Research Article New Solutions for System of Fractional Integro-Differential Equations and Abel s Integral Equations by Chebyshev Spectral Method

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1 Hindawi Mahemaical Problems in Engineering Volume 27, Aricle ID , 3 pages hps://doiorg/55/27/ Research Aricle New Soluions for Sysem of Fracional Inegro-Differenial Equaions and Abel s Inegral Equaions by hebyshev Specral Mehod Hassan A Zedan, Seham Sh Tanawy, 2 and Yara M Sayed 2 Deparmen of Mahemaics, Faculy of Science, Kafrelsheikh Universiy, Kafrelsheikh, Egyp 2 Deparmen of Mahemaics, Faculy of Educaion, Ain Shams Universiy, airo, Egyp orrespondence should be addressed o Hassan A Zedan; hassanzedan23@yahoocom Received 4 Sepember 26; Acceped 4 February 27; Published 3 March 27 Academic Edior: J- orés opyrigh 27 Hassan A Zedan e al This is an open access aricle disribued under he reaive ommons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied hebyshev specral mehod based on operaional mari is applied o boh sysems of fracional inegro-differenial equaions and Abel s inegral equaions Some es problems, for which he eac soluion is known, are considered Numerical resuls wih comparisons are made o confirm he reliabiliy of he mehod hebyshev specral mehod may be considered as alernaive and efficien echnique for finding he approimaion of sysem of fracional inegro-differenial equaions and Abel s inegral equaions Inroducion In recen years, he opic of fracional calculus has araced many scieniss because of is several applicaions in many areas, such as physics, chemisry, biology, and engineering For a deailed survey wih collecions of applicaions in various fields, see, for eample, [ 6] The numerical soluion of differenial equaions of ineger order has been a ho opic in numerical and compuaional mahemaics for a long ime There are many differen mehods and differen basis funcions have been used o esimae he soluion of fracional inegro-differenial equaions or Abel s inegral equaions, such as Adomian decomposiion mehod [7, 8], fracional differenial ransform mehod [9, ], collocaion mehod [, 2], homoopy perurbaion mehod [3, 4], homoopy analysis mehod [5, 6], variaional ieraion mehod [7], discree Galerkin mehod [8], and Haar wavele mehod [9] Specral mehods provide a compuaional approach ha has achieved subsanial populariy over he las four decades They have gained new populariy in auomaic compuaions forawideclassofphysicalproblemsinfluidandheaflow Their fascinaing meri is he high accuracy So, hey have been applied successfully o numerical simulaions of many problems in science and engineering; see [2 24] The operaional mari of fracional derivaives has been deermined for some ypes of orhogonal polynomials, such as hebyshev polynomials [25] and Legendre polynomials [26], and for inegraion has been deermined for several ypes of orhogonal polynomials, such as hebyshev polynomials [27], Laguerre series [28], and Legendre polynomials [29] Recenly, he Bernsein operaional mari approach is developed for solving a sysem of high order linear Volerra Fredholm inegro-differenial equaions in [3] In he presen paper, we use hebyshev specral mehod based on operaional mari o solve sysem of fracional inegro-differenial equaions: Dα u j () =F j (, u (),u 2 (),,u m (),u (l) (), u (l) 2 (),,u(l) m (), K j (, u (),u 2 (),,u m ())d), j=,2,,m, ()

