Convergence Analysis of Piecewise Polynomial Collocation Methods for System of Weakly Singular Volterra Integral Equations of The First Kind

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1 Appe a Coputatoa Mateatcs 28; 7(-): - ttp://wwwscecepubsggroupco//ac o: 648/acs287 ISS: (Prt); ISS: (Oe) Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K Goarea Karaa, Babak Sr, 2, *, Maa Kasf 2 Sa Sattar Aeroautca Uversty of Scece a Tecoogy, Sout Meraba, Tera, Ira 2 Departet of Appe Mateatcs, Uversty of Tabr, Baa Bouevar, Tabr, Ira Ea aress: g_karaa@ustacr (G Karaa), sr@tabruacr (B Sr), aakas@gaco (M Kasf) * Correspog autor To cte ts artce: Goarea Karaa, Babak Sr, Maa Kasf Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K Appe a Coputatoa Mateatcs Speca Issue: Sguar Itegra Equatos a Fractoa Dffereta Equatos Vo 7, o -, 28, pp - o: 648/acs287 Receve: Marc 2, 27; Accepte: Marc 22, 27; Pubse: Apr, 27 Abstract: We stuy reguarty of soutos of weaky sguar Voterra tegra equatos of te frst k We te stuy te uerca aayss of cotuous pecewse poyoa coocato eto for sovg suc systes Te a purpose of ts paper s te ervato of goba coverget a super-coverget propertes of trouce eto o te grae eses We appy reevat eto to a syste of fractoa ffereta equatos a aaye te Te uerca experets cofr te teoretca resuts Keywor: Dscotuous Pecewse Poyoa Spaces, Coocato Meto, Grae Meses, Weaky Sguar Voterra Itegra Equatos Itroucto I ts paper, we coser a syste of weaky sguar Voterra tegra equatos of frst k (SWSVIEFK) of te for (, ) ( t s) t k t s y s f ( t ), t I : [, T ], () were, < <, T R,, f : I R Te oa of te atrx fucto k( t, s): D R, s D I 2 {( t, s) ( t, s), s t T} Aso, we suppose tat k( t, t ) s a osguar atrx for a t I Te y: R R s te ukow vector fucto Te syste () s a Abe's tegra equato f k( t, s ) Te uerca souto of weaky sguar Voterra tegra equatos of frst k as extesvey bee stue (see for exape [2,, 4, 5, 6, 8, 9,,,, 4, 5, 6], but t oes ot ea tat ts subect as copetey bee stue Tere are soe usove probes wc are portat a ee ore caege Oe of te s covergece aayss of coocato eto o te pecewse poyoa spaces for sovg syste (), [] Te a of ts paper s to prove a copete covergece aayss of tese eto for ts syste Te pecewse poyoa coocato eto (PPCM) are easy prograabe a tey ave rap coverget orer for ay equatos cug tegra or ffereta operator Tey ave extesvey bee exae by ay autors We refer ere to [,, 7, 2] a terature gve tere Terefore, t s portat to aaye PPCMs for te syste () Suppose Let q: R R a f : Ω R be scaer a atrx fuctos, respectvey, were Ω s a set I ts paper, by qf we ea ( qf ) qf,, {,, }, a te or we ea te ax or

