Chelyshkov collocation approach to solve the systems of linear functional differential equations

Transcription

1 NTMSCI, No 4, 8-97 (205) 8 New Trends n Mathematcal Scences Chelyshkov collocaton approach to solve the systems of lnear functonal dfferental equatons Cem Oguz, Mehmet Sezer 2 and Arzu Denk Oguz Department of Mathematcs, Faculty of Scence, Ah Evran Unversty, 4000, Krsehr, Turkey 2 Department of Mathematcs, Faculty of Scence, Celal Bayar Unversty, 45047, Mansa, Turkey Department of Mathematcs, Faculty of Scence, Ege Unversty, 500, Ä zmr, Turkey Receved: 2 June 205, Revsed: 8 June 205, Accepted: 6 November 205 Publshed onlne: 26 November 205 Abstract: In ths paper, we present a new collocaton method based on Chelyshkov polynomals for solvng the system of functonal dfferental equatons under the ntal-boundary condtonsby means of Chelyshkov polynomals and collocaton ponts, ths method converts the so-called system nto a matrx equaton, whch nvolves the unknown Chelyshkov coeffcents We gve some llustratve examples, whch arse n physcs, bology, chemstry and mechancs and so on, to ndcate the relablty and effcency of the method Also, a technque based on resdual functons s performed to check the accuracy of the problem Keywords: Systems of delay dfferental-dfference equatons; Chelyshkov polynomals; Numercal solutons; Resdual functons Introducton Mathematcal models, especally those related to systems of delay dfferental and dfferental-dfference equatons, are of the great mportance n the real-lfe such as varous mechancs, physcs, bology, economy, epdemcs, populaton dynamc models, automatc control systems, neural networks, chaotc systems and so on -0] (For more detals, see the references theren) Therefore, durng the last few decades, a number of mathematcal methods that are amed at solvng the so-called systems have appeared n the research lterature such as varatonal teraton method ], Dfferental transformaton method 2], Haar functons method ], homotopy analyss method 4], homotopy perturbaton method 5] and Tau method 6] In addton to these methods mentoned n the lterature, systems of lnear dfferental, ntegral, ntegro-dfferental and dfferental-dfference equatons were solved usng the collocaton methods based on Sezer s matrx methods, whch are derved from specal functons as Taylor, Chebyshev, Legendre, Laguerre, Hermte, Bessel and so on 7-25] The area of orthogonal polynomals s a very actve research area n mathematcs as well as n applcatons n mathematcal physcs, engneerng and computer scence One of the latest set of orthogonal polynomals s the set of the Chelyshkov polynomals C N0 (t),c N (t),,c NN (t),} Recently, these polynomals have created by Chelyshkov n 26-2], whch are orthogonal over the nterval 0,] wth respect to the weght functon w(x) =, and are explctly defned by N n C Nn (t) = j=0 ( )( ) ( ) j N n N + n + j + t n+ j,n = 0,N () j N n Correspondng author e-mal: cemoguz@ahevranedutr

2 84 C Oguz, M Sezer and A Denk Oguz: Chelyshkov collocaton approach to solve the systems of lnear Ths yelds The Rodrgues type representaton d N n ( C Nn (t) = (N n)! t n+ dt N n t N+n+ ( t) N n),n = 0,,,N, (2) and the followng orthogonalty relatons 0 0, p q C N p (t)c Nq (t)dx =, p,q,,n () p+q+, p = q Also t follows from ths relaton that 0 C Nn (t)dx = t n dx = 0 n + By usng The Rodrgues formula and the Cauchy ntegral formula for dervatves of an analytc functon, one can obtan the ntegral relaton C Nn (t) = 2π t n+2 Ω Ω s a closed curve, whch encloses the pont z = t z (N+2+n) ( z) N n (z t ) N n+ dz, Chelyshkov polynomals C Nn (t) have the analogous propertes to those of the classcal orthogonal polynomals In fact, these polynomals are an example of such alternatve orthogonal ones, whch are not solutons of the hypergeometrc type, but can be expressed n terms of the Jacob ones In addton, they can also be connected to hypergeometrc functons, orthogonal expontonal polynomals, and the Jacob polynomals P (α,β) k by the followng relaton, C Nn (t) = t n P (2n,) ( 2t),n = 0,N N n Hence, they keep dstnctvely attrbutes of the classcal orthogonal polynomals and may be facltated to dfferent problems on approxmaton In the famly of orthogonal polynomals C Nn (t)}, every member has degree N wth N-n smple roots Hence, for every N f the roots of the polynomal are chosen as node ponts, then an acurate numercal quadrature can be derved In ths study, we consder the system of functonal dfferental equatons wth varable coeffcents n the form, m k S r=0 = s=0 P r,s subject to the ntal-boundary condtons m ( =0 a n jy () n } j (t)y(r) (λt + µ s ) + Q r j(t)y (r) (t) = g j (t),j =,2,,k, 0 t (4) ) (0)+b n jy () n () = α n, j = 0,,,m, n =,2,,k (5) a n j, bn j,λ, µ and α n, are real or complex constants Meanwhle Pj r (t) and Qr j (t) are contnuous functons defned n 0 t Our am n ths paper s to fnd an approxmate solutons of Eq (4) under the ntal-boundary condtons (5)

