Numerical method for solving system of Fredhlom integral equations using Chebyshev cardinal functions

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1 ) 1-13 Avilble olie t Volume 2014, Yer 2014 Article ID cte-00165, 13 Pges doi: /2014/cte Reserch Article umericl method for solvig system of Fredhlom itegrl equtios usig Chebyshev crdil fuctios Zhr Msouri 1, Seed Htmzdeh-Vrmzyr 2, Esmil Bboli 3 1) Deprtmet of Mthemtics, Khorrmbd Brch, Islmic Azd Uiversity IAU), Khorrmbd, Ir 2) Deprtmet of Electricl Egieerig, Islmshhr Brch, Islmic Azd Uiversity IAU), Tehr, Ir 3) Deprtmet of Mthemtics, Khrzmi Uiversity, Tehr, Ir Copyright 2014 c Zhr Msouri, Seed Htmzdeh-Vrmzyr d Esmil Bboli. This is ope ccess rticle distributed uder the Cretive Commos Attributio Licese, which permits urestricted use, distributio, d reproductio i y medium, provided the origil work is properly cited. Abstrct The focus of this pper is o the umericl solutio of lier systems of Fredhlom itegrl equtios of the secod kid. For this purpose, the Chebyshev crdil fuctios with Guss-Lobtto poits re used. By combitio of properties of these fuctios d the effective Cleshw-Curtis qudrture rule, pplicble umericl method for solvig the metioed systems is formulted. Some error bouds for the method re computed d its covergece rte is estimted. The method is umericlly evluted by solvig some test problems cught from the literture by which the ccurcy d computtiol efficiecy of the method will be demostrted. Keywords: Chebyshev crdil fuctios; Fredholm itegrl equtios system; umericl solutio; Cleshw-Curtis qudrture rule; Error lysis. 1 Itroductio I recet yers, the crdil fuctios hve bee fidig importt role i umericl lysis. Especilly, vluble efforts hve bee spet, by reserchers, o itroducig ovel ides for umericl solutio of vrious fuctiol equtios by usig the superior properties of these fuctios. A umericl techique is preseted i [1] for solutio of prbolic prtil differetil equtio which is derived by expdig the required pproximte solutio s the elemets of the Chebyshev crdil fuctios. [2] presets two umericl techiques for solvig Riccti differetil equtio. These methods use the cubic B-splie sclig fuctios d Chebyshev crdil fuctios. The Chebyshev crdil fuctios re lso used i [3] for solutio of fourth-order itegro-differetil equtios which re reduced to set of lgebric equtios usig the opertiol mtrix of derivtive. A review o pproximte crdil precoditioig methods for solvig prtil differetil equtios by usig rdil bsis fuctios hs bee performed i [4], i which, the uthors hve umericlly compred the relted precoditioers o some umericl exmples of Poisso s, modified Helmholtz, d Helmholtz equtios. [5] uses crdil bses for implemettio of some pseudospectrl methods, the specil cses of differetil d itegro-differetil equtios re solved by these bses. I [6], the crdil iterpoltio of fuctios o the rel lie by splies is determied by certi formul free of solvig lrge or ifiite systems. The uthors obti iterpoltio projectio of the fuctio which symptoticlly mitis the Correspodig uthor. Emil ddress: msouri@yhoo.com

