Some algorthim for solving system of linear volterra integral equation of second kind by using MATLAB 7 ALAN JALAL ABD ALKADER
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1 . Soe lgoi o solving syse o line vole inegl eqion o second ind by sing MATLAB 7 ALAN JALAL ABD ALKADER College o Edcion / Al- Msnsiiy Univesiy Depen o Meics تقديم البحث :-//7 قبول النشر:- //. Absc ( / ) 9 (vesion 7 ) ( MATLAB7 ) In is ppe i is sdy nd odiy soe Qde eod o solve syse o Vole inegl eqion o ode nd sing soe Algoi o ppe seps o solve sing wo ind o Qde eod (Tpezi Sipson /) nd coping beween o eod. Te les sqe eo nd nning ie o ind e esl by sing odiy pog (MATLAB) vesion 7 podc 9. e Sipson's le vey good eod ll less e les sqe eo nning ie 7
2 .. Inodcion: A neicl qde (neicl inegion) les is piy ool sed by enginees nd scieniss o obin ppoie nswes o deinie inegls cn no be solved nlyiclly [5]. Also i is bsis o evey neicl eod o solion o inegl eqion []. By sed qde eods o solve second ind inegl eqions o convolion ype. We is pesen e os ili o neicl inegion e qde les (Tpezoidl le Sipson'sl/ le. And sed o e syse o line VIE's. Q n [ ] w ( ) w ( ) w ( ) w ( ) wi e popey b ( ) d Q[ ] E[ ]... is clled neicl inegion o qde ol Te e E[] is clled e ncion eo o inegion. Te vles { } n e clled e qde nodes nd { e clled e weigs []. n w } n Te ollowing e bsic ols o qde eods. n. Tpezoidl Rle:[5] Te ili nd siple pezoidl le is bsed on ppoiing () on inevl ( b ) by e sig line oining ( ( )) nd ( ( )). Hence sing e e o pezoid wi eig /[U( ) U( )] gives : ( ) d [ ( ) ( )] n ( z) (.) Wee z ( ) wen e leng o inevl [ b] is no sicienly sll e pezoidl le (.) is no o c se. Fo sc n inevl we be i ino sli o inegls ove sll sbinevl nd en pply (.) o ec o ese slle inegls we cll i e coposie pezoidl le. 8
3 . 9 Le b-/n n nd i i * (i n); en i cn be wien s: Using Tpezoidl Rle Using epeed Tpezoidl le in eqion e is sge l we obin e ollowing se o ppoiing eqions:... K wee i i ( ) o ll n. on sipliying e bove we obin: ( ) ( ) ( ) I ollows e sge i ; I n. e obined by solving e syse.
4 . 5 ( ) ( ) ( ) ( ) ( ) ( ) (.) Finlly e Gss-Eliinion pocede is sed o solve e syse (. ) o ec vle o... n o ind ( ); I.... Te lgoi (VIETRAP) Sep :. Asse (b - )/n n N b. Se i i i.... Sep : To cope il i... we se sep (l) nd ping in eq.(. ) nd se Gss-Eliinion pocede o solve e esling syse. Sep : In e se wy s in sep () nd by sing eqion (. ) we cope U l U... ;... n. Sipson's / Rle[] Now e Sipson's / le is sed o ppoie ec inegl e in eqion (.) is sge ppoied ec inegl e in eqion (.) by e Tpezoidl le by sing eq. (.) o ge U l i.... I ollows e sge i e nbe o sbinevls e even we pply e Sipson's / le. Teeoe U i... e obined by solving e syse:
5 . 5 (.) ( ) ( ) ( ) Ten e Gss-Eliinion pocede is sed o solve e syse (. ) o ec vle o... n o ind U i ( ) i.... Also i e nbe o sbinevls e odd we ge cobinion beween Tpezoidl nd Sipson's / les. A e sge U i i... e obined by solving e syse:
