A Convergence Analysis of Discontinuous Collocation Method for IAEs of Index 1 Using the Concept Strongly Equivalent

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1 Appled ad Coputatoal Matheatcs 27; 7(-): do:.648/.ac.s ISSN: (Pt); ISSN: (Ole) A Covegece Aalyss of Dscotuous Collocato Method fo IAEs of Idex Usg the Cocept Stogly Equvalet Gholaeza Kaaal, Babak Sh, *, Elha Sefdga 2 Shahd Satta Aeoautcal Uvesty of Scece ad Techology, South Mehabad, Teha, Ia 2 Atatük Uvesty Faculty of Scece, Depatet of Matheatcs, Ezuu, Tukey Eal addess: g_kaaal@ust.ac. (G. Kaaal), sh@tabzu.ac. (B. Sh), e_sefdga@yahoo.co (E. Sefdga) * Coespodg autho To cte ths atcle: Gholaeza Kaaal, Babak Sh, Elha Sefdga. A Covegece Aalyss of Dscotuous Collocato Method fo IAEs of Idex Usg the Cocept Stogly Equvalet. Appled ad Coputatoal Matheatcs. Specal Issue: Sgula Itegal Equatos ad Factoal Dffeetal Equatos. Vol. 7, No. -, 27, pp do:.648/.ac.s Receved: Apl 3, 27; Accepted: May 2, 27; Publshed: May 3, 27 Abstact: We toduce the cocept stogly equvalet fo tegal algebac equatos (IAEs). Ths defto ad ts coespodg theoes costuct poweful tools fo the classfyg ad aalyzg of IAEs (especally uecal aalyss). The elated theoes wth shot poofs povde poweful techques fo the coplete covegece aalyss of dscetsed collocato ethods o dscotuous pecewse polyoal spaces. Keywods: Voltea Itegal Equato, Voltea Equato, Itegal equato, Dscotuous Pecewse Polyoal Spaces, Collocato Methods. Itoducto Soetes aothe soluto fo a poble ay help us to pogess oe ad to dg deepe scece. Itegal algebac equatos (IAEs) ae xed syste of the fst kd ad the secod kd Voltea tegal equatos. They ae classfed by dex defto. Recetly, the uecal soluto usg collocato ethods o pecewse polyoal spaces has attacted oe atteto to the eseaches. Howeve, thee ae ay questos, usolved o ths subect. The covegece aalyss of cotuous o dscotuous collocato ethod fo a vey estctve cases lke Hessebeg type o low dex IAEs has bee doe oe ecetly (see fo exaple [7,] fo IAEs of dex, [] fo IAEs of dex 2, [] fo IAEs of dex 3 ad [2,3] fo Hessebeg type IAEs of abtay dex). Covegece aalyss fo IAEs of dex, usg cotuous collocato ethods o pecewse polyoal spaces has ot bee povded yet. Howeve, a dect coplete aalyss of lea IAEs of dex usg dscotuous collocato ethods o pecewse polyoal spaces has bee doe by H. Lag ad H. Bue []. The a of ths pape s to get aothe poof fo the covegece aalyss fo IAEs of dex by sepaatg the poble to sple cases. The leas ad theoes toduced hee ca help us to obta covegece aalyss of hghe dex IAEs ad oe coplex ethods lke cotuous collocato ethods. Cosde tegal algebac opeato of the fo t Γ [ A, K, f]( y) = A( t) y( t) + K( t, s, y( s)) ds f( t), () o t I : = [, T], whee A C( I, R ) s a sgula atx wth costat ak fo all t I, f C( I, R ), y C( I, R ), ad K C( D R, R ) wth D : = {( t, s): s t T}. We study Itegal Algebac Equatos (IAEs) of the fo Γ[ A, K, f]( y) (2) wheeys the ukow vecto. If K( t, s, y) = k( t, s) y, whee k C( D, R ), the, the syste () s a lea IAE. The oto of the dex s used to classfy IAEs. Thee ae dffeet otos of dex fo classfcato of IAEs. Gea toduced dffeetal dex fo IAEs [4]. The left dex fo