2 Pge 2 of 13 optiml ccurcy of the bsic crdil iterpoltio o the rel lie. A pproch to idetify multivrible Hmmerstei systems is preseted i [7]. By usig crdil cubic splie fuctios to model the sttic olierities d with pproprite trsformtio, the olier models re prmeterized such tht the olier idetifictio problem is coverted ito lier oe. [8] proposes pseudospectrl method for geertig optiml trjectories of lier d olier costried dymic systems. The method cosists of represetig the solutio of the optiml cotrol problem by usig crdil fuctios. Further iformtio regrdig the crdil fuctios my be foud i [9 12]. A gret del of iterest hs bee focused o the solutio of lier Fredhlom itegrl equtios systems. [13] proposes the Adomi decompositio method for solvig such systems. I [14], umericl solutio of system of lier Fredholm itegrl equtios by mes of the Sic-colloctio method is cosidered d the system is replced by explicit system of lier lgebric equtios. Two other umericl techiques bsed o usig rtiolized Hr fuctios d block-pulse fuctios BPFs) re respectively preseted i [15] d [16]. A direct method to compute umericl solutios of the lier Volterr d Fredholm itegrl equtios system is proposed i [17], where by usig vector forms of trigulr fuctios TFs), solvig of itegrl equtios system reduces to solve system of lgebric equtios. This pper proposes umericl method for solvig system of Lier Fredholm itegrl equtios of the secod kid. For this purpose, the Chebyshev crdil fuctios with Guss-Lobtto poits re used s set of bsis fuctios. By combitio of properties of these fuctios d Cleshw-Curtis qudrture rule, effective umericl method will be formulted for solutio of such systems. The mi dvtges of the preseted method re eough ccurcy the umericl results will be compred with those of other methods), quick covergece, d reltively smll size of clcultios. The orgiztio of this pper is s follows. A review o the crdil fuctios d their properties is provided i sectio 2 d specific Chebyshev crdil fuctios re itroduced. Sectio 3 gives brief resume of Cleshw-Curtis qudrture rule s importt tool for implemettio of the proposed method. Sectio 4 presets the umericl method for solvig system of Lier Fredholm itegrl equtios of the secod kid which is implemeted by combitio of the properties of the Chebyshev crdil fuctios d the Cleshw-Curtis qudrture rule. A error lysis regrdig the proposed method will be doe i sectio 5 where some error bouds re obtied d covergece rte is estimted. Some exmples re cught from the literture d provided i sectio 6 to illustrte the computtiol efficiecy of the method. Commets o the results is the subject of sectio 7 where, by referrig to the obtied results i sectio 6, the method will be compred with other methods i view of ccurcy. Also, the mebsolute errors ssocited with the results obtied by the method will be give to cofirm its quick covergece. Filly, coclusios will be i sectio 8. 2 Crdil fuctios Defiitio 2.1. A crdil fuctio C j t) for specific iterpoltio fuctio d for set of iterpoltio poits t i is defied s [1 3, 8, 18, 19] C j t i ) = δ i, j, i, j = 1,2,...,, 2.1) where is the umber of the iterpoltio poits d δ i, j is Kroecker delt defied s { 1, i = j, δ i, j = 0, i j. Tht is to sy, the crdil fuctios re combitio of the uderlyig bsis trigoometric fuctios, Chebyshev polyomils, or whtever) which re chose so tht the jth fuctio is equl to oe t the jth grid poit d vishes t ll the other grid poits. I this pper specific set of crdil fuctios re cosidered bsed o the zeros of 1 t 2 )Ṫ t), where Ṫ t) = dt t) dt, such tht T t) = cos cos 1 t) ), for t [ 1,1], is the Chebyshev polyomil of degree. We choose these grid iterpoltio) poits s follows: 2.2) t j = cos jπ ), j = 0,1,...,, 2.3)