3 NTMSCI, No 4, 8-97 (205) / wwwntmsccom 85 n the truncated Chelyshkov seres, based on () or (2), form y (t) = N n=0 a,n C N,n (t), =,,k, 0 t (6) so that a,n, n = 0,,2,,N are the unknown Chelyshkov coeffcents Here, N s chosen any postve nteger such that N k,m 2 Fundamental matrx relatons Frst, we can wrte y (t) and ther dervatves n the matrx forms as follows: y (t) = C(t)A or y (t) = T(t)CA, =,,k (7) y () (t) = C () (t)a = T(t)BCA y (2) (t) = C (2) (t)a = T(t)B 2 CA y (r) (t) = C (r) (t)a or y (r) (t) = T(t)B r CA, r =,,m (8) ] ] C(t) = C N0 C N C NN, T(t) = t t N f N s odd, from () and () ( )( ) N N N ( )( ) ( )( ) N N + 2 N N N 0 N C = ( )( ) ( )( ) ( )( ) N 2N N 2N 2N 0 N N N 2 N 0 ( )( ) ( )( ) ( )( ) N 2N + N 2N + 2N + N N N N (N+)x(N+) f N s even ( )( ) N N N ( )( ) ( )( ) N N + 2 N N N 0 N C = ( )( ) ( )( ) ( )( ) N 2N N 2N 2N 0 N N N 2 N 0 ( )( ) ( )( ) ( )( ) N 2N + N 2N + 2N + N N N N (N+)x(N+)

4 86 C Oguz, M Sezer and A Denk Oguz: Chelyshkov collocaton approach to solve the systems of lnear B = 0 N 0 (N+)x(N+) Replacng (λt + µ s ) by t n the relaton (8) we have the matrx form ] T, A = a 0 a a N, =,,k y (r) (λt + µ s ) = C (r) (λt + µ s )A = T(λt + µ s )B r A, r =,,m (9) The relaton between the matrces T(λt + µ s ) and T(t) s T(λt + µ s ) = T(t)M(λ, µ s ) (0) such that, for λ 0 and µ s 0 ] ( ) ( ) ( ) ( ) 0 λ 0 µ 0 s λ 0 µ 2 s λ 0 µ 2 N s λ 0 µ s N 0 ( ) ( ) ( ) 0 λ µ 0 2 s λ µ N s λ µ s N ( ) ( ) M(λ, µ s ) = λ 2 µ 0 N s λ 2 µ s N 2 2 ( ) N 0 λ N µ s N N and for λ 0 and µ s = λ 0 0 M(λ,0) = 0 0 λ λ N We have the followng matrx relaton by substtutng Eq(0) nto Eq(9) y (r) (λt + µ s ) = T(t)M(λ, µ s )B r A, r =,,m and =,,k () By usng the relatons (8),(0) and (),we fnd the followng matrx forms y (r) (t) = T(t)B r CA, r = 0,,,m (2) and y (r) (λt+µ s ) = T(t)M(λ, µ s )B r CA,r = 0,,,m ()