2 2 Goarea Karaa et a: Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K f ax sup f ( t), {,, }, t Ω Te paper s orgae as foow: I secto 2, we revew exstece, uqueess a sootess of te soutos of syste () I secto, we reca appcato of te coocato eto o te cotuous pecewse poyoa spaces I secto 4, we geerae Grawa s equates for atrx fucto equatos I secto 5, we gve te goba covergece of te coocato eto o te cotuous pecewse poyoa spaces I secto 6, we vestgate te stabty fucto trouce te prevous secto Fay, secto 7, we preset uerca experets wc support teoretca resuts 2 Reguarty Propertes Te arguets of ts secto ca be obtae by arguets sar to [] (secto 2) Here, we sou cocer tat te systes we vestgate are of eso greater ta we te syste [] s of eso greater Terefore, stea of vg by a fucto we sou utpy by a verse of correspog atrx fucto G CI a Suppose s( π) G ( ) k (, ) f( t)( t) t (2) π Defe resovet kere assocate wt te gve kere K (, s) as were a R(, s): H (, s) H (, s) H (, v) H ( v, s), 2, H (,s) s s( π) k( s + ( s), s) k (, ) π Te, we obta correspog Voterra s fuaeta resuts tat y( ) G ( ) + R(, s) G ( s) () s a uque souto of syste () ow, we ca arguet about te reguarty of soutos Supposg k ( ) k ( C ( D )), we obta H s C D a (, ) C ( D ) It s stragtforwar te to sow tat R s ( C D ) (, ) Terefore, te reguarty of y epe o te reguarty of G k Teore 2 Let k ( C ( D )), were g ( T ) f ( t) t β g( t) C ( D ),, C ([, ]), +,, a β Te, te syste () as a uque souto a tere exsts q ( ([, T]) ) C suc tat y q + β Proof Itegratg by substtuto t v, we ave Were a ece we obta t g t t t q β + β β : ( ), q v g v v v β t g( t)( t) Sce g ( T ) a cosequety t ( β ) + β () q ( ) + ( + ) q ( ) C ([, ]), we cocue tat ( T ) q C ([, ]), () q ( ) + ( + β) q ( ) C ([, T]) Tus, usg equato (2), tere exsts ( T ) q C ([, ]) + β suc tat G ( ) q ( ) Takg to accout tat q ( ): R(, s) G ( s) 2 + β R(, s) s + β q + β s + β ( C ) R(, v) v q ( v) v ([, T]), a usg te equato (4), we ave wc copetes te proof y q + q + β ( ), 2 Coocato Meto o te Cotuous Pecewse Poyoa Spaces For gve, Let r I { t T : }, (4)

3 Appe a Coputatoa Mateatcs 28; 7(-): - be a grae es, wt grag expoet r Assue tat ( t, t ], [ t, t ], t t a σ σ ax{ :,, } Te, t s stragtforwar to see tat te sequece { } s strcty creasg a < rt (5),,, We use coocato eto to sove syste () recty (wtout trasforg to te seco k tegra equato), o te cotuous pecewse poyoa spaces S ( I ): { v: v π (,,, )}, σ For cotuous pecewse poyoa spaces, et < c < < c, be te coocato paraeters Terefore, te approxate souto u ( I ) for, S, as te u ( t + v ) L ( v) U, v (,],,,, te terva σ ( u u, σ for,, ) Here, L (v) for,,, are Lagrage fuaeta poyoas wt respect to tct coocato paraeters a U,:u (t,) are approxato soutos at te coocato pots t t + c for,, a,, We are :, seekg for a coocato souto u suc tat satsfes te coocato cotos k( t, s) u ( s) f t t,,, ± (6), ( t s), for,, a,, Terefore, t s stragtforwar to sow tat te souto of te syste (6) ca be obtae by sovg recursvey te systes k( t, t v ) L ( v) F t + vu f t (7) +, c, ( t t v ),,, for,,, a,,, were k( t, t + v ) L ( v), F( t ), vu, ( t t v ), s te ag ter I te syste (7), te tegras ca be approxate by te quarature approxatos a c k( t, t + v ) L ( v), v a,,, k t t ( t t v ), (, ) k( t, t + v ) L ( v) v b (,, ) k ( t, t ),,,, ( t t v ), For, were a a, ( ) c, ( v) L ( v) v ( c v) L ( v) b (,, ) v,, t