3 Pge 3 of 13 therefore t = 1 < t 1 < < t 1 < t 0 = ) Defiitio 2.2. Chebyshev crdil fuctio with Guss-Lobtto grids of order i [ 1,1] is defied s [5, 8, 18] with c 0 = c = 2, d c j = 1, for 1 j 1. C j t) = 1) j+1 1 t 2 )Ṫ t) c j 2, j = 0,1,...,, 2.5) t t j ) It is esy to show tht reltio 2.1) is vlid for the crdil fuctio defied by 2.5). Defiitio 2.3. A fuctio f t) c be pproximted i terms of crdil fuctios by the followig series of the form [5, 8, 18]: f t) f t) = f t j )C j t), 2.6) j=0 such tht f t j ) = f t j ), j = 0,1,...,, 2.7) d f t) is uique th-degree iterpoltig polyomil ssocited with the +1 Chebyshev Guss-Lobtto grids. 3 Cleshw-Curtis umericl qudrture A umericl qudrture umericl itegrtio) rule is the bsis of my umericl methods for the solutio of itegrl equtios. I this sectio brief resume of Cleshw-Curtis qudrture which is importt lter o is give. For much fuller tretmet of the subject, see for exmple [20 23]. The Guss-Chebyshev rules re of specil iterest s esy-to-use sequece of Guss rule. However, i prctice the weight fuctio ws) = 1 occurs much more commoly th Chebyshev weight fuctio ws) = 1 s 2 ) 2 1. It is lwys possible to write 1 1 f s) f s)ds = ds, 3.8) s 2 ) 1 2 where f s) = 1 s 2 ) 1 2 f s). However, if f s) is smooth er s = ±1, f s) is ot, d the direct use of Guss-Chebyshev rule o 3.8) will yield results which coverge oly slowly s the qudrture poits or odes icreses. For voidig this slow covergece, we c use very effective Cleshw-Curtis qudrture rule. Here, we preset the Cleshw-Curtis scheme s stdrd itegrtio rule s follows: where 1 1 f s)ds = w k = 4 = =0 eve =0 eve k= ) w k f cos kπ k=0 cos kπ ) kπ ) f cos ), 3.9) cos kπ ), k = 0,1,...,, 3.10) d the ottio mes the first d lst terms re to be hlved before summig. Remrk The resultig formul c be show to be exct if f s) is polyomil of degree 2 1.

4 Pge 4 of The ppret cost of implemetig this rule is high; direct sumtio of 3.10) to compute w k, k = 0,1,...,, ivolves totl of + 1) 2 multiplictios d dditios, compred with oly + 1 to ctully evlutig the sum 3.9). Equtio 3.10) for the weights w k c be viewed s the discrete cosie trsformtio of vector v with etries { 2, eve, v = ) 0, odd. The weights w k c therefore be computed usig Fst Fourier Trsform FFT) techique i O l) opertios; so the rule is resoble i cost d very stble gist roudig errors [20, 24]. 4 umericl solutio of Fredholm itegrl equtios system where Let us cosider the followig system of lier Fredholm itegrl equtios: b Us)Xs) = Fs) + Ks,t)Xt)dt, s [,b], 4.12) Us) = [ u i, j s) ], i, j = 1,2,...,, Fs) = [ f 1 s), f 2 s),..., f s) ] T, Xs) = [ x 1 s),x 2 s),...,x s) ] T, Ks,t) = [ λ i, j k i, j s,t) ], i, j = 1,2,...,, 4.13) d superscript T idictes trspositio. I 4.12), the prmeters λ i, j, the fuctios f i s), u i, j s), d k i, j s,t), for i, j = 1,2,...,, re kow, d x i s), for i = 1,2,...,, re the ukow fuctios to be determied. Also, k i, j s,t) L 2 [,b] [,b] ), d f i s),x i s),u i, j s) L 2 [,b] ), where L 2 is the spce of squre itegrble fuctios. Moreover, we ssume tht t lest oe compoet of y row of mtrix U is o-zero o [,b]. Without loss of geerlity, it is supposed tht = 1 d b = 1, sice y fiite itervl [,b] c be trsformed to itervl [ 1,1] by lier mps. For coveiece, let us cosider the ith equtio of 4.12) whom we c write s u i, j s)x j s) = f i s) + 1 λ i, j 1 k i, j s,t)x j t)dt. 4.14) Approximtig the solutio x j s) by the Chebyshev crdil fuctios from Eqs. 2.5) d 2.6) gives x j s) x j, s) = k=0 j,k C k s), 4.15) where j,k = x j t k ), d t k s, for k = 0,1,...,, re defied by 2.3). Also, subscript deotes pproximte solutio x j, s) i terms of + 1 crdil fuctios. Substitutig 4.15) ito 4.14) yields u i, j s)x j, s) 1 λ i, j 1 k i, j s,t)x j, t)dt f i s). 