6 . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (-) Finlly e Gss-Eliinion pocede is sed o solve e syse (. ) o ec vle o n o ind i ( ); I.... Te Algoi ( VIESIMP/) Sep Asse (b-)/n n N Se i i i Cope i i. by sing (.) wi sepe ( nd ) wen e Nbe o sbinevls Sep Use Gss Eliinion pocede o solve e esling syse 5
7 .. Neicl Solion o Syse o Line VIE 's Using Qde Meods.[7] In is secion we se qde eods o ind e neicl solion o syse o line VIE's o e o: ( ) ( ) i( ) ( ) d I [ b ]i... (.5) wee N ; i... e ssed o be coninos on I nd i ; i... denoes given coninos ncions. Moeove e iniil vles e obined o eq. (.5) by seing gives: i.... Sppose e inevl [ b] is divided ino n eql sbinevls o leng ( b ) sc < <... < n b wi * l n... n nd ( ) o i n cn be deeined sc : ( ) ( ) ( ) ( ) d ( ) ( ) d ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( ) d ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( ) d ( ) ( ).. Neicl Eples: d d d (.) In is secion we es soe o e neicl eples peoed in solving e line syse o Vole inegl eqions. Te ec solion is sed only o sow e cccy o e neicl solion obined wi o eod. 5
8 . Eple Conside e poble: 8 ( ) ( ) d ( ) ( ) d ( ) ( ) d ( ) ( ) d wic is syse o wo line VIE's wi () (). Tble (-) nd (-) pesen copison beween e ec nd qde eod o () nd () especively depending on les sqe eo nd nning ie wi.. Tble (-) X Ec Tp SIMP L.S.E..5e-5.55e -7 Tble (-) X Ec Tp SIMP
9 L.S.E..97e-.85e -8 R.T Eple (): Conside e poble: ( ) e ( ) d ( ) e e ( ) d Wic is syse o wo line VIE's wi () e (). Tble (.) nd (.) pesen copison beween e ec nd neicl solion o ive ypes o qde eods o () nd () especively depending on les sqe eo nd nning ie wi.. Tble (-) X Ec Tp SIMP L.S.E..5e -5.e -7 55
10 . Tble (-) X Ec Tp SIMP L.S.E..5e-5.55e -7 R.T Discssion: In is ppe we ve inodced qde eods o solving syse o line VIE's: Fo ec eod cope pog ws wien nd n eple ws solved. Fo esie copison e solions wee epesened in bl os. We conclde e ollowing es: Te nbe o sbinevls n is esiced o be even o Sipson's / Tog e solion o syse o line VIE's we see Tpezoidl nd Sipson's / gives e bes esls wen i is coped wi oe qde les. [See eple () () ]. Reeences: [] ABD-AL-Heed F.T.; "Neicl Solion o Fedol Inegodieenil Eqion Using Spline Fncion"; M.Sc. esis Univesiy o Tecnology. 5
11 . [] AL-Fo O.M.; "An Appoie Solion o Solving Line Syse o Inegl Eqion wi Applicion on "Si" Pobles"; Univesiy o Tecnology. [] AL-Jboy A.K; "Neicl Tecniqes o Specil Type o Inegl Eqion"; M.Sc. esis Univesiy o Tecnology. [] AL-Nsi R.H.; "Neicl Solion o Vole Inegl Eqion o e Second Kind"; M.Sc. esis Univesiy o Tecnology 999. [5] AL-Rwi S.N.; "Neicl Solion o Fis Kind Inegl Eqion o Convolions Type"; M.Sc. esis Univesiy o Tecnology [] AL-Sli B.F.; "Neicl Solion o Non-Line Vole Inegl Eqion o e Second Kind"; M.Sc. esis Univesiy o Tecnology. [7] AL- Tiee AJ.; "Appoied Meod o Fis ode Vole Inego-dieenil Eqions"; M.Sc. esis Univesiy o Tecnology. 57
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