2 Appled ad Coputatoal Matheatcs 27; 7(-): syste () s aothe oto that was toduced by Russa atheatcas [2, 3]. La [9] toduced v-soothg fo the fst kd Voltea tegal equatos whch s equvalet wth dffeetal dex. The tactable dex s defed by [, 6,, ]. I ths pape we use ak-degee dex [2, 3]. Hee, we wll toduce the cocept Stogly equvalet, IAEs. We wll establsh theoes o the uecal ad aalytcal solutos of the stogly equvalet IAEs, whch educe the covegece aalyss of IAEs. Ths s doe by decoposg the poble to the sple classes. Fo IAEs of dex, we dvde the syste to two faous class of IAEs: A syste of the fst kd Voltea tegal equatos ad a syste of IAEs whch was vestgated [7]. The ext sectos ae ogazed as follows: I secto 2, we ecall ak-degee dex ad the codtos ude whch the syste () -(2) has uque soluto. I secto 3, we toduce the cocept of stogly equvalet fo IAEs ad we show that the stogly equvalet systes have sae solutos. I secto 4, we ecall dscetsed collocato ethods o dscotuous pecewse polyoal spaces ad we show that the appoxate solutos of a stated ethods fo stogly equvalet IAEs ae of the sae ode. I secto 5, we dvde IAEs of dex, wth egad to the stogly equvalet cocept, to two categoes. The, theoes about the exstece of a uque uecal soluto ae stated. I secto 6, a global covegece aalyss of the dscetsed dscotuous collocato ethods (DDCM) solutos s vestgated. I secto 7, we study the olea systes of IAEs. 2. Idex Defto ad the Exstece of Uque Solutos Defto 2. The atx A ( t) atx fo A( t) f t satsfes whch ca be ewtte as wth A( t) A ( t) A( t) = A( t), V( t) A( t ) =, s called se-vese V( t) = I A( t) A ( t) (3) whee I s a detty atx. The followg codtos ae ecessay ad suffcet fo the exstece of a se-vese atx A ( t) p C ([,], R ) [3]: p. The eleets of A( t ) belog to C ([,], R ). 2. aka( t) = cost, t [,]. wth eleets The IAEs ae classfed usg dex oto. Fo ay deftos of dex (. e. [, 2, 3, 4]), we use the followg oe. Defto 2.2 [2, 3] Suppose A C( I, R ) ad K C( D, R ). Let f A A, K k, ( I ) d Λ y = ( A ( t) A ( t)) y + y, dt I + + A A + ( A ( t ) A ( t )) K ( t, t ), K = Λ K, =,, ν. The, we say that the ak degee dex of ( A, K ) sν A ( t) C ( I, R ) fo =,, ν, aka ( t) = cost, t I fo =,, ν, deta =, fo =,, ν, deta ν. Moeove, we say that the ak-degee dex of lea syste (2) s ν ( d = ν ) f addto to the above hypotheses, we have F f, F Λ F, =,, ν, + F C ( I, R ), =,, ν, whee I s a detty opeato. Theoe 2. [2, 3] Suppose the followg codtos ae satsfed fo (2):. d = ν, 2. ( ) (, A t CI R ), ( ) (, F t CIR ), K C( D, R ), A ( t) C ( I, R ), =,, ν, F( t) C ( I, R ), K C ( D, R ), fo 3. A () Aν () Fν () = F () fo =, ν, (cosstecy codtos) 4. I ( λ I I I ) ak( AA ) = degdet ( AA ) + ( AA )' + = c, The the syste (2) has a uque soluto o I. The codto 2 of the Theoe 2. wll ot be used the ext sectos, sce the defto of dex cludes ths codto. 3. Stogly Equvalet I ths secto, we toduce the cocept of stogly equvalet systes. Defto 3. Two systes Γ[ A, K, f]( y) ad Γ[ Aɶ, K, ɶf ]( y) ae called stogly equvalet f thee exst potwse osgula atx fuctos E C( I, R ) ad F C( I, R ) such that