4.16) ow, pproximtig the itegrl opertor i 4.16) by the Cleshw-Curtis qudrture defied by 3.9) d 3.10) follows u i, j s)x j, s) λ i, j w p k i, j s,t p )x j, t p ) f i s). 4.17) p=0

5 Pge 5 of 13 From 2.7) d 4.15) we c write x j, t p ) = x j t p ) = j, p. Therefore u i, j s)x j, s) p=0 Substitutig s = t q, for q = 0,1,...,, defied by 2.3) ito 4.18) follows u i, j t q )x j, t q ) p=0 ow, cosiderig x j, t q ) = j,q d replcig sig with = sig gives [ λ i, j w p k i, j s,t p ) j, p f i s). 4.18) λ i, j w p k i, j t q,t p ) j, p f i t q ), q = 0,1,...,. 4.19) ] u i, j t q ) j,q λ i, j w p k i, j t q,t p ) j, p = f i t q ), q = 0,1,...,. 4.20) p=0 By cosiderig similr procedure for the other equtios of system 4.12) we c filly obti [ ] u i, j t q ) j,q λ i, j w p k i, j t q,t p ) j, p = f i t q ), q = 0,1,...,, i = 1,2,...,. 4.21) p=0 ow, 4.21) is system of + 1) lgebric equtios for the + 1) ukows 1,0, 1,1,..., 1,, 2,0,...,,. Hece, ccordig to 4.15), pproximte solutio X s) = [ x 1, s),x 2, s),...,x, s) ] T is obtied for Fredholm itegrl equtios system 4.12). 5 Error lysis d covergece evlutio Without loss of geerlity, system 4.12) c be cosidered s follows b Xs) = Fs) + Ks,t)Xt)dt, s [,b] = [ 1,1], 5.22) where Xs), Fs), Ks,t) re defied by 4.13). Equtio 5.22) c be rewritte i the followig form: where I is idetity opertor d opertor K is defied s Let us cosider j=0 I K )X = F, 5.23) b K X)s) = Ks, t) Xt) dt. 5.24) w j Ks,t j )Xt j ) = K X)s) E t Ks,t)Xt) ), 5.25) i which E t idictes the error fuctiol for the Cleshw-Curtis qudrture rule opertig o Ks,t)Xt), viewed s fuctio of t for fixed s [20]. From 5.22) we c defied E t Ks,t)Xt) ) s vector of the form ) E t Ks,t)Xt) = Et 1,1 Et 2,1. E,1 t k1,1 s,t)x 1 t) ) + Et 1,2 k1,2 s,t)x 2 t) ) + + Et 1, k1, s,t)x t) ) k2,1 s,t)x 1 t) ) + Et 2,2 k2,2 s,t)x 2 t) ) + + Et 2, k2, s,t)x t) ) k,1 s,t)x 1 t) ) + Et,2 k,2 s,t)x 2 t) ) + + Et, k, s,t)x t) ). 5.26)

6 Pge 6 of 13 Suppose X be the pproximte solutio of 5.22) obtied by the preseted method. It stisfies the followig equtio: X s) = Fs) + j=0 w j Ks,t j )X t j ). 5.27) If we cosider s = t i = cos iπ ), for i = 0,1,...,, the 5.27) is equivlet to lgebric system 4.21). ow, the error of the preseted method c be defied s e s) = Xs) X s), 5.28) where e s) is vector such tht its ith compoet is the error correspodig to solvig the ith itegrl equtio of system 5.22). From 5.25) we obti Subtrctig 5.29) form 5.22) gives X s) = Fs) + b e s) = E t Ks,t)X t) ) + Ks,t)X t)dt E t Ks,t)X t) ). 5.29) b Ks,t)e t)dt. 5.30) Thus, the vector of the error fuctios e s) stisfies Fredholm itegrl equtios system with the sme kerels s 5.22), but with differet drivig terms. Equtio 5.30) c lso be writte i the form This equtio hs the solutio I K )e s) = E t Ks,t)X t) ). 5.31) e s) = I K ) 1 E t Ks,t)X t) ) = I + H )E t Ks,t)X t) ), where H is the resolvet opertor see [25, 26]). Hece, tkig rbitrry orms we fid 5.32) e 1+ H ) E t Ks,t)X t) ). 