Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM

Transcription

1 Appl. Math. If. Sc. 7, No. 5, (2013) 2011 Appled Matheatcs & Iforato Sceces A Iteratoal Joural Nuercal Study for the Fractoal Dfferetal Equatos Geerated by Optzato Proble Usg Chebyshev Collocato Method ad FDM M. M. Khader 1,2,, N. H. Swela 3 ad A. M. S. Mahdy 4 1 Departet of Matheatcs ad Statstcs, College of Scece, Al-Ia Mohaed Ib Saud Islac Uversty, Ryadh, Saud Araba 2 Departet of Matheatcs, Faculty of Scece, Beha Uversty, Beha, Egypt 3 Departet of Matheatcs, Faculty of Scece, Caro Uversty, Gza, Egypt 4 Departet of Matheatcs, Faculty of Scece, Zagazg Uversty, Zagazg, Egypt Receved: 3 Feb. 2013, Revsed: 4 Ju. 2013, Accepted: 5 Ju Publshed ole: 1 Sep Abstract: Ths paper s devoted wth uercal soluto of the syste fractoal dfferetal equatos (FDEs) whch are geerated by optzato proble usg the Chebyshev collocato ethod. The fractoal dervatves are preseted ters of Caputo sese. The applcato of the proposed ethod to the geerated syste of FDEs leads to algebrac syste whch ca be solved by the Newto terato ethod. The ethod troduces a prosg tool for solvg ay systes of o-lear FDEs. Two uercal exaples are provded to cofr the accuracy ad the effectveess of the proposed ethods. Coparsos wth the fractoal fte dfferece ethod (FDM) ad the fourth order Ruge-Kutta (RK4) are gve. Keywords: No-lear prograg, pealty fucto, dyac syste, Caputo fractoal dervatves, Chebyshev approxatos, fte dfferece ethod, Ruge-Kutta ethod. Noeclature D α : The Caputo fractoal dervatve of order α; N: The set of all ature ubers; α : The celg fucto to deote the sallest teger greater tha or equal to α; R: The set of all real ubers; h(x): The gradet of the fucto h(x); µ: A auxlary pealty varable; θ: A costat; T (x): The Chebyshev polyoal of degree ; 1 Itroducto I last decades, fractoal calculus has draw a wde atteto fro ay physcsts ad atheatcas, because of ts terdscplary applcato ad physcal eag [1, 2]. Fractoal calculus deals wth the geeralzato of dfferetato ad tegrato of o-teger order. Fractoal dfferetal equatos have bee the focus of ay studes due to ther frequet appearace varous applcatos flud echacs, bology, physcs ad egeerg [3]. Cosequetly, cosderable atteto has bee gve to the solutos of FDEs ad tegral equatos of physcal terest. Most FDEs do ot have exact aalytcal solutos, so approxate ad uercal techques [4, 5] ust be used. Several uercal ethods to solve FDEs have bee gve such as, hootopy perturbato ethod [5], hootopy aalyss ethod [6], collocato ethod [7, 14] ad others [12]. Represetato of a fucto ters of a seres expaso usg orthogoal polyoals s a fudaetal cocept approxato theory ad for the bass of the soluto of dfferetal equatos [15, 16]. Chebyshev polyoals are wdely used uercal coputato. Oe of the advatages of usg Chebyshev polyoals as a tool for expaso fuctos s the good represetato of sooth fuctos by fte Chebyshev expaso provded that the fucto y(x) s ftely dfferetable. Correspodg author e-al: ohaedbd@yahoo.co

