deoe y pove co uc or ll. eore o eece d uquee e pce 7 For e Fredol erl equo we ue : ; ere e > uc u u u v v v u v u v u v or ll u v ;. Uder ee codo e erl equo uque oluo * c e oed y e ucceve pproo eod r y elee. oreover e - ucceve pproo e we ve e ollow eo: *. eore o eece d uquee e pere B ; r 7 For e Fredol erl equo we ue : J J cloed ervl ; ere e > uc u u u v v v u v u v u v or ll u v J ;. I ere e r > uc B ; r J d e ollow codo ullled: ISB:

2 ece Advce Appled d eorecl ec v r were pove co uc or e rerco J cloed ervl we ve: J u v w or ll u v w J e e erl equo uque oluo * B ; r c e oed y e ucceve pproo eod r y elee B ; r. oreover e - ucceve pproo e e eo e. e erl equo o lr ype ve ee uded e uercl ly o erl equo co ele o eod or ppro e oluo o e uded equo. Uder e codo o eore preeed ove e purpoe o pper o develop wo eod or ppro e oluo o e erl equo u e ucceve pproo eod oeer w e ollow wo qudrure orul: e rpezod orul d e recle orul were ued e pproe clculo o e erl ro e er o e ucceve pproo equece. e ee o e prole o el ee procedure or ppro e oluo o e erl equo were ued e reul ve y G o I. u G. Pvel d I. A. u. V. Ioecu 9 I. A. u 5 V. ureș d Geore recu. We uppoe e codo o eore re ullled d ereore e erl equo uque oluo e pere B ; r. We deoe oluo y * B ; r d c e oed y e ucceve pproo eod r y elee B ; r. I ddo e - ucceve pproo e e eo rue. ereore or e deero o * we pply e ucceve pproo eod. o e eer reul codered equd dvo o e ervl rou e po.... ow we ve e equece o ucceve pproo: B ; r d d d e e ollow wo eco we pree wo eod or ppro e oluo o erl equo oed y pply e ucceve pproo eod d u lo e rpezod orul d e recle orul or e pproe clculo o e erl pper e er o e ucceve pproo equece. Approo o e oluo u e rpezod orul o eec we uppoe J J cloed ervl d. We wll pproe e er o e ucceve pproo equece u e rpezod orul: d 5 were... equd dvo o e ervl d e re o orul 5 v e eo:. 6 For e clculo o e erl pper e er o e ucceve pproo equece e rpezod orul 56 o o e ppled. I e eerl ce or we ve: ISB:

3 7 w e eo o e re ". 8 Sce d J reul J d ere e e dervve o e uco ro e epreo o d ereore o e clculed. So we ve: d d d d d ereore " d d d d ". I we e o ccou e epreo o e dervve o - d we deoe u w v u e we o e ollow eo or - d dervve: " wle or e dervve o uco we ve e eo: " 5 4. I ovou doe deped o d ereore e eo o e re : 9 were d u we o orul or e pproe clculo o e erl pper e er o e ucceve pproo equece. U e eod o ucceve pproo d e orul 7 w e eo o e re reuled ro 9 we ue urer o lor order o olve e erl equo pproely. o ed we wll clcule pproely e er o e ucceve pproo equece. u we ve: d ece Advce Appled d eorecl ec ISB:

4 d were. reo coue or d rou duco we o: d { } L. Sce e codo o eore o eece d uquee o e oluo o erl equo re ullled reul d we ve e eo:. u we ve oed ew equece ee e ucceve pproo equece u equd dvo o e ervl :... w e ollow error clculo: wc u e eyev or ecoe:. ow u e ee d o e ollow reul. eore Suppoe e codo o eore re ullled. I ddo we ue e ec oluo * o e erl equo pproed y e equece o e ode o e equd dvo o e ervl u e ucceve ece Advce Appled d eorecl ec ISB:

5 pproo eod 4 d e rpezod orul 56. Uder ee codo e error o pproo ve y e ollow evluo: *. 4 Approo o e oluo u e recle orul o eec we uppoe J J cloed ervl. We wll pproe e er o e ucceve pproo equece u e recle orul coder e eredry po o e dvo o e ervl o e le ed o e prl ervl ξ I ely d 4 or coder e eredry po o e dvo o e ervl o e r ed o e prl ervl ξ I ely d 4 w e eou dvo o e ervl :... d we ve ued. e re o e orul e eo:. 5 For e clculo o e erl pper e er o e ucceve pproo equece e recle orul 45 o o e ppled. I e eerl ce or we ve: 6 w e eo o e re d d. 7 Sce d J reul J d ere e e dervve o e uco ro e epreo o d ereore o e clculed. So we ve: d d d ereore d d d. I we e o ccou e epreo o e dervve o - d we deoe u w v u e we o e ollow eo or - d dervve: wle or e dervve o uco we ve e eo: ece Advce Appled d eorecl ec ISB:

6 . I ovou doe deped o d o e eo o e re : 8 were d u we o orul or e pproe clculo o e erl pper e er o e ucceve pproo equece. U e eod o ucceve pproo d e orul 6 w e eo o e re reuled ro 8 we ue urer o lor order o olve e erl equo pproely. o ed we wll clcule pproely e er o e ucceve pproo equece d we o: d were. reo coue or d rou duco we o d { } L. Sce e codo o eore o eece d uquee o e oluo o erl equo re ullled reul d we ve e eo:. 9 u we ve oed e equece ee e ucceve pproo equece u equd dvo o e ervl :... w e ollow error clculo: wc u e eyev or ecoe: ece Advce Appled d eorecl ec ISB:

7 ece Advce Appled d eorecl ec. ow u e ee d o e ollow reul. eore 4 Suppoe e codo o eore re ullled. I ddo we ue e ec oluo * o e erl equo pproed y e equece o e ode o e equd dvo o e ervl u e ucceve pproo eod 4 d e recle orul 45. Uder ee codo e error o pproo ve y e evluo: *. 5 ocluo erd o e wo eod preeed pper we oerve e ollow: I o ce we ued e eod o ucceve pproo; e er o ucceve pproo equece were pproed u wo qudrure orul: e rpezod orul 56 d e recle orul 45 repecvely. c I o ce we ued equd dvo o e ervl rou e po... ; lo e o ce we o ew equece ee e ucceve pproo equece ; d We oed e ollow ee o e error o pproo o ode o e er o ucceve pproo equece: - u e rpezod orul: - u e recle orul: d o vlue d ro evluo o re re depede o d ; e Flly e error o pproo o e ec oluo o e erl equo rou e er o e ew r ve y e relo we we ued e rpezod orul d y e relo we we ued e recle orul. O e wo ee o reder ro ove we deduce e pproo error o e oluo oed y pply e ucceve pproo eod le e rpezodl orul ued e recle orul ued. Flly ould e oed ll rcle d oo repecvely ro reerece coued por reerc erl prepro o rcle. eerece: Aro. Approo o e oluo o erl equo w oded rue Sud Uv. Beș-Boly ec 978 pp. 6- o. Adrá Sz. Fredol-Volerr erl equo Pure. Appl. : pp.-. o G. u I. Pvel G. d u I. A. Iroduco e Operol Equo eory c luj-poc 976 o. 4 orţou. Aly o erl equo w oded rue Sud Uv. Beş- Boly Vol.5 o. 6 pp orțou. Ierl Equo w oded Arue luj Uvery Pre luj-poc 9 o. 6 orţou. orţou A-. A ppro lor or e oluo o erl equo ro epdec Al dell Uver d Ferrr Vol. 56 Iue pp OI.7/ orțou. A l o oler Ierl Equo rylv Jourl o ec d ecc Vol. 4 o. pp Hcuc W. Ierl equo Bräuer Berl Ioecu. V. uercl Qudrure ecă Bucre 957 o. ISB:

8 ece Advce Appled d eorecl ec recu G. ecl Aly Vol. I dccă ş Pedocă Bucre 98 o. ure V. Fucol-Ierl Equo edr luj-poc. Poly A.. zrov A. V. Hdoo o erl equo Pre Lodo 998. Precup. oler Ierl Equo Be-Boly Uvery o luj-poc 99 o. 4 Precup. eod oler erl equo luwer Acdec Puler. 5 u I. A. Prcple d Applco o e Fed Po eory c luj-poc 979 o. 6 o A. e eerlzed oluo o e oudry-vlue prole rerd e ed o e elc rod o elc oudo. I. e ye o eerlzed equo Proceed o e o Acdey ere A: ec Pyc eccl Scece Ioro Scece o./8. 7 o A. e eerlzed equo o rverl vro o vcoelc rod o vcoelc oudo. epor pp.5-4. ISB:

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