5 NTMSCI, No 4, 8-97 (205) / wwwntmsccom 87 y (r) (t) y (r) y (r) y (r) 2 (t) = (t) (λt+µ s),y (r) y (r) 2 (λt+µ s ) = (λt+µ s),a = y (r) k (t) y (r) k (λt+µ s) A A 2 A k T(t) T(t) 0,T(t) =, 0 0 T(t) M(λ, µ s ) 0 0 B r 0 0 C M(λ, µ s ) 0 M(λ, µ s ) =,B r 0 B r 0 0 C 0 = and C = 0 0 M(λ, µ s ) 0 0 B r 0 0 C Method of soluton In ths secton, we convert the system (4) to lnear systems of matrx equatons whch can be easly solved Frstly, by means of the matrx relatons (2) and (), we can wrte the system (4) n the matrx form P r,s (t) = m S r=0 s=0 P r,s P r,s 2 (t) Pr,s 2 (t) Pr,s 22 P r,s k } P r,s (t)y (r) (λt + µ s ) + Q r (t)y (r) (t) = g(t), (4) (t) Pr,s k (t) Q r (t) Pr,s 2k (t) (t) Qr 2 (t) Qr k (t) g (t) Q r 2,Q r (t) = (t) Qr 22 (t) Qr 2k (t) g 2 (t),g(t) = (t) Pr,s kk (t) Q r k (t) Qr k2 (t) Qr kk (t) g k (t) (t) Pr,s k2 By substtutng the collocaton ponts defned by t q = q, q = 0,,,N N nto Eq (4), the system of the matrx equatons s obtaned as or brefly expressed as follows m S r=0 s=0 } P r,s (t q )y (r) (λt q + µ s ) + Q r (t q )y (r) (t q ) = g(t q ), m S r=0 s=0 P rs Y (r) + Q r Y (r)} = G, (5) P r,s (t 0 ) 0 0 y (r) (λt 0 + µ ) y (r) (t 0 ) 0 P r,s (t ) 0 P rs =,Y (r) y (r) (λt + µ ) =,Y (r) y (r) (t ) =, 0 0 P r,s (t N ) y (r) (λt N + µ ) y (r) (t N )

6 88 C Oguz, M Sezer and A Denk Oguz: Chelyshkov collocaton approach to solve the systems of lnear Q r (t 0 ) 0 0 g(t 0 ) 0 Q r (t ) 0 g(t ) Q r =,G= 0 0 Q r (t N ) g(t N ) Usng relatons (2) and () n Eq(5), we have the fundamental matrx equaton m S r=0 s=0 Prs TM(λ, µ s )B r C + Q r TB r C } = G, (6) Brefly, we can wrte Eq(6) n the form ] T T = T(t 0 ) T(t ) T(t N ) WA = G W;G] (7) whch corresponds to a system of the lnear algebrac equatons wth the unknown Chelyshkov coeffcents elements a,n, =,2,,k,n = 0,,,N us fnd a matrx representaton of the condtons gven n (5) Usng the relaton (2), the matrx representaton of the ntal and boundary condtons whch depend on the Chelyshkov coeffcents matrx s obtaned as m aj T(0) + b j T() } B j CA = α, (8) j=0 a j 0 0 b j a 2 j 0 0 b 2 j 0 a j =,b j = and a j = 0 0 a k k j 0 0 b b j = j Thus, the matrx form (8) for the condtons becomes UA = α or U;α];U = ] T α= α α 2 α k a 0 j a j a m j b 0 j b j b m j m a j T(0) + b j T() } B j C (9) j=0 Lastly, by replacng the last rows of the augmented matrx (7) by the rows of matrx U;α], we have the new augmented matrx WÃ = G or W; G] whch s a lnear system of algebrac equatons The unknown Chelyshkov coeffcents can be found by solvng ths system When the unknown Chelyshkov coeffcents a,0, a,,,a,n are substtuted n Eq (6), we obtan the Chelyshkov polynomal soluton y (t) = N n=0 a,n C N,n (t), =,,k, 0 t On the other hand, when W = 0, f rank W = rank W; G ] < k(n + ), then we may fnd a partcular soluton Otherwse there s not a soluton We can easly check the accuracy of ths soluton as follows: ] T ] T (20) Snce the truncated Chelyshkov seres (6) s approxmate soluton of (4), when the functon y,n (t), =,2,,k and ther

