Research Article Collocation Method Based on Genocchi Operational Matrix for Solving Generalized Fractional Pantograph Equations

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1 Hndaw Internatonal Dfferental Equatons Volume 2017, Artcle ID , 10 pages Research Artcle Collocaton Method Based on Genocch Operatonal Matrx for Solvng Generalzed Fractonal Pantograph Equatons Abdulnasr Isah, 1,2 Chang Phang, 2 and Pau Phang 3 1 Department of Mathematcs, Ahmadu Bello Unversty, Zara, gera 2 Department of Mathematcs and Statstcs, Faculty of Scence, Technology and Human Development, UnverstTunHussenOnnMalaysa,BatuPahat,Malaysa 3 Faculty of Computer Scence and Informaton Technology, Unverst Malaysa Sarawak, Sarawak, Malaysa Correspondence should be addressed to Pau Phang; pphang@unmasmy Receved 2 January 2017; Revsed 16 March 2017; Accepted 16 May 2017; Publshed 13 June 2017 AcademcEdtor:PatrcaJYWong Copyrght 2017 Abdulnasr Isah et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted An effectve collocaton method based on Genocch operatonal matrx for solvng generalzed fractonal pantograph equatons wth ntal and boundary condtons s presented Usng the propertes of Genocch polynomals, we derve a new Genocch delay operatonal matrx whch we used together wth the Genocch operatonal matrx of fractonal dervatve to approach the problems The error upper bound for the Genocch operatonal matrx of fractonal dervatve s also shown Collocaton method based on these operatonal matrces s appled to reduce the generalzed fractonal pantograph equatons to a system of algebrac equatons The comparson of the numercal results wth some exstng methods shows that the present method s an excellent mathematcal tool for fndng the numercal solutons of generalzed fractonal pantograph equatons 1 Introducton Fractonal calculus, the calculus of dervatve and ntegral ofanyorder,susedasapowerfultoolnscenceand engneerng to study the behavors of real world phenomena especally the ones that cannot be fully descrbed by the classcal methods and technques [1 Dfferental equatons wth proportonal delays are usually referred to as pantograph equatons or generalzed pantograph equatons The name pantograph was orgnated from the study work of Ockendon and Tayler [2 Many researchers have studed dfferent applcatons of these equatons n appled scences such as bology, physcs, economcs, and electrodynamcs [3 5 Solutons of pantograph equatons were also studed by many authors numercally and analytcally Bhrawy et al proposed a new generalzed Laguerre-Gauss collocaton method for numercal soluton of generalzed fractonal pantograph equatons [1 Tohd et al n [6 proposed a new collocaton scheme based on Bernoull operatonal matrx for numercal soluton of generalzed pantograph equaton Yusufoglu [7 proposed an effcent algorthm for solvng generalzed pantograph equatons wth lnear functonal argument In [8, Yang and Huang presented a spectral-collocaton method for fractonal pantograph delay ntegrodfferental equatons and n [9 Yüzbas and Sezer presented an exponental approxmaton for solutons of generalzed pantograph delay dfferental equatons Chebyshev andbesselpolynomalsare,respectvely,usedn[10,11to obtan the solutons of generalzed pantograph equatons Operatonal matrces of fractonal dervatves and ntegraton have become very mportant tool n the feld of numercal soluton of fractonal dfferental equatons In ths paper, a member of Appell polynomals called Genocch polynomals s used; although ths polynomal s not based on orthogonal functons, t possesses operatonal matrces of dervatves wth hgh accuracy It s very mportant to note that ths polynomal shares some great advantages wth Bernoull and Euler polynomals for approxmatng an arbtrary functon over some classcal orthogonal polynomals; we refer the reader to [6 for these advantages On top of that, we