3 4 Gholaeza Kaaal et al.: A Covegece Aalyss of Dscotuous Collocato Method fo IAEs of Idex Usg the Cocept Stogly Equvalet Aɶ ( t) = E( t) A( t) F( t), K ( t, s, y( s)) = E( t) K( t, s, F( s) y( s)), ɶf ( t) = E( t) f( t) If ths s the case, we wte [ A, K, f ]~[ Aɶ, K, ɶf ]. Moeove, ν ν f E C ( I, R ) ad F C ( I, R ) we ca wte stogly ν -equvalet stead of stogly equvalet. Theoe 3. Let [ A, K, f ]~[ Aɶ, K, ɶf ]. The, x s a soluto of Γ[ A, K, f]( x) ff F x s a soluto of Γ[ Aɶ, Kɶ, ɶ f]( y). Ths eas f oe of the stogly equvalet systes has a uque soluto, aothe, has also a uque soluto. Poof. Let Γ[ A, K, f]( x). Multplyg ths equato by E ad usg x FF x, we obta Γ[ EA, EK, Ef]( FF x). Hece, fo t I : = [, T], ( ) ( ) ( ) ( ) ( ) t E t A t F t F t x t + E( t) K( t, s, F( s) F ( s) x( s)) ds E( t) f ( t) =, ad thus ɶ ɶ Γ[ A, K, f]( F x), (It should ot be cofused wth the substtuto ule calculus). Covesely, suppose ɶ ɶ Γ[ A, K, f]( F x). We ca ultply ths equato by E to obta Γ[ A, K, f]( x), whch poves the theoe. Ths theoe s ot tue fo stogly equvalet te vaat DAEs, theefoe, Kukel ad Meha [8] has defed globally equvalet cocept. 4. Dscetsed Collocato Methods o Pecewse Polyoal Space Let Ih : = { t : = t < t <... < tn = T}, be a gve (ot ecessaly ufo) patto of I, ad set σ : = ( t, t+ ], : = [ t, t ], wth h = t t fo =,..., N ad σ + + daete of ths patto be h = ax{ h : N}. Defto 4. [] Fo a gve esh I h, the pecewse polyoal space, wth µ >, d µ, s gve by (4) ( d ) d S ( Ih): = { v C ( I ): v π ( =,,..., N )}. (5) µ σ µ Hee, π µ deotes the space of (eal) polyoals of degee ot exceedg µ (also, C ( I ) s the space of absolutely cotuous fuctos). I ths pape, we oly cosde d = whch the coespodg spaces s called dscotuous space. By defg u = uh σ ( π ), the dese output of appoxate soluto ( ) u h h whee the polyoals S ( I ) ca be obtaed by u ( t + sh ) = L ( s) U, s (,], (6), v ck L( v): =, =,,, c c k = k deote the Lagage fudaetal polyoals wth espect to the dstct collocato paaetes < c < c2 < < c. The ukows U, : = u( t, ), ca be obtaed by applyg dscetsed dscotuous collocato ethods (DDCM). Ipleetg DDCM to the IAE (2), we obta U, by solvg followg syste (see [, 2]): k A( t ) U + F + h a K( t, t, U ) = f( t ) (7),,,,,,, fo =,,, whee the lag te s defed by F = h b K( t, t + sh, L ( s) U ),,, l l l, l = ad t, = t + ch. Hee, b a = L ( s) ds ad = c L ( s) ds fo =,, ad =,,, [,2]. Theoe 4.. Let uh be the uque appoxate soluto of applyg DDCM to the IAE [ A, K, f ]~[ Aɶ, K, ɶf ]. The, uh ɶ, the appoxate soluto of applyg DDCM to the IAE Γ[ Aɶ, K, ɶf ]( y), s uque ad h,, h, uɶ ( t ) = F ( t ) u ( t ), =,, N, (8) whee =,,. Poof. The poof s by ducto o. Suppose =. We show the syste A( t ) u ɶ ( t ) + h a K ( t, t, u ɶ ( t )) = ɶf ( t ), (9), h,,, h,, fo =,, has a uque soluto. Multplyg the lefthad sde of equato (9) by E ( t ), ad left-had sde of, the te ɶ uh( t, ) by F ( t, ) F( t, ), we obta ɶ E ( t, ) A( t, ) F ( t, ) F( t, ) uɶ h( t, ) h a E ( t, ) K( t,, t,, F ( t, ) F( t, ) uɶ h( t, )) E ( t ) ɶf ( t ), =,.,, ()