2 2 Mahemaical Problems in Engineering wih iniial condiions u ( ) j () =a j, =,,2,,n, j=,,m, m N, where n <α n, l=,,,n, n N And we use Abel s inegral equaion: (2) u () λu () =f() + d,, L, (3) where λ=or λ=, f() is a coninuous funcion, and L is consan 2 Basic Definiions In his secion, we summarize some basic definiions and properies of fracional calculus heory Definiion Arealfuncionf(), >, issaidobeinhe space μ, μ R, if here eiss a real number p>μ,suchha f() = p f (), wheref () ([, )) learly μ β if β μ Definiion 2 Afuncionf(), >, issaidobeinspace n μ, n N,iff(n) μ Definiion 3 The Riemann-Liouville fracional inegral operaor of order α(α ),ofafuncionf μ, μ,is defined as J α a f () = Γ (α) ( s) α f (s) ds, >a, α>, a (4) J a f () =f() Definiion 4 The apuo fracional derivaives of order α are defined as a Dα f () =Jn α a D n f () = Γ (n α) ( s) n α d n a ds n f (s) ds, >a, where n < α n and D n is he classical differenial operaor of order n For apuo derivaive, we have (5) Dα β =, { for β N, β < α, { Γ(β+) { Γ(β+ α) β α, for β N, β α or β N, β > α (6) We use he ceiling funcion α denoing he smalles ineger greaerhanorequaloα and he floor funcion α denoing he larges ineger less han or equal o α Also N = {, 2, 3, } and N = {,, 2, } Recallha,forα N, heapuo differenial operaor coincides wih he usual differenial operaor of ineger order More properies of he fracional derivaives and he fracional inegral can be found in [3, 4] 3 Some Properies of he Shifed hebyshev Polynomials The well-known hebyshev polynomials {T i (); i =,,} are defined on he inerval (, ) and can be deermined wih he aid of he following recurrence formula: T i+ () =2T i () T i (), i=,2,, (7) where T () = and T () = The hebyshev polynomials areorhogonalonheinerval(, ) wih respec o he weigh funcion w() = / 2 These polynomials saisfy he relaion T i () T k () w () d = δ ik h k, (8) where h k =(ε k /2)π, ε =2, ε k =, k The analyic form of he hebyshev polynomial of degree i is given by [i/2] T i () =i k= ( ) k 2 i 2k (i k )! (k)! (i 2k)! i 2k, (9) where [i/2] denoes he ineger par of i/2 The zeros of T i () are denoed by k = cos (( 2k + )π), k=,,2,,i () 2i In order o use hese polynomials on he inerval (, L), we defined he so-called shifed hebyshev polynomials by inroducing he change of variable =2/L Le he shifed

3 Mahemaical Problems in Engineering 3 hebyshev polynomials T i (2/L ) be denoed by T L,i (), saisfying he orhogonaliy relaion L T L,i () T L,k () w L () d = δ ik h k, () where w L () = / L 2 The shifed hebyshev polynomials are defined as T L,i () = T i (2/L ) = T 2i ( /L) and he analyic form is given by T L,i () =i i k= ( ) i k 2 2k (i+k )! (i k)! (2k)!L k k, (2) where T L,i () = ( ) i and T L,i (L) = In his form, T L,i () may be generaed wih he aid of he following recurrence formula: T L,i+ () =2( 2 L )T L,i () T L,i (), i=,2,, (3) where T L, () = and T L, () = 2/L The zeros of T L,i () are denoed by k = L 2 + L 2 + cos ((2k )π), 2i k=,,2,,i (4) Afuncionu(),squareinegrablein(, L), may be epressed in erms of he shifed hebyshev polynomials as u () = j= where he coefficiens c j are given by c j T L,j (), (5) c j = L u () T h L,j () w L () d, j=,,2, (6) j In pracice, only he firs (N + )-erms shifed hebyshev polynomials are considered Hence, if we wrie u N () N j= c j T L,j () = T φ (), (7) where he shifed hebyshev coefficien vecor and he shifed hebyshev vecor φ() are given by T =[c,c,,c N ], φ () =[T L, (),T L, (),,T L,N ()] T, hen he derivaive of he vecor φ() canbeepressedby (8) dφ () d = D() φ (), (9) where D () is he (N + ) (N + ) operaional mari of derivaive given by 4i D () { =(d ij )= ε j L, j=,,,i=j+k,{ k=,3,5,,n, { k=,3,5,,n, { { {, oherwise; if N is odd, if N is even, (2) for eample, for even N,wehave D () = 2 L ( ) N 2N 2 2N 2 ( 2N 2N 2N ) (2) 4 The Shifed hebyshev Operaional Mari (OM) Fracional Derivaives The main objecive of his subsecion is o generalize he OM of derivaives for he fracional calculus By using (9), i is clear ha d n φ () d n =(D () ) n φ (), (22) where n N and he superscrip, in D (), denoes mari powers Thus D (n) =(D () ) n, n=,2, (23) Lemma 5 Le T L,i () be a shifed hebyshev polynomial; hen Dα T L,i () =, i=,,, α, α> (24)