2 2 Internatonal Dfferental Equatons hadsuccessfullyappledtheoperatonalmatrxvagenocch polynomals for solvng nteger-order delay dfferental equatons [12 and fractonal optmal control problems [13, and the numercal solutons obtaned are comparable or even more accurate compared to some exstng well-known methods Motvated by these advantages, n ths paper, we ntend to extend the result for nteger-order delay dfferental equatons n [12 to fractonal delay dfferental equatons or so-called generalzed fractonal pantograph equatons To the best of our knowledge, ths s the frst tme that the operatonal matrx based on Genocch polynomals s appled to solve the fractonal pantograph equatons On the other hand, some other types of polynomals were employed to solve some specal type of fractonal calculus problems; for example, Bessel polynomals were used for the soluton of fractonal-order logstc populaton model [14; Bernsten polynomalswerealsousedforthesolutonofrccattype dfferental equatons [15 In ths paper, we use the new operatonal matrx of fractonal-order dervatve va Genocch polynomals to provde approxmate solutons of the generalzed fractonal pantograph equatons of the followng form [1: D α y (t) = J m 1 j=0 n=0 subject to the followng condtons: m 1 n=0 p j,n (t) D β n y(λ j,n t+μ j,n )+g(t), 0 t 1 (1) a n, y (n) (0) =d, =0,1,,m 1, (2) where a n,, λ j,n,andμ j,n are real or complex coeffcents; m 1<α<m, 0<β 0 <β 1 < < β m 1 <α, whle p j,n (t) and g(t) aregvencontnuousfunctonsnthenterval[0, 1 The rest of the paper s organzed as follows: Secton 2 ntroduces some mathematcal prelmnares of fractonal calculus In Secton 3, we dscuss some mportant propertes of Genocch polynomals In Secton 4, we derve the Genocch delay operatonal matrx and we apply the collocaton method for solvng fractonal pantograph equaton (1) usng the Genocch operatonal matrx of fractonal dervatve and the delay operatonal matrx n Secton 5 In Secton 6, the proposed method s appled to several examples and concluson s gven n Secton 7 2 Prelmnares 21 Fractonal Dervatve and Integraton We recall some basc defntons and propertes of fractonal calculus that we wll use There are varous competng defntons for fractonal dervatves [16, 17 The Remann-Louvlle defnton played a vtal role n the development of the theory of fractonal calculus However, there are certan dsadvantages of usng ths defnton when modelng real world phenomena To cope wth these dsadvantages, Caputo defnton was ntroduced whch s found to be more relable n applcaton So we use ths defnton of fractonal dervatves We begn wth the defnton of Remann-Louvlle ntegral, n whch the fractonal ntegral operator I of a functon f(t) s defned as follows Defnton 1 The Remann-Louvlle ntegral I of fractonalorder α of f(t) s gven by I α f (t) = 1 t Γ (α) (t τ) α 1 f (τ) dτ, 0 t>0, α R +, where Γ( ) sthegammafunctonthefractonaldervatve of order α>0duetoremann-louvllesdefnedby (D α l f) (t) =(d dt ) m (I m α f) (t), (α >0, m 1<α<m) The followng are mportant propertes of Remann-Louvlle fractonal ntegral I α : I α I β f (t) =I α+β f (t), I α t β = Γ(β+1) Γ(β+α+1) tβ+α α>0, β>0, Defnton 2 The Caputo fractonal dervatve D α of a functon f(t) s defned as D α f (t) = t 1 Γ (n α) 0 f (n) (τ) (t τ) α n+1 dτ, n 1<α n, n Some propertes of Caputo fractonal dervatves are as follows: (3) (4) (5) (6) D α C=0, (C s constant), 0, { β {0}, β < α D α t β = { Γ(β+1) { Γ(β+1 α) tβ α, β {0}, β α or β, β > α, (7)