4 Appled ad Coputatoal Matheatcs 27; 7(-): as Settg X : = F( t, ) uɶ h( t, ), the syste () ca be wtte A( t ) X + h a K( t, t, X ) = f ( t ), =,. (),,,, whch has uque soluto X = uh( t, ) = F( t, ) uɶ h( t, ), by hypotheses of the theoe. Now, assue that (8) s tue fo. we wll show that t s tue fo +. Hece we show that ɶ uh t+, F t+, uh t+, ( ) = ( ) ( ) s a uque soluto of the syste Aɶ ( t ) uɶ ( t ) + h a K ( t, t, uɶ ( t )) +, h +, +, +, h +, = h b K ( t, t + c h, uɶ ( t )) + ɶf ( t ), l= +, l l h l, +, (2) fo =,. Multplyg the left-had sde of equato (2) by E( t +, ), ad left-had sdes of the tes uɶ h( tl, ) by F ( t ) F( t ), fo l =,, +, we obta l, l, ɶ E( t, ) A( t, ) F ( t, ) F( t, ) uɶ h( t+, ) h a E( t, ) K( t,, t,, F ( t, ) F( t, ) uɶ h( t+, )) (3) h b ( E t+, ) K( t+,, tl + chl, F ( tl, ) F( tl, ) uɶ h( tl, )) l = + E( t ) ɶf ( t ), =,. +, +, Settg X : = F( t, ) uɶ + h( t+, ), ad usg the ducto hypothess ɶ u ( t ) = F ( t ) u ( t ) fo l =,, the syste (3) ca be wtte as h l, l, h l, A( t ) X + h a K( t, t, X ) +, +, +, l= = h b K( t, t + c h, u ( t )) + f ( t ), +, l l h l, +, (4) whch has uque soluto X = uh( t+, ) = F( t, ) uɶ + h( t+, ), by hypotheses of the theoe. Ths copletes the poof of the theoe. Coollay 4. Let u h ad u ɶ h be the appoxate solutos of applyg DDCM to the lea IAEs Γ[ A, K, f]( y) ad Γ[ A, K, f]( ɶy ), whch have uque soluto y ad ɶ y, espectvely. Let [ A, K, f ]~[ Aɶ, K, ɶf ]. The, thee exst two postve costatsc ad c 2 such that c yɶ uɶ < y u < c yɶ uɶ 2 whee f s the ax o, ax, { f( t, ) }, fo =,, N, =,,. Poof. By Theoes 3. ad 4., thee exsts a potwse osgula atx fucto F C( I, R ) such that ɶ y( t) = F ( t) y( t) ad ɶ u ( t ) F ( t ) u ( t ). Theefoe, ad h,, h, yɶ uɶ F y u y u F y u. Sce, det F( t), t I ad adf( t) F ( t) =, hece det F( t) F (, CI R ). Thus, both fuctos F( t ) ad F ( t) ae bouded ad thee exst eal ubes c > ad c 2 > such that F < c2 ad F <. Cosequetly, we have c ad c yɶ uɶ y u y u c yɶ uɶ 2, whch pove the theoe. Coollay 4. shows that the appoxate solutos of the stated ethods fo stogly equvalet systes Γ[ A, K, f]( y) ad Γ[ A, K, f]( ɶy ), ae of the sae ode, whch s the key pot the uecal aalyss of IAEs. 5. IAEs of Idex By usg Theoe 3., the lea IAEs (2) of dex, ca be dvded to two categoes. Theoe 5.. Fo all lea IAEs of dex : (I) thee exsts a potwse osgula atx fucto ˆk of deso, ad a vecto fucto f ˆ, such that [ ˆ ˆ] whee O s a zeo atx fucto, o [ A, k, f ] ~ O, k, f (5) (II) thee exst atx fuctos k ɶ ad f, fo, {,2}, such that