7 NTMSCI, No 4, 8-97 (205) / wwwntmsccom 89 dervatves are substtuted n Eq(4), the resultng equaton must be satsfed approxmately; that s, for t q 0,],q = 0,, E j,n (t q ) = m k S r=0 = s=0 } j (t q)y (r) (λt q + µ s ) + Q r j(t q )y (r) (t q ) g j (t q ) = 0,j =,2,,k (2) P r,s or E j,n (t q ) 0 r p (r p s any postve number) If max(0 r p ) =0 r (rsanypostventeger) s prescrbed, then the truncaton lmt N s ncreased untl the dfferences E j,n (t q ) at each of the ponts become smaller than the prescrbed 0 r, see 4-8] If when N s suffcently large enough, then the error decreases On the other hand, the error can be estmated by system, E j,n (t) = m k S r=0 = s=0 } j (t)y(r) (λt + µ s ) + Q r j(t)y (r) (t) g j (t), j =,,2,,k P r,s 4 Illustratons In ths secton, some numercal examples on the problem (4) are gven to llustrate the accuracy and effectveness propertes of the method Example 2] Let us consder the followng lnear system of second-order advanced dfferental-dfference equatons, y (2) (t /2) + 2ty() (t /4) ty() y (2) 2 y (2) and the ntal condtons (t + /) ty() (t + /) ty 2 (t /2) +t 2 y (t ) = g (t) (t + /5) ty 2 (t /6) + 5y (t /2) = g 2 (t) (t /6) + y() (t /) + y (t + /4) + 2ty 2 (t + /) = g (t),0 t (22) y (0) = 0,y (0) =,y 2(0) =,y 2 (0) = 0,y (0) =,y (0) = l]g (t) = sn(t /2) + 2t cos(t + /) t cos(t /2) +t 2 e t, g 2 (t) = cos(t /4) t cos(t + /5) +t cos(t /6) + 5e t /2, g (t) = t cos(t /6) +te t / + sn(t + /4) + 2t cos(t + /) + e t+/, and the exact solutons are y (t) = snt,y 2 (t) = cost and y (t) = e t For N =, the approxmate solutons by the truncated Chelyshkov seres and the collocaton ponts are, respectvely, gven by y (t) = n=0 a n C,n (t), =,2, and t 0 = 0,t = /,t 2 = 2/,t = The fundamental matrx equaton of the problem s as follows, P 0 TM, C + P 04 TM 2, C + P 05 TM, C + P 06 TM 6, C + P 07 TM, C + P 2 TM 4, BC +P 4 TM, BC + P 6 TM 6, BC + P 8 TM, BC + P 2 TM 5, B 2 C + P 2 TM 2, B 2 C + P 26 TM 4, B 2 C A = G,

8 90 C Oguz, M Sezer and A Denk Oguz: Chelyshkov collocaton approach to solve the systems of lnear ] P j = dag P,j (0), P,j (/), P,j (2/), P,j (), = 0,,2 and j =,2,,8 0 t t 2 P 0, (t) = 0 0 5,P 0,4 (t) = 0 t 0,P 0, (t) =,P 0,6 (t) =, 0 2t 0 2t 0 0 P 0,7 (t) = 0 0,P,2 (t) =,P,4 (t) =,P,6 (t) =, 0 0 t t P,8 (t) = t 0 5,P 2, (t) =,P 2, (t) = 0 0,P 0,6 (t) =, 0 0 ] M,a = dag M,a, M,a, M,a, a = / 6, / 2,± / 4,± /, / 5, / 2 / 4 / 8 M, /2 = 0 / / 6 / 6 / / 2,M, /6 = 0 / / / 2, M, = , ±/ /9 ±/27 ±/4 /6 ±/64 /5 /25 /25 M,±/ = 0 ±2/ / 0 0 ±, M,±/4 = 0 ±/2 /6 0 0 ±/4 M,/5 = 0 2/5 / /5 T(0) ( T / ) ] T = ( T 2 / ), T(t s ) = dag T(t s ), T(t s ), T(t s ), T() T(0) = ( / ) T T ( 2 / T() = B 0 0 B = 0 B 0,B = ,C = B g(0) 50/045 ( ) /40 G = g(/) g(2/),g(0) = 975/957,g = 80/ / /282 g() ], = / / 9 / ] 27, ) = 2/ 4/ 9 8/ ] 27, ],g ( ) 2 677/4 = 247/48 085/76, and 98/745 g() = 2065/264,A = 58/427 A A 2 A ] A = a 0 a a 2 ] and A 2 = a 20 a 2 a 22 ] A = a 0 a a 2