4 4 Mahemaical Problems in Engineering Theorem 6 Le φ() be he shifed hebyshev vecor defined in (8) and suppose α>;hen Dα φ () D(α) φ (), (25) where D (α) is he (N+) (N+)OM of derivaives of order α in he apuo sense and is defined as follows: D (α) S α ( α,) S α ( α,) S α ( α,2) S α ( α,n) =, S α (i, ) S α (i, ) S α (i, 2) S α (i, N) ( ) ( S α (N, ) S α (N, ) S α (N, 2) S α (N, N) ) (26) Also, by subsiuing (22) and (28) in (2), we obain u ( ) j () = T j D( ) φ () =a j, =,,,n (32) Then we have o collocae (3) a he (N n + ) shifed hebyshev roos L,N n+,k, k =,,,N n, j =,2,,m These equaions ogeher wih (32) generae m(n + ) of algebraic equaions which can hen be solved for he unknown coefficiens of he vecors j, j=,,m, using a suiable mehod onsequenly, he approimae soluion u j (), j=,2,,m,canbeobained In our compuaions we used he Gaussian eliminaion mehod o solve he resuling linear sysem of algebraic equaions and Newon s ieraion mehod o solve he resuling nonlinear sysem of algebraic equaions Now, we can presen he following problems Eample 8 onsider he following sysem of fracional inegro-differenial equaions [7, 5]: where S α (i, j) = i k= α ( ) i k 2i (i+k )!Γ (k α+/2) (27) ε j L α Γ (k+/2)(i k)!γ(k α j+)γ(k+j α+) Noe ha, in D (α),hefirs α rows are all zero Proof See [25] Dα Dα 2 () =++ 2 y() y () = +() wih he iniial condiions ( (τ) +y(τ))dτ, ( (τ) y(τ))dτ, <α <α 2, (33) Remark 7 If α=n N, hen Theorem 6 gives he same resul as (22) 5 Sysem of Fracional Inegro-Differenial Equaions In order o use OM for sysem of fracional inegrodifferenial equaions of he form (), we firs approimae u j (), Dα u j(), andu (l) j () by he shifed hebyshev polynomials as u j () N i= c j,i T L,i () = T j φ (), (28) () =, y () = The eac soluions, when α =α 2 =,are () =+e, y () = e (34) (35) We use hebyshev specral mehod; we may wrie he approimae soluion Dα u j () T j Dα φ () T j D(α) φ (), (29) u (l) j () T j D(l) φ () T j D(l) φ () (3) By subsiuing hese equaions in (), we ge T j D(α) φ () F j (, T φ (),T 2 φ (),,T mφ (), T D(l) φ (), T 2 D(l) φ (),, T m D(l) φ (), K j (, T φ (),,T mφ ())d) (3) where () y () 8 i= 8 i=,i T,i () = T φ (), 2,i T,i () = T 2 φ (), (36) T =(c,c,c 2,c 3,c 4,c 5,c 6,c 7,c 8 ), T 2 =(c 2,c 2,c 22,c 23,c 24,c 25,c 26,c 27,c 28 ) (37)

5 Mahemaical Problems in Engineering 5 Subsiuing (36) in (33) and (34), for α =α 2 =,wege T D() φ () + T 2 φ () + 2 =, T 2 D() φ () T φ () + ++=, T φ () =, ( T φ (τ) +T 2 φ (τ))dτ ( T φ (τ) T 2 φ (τ))dτ (38) T 2 φ () = (39) The roos of he shifed hebyshev polynomial T,8 () are given by k = cos ((2k+)π), k=,,2,,7 (4) 6 Now, for calculaing he shifed hebyshev coefficien for N=8,subsiue(4)in(38),andsolvingheresulinglinear sysem of equaions and (39), we ge c = , c = 3539, c 2 = 529, c 3 = 8722, c 4 = , c 5 = 2754, c 6 = 283 6, c 7 = , c 8 = , c 2 = 25339, c 2 = 35392, c 22 = 529, c 23 = 8722, c 24 = , c 25 = , c 26 = 284 6, c 27 = , c 28 = (4) Thus using (36), we ge () = , y () = (42) Tableshowshecomparisonbeweenheeacsoluion and he approimae soluion wih he absolue error a N= 8,andTable2showshemaimumofabsolueerrorbeween eac soluions and approimae soluions for various choices of N Figure shows he graph of he eac soluion and he approimae soluion a N=8, α =α 2 =Figure2shows he graph of he eac soluions and he approimae soluions a N=8, α =α 2 = 9, 85,and75 Eample 9 onsider he following nonlinear fracional sysem of inegro-differenial equaions [5]: Dα () = y 2 () + 2 ( 2 (τ) +y 2 (τ))dτ, Dα 2 y () = + 2 () + 4 ( 2 (τ) y 2 (τ))dτ, wih he iniial condiions () =, () =2, y () =, y () = The eac soluions, when α =α 2 =2,are () =+e, y () = e <α,α 2 2, (43) (44) (45) We use hebyshev specral mehod; we may wrie he approimae soluion () y () 3 i= 3 i=,i T,i () = T φ (), 2,i T,i () = T 2 φ (), (46)