t for, {,, } Terefore, te fuy cretse verso ca be obtae by sovg te systes + ˆ,,,,,, Fˆ ( t ) a k( t, t ) U f ( t ), (8) recursvey, for,, a,,, were ˆ,,,, Fˆ ( t ) b (,, ) k( t, t ) U, for,,, a U ˆ are te coocato souto at, t, for,, a,, ow, te ese output approxate souto ca be approxate by uˆ ( t + v ) L ( v) Uˆ, v (,],,,, We ote tat ts cretse verso s sgty fferet fro cretse verso trouce [] However, t reuce te coputato copexty by a factor of, a as we w see t oes ot reuce te orer of te coocato eto Settg ( A) : a, ˆ ɵ, ˆ T U : [ U,, U ],, [ ˆ,, ˆ T F f t F t f t F( t )], a takg to accout tat,,,, k( t, t ) k( t, t ) +O,,, we ca wrte (8) te atrx for A ( k( t, t ) + O ( )) Uɵ F ow, sce A s vertbe by Teore 6, a k( t, t ) s vertbe by ypoteses of te troucto, te atrx A ( k( t, t ) +O ( )) s vertbe a tere exsts a uque souto for syste (8), for suffcety sa Cosequety, te fuy cretse coocato eto s we-efe I a sar aer te syste (7) as a uque souto for suffcety sa, a te trouce coocato eto s we-efe

4 4 Goarea Karaa et a: Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K 4 Grawa s Iequates Frst, we reca te Grawa s equates [] ote tat, we wrte v O( ) weever v O( ) Lea 4 (Growa's equaty) Assue tat { k }, ( ) s a gve o-egatve sequece a te sequece { } satsfes a + q + k wt a q for( ) Te exp + q k I ts paper, we ee a geeraato of Growa s equaty for te atrx fuctos Tus, we coser R + K + (9) were R,, K a for,,, are atrx fuctos We suppose tat R s a agoaabe atrx e, R P DP, were D ag( λ,, λ ) r s a agoa atrx a P s a osguar atrx Aso, we suppose tat λ [,) Te, It s stragtforwar to sow tat tere exsts a postve costat C suc tat ( + C ) exp C K 5 Goba Covergece 5 Dscotuous Coocato Meto Lea 5 For r, we ave cases ca be obtae by appyg te Hopta s rue k Teore 52 Let k ( C ( D )), ( C ( D )),, f( t) t β g( t) were g ( T ) C ([, ]), +,, a β Te te approxate souto u of te cotuous coocato eto for syste () wt coocato paraeters < c < c < < c 2 a te grag expoet r coverges to te souto f te egevaues of te stabty atrx R be te terva [,) Furterore, te coocato error satsfes y( t) u ( t) c r( + β ) +, r, + β, r, + β for a costat c > a suffcety arge Proof Suppose tat te assuptos of Teore \ref{t} o Let e ( t) y( t) u ( t) A appcato of Teore 2 pes tat ( y C ([, T]) ) for a > Terefore, by Peao s teore ([], Secto 8) λ + +, e ( t v ) L ( v) E R ( v) () were E e ( t + c ) for,,,,,, te, reaer R ( v ) s a boue fucto a β +,, λ, Oterwse () By subtractg equato () (at t t, ) fro equato (6) we obta k( t, s) e ( s) t,,, (2) ( t s), for,,,,, a ece, Hece r r ( ) O + O( ) r r ( + ) O( ), t + k( t, s) e ( s), ( s) t t, k( t, s) e ( s) t,, + t ( t s), () Lettg, substtutg s v, a usg () we ave a sary O ( ) O( ) Proof Oe ca easy observe te resuts by expag te poyoas ( + ) r a ( ) r for case r Te oter c k( c, v ), L v ve ( c v) λ c R ( v) v, (4) for,, By Tayor's teore for utvarabe atrx fuctos, k( c, v ) k(,) +O ( ) a te syste (4) ca