5.33) However, if K < 1 we fid, o tkig orms i 5.30) d rerrgig, the simpler boud e E t Ks,t)X t) ). 5.34) 1 K ) The bouds of the error vector show tht e is directly relted to the error of the qudrture rule. The simplest d most commoly used procedure for estimtig the chieved ccurcy is to choose fmily of qudrture rules R d to compute pproximte solutio for members of this fmily with icresig, d hece decresig error, util the results pper to hve settled dow to the required ccurcy. It is usully strightforwrd to gurtee covergece; tht is, to esure tht [20] Sufficiet coditios for this re give i [20, 27]. ow, we try to compute the rte t which covergece is chieved. lim e R = ) Lemm 5.1. The error estimtes for poit qudrture rules of vrious types hve the form E t f b = f t)dt w i f t i ) C f ) p, 5.36) i=1 where C f ) is costt depedet o the fuctio f t) d the expoet p depeds either o the degree of the rule if f t) is smooth eough) or o the cotiuity properties of f t) if the degree of the rule is high eough). We refer to p s the order of covergece.

7 Pge 7 of 13 Proof. [20]. ow, cosiderig Lemm 5.1 d usig 5.33) we obti e 1+ H ) C Ks,t)X t) ) p, 5.37) where C Ks,t)X t) ) is vector of the form C 1,1 k1,1 s,t)x 1 t) ) +C 1,2 k1,2 s,t)x 2 t) ) + +C 1, k1, s,t)x t) ) C Ks,t)Xt) ) C 2,1 k2,1 s,t)x 1 t) ) +C 2,2 k2,2 s,t)x 2 t) ) + +C 2, k2, s,t)x t) ) =. C,1 k,1 s,t)x 1 t) ) +C,2 k,2 s,t)x 2 t) ) + +C, k, s,t)x t) ). 5.38) If C i, j, for i, j = 1,2,...,, is uiformly bouded i s such tht C i, j ki, j s,t)x j, t) ) Mi, j, s b. 5.39) The, tkig the mximum orm o 5.37) gives where C Ks,t)X t) ) M. e 1+ H ) M p, 5.40) The estimtes of covergece rte bove show tht the preseted error will be rpidly coverget to zero if the degree of the qudrture rule is high eough d if for every fixed s, Ks,t)Xt) is vector of smooth fuctios with respect to t. Referrig to the fct tht if F d K i 5.22) re cotiuous the so is X see [25]), we pose the followig theorem s the extesio of Theorem i [20]. Theorem 5.1. Let i 5.22) f i s) C p) [,b], for i = 1,2,...,, d k i, j s,t) C p) [,b] C p) [,b], where C p) is spce of cotiuous d differetible fuctios of order p. The, x i s) C p) [,b], for i = 1,2,...,. Proof. We hve whece x i s) = f i s) + x is) = f i s) + b λ i, j b λ i, j k i, j s,t)x j t)dt, i = 1,2,...,, 5.41) s k i, j s,t)x j t)dt, i = 1,2,...,. 5.42) But, for p 1, s k i, j s,t) is cotiuous by hypothesis, d the itegrls b s k i, j s,t)x j t)dt re cotiuous fuctios of s, for i, j = 1,2,...,. So, λ b i, j s k i, j s,t)x j t)dt is cotiuous fuctio. Therefore, if f i s), for i = 1,2,...,, re cotiuous, the ccordig to 5.42) x i s), for i = 1,2,...,, will be cotiuous. Similrly for y q p x q) i s) = f q) b i s) + q λ i, j s q k i, j s,t)x j t)dt, i = 1,2,...,, 5.43) whece, proceedig iductively, it follows tht x i s) C p) [,b], for i = 1,2,...,. 6 umericl results Some exmples re ivestigted by the proposed method i this sectio. Most of these exmples re cught from vrious refereces, such tht we re ble to compre the umericl results obtied by the proposed method with both the exct solutio d those preseted i the relted refereces. The umericl results re give for eleve poits s i itervl [,b]. These poits re set by dividig the itervl to te equl segmets ccordig to the relted literture).