6 6 Mahemaical Problems in Engineering Table i Eac Appro Eac Appro y Eac y Appro y Eac y Appro N Table 2 Maimum of absolue error of () Maimum of absolue error of y() where The roos of he shifed hebyshev polynomial T,2 () are = 4 (2 + 2), = 4 (2 2) (5) Now,forcalculaingheshifedhebyshevcoefficienforN= 3,subsiue(5)in(48),andsolvingheresulingnonlinear sysem of equaions and (49), we ge T =(c,c,c 2,c 3 ), (47) T 2 =(c 2,c 2,c 22,c 23 ) Subsiuing (46) in (43) and (44), for α =α 2 =2,wege T D(2) φ () + 2 [T 2 D() φ ()] 2 c = , c = , c 2 = , c 3 = , c 2 = , c 2 = , (5) 2 ([ T φ (τ)]2 +[ T 2 φ (τ)]2 )dτ =, T 2 D(2) φ () + T φ () 4 ([ T φ (τ)]2 [ T 2 φ (τ)]2 )dτ+ 2 =, T φ () =, T D() φ () =2, T 2 φ () =, T 2 D() φ () = (48) (49) Thus using (46), we ge c 22 = , c 23 = () = , y () = (52) Table 3 shows he comparison beween he eac soluion and he approimae soluion wih he absolue error a N= 3 Figure 3 shows he graph of he eac soluion and he approimae soluion a N=3, α =α 2 =2Figure4shows he graph of he eac soluions and he approimae soluions a N=3, α =α 2 = 9, 8,and7

7 Mahemaical Problems in Engineering 7 () y() Eac sol of () Appro a α =α 2 = (a) Eac sol of y() Appro a α =α 2 = (b) Figure : The graphs of he eac soluions and he approimae soluions a N=8, α =α 2 = () 25 y() Eac sol of () Appro a α =α 2 = 9 Appro a α =α 2 = 85 Appro a α =α 2 = 75 (a) Eac sol of y() Appro a α =α 2 = 9 Appro a α =α 2 = 85 Appro a α =α 2 = 75 (b) Figure 2: The graphs of he eac soluions and he approimae soluions a N=8, α =α 2 = 9, 85, and 75 Table 3 i Eac Appro Eac Appro y Eac y Appro y Eac y Appro

8 8 Mahemaical Problems in Engineering () y() Eac sol of () Appro a α =α 2 =2 (a) Eac sol of y() Appro a α =α 2 =2 (b) Figure 3: The graphs of he eac soluions and he approimae soluions a N=3, α =α 2 =2 () y() Eac sol of () Appro a α =α 2 = 9 Appro a α =α 2 = 8 Appro a α =α 2 = 7 (a) Eac sol of y() Appro a α =α 2 = 9 Appro a α =α 2 = 8 Appro a α =α 2 = 7 (b) Figure 4: The graphs of he eac soluions and he approimae soluions a N=3, α =α 2 = 9, 8, and 7 6 Abel s Inegral Equaion In order o use OM for Abel s inegral equaion of he form (3), we firs approimae u() by he shifed hebyshev polynomials as u () N i= By subsiuing (53) in (3), we ge λ T φ () =f() + c i T L,i () = T φ () (53) T φ () d (54) Then we have o collocae (54) a he (N + ) shifed hebyshev roos L,N+,k, k =,,,NTheseequaions generae (N + ) linear algebraic equaions which can be solved for he unknown coefficiens of he vecor, using a suiable mehod onsequenly, u() given in (53) can be calculaed, which gives a soluion of (3) Eample onsider Abel s inegral equaion of he firs kind [3, 32] u () d = 2 5 ( ), which has he eac soluion u() = [, ] (55)