5 Appe a Coputatoa Mateatcs 28; 7(-): - 5 be wrtte te atrx for ( A k(,) + O( )) E O ( λ ) were E E E,, [,, ] T a L ( v) ( A ) v,, {,, } c (5) ( c v) Sce, A a k (, ) are vertbe atrces, we ca cocue tat E O ( λ ), for suffcety sa We ca, ow procee to obta a estate for e, e σ Substtutg s t + v,,,, to correspog tegras () a usg () we obta 2, ( t t s ) ( t t s ),, + + Rˆ k( t, t + s ) k( t, t + s ),, L ( s) k( t, t + s ) L ( s),, ( t t s ), k( t, t + s ) L ( s) c,, ( t t s ), c k( t, t + s ) L ( s),, ( t t s ), (6) Rewrtg (6) wt repace by a a subtractg t fro (6), we obta k( t, t + s ) k( t, t + s ),, 2 L s, ( t t s ) ( t t s ),, were + + Rˆ Rˆ k( t, t + s ) L ( s),, ( t t s ), c k( t, t + s ) L ( s), ( t t s ), c, k( t, t + s ) L ( s),, ( t t s ), k( t, t + s ) k( t, t + s ) 2,, λ ( t t s ) ( t t s ),, (7) R ( s) + + c λ k( t, t + s ) R ( s), ( c + s) λ k( t, t + s ) R ( s), ( c s) c O + λ k( t, t + s ) R ( s), ( c s) λ ( ), (8) by Lea 5 a foowg Reark Reark 5 We ote tat, k( x, t + s ) ( x, s) ( C ([ t, t ] [,]),, ), for ( x t s ),, 2 Terefore, f we appy ea vaue teore for eac copoets of te above atrx fucto, te we ave k( t, t + s ) k( t, t + s ) ( t t s ) ( t t s ),,,, k ( t + s ) were tere exst t θ t suc tat, pq,, ( k ( t + s )) pq + ( c ( c ) ) kpq (, t s ) pq, θ + ( θ t s ) pq, for p, q {,, },,, a,, 2 (9) k By our assuptos s boue a tere exsts M > k suc tat ( c + ( c ) ) < M Terefore, ( k ( t + s )) L ( s) pq M L ( s) ( θ t s ) pq, M L ( s) ( t t s ) M2, ( ), (2) for,, 2 Te ast equaty s obtae by Lea (62) [] Aso, settg M2 γ, ( ) we cocue tat

6 6 Goarea Karaa et a: Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K ( k ( t s )) L ( s) + γ ( ) (2) Furterore, to obta equato (8), we use te fact tat 2 te su ( ) s boue Substtutg (9) to (7), a usg Tayor's seres for oter ters a copoets we obta 2 k ( t + s ) L ( s), + + ( k( t, t ) + c ( s) + c ( s) ) L ( s) 2, ( c + s) c c Rˆ ( k( t, t ) + c ( s) ) L ( s) ( s), ( k( t, t ) + c ( s) ) L ( s) c 4, ( c s) (22) were c( s) for,,4 are boue atrx fuctos Mutpyg (2) by k k( t, t ) a vg by, we obta 2 k ( t + s ) L ( s), + k k ( + c ( s) + c ( s) ) L ( s) 2 ( c + s), c k c4( s) L( s), ( c s) k c ( s) L ( s) c (2) ( K ) :,, ( c s) a we ca rewrte syste (22) te atrx for 2 B E () ( O( ) 2 ) + A + A + K IE Rˆ + + ( (2) ) A K IE, (24) were I s etty atrx of eso Mutpyg equato (2) by ( ) ( A + K ) I ( A + K ) I (2) (2) (wc exsts for suffcety sa, by Teore 6), we obta E ( R + O( )) E 2 λ + C E + O( + ) (2) were C ( A + K ) I B a R ( A ) A A I 2 (25) + Deotg, ( + k c ( s) ) L ( s) c s c 4 ( c s) Rˆ c, ( + k c ( s) ) L ( s), ( B ) :,,, 2,, k t + s L s L ( s) ( A ) :,, ( c + s) c L ( s) ( A ) :, 2, ( c s) c L ( s) ( A ) :,, ( c s) ( k c ( s) + k c ( s)) L ( s) 2 (), ( c + s) ( K ) : (2) Usg Lea 5 we ca cocue tat E a by (5) we ave O E ( R + ( )) 2 λ C E O + + ( + ) E R E + O( ) E 2 λ + C E + O( + ) (26) If egevaues of te stabty atrx R be te terva [,), we ca voke geerae Grawa's equaty to cocue tat λ C ( ) E E + O exp C C O( ) (27) were C > s a costat Reark 54 By cotuty a o-sguarty assuptos, for suffcety sa (arge ), we ca f