3 Internatonal Dfferental Equatons 3 where α denotes the smallest nteger greater than or equal to α and α denotes the largest nteger less than or equal to α Smlar to the nteger-order dfferentaton, the Caputo fractonal dfferental operator s a lnear operator; that s, D α (λf (t) +μg(t)) =λd α f (t) +μd α g (t) (8) for λ and μ constants 3 Genocch Polynomals and Some Propertes Genocch polynomals and numbers have been extensvely studed n many dfferent contexts n branches of mathematcs such as elementary number theory, complex analytc number theory, homotopy theory (stable homotopy groups of spheres), dfferental topology (dfferental structures on spheres), theory of modular forms (Esensten seres), and quantum physcs (quantum groups) The classcal Genocch polynomal G n (x) s usually defned by means of the exponental generatng functons [ te xt e t +1 = G n (x) tn, ( t <π), (9) n! n=0 where G n (x) s the Genocch polynomal of degree n and s gven by G n (x) = n k=0 ( n k )G n kx k (10) G n k here s the Genocch number Some of the mportant propertes of these polynomals nclude 1 0 G n (x) G m (x) dx = 2 ( 1)n n!m! G (m+n)! m+n n, m 1, (11) dg n (x) dx =ng n 1 (x), n 1, (12) G n (1) +G n (0) =0, n>1 (13) Before we move to the next level, we need the followng lnear ndependence on whch the rest of theoretcal results are based Lemma 3 The set A={G 1 (t), G 2 (t),,g (t)} L 2 [0, 1 s alnearlyndependentsetnl 2 [0, 1 Proof To show that A s the set of lnearly ndependent elements of L 2 [0, 1, tsenoughtoshowthatthegram determnant s not zero That s, Gram (G 1,G 2,,G ) =0, (14) where Gram (G 1,G 2,,G ) = G 1,G 1 G 1,G 2 G 1,G G 2,G 1 G 2,G 2 G 2,G G n,g 1 G n,g 2 G n,g (15) ow, to prove that ths determnant s not equal to zero, we frst reduce the Gram matrx to an upper trangular matrx by Gaussan elmnaton and t s not dffcult to see that the elementsofthedagonalofthereducedmatrxaregvenby (n! (n+1)!)2 a (n) =, n (16) (2n!)(2n + 1)! Clearly, one can see that, for any, a(n) =0Consequently, the determnant gven by n=1 a (n) (17) s not equal to zero Therefore, the set A s the set of lnearly ndependent sets 31 Functon Approxmaton Assume that {G 1 (t), G 2 (t),,g (t)} L 2 [0, 1 s the set of Genocch polynomals and Y=Span{G 1 (t), G 2 (t),,g (t)} Letf(t) be arbtrary element of L 2 [0, 1;snceY s a fnte dmensonal subspace of L 2 [0, 1 space, f(t) has a unque best approxmaton n Y,say f (t),suchthat f (t) f (t) 2 f (t) y(t) 2, y(t) Y (18) Ths mples that, y(t) Y, f (t) f (t),y(t) =0, (19) where denotes nner product Snce f (t) Y, there exst the unque coeffcents c 1,c 2,,c such that f (t) f (t) = n=1 c n G n (t) = C T G (t), (20) where C = [c 1,c 2,,c T, G(t) = [G 1 (t), G 2 (t),, G (t) T Usng (19), we have f (t) C T G (t),g (t) =0 =1,2,,; (21) for smplcty, we wrte C T G (t), G (t) = f (t), G (t), (22) where G(t), G(t) s an matrx Let W= G(t), G(t) = 1 0 G(t)GT (t)dt The entres of the matrx W can be calculated from (11) Therefore, any functon f(t) L 2 [0, 1 can be expanded by Genocch polynomals as f(t) = C T G(t),where C =W 1 f (t), G (t) (23)

4 4 Internatonal Dfferental Equatons 4 Genocch Operatonal Matrx where In ths secton, we derve the operatonal matrces for the delay and that of fractonal dervatve based on Genocch polynomals for the soluton of fractonal pantograph equatons 41GenocchDelayOperatonalMatrx The Genocch delay vector G(t μ) canbeexpressedas G (t μ) = RG (t), (24) where R s the operatonal delay matrx gven by M= (28) [ [ Thus, M s operatonal matrx of dervatve It s not dffcult to show nductvely that the kth dervatve of G(t) can be gven by R =W 1 W μ μ 2 3μ = 4μ 3 6μ 2 4μ 1 0 0, 5μ 4 10μ 3 10μ 2 5μ 1 0 [ [ b n (1) b n (2) b n (3) b n (4) b n (5) 1 (25) d k G (t) T = G (t) (M T ) k (29) dt k In the followng theorem, the operatonal matrx of fractonal-order dervatve for the Genocch polynomals s gven Theorem 5 (see [21) Suppose that G(t) s the Genocch