9 Mahemaical Problems in Engineering 9 By applying he hebyshev specral mehod, we may wrie he approimae soluion u () = 3 i= Subsiuing (56) in (55), we ge i T,i () = T φ () (56) T φ () d 2 5 ( )= (57) The roos of he shifed hebyshev polynomial T,4 () are = 4 (2 2 2), = 4 ( ), 2 = 4 (2 2+ 2), 3 = 4 ( ) (58) Now, calculaing he shifed hebyshev coefficien for N=3 by subsiuing (58) in (57) and solving four equaions yields Therefore, we have u () =( 5 6, 32, 6, 32 )( = 3 2 +, which is he eac soluion c = 5 6, c = 32, c 2 = 6, c 3 = ) (59) (6) Eample onsider Abel s inegral equaion of he firs kind [32] Table 4 i u Eac u Appro u Eac u Appro Table 5 N Maimum of absolue error Table4showshecomparisonbeweenheeacsoluion and he approimae soluion wih he absolue error a N= 2,andTable5showshemaimumofabsolueerrorbeween eac soluion and approimae soluion for various choices of N Figure 5 shows he graph of he eac soluion and he approimae soluion a N=2 Eample 2 onsider Abel s inegral equaion of he second kind [32, 33] u () = 4 u () 3 3/2 + d, [, L] (62) which has he eac soluion u() = By applying he hebyshev specral mehod, we may wrie he approimae soluion u () 2 i= i T L,i () = T φ () (63) Subsiuing (63) in (62), we ge T T φ () φ () d = (64) The roos of he shifed hebyshev polynomial T L,3 () are = L 2, u () d = 3π 8 2, [, ] (6) which has he eac soluion u() = = L 4 (2 3), 2 = L 4 (2 + 3) (65)

10 Mahemaical Problems in Engineering u() Eac soluion Appro soluion Figure 5: The graph of he eac soluion and he approimae soluion a N=2 Now, calculaing he shifed hebyshev coefficien for N=2 by subsiuing (65) in (64) and solving hree equaions yields Therefore, we have c = L 2, c = L 2, c 2 = u () =( L 2, L 2,)( which is he eac soluion (66) 2 L )=, (67) 8 2 L 2 8 L + Eample 3 onsider Abel s inegral equaion of he second kind [32, 34] u () = u () 5 5/2 d, [, L] (68) which has he eac soluion u() = 2 Weusehebyshevspecralmehod;wemaywriehe approimae soluion u () 2 i= Subsiuing (69) in (68), we ge i T L,i () = T φ () (69) T T φ () φ () d /2 = (7) Table 6 i u Eac u Appro u Eac u Appro The roos of he shifed hebyshev polynomial T L,3 () are = L 2, = L 4 (2 3), 2 = L 4 (2 + 3) (7) Now, calculaing he shifed hebyshev coefficien for N=2 by subsiuing (7) in (7) and solving hree equaions yields Therefore, we have c = 3L2 8, c = L2 2, c 2 = L2 8 u () =( 3L2 8, L2 2, L2 8 )( which is he eac soluion 2 L 8 2 L 2 8 L + (72) )= 2, (73) Eample 4 onsider Abel s inegral equaion of he second kind [32, 34] u () = 2 arc sinh ( ) u () d, which has he eac soluion u() = /( + ) [, ] (74) Table 6 shows he comparison beween he eac soluion and he approimae soluion wih he absolue error a