7 Appe a Coputatoa Mateatcs 28; 7(-): - 7 M > suc tat ( A + K ) I (2) M (2) ( A + K ) I γ M γ by (2), ( B ) γ ( ) a terefore, M( ), C (28) Cosequety, te equato (28) ye 2 2 C M ( ) 2 + M ( s) M ( s) + ( ) M + ( ) rtm rtm (29) Takg to accout te equatos (27) a (29), we obta te a resuts of ts secto: E O T + ( rt) rλ MrT C + O exp 52 Dscretse Dscotuous Coocato Meto () ow, we ca state a error bou for soutos of cretse cotuous coocato eto k Teore 55 Let k ( C ( D )), ( C ( D )),, f( t) t β g( t) were g ( T ) C ([, ]), +,, a β Te te approxate souto u of te cretse cotuous coocato eto for syste () wt coocato paraeters < c < c < < c 2 a te grag expoet r coverges to te souto f te egevaues of te stabty atrx R be te terva [,) Furterore, te coocato error satsfes y( t) u ( t) c r( + β ) +, r, + β, r, + β for a costat c >, a suffcety arge Proof By parttog te oa of tegra () a substtutg s t + v we obta k( t, t + v ) y( t + v ) v ( t t + s ) t t k t t + v y t + v ( t t + s ) (, ) + v f( t) (2) for,, a,, Appyg Teore 2 a Peao s teore, we obta k( t, t + v ) y( t + v ) λ,, k( t, t ) y( t ) L ( v) + R ( t, v), were λ s efe by (\ref{aba}) a R ( t, v ) for,, are ufory boue fuctos o D Settg t t,, a usg (), we obta were vk t t y t,,, ( t t + s ), c ( t t + s ), (, ),,,,, L ( v) (, ) L ( v) + vk t t y t + Rɶ () f( t ), R ( t, v) Rɶ v, λ, ( t t + s ), λ c R ( t, v), + v ( t t + s ), (4) for,, a,, ow, we ca use te otato of prevous sectos to wrte te equato () te for b (,, ) k( t, t ) y( t ),,, + a k( t, t ) y( t ) + Rɶ f ( t ),,,,, By subtractg equato (2) fro equato (8) we ave b (,, ) k( t, t ) Eˆ,,, + a k( t, t ) Eˆ + Rɶ,,,,, (5) (6) were Eˆ y( t Uˆ ) for,, a,,,,, Rewrtg (6) wt repace by a a subtractg t fro (6), we obta

8 8 Goarea Karaa et a: Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K 2 ( (,, ) (, ) (,, ) (, ) ) b k t t b k t t Eˆ,,,,, + b (,, ) k( t, t ) Eˆ,,, a k( t, t ) Eˆ,,, + a k( t, t ) Eˆ + Rɶ Rɶ,,,,,, (7) for,, a,,, were t s stragtforwar to sow tat Rɶ λ, Rɶ, O ( + ) Aso, by Reark 5, we ave b (,, ) k( t, t ) b (,, ) k( t, t ),,,, k ( t ), Usg ea vaue teore for oter ters a copoets of equato (7), tere exst boue atrx fuctos cɶ ( s), for,,4 suc tat 2 k ( t ) L ( s),, + + ( k( t, t ) + cɶ ( s) + cɶ ( s) ) L ( s) 2, ( c + s) c c,, ( c s),, ( k( t, t ) + cɶ ( s) ) L ( s) ( s) c ( k( t, t ) + cɶ ( s) ) L ( s) 4 + Rɶ Rɶ,,,,,, (8) Te equato (8) s sar to (22) a ece te rest of te proof s sar to te es after ts equato 6 Stabty Matrx We ca use te we-kow forua of terpoato to obta c L ( s) s,,,,, b L ( s) b s c ( s) ( s) (9) Terefore, efg foowg atrces a tat ( D), c c C:, (4) c c s ( D ) :,, ( c + s) c s ( D ) :, 2, ( c s) s c for, {,, }, ( c s) 2 2 we ca sow A C D, A C D, A C D (4) Te atrx C s a Vaeroe atrx, a s vertbe Tus, A s vertbe f a oy f D be vertbe Te eeets of D, D a D ca be copute by te 2 foowg forua s ( s) b b + a for D we ave + + ( ) + c c + D + c c + Teore 6 Te atrces A a D are vertbe atrces Proof By prevous arguet, t reas to prove tat D s vertbe It s ot eoug to sow tat te cous of D are epeets Let D ( ) be te -t cou of te atrx D a et D, were R Terefore, x x k k k k for x c,, c Hece, te poyoa k p( x) x k k k s of egree, a as roots, ( p( c ) for,, ) a s ero poyoa by fuaeta teore

