New approach for numerical solution of Fredholm integral equations system of the second kind by using an expansion method

Transcription

1 Ieraoal Reearch Joural o Appled ad Bac Scece Avalable ole a wwwrabcom ISSN 5-88X / Vol : 8- Scece xplorer Publcao New approach or umercal oluo o Fredholm eral equao yem o he ecod d by u a expao mehod Nare Mahmood Dara Behra Mohammad halhala Faemeh oucha Nezhad Mad Youehohbah* 4 Deparme o Mahemac Malayer Brach Ilamc Azad Uvery Malayer Ira M Sc Deree o Mechacal eer Ferdow Uvery o Mahhad Ira M Sc Deree o Mahemac Kara Brach Ilamc Azad Uvery Alborz Ira 4 You Reearcher Club Hameda Brach Ilamc Azad Uvery Hameda Ira Correpod auhor emal: hohbah@auhacr ABSTRACT: A expao mehod ued or reame o ecod d Fredholm eral equao yem Th mehod ve a aalyc oluo or he yem The mehod reduce he yem o eral equao o a lear yem o ordary dereal equao Aer coruc boudary codo h yem reduce o a yem o equao ha ca be olved ealy wh ay o he uual mehod Fally or how he ececy o he mehod we ue ome umercal example Keyword: Syem o Fredholm eral equao; Taylor-ere expao; Boudary codo; Approxmao; Covoluo INTRODUCTION I h paper we ue a moded Taylor-ere expao mehod or olv Fredholm eral equao yem o he ecod d Th mehod wa r preeed Delve ad Mohamed 985 or olv Fredholm eral equao o ecod d ad he Maleead ad Ahazadeh 5; Re e al 999 or olv Volerra eral equao o ecod d ad Fredholm eral yem o ecod d Coder he ecod d Fredholm eral equao yem o he orm: Where F F K G K F d T [ ] T [ ] G [ ] I q he uco K ad G are ve ad F he oluo o be deermed Ad alo we aume ha q ha a uque oluo Coder he h equao o q We have d A Taylor-ere expao ca be made or he oluo he eral q a ollow:

2 Il Re J Appl Bac Sc Vol 8- Where deoe he error bewee ad Taylor-ere expao q Ideed we have I we ue he r m erm o he Taylor-ere expao q ad elec he erm q he by ubu q or he eral q we wll e: m! r r r r d I he eral q 4 ca be olved aalycally he he quae brace are he uco o aloe So q 4 become a lear yem o ordary dereal equao ha ca be olved However h dereal equao requre approprae boudary codo Boudary codo I order o coruc boudary codo we r dereae boh de o q So we oba he ollow dereal equao: U q ubue Where ad 6 Now we have: To oba or ad : d or d 4 he eral q 5 < < 8 m m m m d d Now combao o q 4 wh q 7-8 become a mh order lear yem o alebrac equao ha ca be olved For urher llurao we ow ue h mehod wh ad m or olv a ecod d Fredholm eral equao yem Coder he ollow ecod d Fredholm eral equao yem: F [ ] d m m F [ ] d A Taylor-ere expao ca be made or ad a ollow: m d d < <

3 Il Re J Appl Bac Sc Vol 8-!! I we ubue ad q9 he we e he ollow yem: d d d d q become a lear yem o ordary dereal equao ad ca be olved aer produc boudary codo For produc boudary codo we r dereae boh de o q 9 The we oba he ollow yem: Subue or he eral q ve: d d d d d d d d Combao o q wh q become a lear wo order yem o alebrac equao ha ca be olved ealy

4 Il Re J Appl Bac Sc Vol 8- Numercal example xample For he r example coder he ollow Fredholm yem o eral equao: Wh d 4 d / / 5 / / 5 4/ 6 / Ad wh he exac oluo / d / d 4 / ad Numercal reul are ve Table Table Numercal reul or xample wh m Approxmaed Approxmaed xample For ecod example coder he ollow Fredholm yem o eral equao: co d d e d d Wh co e co co Ad wh he exac oluo ad co Numercal reul are ve Table Table Numercal reul or xample wh m4 Approxmaed Approxmaed

5 Il Re J Appl Bac Sc Vol 8- CONCLUSION I h paper we ue moded Taylor-ere expao mehod or olv Fredholm eral equao yem o ecod d I eem ha h mehod alo applcable or ero-dereal equao ad eral equao o he ecod d wh mooh ad wealy ular erel all compuao were carred ou u Mahemac 7 o a peroal compuer RFRNCS Delve LM Mohamed JL 985 Compuaoal Mehod or Ieral quao Cambrde Uvery Pre Cambrde Maleead K Ahazadeh N 5 Numercal oluo o Volerra eral equao o he ecod d wh covoluo erel by u Taylorere expao mehod Appl Mah Compu 6: 95-9 Re K Zha B Qao H 999 A mple Taylor ere expao mehod or a cla o ecod d eral equao J Compu Appl Mah : 5-4