11 Mahemaical Problems in Engineering 9 8 u() Eac soluion Appro soluion Figure 6: The graph of he eac soluion and he approimae soluion a N=2 8 8 () 6 4 y() Eac sol of () Appro sol of () (a) Eac sol of y() Appro sol of y() (b) Figure 7: The graphs of he eac soluions and he approimae soluions a N=2 N=2,andTable7showshemaimumofabsolueerror beween eac soluion and approimae soluion for various choices of N Figure 6 shows he graph of he eac soluion and he approimae soluion a N=2 Eample 5 onsiderhelinearsysemofsingularvolerra inegral equaions [] π () s [ (s) +y(s)]ds= π /2, y () The eac soluions are s (s) ds = () =, y () = (75) (76) Table 7 N Maimum of absolue error Table 8 shows he comparison beween he eac soluion and he approimae soluion wih he absolue error a N=2,andTable9showshemaimumofabsolueerror beween eac soluion and approimae soluion for various choices of N Figure 7 shows he graph of he eac soluion and he approimae soluion a N=2

12 2 Mahemaical Problems in Engineering Table 8 i Eac Appro Eac Appro y Eac y Appro y Eac y Appro N Table 9 Maimum of absolue error of () Maimum of absolue error of y() onclusion In his aricle, we develop he hebyshev specral mehod based on operaional mari for solving linear and nonlinear sysem of fracional inegro-differenial equaions and Abel s inegral equaions Our approach was based on he shifed hebyshev collocaion mehods I can be concluded ha he hebyshev specral mehod is very powerful and efficien echnique for finding eac soluions for wide classes of problems onflics of Ineres The auhors declare ha here are no conflics of ineres regarding he publicaion of his paper References [] K S Miller and B Ross, An Inroducion o he Fracional alculus and Fracional Differenial Equaions,JohnWiley&Sons, New York, NY, USA, 993 [2] R Hilfer, Applicaions of Fracional alculus in Physics, World Scienific, Singapore, 2 [3] SGSamko,AAKilbas,andOIMarichev,Fracional Inegrals and Derivaives: Theory and Applicaions, Gordonand Breach, London, UK, 993 [4] I Podlubny, Fracional Differenial Equaions,vol98ofMahemaics in Science and Engineering, Academic Press, New York, NY, USA, 999 [5] ADebboucheandJJNieo, Sobolevypefracionalabsrac evoluion equaions wih nonlocal condiions and opimal muli-conrols, Applied Mahemaics and ompuaion, vol 245, pp 74 85, 24 [6] A Debbouche and J J Nieo, Relaaion in conrolled sysems described by fracional inegro-differenial equaions wih nonlocal conrol condiions, Elecronic Differenial Equaions,vol25,no89,pp 8,25 [7] S Momani and R Qaralleh, An efficien mehod for solving sysems of fracional inegro-differenial equaions, ompuers and Mahemaics wih Applicaions,vol52,no3-4,pp459 47, 26 [8] R H Khan and H O Bakodah, Adomian decomposiion mehod and is modificaion for nonlinear Abel s inegral equaion, Inernaional Mahemaical Analysis, vol 7,no45-48,pp ,23 [9] A Arikoglu and I Ozkol, Soluion of fracional inegrodifferenial equaions by using fracional differenial ransform mehod, haos, Solions & Fracals,vol4,no2,pp52 529, 29 [] HTaghvafardandGHErjaee, Onsolvingasysemofsingular Volerra inegral equaions of convoluion ype, ommunicaions in Nonlinear Science and Numerical Simulaion,vol6,no 9, pp , 2 [] J Zhao, J Xiao, and N J Ford, ollocaion mehods for fracional inegro-differenial equaions wih weakly singular kernels, Numerical Algorihms, vol 65, no 4, pp , 24 [2] RKPandey,SSharma,andKKumar, ollocaionmehod for generalized Abel s inegral equaions, ompuaional and Applied Mahemaics,vol32,pp8 28,26 [3] K Sayevand, M Fardi, E Moradi, and F Hemai Boroujeni, onvergence analysis of homoopy perurbaion mehod for Volerra inegro-differenial equaions of fracional order, Aleandria Engineering Journal,vol52,no4,pp87 82,23 [4] S Kumar, O P Singh, and S Dii, Homoopy perurbaion mehod for solving sysem of generalized Abel s inegral equaions, Applicaions and Applied Mahemaics, vol6,no,pp , 2 [5] M Zuriga, S Momani, and A Alawneh, Homoopy analysis mehod for sysems of fracional inegro-differenial equaions, Neural, Parallel and Scienific ompuaions, vol7,no2,pp 69 86, 29 [6] F Mirzaee, M Komak Yari, and M Paripour, Solving linear and nonlinear Abel fuzzy inegral equaions by homoopy analysis mehod, Taibah Universiy for Science, vol 9, no, pp 4 5, 25