2 2012 M. M. Khader et al: Nuercal Study for the Fractoal Dfferetal Equatos... The coeffcets Chebyshev expaso approach zero faster tha ay verse power as goes to fty. Optzato theory s aed to fd out the optal soluto of probles whch are defed atheatcally fro a odel arse wde rage of scetfc ad egeerg dscples. May ethods ad algorths have bee developed for ths purpose sce The pealty fucto ethods are classcal ethods for solvg o-lear prograg (NLP) proble [17, 18]. Also, dfferetal equato ethods are alteratve approaches to fd solutos to these probles. I ths type of ethods the optzato proble s forulated as a syste of ordary dfferetal equatos so that the equlbru pot of ths syste coverges to the local u of the optzato proble [19, 21]. I ths artcle, we wll copare our approxate soluto wth those uercal obtaed usg the plct fte dfferece ethod. It has bee show that FDM s a powerful tool for solvg varous kds of probles [22, 23]. Also, ths techque reduces the proble to a syste of algebrac equatos. May authors have poted out that the FDM ca overcoe the dffcultes arsg the calculato of soe uercal ethods, such as, fte eleet ethod. The a a of the preseted paper s cocered wth the applcato of the Chebyshev collocato ethod ad fractoal fte dfferece ethod to obta the uercal soluto of the syste of FDEs whch s geerated fro the o-lear prograg probles ad study the covergece aalyss of the proposed ethod. The structure of ths paper s arraged the followg way: I secto 2, we troduce soe basc deftos about Caputo fractoal dervatves, the defto of the optzato proble ad ts geerated syste of FDEs. I secto 3, we derve a approxate forula for fractoal dervatves usg Chebyshev seres expaso ad estate a upper boud of the resultg error of the proposed forula. I secto 4, uercal exaples are gve to solve the syste of FDEs whch obtaed fro the o-lear prograg proble ad show the accuracy of the preseted ethods. Fally, secto 5, the paper eds wth a bref cocluso ad soe rearks. 2 Prelares ad otatos I ths secto, the forulato of the optzato proble ad ts correspodg syste of FDEs are gve ad we preset soe ecessary deftos ad atheatcal prelares of the fractoal calculus theory requred for our subsequet developet. 2.1 The fractoal dervatve the Caputo sese The Caputo fractoal dervatve operator D α of order α s defed the followg for D α f(x)= 1 x f () (ξ) dξ, α > 0, Γ( α) 0 (x ξ) α +1 where 1<α, N, x>0. Slar to teger-order dfferetato, Caputo fractoal dervatve operator s lear D α (c 1 p(x)+c 2 q(x))=c 1 D α p(x)+c 2 D α q(x), where c 1 ad c 2 are costats. For the Caputo s dervatve we have D α C=0, C s a costat, (1) { 0, for N0 D α x ad < α ; = Γ(+1) Γ(+1 α) x α, for N 0 ad α. (2) We use the celg fucto α to deote the sallest teger greater tha or equal to α ad N 0 = {0,1,2,...}. Recall that for α N, the Caputo dfferetal operator cocdes wth the usual dfferetal operator of teger order. For ore detals o fractoal dervatves deftos ad ther propertes see ( [15], [24]). 2.2 Optzato proble ad ts correspodg syste of FDEs Cosder the o-lear prograg proble wth equalty costrats defed by ze f(x), subect to x M, (3) wth M = {x R : h(x) = 0}, where f : R R ad h=(h 1,h 2,...,h p ) T : R R p (p ). It s assued that the fuctos the proble are at least twce cotuously dfferetable, that a soluto exsts, ad that h(x) has full rak. To obta a soluto of (3), the pealty fucto ethod solves a sequece of ucostraed optzato probles. A well-kow pealty fucto for ths proble s gve by F(x, µ)= f(x)+ µ 1 θ p l=1 (h l (x)) θ, (4) where θ > 0 s a costat ad µ > 0 s a auxlary pealty varable. The correspodg ucostraed optzato proble of (3) s defed as follows ze F(x, µ) s.t. x R. (5)