8 Pge 8 of 13 Exmple 6.1. For the followig lier Fredholm itegrl equtios system [13, 14, 17]: x 1 s) = 18 s s+t 3 x1 t) + x 2 t) ) dt, x 2 s) = s s st x 1 t) + x 2 t) ) dt, 6.44) the preseted method i this pper gives the exct solutios x 1 s) = s +1 d x 2 s) = s 2 +1 for = 2. Tble 1 shows the umericl results obtied by the method d those give i [13, 17]. Tble 1: umericl results for Exmple 6.1 s Exct solutio Preseted method, Direct method [17], Decompositio method [13], = 2 m = 32 k = 11 Results for x 1 s) Results for x 2 s) Exmple 6.2. For the followig lier Fredholm itegrl equtios system [16]: x 1 s) = 11 6 s s +t)x 1t)dt 1 0 s + 2t2 )x 2 t)dt, x 2 s) = 4 5s s 1 0 st2 x 1 t)dt 1 0 s2 t x 2 t)dt, 6.45) the preseted method gives the exct solutios x 1 s) = s d x 2 s) = s 2 for = 4. Tble 2 shows the umericl results obtied by it d those give i [16].

9 Pge 9 of 13 Tble 2: umericl results for Exmple 6.2 s Exct solutio Preseted method, BPFs method [16], = 4 m = 32 Results for x 1 s) Results for x 2 s) Exmple 6.3. Cosider the followig Fredholm itegrl equtios system [14 17]: x 1 s) = 2e s + es+1 1 s es t x 1 t)dt 1 0 es+2)t x 2 t)dt, x 2 s) = e s + e s + es+1 1 s est x 1 t)dt 1 0 es+t x 2 t)dt, 6.46) with the exct solutios x 1 s) = e s d x 2 s) = e s. The umericl results re show i Tble 3.

10 Pge 10 of 13 Tble 3: umericl results for Exmple 6.3 s Exct solutio Preseted method, Preseted method, Direct method [17], Rtiolized Hr method [15], BPFs method [16], = 4 = 6 m = 32 k = 32 m = 32 Results for x 1 s) Results for x 2 s) Exmple 6.4. This exmple icludes Fredholm itegrl equtios system with vrible coefficiets with respect to s s follows: s 2 x 1 s) + s + 1)x 2 s) = y 1 s) 1 0 sis t)x 1t)dt 1 0 coss t)x 2t)dt, sx 1 s) + 1 2s 2 )x 2 s) = y 2 s) 1 0 sis +t)x 1t)dt 6.47) 1 0 coss +t)x 2t)dt, with the exct solutios x 1 s) = s d x 2 s) = 1 s 2 d suitble y 1 s) d y 2 s). Figure 1 shows the umericl results for this problem for = 6. ) Figure 1: umericl results for Exmple 6.4 obtied by the proposed method. ) Results for x 1 s). b) Results for x 2 s). b)

11 Pge 11 of 13 7 Commets o the results Four test problems were illustrted bove for evlutig the pplicbility d ccurcy of the proposed method. Exmple 6.1 hs bee solved i [13], [14], d [17], too. [13] proposes the decompositio method d [17] itroduces direct method usig the trigulr fuctios to solve the problem. Also, Sic-colloctio method is cosidered i [14] for umericl