9 NTMSCI, No 4, 8-97 (205) / wwwntmsccom 9 The augmented matrx for ths fundamental matrx equaton s calculated as ; 50/ /2 8/2 45/2 /2 55/8 605/8 95/8 5/8 ; 975/957 57/64 8/64 5/64 / ; 206/965 45/9 247/ 6/9 /9 29/648 87/26 4/648 / /24 26/8 28/24 8/24 ; /40 62/45 4/25 2/225 64/225 66/ / /648 2/ /26 95/72 25/26 5/26 ; 80/245 W;G] = 809/92 77/64 95/64 09/92 6/8 8/27 2/8 6/8 0 82/ 6 4 ; 529/ / /24 67/08 29/24 /24 272/24 268/8 00/24 4/24 ; 677/4 46/45 266/75 806/225 8/225 /4 9/2 7/2 29/2 545/26 5/72 45/26 5/26 ; 247/48 5/ /64 89/576 04/576 4/ 4/ 4/ 4/ 820/ /9 56/9 ; 085/76 295/ 87 55/ 4/ /8 /8 5/8 /8 4 ; 98/745 86/5 78/25 96/25 08/25 889/ /72 427/26 847/26 5/8 5/8 25/8 5/8 ; 2065/264 4/92 05/64 595/92 725/92 664/27 76/9 20/27 28/27 470/ 06 6/ 28/ ; 58/427 From Eq(9), the matrx form for ntal condtons s computated as : ; ; U,α] = ; ; ; Hence, the new augmented matrx based on condtons from systems (20) can be obtaned as follows ; 50/ /2 8/2 45/2 /2 55/8 605/8 95/8 5/8 ; 975/957 57/64 8/64 5/64 / ; 206/965 45/9 247/ 6/9 /9 29/648 87/26 4/648 / /24 26/8 28/24 8/24 ; /40 62/45 4/25 2/225 64/225 66/ / /648 2/ /26 95/72 25/26 5/26 ; 80/245 W; G ] = 809/92 77/64 95/64 09/92 6/8 8/27 2/8 6/8 0 82/ 6 4 ; 529/ ; ; ; ; ; ; By solvng ths system, substtutng the resultng unknown Chelyshkov coeffcents matrx nto Eq(6) we obtan the approxmaton solutons for N = as y, (t) = t t 2 +t, y 2, (t) = t t 2 +, y, (t) = t t 2 +t + By the help of smlar process for N = 5, 8 we get the approxmate solutons, respectvely, as y,5 (t) = t t t t 2 +t, y 2,5 (t) = t t t t 2 +, y,5 (t) = t t t t 2 +t +,

10 92 C Oguz, M Sezer and A Denk Oguz: Chelyshkov collocaton approach to solve the systems of lnear and y,8 (t) = t t t t t t t 2 +t, y 2,8 (t) = t t t t t t t 2 +, y,8 (t) = t t t t t t t 2 + Tables - show the comparson of some numercal values of the absolute errors of Chelyshkov polynomal solutons for N =,5,8 and, and also Fgs a, b and c dsplay the exact and approxmate solutons of Eq (22) From tables, we see that the errors decrease rapdly as N ncreases Table : Comparsons of the absolute error functons E,N (t) of Eq (22) t N= N=5 N=8 N= e, (t ) e,5 (t ) e,8 (t ) e, (t ) 0, 4,8685e- 6,500758e-5 2,858860e-8 6,60e-0 0,,92252e- 2,566262e-5,496e-8,8050e-0 0,5,404e-2 2,28245e-5,08000e-2 2,00859e-9 0,7,2758e-2,069425e-5 8,40749e-8,5740e-8 0,9,26244e-2 8,5644e-4 7,440406e-6,59974e-7 Table 2: Comparsons of the absolute error functons E 2,N (t) of Eq (22) t N= N=5 N=8 N= e 2, (t ) e 2,5 (t ) e 2,8 (t ) e 2, (t ) 0,,9804e-2 8,724e-5 4,29270e-9,890e-0 0,,52920e-2 4,580208e-5,88750e-9,88928e-0 0,5,82e- 6,625824e-5 4,00000e-2,7764e-9 0,7,04490e- 4,02964e-4 5,88580e-9 4,54605e-9 0,9 5,2444e-,685e- 2,056825e-7,9994e-9 Table : Comparsons of the absolute error functons E,N (t) of Eq (22) t N= N=5 N=8 N= e, (t ) e,5 (t ) e,8 (t ) e, (t ) 0 2,79488e-2,8574e-4,6955e-8 4,5980e-9 0,27292e-2,08602e-4 6,474959e-9 2,46228e-8 05,67859e-,76777e-4 2,48560e-0,226285e ,07659e-,2995e- 5,06500e-8 4,90669e ,7606e-,09e-2 4,40259e-6,68757e-6