13 Mahemaical Problems in Engineering 3 [7] H A Zedan, S S Tanawy, and Y M Sayed, onvergence of he variaional ieraion mehod for iniial-boundary value problem of fracional inegro-differenial equaions, Fracional alculus and Applicaions, vol5,supplemen3,pp 4, 24 [8] P Mokhary, Discree Galerkin mehod for fracional inegrodifferenial equaions, Aca Mahemaica Scienia Series B English Ediion, vol 36, no 2, pp , 26 [9] H Saeedi, N Mollahasani, M Mohseni Moghadam, and G N huev, An operaional Haar wavele mehod for solving fracional Volerra inegral equaions, Inernaional Applied Mahemaics and ompuer Science, vol2,no3,pp , 2 [2]anuo,MYHussaini,AQuareroni,andTAZang, Specral Mehods in Fluid Dynamics, Springer, New York, NY, USA, 988 [2] EAousias,THagsrom,andDTorres, Anefficienspecral mehod for ordinary differenial equaions wih raional funcion coefficiens, Mahemaics of ompuaion,vol65,no24, pp6 635,996 [22] YYang,Yhen,andYHuang, Specral-collocaionmehod for fracional Fredholm inegro-differenial equaions, Journal of he Korean Mahemaical Sociey, vol5,no,pp23 224, 24 [23]EHDoha,AHBhrawy,andMASaker, Inegralsof Bernsein polynomials: an applicaion for he soluion of high even-order differenial equaions, Applied Mahemaics Leers, vol 24, no 4, pp , 2 [24] E H Doha and A H Bhrawy, Efficien specral-galerkin algorihms for direc soluion of fourh-order differenial equaions using Jacobi polynomials, Applied Numerical Mahemaics,vol 58, no 8, pp , 28 [25] E H Doha, A H Bhrawy, and S S Ezz-Eldien, A hebyshev specral mehod based on operaional mari for iniial and boundary value problems of fracional order, ompuers & Mahemaics wih Applicaions, vol62,no5,pp , 2 [26] A Saadamandi and M Dehghan, A new operaional mari for solving fracional-order differenial equaions, ompuers and Mahemaics wih Applicaions An Inernaional Journal,vol59, no 3, pp , 2 [27] A H Bhrawy and A S Alofi, The operaional mari of fracional inegraion for shifed hebyshev polynomials, Applied Mahemaics Leers,vol26,no,pp25 3,23 [28] Hwang and Y P Shih, Parameer idenificaion via Laguerre polynomials, Inernaional Sysems Science, vol 3, no 2, pp 29 27, 982 [29] P N Paraskevopoulos, Legendre series approach o idenificaion and analysis of linear sysems, IEEE Transacions on Auomaic onrol,vol3,no6,pp ,985 [3] K Maleknejad, B Basira, and E Hashemizadeh, A Bernsein operaional mari approach for solving a sysem of high order linear Volerra Fredholm inegro-differenial equaions, Mahemaical and ompuer Modelling, vol55,no3-4,pp , 22 [3] S Sohrabi, omparison hebyshev waveles mehod wih BPFs mehod for solving Abel s inegral equaion, Ain Shams Engineering Journal,vol2,no3-4,pp ,2 [32] Z Avazzadeh, B Shafiee, and G B Loghmani, Soluion of Abels inegral equaions using legendre polynomials and fracional calculus echniques, Inernaional Mahemaical Archive (IJMA), vol 2, no 8, pp , 2 [33] E A Galperin and E J Kansa, Applicaion of global opimizaion and radial basis funcions o numerical soluions of weakly singular Volerra inegral equaions, ompuers & Mahemaics wih Applicaions,vol43,no3-5,pp49 499,22 [34] R K Pandey, O P Singh, and V K Singh, Efficien algorihms o solve singular inegral equaions of Abel ype, ompuers & Mahemaics wih Applicaions,vol57,no4,pp ,29

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