5 6 Gholaeza Kaaal et al.: A Covegece Aalyss of Dscotuous Collocato Method fo IAEs of Idex Usg the Cocept Stogly Equvalet [ A, k, f]~ I kɶ ( t, s) kɶ 2( t, s) f( t),, kɶ 2( t, s) kɶ 22( t, s) f2( t) (6) whee k ɶ 22( t, t ) s a potwse osgula atx fucto of deso ( ) ( ). Poof. If, A. the, det k( t, t) o I, ad ths s the fst kd Voltea tegal equato, (case (I)). Thus, suppose < aka = = cost <. Theefoe, thee exst osgula atx fuctos E ad F such that sce aka( t) = cost. Hece, Let ad Aɶ I Aɶ E ( t) A( t) F ( t), (7) = = I, = A Fɶ =. A E kɶ ( t, s) kɶ 2( t, s) kɶ ( t, s) = E( t) k( t, s) F ( s) =, kɶ 2( t, s) kɶ 22( t, s) The, we have f ( t) f( t) = E( t) f( t) = f 2( t) E ( t)( A( t) + ( I AA ) k( t, t)) F ( t) = Aɶ + ( I AA ɶɶ ) kɶ ( t, t) I =. k ɶ ( t, t) kɶ 2( t, t) kɶ 22( t, t) O the othe had, we see that usg the defto 2.2, the atx fucto A( t) + ( I AA ) k( t, t) s osgula o I. Hece k ɶ 22( t, t ) s also a osgula atx fucto o I. Reak 5. It s staghtfowad to see that the stogly equvalecy of the equato (6) ca be eplaced by the stogly ν -equvalecy f the ν th devatves of the atx fuctos A, k ad f wth espect to the vaables, ae cotuous. 6. Global Covegece Aalyss The global covegece of the DDCM fo the cases (I) ad (II) has bee studed [] (Secto 2.4) as syste of deso =, ad [7] as syste of deso = 2. Fo syste of abtay deso, oe ca see [2] (Theoe 2, wth = ). The coplete global covegece aalyss of the ethod DDCM s vestgated []. Aothe poof ca be obtaed as follow: Theoe 4.. Let the lea syste (2) be of dex ad + f ( t) C ( I, R ), A( t) C ( I, R ), k( t, s) k( t, s), C ( D, R ), t fo N. The the appoxate soluto of applyg the DDCM fo suffcetly sall h, say, uhwth dstct collocato paaetes c,, c (,] coveges to the soluto y, fo, as h, wth Nh < cost., f ad oly f c λ : = ( ). c = Moeove, the followg eo estates holds: O( h ), f λ [,), y uh = O( h ), f λ =, as h, wth Nh < cost. Poof. Usg Coollay 4., ad Theoe 5. t s suffcet to pove ths theoe fo the cases (I) ad (II) of the Theoe 5.. The case (I) ca be obtaed by takg = [2] (Theoe 2), fo deso =, see [] (Secto 2.4). Fo case (II), the equed aalyss exsts oly fo deso = 2, [7], ad a sla poof ca be povded fo abtay deso. Reak 6. The suppecovegece esult of [7] ca be expessed fo the case (II), as a dect esult of coollay 4. (see also []). 7. Nolea Systes Assue that the syste (2) has a uque soluto y C( I, R ). Ths assupto s potat, sce ay olea tegal equatos ae ll-posed. Suppose K has cotuous devatve wth espect to s. To geealze the dex deftos fo olea systes gve [2, 3], we toduce followg defto Defto 7. We say that the dex fo the olea syste (2) s ν, f thee exsts a eghbohood of the exact ν soluto y, Nε( y) = { η C ( I, R ): η y ε}, ε >, whch the dex of lea syste t A( t) y( t) + K ( t, s, η ( s)) u( s) ds = R( t) y (8) ν be ν, fo all η N ε ( y) ad fo a fucto R C ( I, R ). Moeove, we say that the dex of ( A, K ) s ν, f thee exsts a eghbohood of the exact soluto y, Nε( y) whch the dex of ( A, Ku) be ν, fo all η N ε ( y). By usg the defto (2.2), f the dex of the syste ν ν (8) fo oe R C ( I, R ) beν fo othe R C ( I, R ) s also ν. Thus, the defto of dex s well defed. Now, we ca