9 Appe a Coputatoa Mateatcs 28; 7(-): - 9 of agebra Hece,, for,,, wc copetes te proof Base o te Teore 52, te covergece of te trouce coocato eto epe o te egevaues of te stabty atrx R ( A ) A A I 2 Sce, A s vertbe, te stabty atrx R s weefe Aso, te atrces R a te atrx ( A ) ( A A 2 ) ave te sae egevaues, a wtout oss of geeraty, we ca reefe te stabty atrx as R : ( A ) ( A A ) Moreover, we ca use equato (4) to 2 obtac R C ( D ) ( D D ) 2 Terefore, R a ( D ) ( D D ) are sar a ave te 2 sae egevaues Ts fact ca ep us to copute te egevaues of R For case, we ave 2 c ( + c ) ( D ) ( D D ) 2 c For,, t agrees wt te resuts of [] wc s te Voterra tegra equato of frst k For 2, oe ca recty copute te egevaues of ( D ) ( D D ) for a 2 gve coocato pots 7 uerca Experets We gve soe exapes to sow effcecy a effectveess of te trouce eto I tese exapes, we obta te absoute error wt respect to te paraeters a t, we eote t by ax u ( t ) y ( t ),,,,,,,, for, were y ( t ) a u ( t ) are exact a uerca soutos of te -t copoets, respectvey We report te approxato of coverget orer usg te forua o g, We appy te foowg eto 2 2 for te ext exapes: Meto, a c 5 Meto 2, c 5 a c 2 Meto 2 Roots of sfte Legere poyoa of egree + 2: c, c Meto Roots of sfte Cebysev poyoa of egree 2: c, c Meto 4 Roots of sfte Cebysev poyoa of egree : c, c, c Meto 5 Roots of sfte Legere poyoa of egree : c, c, c Set λ ax λ, were λ for,2, are te,2, egevaues of stabty atrx R Tabe sows te vaues of λ for eto -5, a fferet vaues of Ts sures us tat te uerca souto of te correspog eto s coverget to te exact souto, for gve exapes Tabe λ ax λ, for eto -5 a fferet vaues of,2, M M M M M a ow, coser syste () wt f( t) t s e k( t, s) Γ ( ) ( s t) e ( s + t) e + s s e 2t s Γ ( β + ) t Γ ( β + + 2) Γ ( β + + ) 2 e + + t Γ ( + β) Γ ( + β) Γ (2 + β) β t Γ ( β + + ) 2t Γ ( β + ) Γ ( β + + ) te + + t Γ (2 + β) Γ ( + β) Γ (2 + β) Te, te exact souto of te two esoa syste () s β + β + (, ) y( s) e s e s s s T We ca costruct te foowg exapes by aterg te paraeters of te above syste Exape 7 Let T, β a 2 4,, te appy 5 5 te eto Exape 7 2 Let T 5, β a,,, te appy te eto 4-5 Exape 7 Let T, β,5,2 a 9, te appy te eto -5 By Teore 52, we expect te error estate e ( t) c r, r,, r, + β Exapes 7-72 Tabes, 2-5 cofr ts error estate Tabe 2 Te absoute error a correspog coverget orer Exape wt r 4 8 M 64 e-2-688e-4 -