vector gvenn(20)andletα>0then, D α G (t) T =P α G (t) T, (30) where P α s operatonal matrx of fractonal dervatve of order α n Caputo sense and s defned as follows: where W 1 = 1 0 G(t μ)gt (t)dt and b n () = ( 1) n (( n )) μ n, =1,2,,n Also, for any delay functon f(t μ), we can express t n terms of Genocch polynomals as shown n (26): f (t μ) = =1 c G (t μ) = C T RG (t), (26) where C s gven n (23) The followng lemma s also of great mportance Lemma 4 Let G (t) be the Genocch polynomals; then D α G (t) = 0,for = 1,, α 1, α > 0 Theproofofthslemmasobvous;onecanuse(7)and (8) on (10) 42 Genocch Operatonal Matrx of Fractonal Dervatve If we consder the Genocch vector G(t) gven by G(t) = [G 1 (t), G 2 (t),,g (t), then the dervatve of G(t) wth the ad of (12) can be expressed n the matrx form by dg (t) T =MG (t) T, (27) dt P (α) = [ [ α ρ α,k,1 ρ,k,1 ρ,k,1 where ρ,k,j s gven by ρ,k,j = α ρ α,k,2 ρ,k,2 ρ,k,2 α ρ α,k, ρ,k, ρ,k,, (31)!G k ( k)!γ (k+1 α) c j (32) G k s the Genocch number and c j canbeobtanedfrom (23) Proof For the proof, see [21

5 Internatonal Dfferental Equatons 5 43 Upper Bound of the Error for the Operatonal Matrx of Fractonal Dervatve P α Webegnherebyprovngtheupper bound of the error of arbtrary functon approxmaton by Genocch polynomals n the followng Lemma Lemma 6 Suppose that f(t) C n+1 [0, 1 and Y = Span{G 1 (t), G 2 (t),,g (t)}; fc T G(t) sthebestapproxmaton of f(t) out of Y,then f (t) CT G (t) R (n+1)! 2n + 3, (33) where R=max t [0,1 f (n+1) (t) To see ths, we set {1,t,,t n } as a bass for the polynomal space of degree n Defne y 1 (t) = f(0) + tf (0) + (t 2 /2!)f (0) + + (t n /n!)f (n) (0) From Taylor s expanson, one has f(t) y 1 (t) = (t n+1 /(n+ 1)!)f (n+1) (ξ t ),whereξ t (0, 1) Snce C T G(t) s the best approxmaton of f(t) out of Y and y 1 (t) Y,from(18),onehas f (t) 2 CT G (t) 2 f (t) y 1 (t) f (t) y 1 (t) 2 0 dt 1 = ( tn (n+1)! )2 f(n+1) (ξ t ) dt R 2 1 ((n+1)!) 2 t 2n+2 R 2 dt = 0 ((n+1)!) 2 (2n + 3) Takng the square root of both sdes, one has f (t) CT G (t) R (n+1)! 2n + 3 whch s the desred error bound We use the followng theorem from [22 (34) (35) Theorem 7 (see [22) Suppose that H s a Hlbert space and Y s a closed subspace of H such that dm Y < and y 1,y 2,,y n s a bass for Y Letf be an arbtrary element n H and let y 0 betheunquebestapproxmatonoff out of Y Then, where f y 0 2 = Gram (f, y 1,y 2,,y n ) Gram (y 1,y 2,,y n ), (36) Gram (y 1,y 2,,y n ) = y 1,y 1 y 1,y 2 y 1,y n y 2,y 1 y 2,y 2 y 2,y n y n,y 1 y n,y 2 y n,y n (37) Theorem 8 Suppose that f(t) L 2 [0, 1 s approxmated by f n (t) as then, f n (t) = n =1 c G (t) = C T G (t) ; (38) lm n f (t) f n (t) =0 (39) The proof of ths theorem obvously follows from Lemma 6 The operatonal matrx error vector E α s gven by where from Theorem 7, we get f 0 (t) j=1 E α =P α G (t) D α G (t), (40) c j G j (t) E α = [ [ e α 1 e α 2 e α n ; (41) =( Gram (f (t),g 1/2 1 (t),,g (t)) ) Gram (G 1 (t),,g (t)) Thus, accordng to equatons (29) and(30) n[21, one has eα = D α G (t) j=1 ( j=1!g k ( k)!γ (k+1 α) c j)g j (t)!g k ( k)!γ (k+1 α) tk α f 0 (t) c j G j (t)!g k ( k)!γ (k+1 α) tk α ( Gram (f (t),g 1/2 1 (t),,g (t)) ) Gram (G 1 (t),,g (t)) (42) (43) By consderng Theorem 8 and (43), we can conclude that by ncreasng the number of the Genocch bases the vector e α tends to zero For comparson purpose n Table 1, we show below the errors of operatonal matrx of fractonal dervatve based on Genocch polynomals and shfted Legendre polynomals derved n [23, 24 when =10and α = 075 at dfferent