6 Appled ad Coputatoal Matheatcs 27; 7(-): aalyze the systes of olea IAEs usg DDCM. We use Peao tepolato foula k( t, s, y( s)) y( t + sh) = = +,,, L ( s) k( t, t, y( t )) h S ( t), whee S, ( t) s the Peao ede te, to obta A( t ) y( t ),, + + h bk( t,, tl,, y( tl, )) + h S, ( t, ) ds l = c +,,,,,, + h a K( t, t, y( t )) + h h S ( t ) = f ( t ) (9) (2) fo (2). Usg ea value theoe, thee exsts θ ( s) betwee y( s) ad u( s) such that K( t, s, u ( s)) K( t, s, y( s)) = K ( t, s, θ( s))( u ( s) y( s)). h u h Subtactg syste (2) fo (7) we obta,,, u,, = l = = + A( t ) e( t ) + h a K ( t, t, η( t )) = O( h ) + h b K ( t, t, θ( t )) e( t ) u, l, l, l, (2) whee, e = uh y. Note that, f we apply the DDCM to the lea syste t A( t) y( t) + Ky( t, s, θ ( s)) u( s) ds = R( t), (22) the we wll obta a lea syste of the eo fucto sla to the syste (2). Thus, the ode of the eo fuctos the olea systes of dexν s equal to ts coespodg lea syste of dex ν. Reak 7. Note that, the above aguet, we do ot kow aythg about the cotuty o the dffeetablty of the fucto θ, hece of the fucto K( t, s) = K ( t, s, θ( s)) wth espect to s. Howeve, t does ot daage ou expessed easos, sce the above aalyss we oly eed the + tes dffeetablty of the soluto (to use the Peao tepolato foula), whch follows fo the assupto + K( t, s, y) C ( D R, R ). Now, we ca state the followg theoe whch s the dect esult of the expessed facts. Theoe 7.. Let the olea syste (2) be dex ad + f ( t) C ( I, R ), A( t) C ( I, R ), + K( t, s, y( s)), C ( D R, R ), fo N. The the appoxate soluto of applyg the DDCM fo suffcetly sall h, say, uh wth dstct collocato y paaetes c,, c (,] coveges to the soluto y, fo, as h, wth Nh < cost., f ad oly f c λ : = ( ). c = Moeove, the followg eo estates holds: as h, wth Nh < cost. Refeeces O( h ), f λ [,), y uh = O( h ), f λ =, [] H. Bue, Collocato Methods fo Voltea Itegal ad Related Fuctoal Equatos, Cabdge uvesty pess, 24. [2] M. V. Bulatov, Tasfoatos of dffeetal-algebac systes of equatos, hual Vychsltel'o Mateatk Mateatchesko Fzk, 996. [3] V. F. Chstyakov, Algebo-Dffeetal Opeatos wth Fte- Desoal Coe, Novosbsk: Naukka, Sbea Publshg Copay RAS., 996. [4] C. W. Gea, Dffeetal algebac equatos, dces ad tegal algebac equatos, SIAM J. Nue. Aal., 99, 27(6), [5] F. Ghoesh, M. Hadzadeh ad S. Pshb, O the covegece aalyss of the sple collocato ethod fo syste of tegal algebac equatos of dex-2, It. J. Coput. Methods, 22, 9(4), [6] M. Hadzadeh, F. Ghoesh ad S. Pshb, Jacob spectal soluto fo tegal algebac equatos of dex-2, Appled Nuecal Matheatcs, 2, 6(), [7] J. P. Kauthe, The uecal soluto of tegal-algebac equatos of dex by polyoal sple collocato ethods, Math. Cop., 2, 7(236), [8] P. Kukel ad Meha, Dffeetal-algebac equatos: aalyss ad uecal soluto, Euopea Matheatcal Socety, 26. [9] P. K. La, A suvey of egulazato ethods fo fstkd Voltea equatos, Spge Vea. 2. [] H. Lag, ad H. Bue, Itegal-Algebac Equatos: Theoy of Collocato Methods I, SIAM Joual o Nuecal Aalyss, 23, 5(4), [] S. Pshb, Optal covegece esults of pecewse polyoal collocato solutos fo tegal algebac equatos of dex-3, J. Coput. Appl. Math, 25, 279(), [2] B. Sh, Nuecal soluto of hghe dex olea tegal algebac equatos of Hessebeg type usg dscotuous collocato ethods, Matheatcal Modellg ad Aalyss, 24, 9(), [3] B. Sh, S. Shahoad ad G. Hoat, Covegece aalyss of pecewse cotuous collocato ethods fo hghe dex tegal algebac equatos of Hessebeg type, AMCS, 23, 23(2),