10 Goarea Karaa et a: Covergece Aayss of Pecewse Poyoa Coocato Meto for Syste of Weaky Sguar Voterra Itegra Equatos of Te Frst K M 2 M e e e e e e e e e e e e e e e e Tabe Te absoute error a correspog coverget orer Exape wt M M 2 M 2 r e e e e e e e-4-55e e e e e e e e e e e-6 92 Tabe 4 Te absoute error a correspog coverget orer Exape 2 wt r M 4 M e-4-76e e e e e e-4-445e e e e e Tabe 5 Te absoute error a correspog coverget orer Exape 2 wt M M 2 r e-6-24e e e e e e-6-26e e e e-8 4 e-8 98 Aso, by Teore 52, we expect te estate e ( t) c ( β ) r, r, β, r, + β Exape 7 Tabes, 6-7 sow ts estate I a tabes, we reporte estate of te orer for te copoet of te syste wc s ess Fay, tese exapes sow tat te obtae coverget resut s opta a ca t be prove for our vestgate cass of SWSVIEFKs Tabe 6 Te absoute error a correspog coverget orer Exape wt r M M 2 M M 4 M 5 β β e-4-598e e e e-4-272e e e e-4-986e e e e e e e e-5-494e e e Tabe 7 Te absoute error a correspog coverget orer Exape wt r M M 2 M M 4 M 5 β β β e-7-597e e e e-6-69e e e e-5-5e e e e-8-697e e e e-8-697e e e Cocuso A covergece aayss of te coocato eto for SWSVIEFKs o cotuous pecewse poyoa spaces as bee vestgate Base o ts aayss, te orer of te eto oes ot cage by creasg coocato paraeters o ufor es However, t ca be crease up to by usg grae es Our aayss sta o te egevaues of stabty atrx We obtae a cose for of ts egevaues for case However, for cases >, we obtae te egevaues of stabty atrx for prescrbe coocato paraeters Refereces [] K E Atkso, Te uerca souto of tegra equatos of te seco k, Vo 4, Cabrge uversty press, 997 [2] H W Braca, Te oear Voterra equato of abe's k a ts uerca treatet, Coput, 978, 2, 7-24 [] H Bruer, Coocato eto for Voterra tegra a reate fuctoa ffereta equatos, Vo 5, Cabrge Uversty Press, 24

11 Appe a Coputatoa Mateatcs 28; 7(-): - [4] H Bruer, A Peas a G Vakko, Pecewse poyoa coocato eto for ear Voterra tegro-ffereta equatos wt weaky sguar keres, SIAM Joura o uerca Aayss, 2, 9, [5] R Caero a S McKee, Prouct tegrato eto for seco-k Abe tegra equatos, Joura of Coputatoa a Appe Mateatcs, 984,, - [6] P Eggerot, A ew aayss of te trapeoacretato eto for te uerca souto of Abe-type tegra equatos, Joura of Itegra Equatos a Appcatos, 98,, 7-2 [7] R Kress, V Ma'ya a V Koov, Lear tegra equatos, Vo 82, Sprger, 989 [8] C Lubc, Fractoa ear utstep eto for Abe- Voterra tegra equatos of te frst k, IMA Joura of uerca Aayss, 987, 7, 97-6 [9] A Peas a E Tae, O te covergece of spe coocato eto for sovg fractoa ffereta equatos, Joura of Coputatoa a Appe Mateatcs, 2, 25, [] H Te Ree a P Scroevers, A coparatve survey of uerca eto for te ear geerae Abe tegra equato, ZAMM- Joura of Appe Mateatcs a Mecacs, 986, 66, 6-7 [] A Saaata a M Dega, A coocato eto for sovg Abe s tegra equatos of frst a seco k, Zetscrft für aturforscug A, 28, 6, [2] B Sr, uerca souto of ger ex oear tegra agebrac equatos of Hesseberg type usg cotuous coocato eto, Mateatca Moeg a Aayss, 24, 9, 99-7 [] V Voterra, Sua versoe eg tegra eft, Att ea Accaea ee scee Toro,896,, -2 [4] R Wess, uerca proceures for voterra tegra equatos, Buet of te Austraa Mateatca Socety, 97, 8, [5] R Wess, Prouct tegrato for te geerae Abe equato, Mateatcs of Coputato, 972, 26, 77-9 [6] R Wess a R Aersse, A prouct tegrato eto for a cass of sguar frst k Voterra equatos, uerca Mateatcs, 97, 8,