3 Appl. Math. If. Sc. 7, No. 5, (2013) / For ore detals about NLP proble ca be foud ( [12 14], [17], [18]). We ca wrte the NLP proble a syste of fractoal dfferetal equatos as follows: Cosder the ucostraed optzato proble (5), a approach based o fractoal dyac syste ca be descrbed by the followg FDEs D α x(t)= x F(x, µ), 0<α 1, (6) wth the tal codtos x(t 0 )=c, =1,2,...,. Note that, a pot x e s called a equlbru pot of (6) f t satsfes the rght had sde of Eq.(6). Also, we ca rewrte the fractoal dyac syste (6) ore geeral for as follows D α x (t)=g (t, µ,x 1,x 2,...,x ), =1,2,...,. (7) The steady state soluto of the o-lear syste of FDEs (7) ust be cocded wth local optal soluto of the NLP proble (3). 3 Dervato a approxate forula for fractoal dervatves usg Chebyshev seres expaso The well kow Chebyshev polyoals [25] are defed o the terval [ 1,1] ad ca be detered wth the ad of the followg recurrece forula T +1 (z)=2zt (z) T 1 (z), T 0 (z)=1, T 1 (z)=z, =1,2,... The aalytc for of the Chebyshev polyoals T (z) of degree s gve by [ 2 ] T (z)= ( 1) ( 1)! ()!( 2)! z 2, (8) where [/2] deotes the teger part of /2. The orthogoalty codto s 1 T (z)t (z) π, for = = 0; dz= π 1 1 z 2 2, for = 0; 0, for. I order to use these polyoals o the terval [0,L] we defe the so called shfted Chebyshev polyoals by troducg the chage of varable z= L 2 t 1. The shfted Chebyshev polyoals are defed as T (t) = T ( L 2t 1) = T 2( t/l). The aalytc for of the shfted Chebyshev polyoals T (t) of degree s gve by T (t)= k=0 ( 1) k 22k (+k 1)! L k (2k)!( k)! tk, =2,3,... (9) The fucto x(t), whch belogs to the space of square tegrable fuctos o [0, L], ay be expressed ters of shfted Chebyshev polyoals as x(t)= c T (t), (10) where the coeffcets c are gve by (for =1,2,...) L x(t)t0 (t) dt, c = 2 L Lt t 2 π 0 x(t)t (t) Lt t 2 dt. c 0 = 1 π 0 (11) I practce, oly the frst ( + 1)-ters of shfted Chebyshev polyoals are cosdered. The we have x (t)= c T (t). (12) Theore 3.1 (Chebyshev trucato theore) [25] The error approxatg x(t) by the su of ts frst ters s bouded by the su of the absolute values of all the eglected coeffcets. If the x (t)= k=0 E T () x(t) x (t) c k T k (t), (13) k=+1 c k, (14) for all x(t), all, ad all t [ 1,1]. The a approxate forula of the fractoal dervatve of x (t) s gve the followg theore. Theore 3.2 Let x(t) be approxated by Chebyshev polyoals as (12) ad also suppose α > 0, the where w (α),k D α (x (t))= s gve by = α c w (α),k tk α, (15) w (α),k =( 1) k 2 2k (+k 1)!Γ(k+ 1) L k ( k)!(2k)!γ(k+ 1 α). (16) Proof. Sce the Caputo s fractoal dfferetato s a lear operato we have D α (x (t))= c D α (T (t)). (17) Eployg Eqs.(1) ad (2) o the forula (9) we have D α T (t)=0,,1,..., α 1, α > 0. (18)