solutio of Exmple 6.1. Our method gives the exct solutio for this problem for very smll size of discretiztio = 2). The umericl results obtied by the decompositio d direct methods show i Tble 1 of this rticle d lso the error vlues give i Tble 2 of [14] cofirm the superiority of the proposed method over the three metioed methods i view of ccurcy. O the other hd, the umber of clcultios i the decompositio method is higher. For Exmple 6.2, [16] gives pproximte solutio by usig the block-pulse fuctios. The relted results for m = 32 re show i Tble 2. Our method gives the exct solutio for this exmple for = 4, whece it follows tht this method is much more ccurte th the BPFs method. [15] proposes umericl method bsed o usig rtiolized Hr fuctios for lier Fredholm itegrl equtios system. This method together with those preseted i [14, 16, 17] obti umericl solutio for Exmple6.3. The umericl results i Tble 3 of this pper d lso the error vlues give i Tble 1 of [14] still cofirm good ccurcy of the method proposed i this pper. For further evlutio of the computtiol efficiecy of the method we give the me-bsolute errors ssocited with it withi solvig two of the exmples. The me-bsolute error is clculted by cosiderig the errors t 0 poits s [,b] d by usig the followig reltio: E 0) j, = i=1 x j s i ) x j, s i ), 7.48) where E 0) j, is the me-bsolute error, d x js) d x j, s) re the jth exct d pproximte solutios, respectively. For Exmples 6.3 d 6.4, these errors for 0 = 11 poits s i = + b 10 i, i = 0,1,...,10, d = 2,4,...,12 re illustrted i Tble 4. Obviously, these results cofirm very quick covergece of the proposed method mewhile emphsis gi o its excellet ccurcy. Tble 4: Me-bsolute errors Me-bsolute errors for Exmple 3 Me-bsolute errors for Exmple 4 Results for x 1 s) Results for x 2 s) Results for x 1 s) Results for x 2 s) 2 1.4E 2 6.0E 3 4.0E 1 1.9E E 5 1.4E 5 1.0E 1 4.1E E 8 1.7E 8 5.7E 5 3.1E E E E 7 7.3E E E E E E E E E 13 8 Coclusio A effective d ccurte umericl pproch for solvig lier systems of Fredhlom itegrl equtios of the secod kid ws proposed by usig the Chebyshev crdil fuctios with Guss-Lobtto poits d lso the Cleshw- Curtis qudrture rule. Moreover, two error bouds were computed for the method i terms of the error of the Cleshw-Curtis qudrture rule d its covergece rte ws estimted. Some test problems were solved by the preseted method which showed tht it is pplicble d ccurte i solvig of the metioed systems.