6 6 Internatonal Dfferental Equatons Table 1: Comparson of the operatonal matrx errors for the GPOMFD and SLPOMFD E α x=1 x=0 x=05 GPOMFD SLPOMFD GPOMFD SLPOMFD GPOMFD SLPOMFD e e e e e e e e e e ponts on [0, 1 From ths table, t s clear that the accuracy of Genocch polynomals operatonal matrx of fractonal dervatve (GPOMFD) s better than the shfted Legendre polynomals operatonal matrx of fractonal dervatves (SLPOMFD) We beleve that ths s the case for any value of because the Genocch polynomals have smaller coeffcents of ndvdual terms compared to shfted Legendre polynomals 5 Collocaton Method Based on Genocch Operatonal Matrces In ths secton, we use the collocaton method based on Genocch operatonal matrx of fractonal dervatves and Genocch delay operatonal matrx to solve numercally the generalzed fractonal pantograph equaton We now derve an algorthm for solvng (1) To do ths, let the soluton of (1) be approxmated by the frst terms Genocch polynomals Thus, we wrte y (t) n=1 c n G n (t) =G(t) C, (44) where the Genocch coeffcent vector C and the Genocch vector G(t) are gven by C T =[c 1,c 2,,c, G (t) =[G 1 (t),g 2 (t),,g (t); (45) thus, D α y (t) and D β n y (t), n = 0,1,,m 1,canbe expressed, respectvely, as follows: D α y (t) =G(t) (P T ) α C, D β n y (t) =G(t) (P T ) β n C, n=0,1,,m 1 (46) Substtutng (44) and (46) n (1), we have G (t) (P T ) α C = J m 1 j=0 n=0 p j,n (t) G(λ j,n t+μ j,n )(P T ) β n C+g(t), (47) where G(λ j,n t+μ j,n )=[G 1 (λ j,n t+μ j,n ), G 2 (λ j,n t+μ j,n ),, G (λ j,n t+μ j,n ) Also the ntal condton wll produce m other equatons: m 1 n=0 a n, G (0) (P T ) C=d, =0,1,,m 1 (48) To fnd the soluton y (t) we collocate (47) at the collocaton ponts t j =j/( m), j=1,2,, m,toobtan G(t j )(P T ) α C = J m 1 j=0 n=0 +g(t j ) p j,n (t j )G(λ j,n t j +μ j,n )(P T ) β n C (49) for j = 1,2,, m Addtonally, one can also use both the operatonal matrx of fractonal dervatve and delay operatonal matrx to solve problem (1) Accordng to (44), we can approxmate the delay functon y(λ j,n t j +μ j,n ) and ts fractonal dervatve usng the operatonal matrces P and R as follows: y (λ j,n t j +μ j,n )=RC T G (t), D β n y (λ j,n t j +μ j,n )=RC T P β n G (t) (50) Puttng ths approxmaton together wth (44) n (1), we have G (t) (P T ) α C= J m 1 j=0 n=0 p j,n (t) RC T P β n G (t) +g(t) (51)

7 Internatonal Dfferental Equatons 7 Table 2: Comparson errors obtaned by the present method and those obtaned n [25 when α = 06 and τ = 03 for Example 1 t Error (new method) [25 Error (FAM) [25 Error (present method) E E E E E 02 Thus, collocatng (51) at the same collocaton pont as that n (47), we get subject to y (0) =0, G(t j )(P T ) α C= J m 1 j=0 n=0 +g(t j ) p j,n (t j ) RC T P β n G(t j ) (52) Hence, (49) or (52) s mnonlnear algebrac equaton Any of these equatons together wth (48) makes algebrac equatons whch can be solved usng ewton s teratve method Consequently, y (x) gven n (44) can be calculated 6 umercal Examples In ths secton, some numercal examples are gven to llustrate the applcablty and accuracy of the proposed method All the numercal computatons have been done usng Maple 18 Example 1 Consder the followng example solved n [25: subject to D α y (t) = 2 Γ (3 α) y1 α/2 (t) +y(t τ) y(t) (53) +2τ y (t) τ 2 y (t) =0, t 0 (54) The exact soluton for ths example s gven by y(t) = t 2 We solve the example when α = 06 and τ = 03 InTable2,wecomparetheerrorsobtanedbyourmethod wth those obtaned usng FAM and new approach n [25 As reported n [25, the tme requred for the new method s seconds and for the FAM the tme taken s seconds for completng the same task, whereas n our method we only need seconds to complete the computatons Example 2 (see [1) Consder the followng generalzed fractonal pantograph equaton: D 5/2 y (t) = y(t) y(t 05) +g(t), t [0, 1 (55) where y (0) =0, y (0) =0, (56) g (t) = Γ (4) Γ (3/2) t1/2 +t 3 + (t 05) 3 (57) The exact soluton of ths problem s known to be y(t) = t 3 Ths