11 NTMSCI, No 4, 8-97 (205) / wwwntmsccom 9 (a) Comparson of the exact soluton snt and the approxmate solutons y,n (t) (b) Comparson of the exact soluton e t and the approxmate solutons y,n (t) (c) Comparson of the exact soluton cost and the approxmate solutons y 2,N (t) Fg : Graphs of exact and numercal solutons of equaton (22) for N =,5,8 and Example 2 24] Let us consder system of the lnear dfferental dfference equatons wth varable coeffcents gven by y (2) (t) y (t) + y 2 (t) y (t 02) = e t 02 + e t y 2 (2) (t) + y (t) y 2 (t) y 2 (t 02) = e t+02 + e t, 0 t (2) wth the ntal condtons y (0) =,y (0) =,y 2(0) =,y 2 (0) = and the exact solutons y (t) = e t,y 2 (t) = e t Here, Q 0, (t) =,Q0,2 (t) =,Q0 2, (t) =,Q0 2,2 (t) =,Q2, (t) =,Q2,2 (t) = 0 = Q2 2, (t),q2 2,2 (t) =, P 0,0, (t) =,P0,0,2 (t) = 0,P0,0 2, (t) = 0,P0,0 2,2 (t) =,g (t) = e t 02 + e t andg 2 (t) = e t+02 + e t From Eq(6), the fundamental matrx equaton of the problem s Q 0 TC + P 00 TM, 02 C + Q 2 TB 2 C}A = G

12 94 C Oguz, M Sezer and A Denk Oguz: Chelyshkov collocaton approach to solve the systems of lnear Usng the procedure n Secton, we get the approxmate solutons by the Chelyshkov polynomals of the problem for N =,6,0 y, (t) = t +t, y 2, (t) = t 054t 2 +, y,6 (t) = t t t t +t, y 2,6 (t) = t t t t 05t 2 +, and y,0 (t) = t t t t t t t t +t, y 2,0 (t) = t t t t t t t t 05t 2 + Table 4: Numercal results of the exact and the approxmate solutons y,n (t) for N=,6,0 t Exact Value Approxmaton solutons e t y, (t ) y,6 (t ) y,0 (t ) , , , , , , , , , , Table 5: Numercal results of the exact and the approxmate solutons y 2,N (t) for N=,6,0 t Exact Value Approxmaton solutons e t y 2, (t ) y 2,6 (t ) y 2,0 (t ) , , , , , , , , , , Table 6: The maxmum errors E,N (t) and E 2,N (t) of Eq (2) N 6 0 E,N (t),04e-2 2,04e-6,688e-0 E,N (t) 4,9495e- 9,6458e-7 6,6667e-0 In Tables 4 and 5, t s gven a comparson of numercal results of the approxmate solutons obtaned by the presented method for N =,6 and 0 wth the exact solutons of Eq (2) In addton, the absolute error functons are shown n

13 NTMSCI, No 4, 8-97 (205) / wwwntmsccom 95 Fg 2 As seen from Table 6, the resultng solutons from Chelyshkov polynomal method for N = 0 are almost the same as the results of the exact solutons and we see that the errors decrease rapdly as N ncreases (a) Comparson of the absolute errors functons e,n (t) for N =,6 and 0 (b) Comparson of the absolute errors functons e 2,N (t) for N =,6 and 0 Fg 2: The Graph of the functon n equaton (2) for N =,6 and 0 Example 25] Let us consder the system of ntal value problems gven by y (t ) + y 2 (t ) = 2t, y (0) = 0 y (t ) y (t ) = 2t, y 2(0) = 0, 0 t y (t ) + y (t ) = t, y (0) = 0 (24) Usng the present method for N = as n Example, we obtan solutons of the problem as y (t) = t 2,y 2 (t) = 2t and y (t) = t whch are the exact solutons Moreover, f hgher values of N be chosen, we obtan the exact soluton agan 5 Conclusons In ths paper, we have presented a new collocaton method and used t for the systems of the mentoned lnear functonal dfferental equatons wth varable coeffcents The comparson of the results shows that the present method s a powerful mathematcal tool for fndng the numercal solutons of these type systems One of the consderable advantages of the method s that the approxmate solutons are found very easly by usng avable software such as maple or matlab snce the method s based on matrx operatons Moreover, the method proposed n ths work can be extended to solve the systems of nonlnear equatons whch play an mportant role n physcs and engneerng

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