12 Pge 12 of 13 Refereces [1] M. Lkesti, M. Dehgh, The use of Chebyshev crdil fuctios for the solutio of prtil differetil equtio with ukow time-depedet coefficiet subject to extr mesuremet, Jourl of Computtiol d Applied Mthemtics, ) [2] M. Lkesti, M. Dehgh, umericl solutio of Riccti equtio usig the cubic B-splie sclig fuctios d Chebyshev crdil fuctios, Computer Physics Commuictios, ) [3] M. Lkesti, M. Dehgh, umericl solutio of fourth-order itegro-differetil equtios usig Chebyshev crdil fuctios, Itertiol Jourl of Computer Mthemtics, 87 6) 2010) [4] D. Brow, L. Lig, E. Ks, J. Levesley, O pproximte crdil precoditioig methods for solvig PDEs with rdil bsis fuctios, Egieerig Alysis with Boudry Elemets, ) [5] A. Aliph, Spectrl methods usig crdil fuctios, Ph.D. Thesis i Applied Mthemtics, Amirkbir Uiversity of Techology, 2006). [6] G. Viikko, Crdil Approximtio of Fuctios by Splies o Itervl, Mthemticl Modellig d Alysis, 14 1) 2009) [7] K. H. Ch, J. Bo, W. J. White, Idetifictio of MIMO Hmmerstei systems usig crdil splie fuctios, Jourl of Process Cotrol, ) [8] G. Elgr, M. A. Kzemi, Pseudospectrl Chebyshev optiml cotrol of costried olier dymicl systems, Computtiol Optimiztio d Applictios, ) [9] J. P. Boyd, The er-equivlece of five species of spectrlly-ccurte rdil bsis fuctios RBFs): Asymptotic pproximtios to the RBF crdil fuctios o uiform, ubouded grid, Jourl of Computtiol Physics, ) [10] J. P. Boyd, L. Wg, A lytic pproximtio to the crdil fuctios of Gussi rdil bsis fuctios o ifiite lttice, Applied Mthemtics d Computtio, ) [11] G. Wu, D. Li, H. Xio, Z. Liu, The M-bd crdil orthogol sclig fuctio, Applied Mthemtics d Computtio, ) [12] G. Wu, Z. Cheg, X. Yg, The crdil orthogol sclig fuctio d smplig theorem i the wvelet subspces, Applied Mthemtics d Computtio, ) [13] E. Bboli, J. Bizr, A. R. Vhidi, The decompositio method pplied to systems of Fredholm itegrl equtios of the secod kid, Applied Mthemtics d Computtio, )

13 Pge 13 of 13 [14] J. Rshidii, M. Zrebi, Covergece of pproximte solutio of system of Fredholm itegrl equtios, Jourl of Mthemticl Alysis d Applictios, ) [15] K. Mlekejd, F. Mirzee, umericl solutio of lier Fredholm itegrl equtios system by rtiolized Hr fuctios method, Itertiol Jourl of Computer Mthemtics, 80 11) 2003) [16] K. Mlekejd, M. Shhrezee, H. Khtmi, umericl solutio of itegrl equtios system of the secod kid by Block-Pulse fuctios, Applied Mthemtics d Computtio, ) [17] E. Bboli, Z. Msouri, S. Htmzdeh-Vrmzyr, A direct method for umericlly solvig itegrl equtios system usig orthogol trigulr fuctios, Itertiol Jourl of Idustril Mthemtics, 1 2) 2009) [18] J. P. Boyd, Chebyshev d Fourier Spectrl Methods, Dover Publictios, Ic., 2000). [19] J. P. Boyd, Multipole expsios d pseudospectrl crdil fuctios: ew geerliztio of the fst Fourier trsform, Jourl of Computtiol Physics, ) [20] L. M. Delves, J. L. Mohmed, Computtiol Methods for Itegrl Equtios, Cmbridge Uiversity Press, Cmbridge, 1985). [21] P. J. Dvis, P. Rbiowitz, Methods of umericl Itegrtio, Acdemic Press, ew York, 1975). [22] A. H. Stroud, D. Secrest, Gussi Qudrture Formuls, Pretice-Hll, Eglewood Cliffs, ew Jersey, 1966). [23] C. W. Cleshw, A. R. Curtis, A method for umericl itegrtio o utomtic computer, umericl Mthemtics, ) [24] W. Getlem, Implemetig Cleshw-Curtis qudrture, I methodology d experiece, Commuictios of the ACM, ) [25] F. Smithies, Itegrl Equtios, Cmbridge Uiversity Press, Cmbridge, 1965). [26] P. K. Kythe, P. Puri, Computtiol Methods for Lier Itegrl Equtios, Birkhäuser, Bosto, 2002). [27] C. T. H. Bker, The umericl Tretmet of Itegrl Equtios, Oxford Uiversity Press, 1977).