problem s solved n [1 usng generalzed Laguerre- Gauss collocaton scheme We apply our technque wth = 4 Approxmatng (55) wth Genocch polynomals, we have G (t) (P T ) 5/2 C= G(t) C+G(t 05) C+g(t) (58) Also from the ntal condtons we have G (0) C=0, G (0) (P T )C=0, G (0) (P T ) 2 C=0 Thus, collocatng (58) at t = , weget c 4 +2c c c 2 =0, and (59) gves c 1 c 2 +c 4 =0, 2c 1 3c 2 =0, 6c 3 12c 4 =0 Solvng these equatons, we have c 1 = , c 2 = , c 3 = , c 4 = (59) (60) (61) (62) Thus, y(t) = G(x)C s calculated and we have t 3 whch s almost the exact soluton In Table 3, we compare the absolute errors obtaned by our method (wth only few terms =4) and the absolute errors obtaned n [1 when =22wth dfferent Laguerre parameters β

8 8 Internatonal Dfferental Equatons Table 3: Comparson of the absolute errors obtaned by the present method and those obtaned n [1 for Example 2 t =22[1 =4 β=2 β=3 β=5 Present method E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 06 Table 4: Comparson of the absolute errors obtaned by the present method and those n [26 for Example 4 t Absolute error [26 =4 Absolute error (present method) = E E E E E E E E E E E E E E E E E E E E 09 Example 3 Consder the followng fractonal pantograph equaton: D 1/2 y (t) =2y( 3t )+8t3/2 2 3 π 9t2, t [0, 1 (63) 2 subject to y (0) =0, y (1) =1 (64) The exact soluton of ths problem s known to be y(t) = t 2 We solve (63) usng our technque wth =3only As n Example2,weobtanedthevaluesofthecoeffcentstobe c 1 = 1 2, c 2 = 1 2, c 3 = 1 3 (65) Thus, y (t) = G(t)C s calculated to be t 2 whch s the exact soluton and so there s nothng to compare for the error s zero Example 4 Consder the followng fractonal pantograph equaton solved n [26: subject to D 2 y (t) +D 3/2 y (t) +y(t) =y( t 2 )+3t t π +2, t [0, 1 (66) y (0) =0, y (1) =1 (67) The exact soluton of ths problem s known to be y(t) = t 2 AsnExample3,wesolve(66)usngourtechnquewth =3and the values of the coeffcents obtaned are c 1 = , c 2 = , c 3 = (68) Thus, y (t) = G(t)C s calculated and compared wth the exact soluton Ths problem s solved usng Taylor collocaton method n [26 when =4,5, and 6 In Table 4, we compare the absolute errors obtaned by present method when =3wth the errors obtaned when =4n [26

9 Internatonal Dfferental Equatons 9 7 Concluson In ths paper, a collocaton method based on the Genocch delay operatonal matrx and the operatonal matrx of fractonal dervatve for solvng generalzed fractonal pantograph equatons s presented The comparson of the results shows that the present method s an excellent mathematcal tool for fndng the numercal solutons delay equaton The advantage of the method over others s that only few terms are needed and every operatonal matrx nvolves more numbers of zeroes; as such the method has less computatonal complexty and provdes the soluton at hgh accuracy Conflcts of Interest The authors declare that there are no conflcts of nterest regardng the publcaton of ths manuscrpt Acknowledgments Ths work was supported n part by MOE-UTHM FRGS Grant Vot 1433 The authors also acknowledge fnancal support from UTHM through GIPS U060 The thrd author would lke to acknowledge Faculty of Computer Scence and Informaton Technology, UIMAS, for provdng contnuous support 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10 10 Internatonal Dfferental Equatons [25 V Daftardar-Gejj, Y Sukale, and S Bhalekar, Solvng fractonal delay dfferental equatons: a new approach, Fractonal Calculus and Appled Analyss, vol 18, no 2, pp , 2015 [26 A Anapal, Y Öztürk, and M Gülsu, umercal Approach for Solvng Fractonal Pantograph Equaton, Internatonal Journal of Computer Applcatons,vol113,no9,pp45 52,2015

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