4 2014 M. M. Khader et al: Nuercal Study for the Fractoal Dfferetal Equatos... Also, for = α, α + 1,...,, ad by usg Eqs.(1) ad (2), we get D α T (t)= = ( 1) k 22k (+k 1)! L k ( k)!(2k)! Dα t k ( 1) k 2 2k (+k 1)!Γ(k+ 1) L k ( k)!(2k)!γ(k+ 1 α) tk α (19) A cobato of Eqs.(18), (19) ad (16) leads to the desred result (15) ad copletes the proof of the theore. Theore 3.3 The Caputo fractoal dervatve of order α for the shfted Chebyshev polyoals ca be expressed ters of the shfted Chebyshev polyoals theselves the followg for D α (T (t))= where (for,1,...) Θ,,k = Θ,,k T (t), (20) ( 1) k 2(+k 1)!Γ(k α+ 1 2 ) h Γ(k+ 2 1 )( k)!γ(k α +1)Γ(k+ α+1)lk Proof. We cocer the propertes of the shfted Chebyshev polyoals [25] ad expadg t k α Eq.(19) the followg for t k α = c k T (t), (21) where c k ca be obtaed usg (11) where x(t) = t k α the c k = 2 L t k α T (t) h π dt, h 0 = 2, h = 1, = 1,2,... 0 Lt t 2 At = 0 we fd c k0 = 1 π L 0 t k α T 0 (t) Lt t 2 dt = 1 π Γ(k α+ 1/2) Γ(k α+ 1), also, at ay ad usg the forula (9) we ca fd that c k = π r=0 ( 1) r ( +r 1)!2 2r+1 Γ(k+r α+1/2) ( r)!(2r)!γ(k+r α+1)l r, for = 1,2,... Eployg Eqs.(19) ad (21) gves D α (T (t))= Θ,,k T (t), = α, α +1,..., where Θ,,k = ( 1) k (+k 1)!2 2k k!γ(k α+ 1 2 ) ( k)!(2k)!, = 0; π(γ(k+1 α)) 2 ( 1) k (+k 1)!2 2k+1 k! πγ(k+1 α)( k)!(2k)! ( 1) r ( +r 1)!2 2r Γ(k+r α+ 1 2 ) r=0, = 1,2,... ( r)!(2r)!γ(k+r α+1)l r After soe legthly apulato Θ,,k ca put the followg for (for = 0,1,...) ( 1) Θ,,k = k 2(+k 1)!Γ(k α+ 2 1) h Γ(k+ 2 1 )( k)!γ(k α +1)Γ(k+ α+1)lk, (22) ad ths copletes the proof of the theore. Theore 3.4 The error E T () = D α x(t) D α x (t) approxatg D α x(t) by D α x (t) s bouded by E T () =+1 c ( Θ,,k ). (23) Proof. A cobato of Eqs.(10), (12) ad (20) leads to E T () = D α x(t) D α x (t) = =+1 c ( but T (t) 1, so, we ca obta E T () =+1 c ( Θ,,k T (t)), Θ,,k ), ad subtractg the trucated seres fro the fte seres, boudg each ter the dfferece, ad sug the bouds copletes the proof of the theore. 4 Nuercal pleetato I order to llustrate the effectveess of the proposed ethod, we pleet the to solve the followg syste of FDEs whch s geerated fro the o-lear prograg proble. 4.1 Optzato proble 1: Cosder the followg o-lear prograg proble [26] ze f(x)=100(u 2 v) 2 +(u 1) 2, subect to h(x)=u(u 4) 2v+12=0. (24) The optal soluto s x = (2,4), where x = (u,v). For solvg the above proble, we covert t to a ucostraed optzato proble wth quadratc pealty fucto (4) for θ = 2, the we have F(x, µ)=100(u 2 v) 2 +(u 1) µ(u(u 4) 2v+12)2, where µ R + s a auxlary pealty varable. The

5 Appl. Math. If. Sc. 7, No. 5, (2013) / correspodg o-lear syste of FDEs fro (6) s defed as D α u(t)= 400(u 2 v)u 2(u 1) µ(2u 4)(u 2 4u 2v+12), D α v(t)=200(u 2 v)+2µ(u 2 4u 2v+12), 0<α 1, (25) wth the followg tal codtos u(0) = 0 ad v(0)=0. 1.I: Ipleetato of Chebyshev approxato Cosder the syste of fractoal dfferetal equatos (25). I order to use the Chebyshev collocato ethod, we frst approxate u(t) ad v(t) as u (t)= a T (t), v (t)= Fro Eqs.(26) ad Theore 3.2 we have = α ( (( = α + 2µ(( a w (α),k tk α = 400(( a T (t)) 2( a T (t)) 2 4( a T (t)) 2 a T (t) 1) µ(2 b w (α),k tk α = 200(( a T (t)) 2 4( a T (t)) 2( (t). (26) a T (t)) 2 a T (t)) 2( (t)) a T (t) 4) (t))+12), (27) (t)) (t))+12). (28) We ow collocate Eqs.(27) ad (28) at (+1 α ) pots t p (p=0,1,...,+1 α ) as = α ( 4)(( = α a w (α),k tk α p a T (t p )) 2( 2( a T (t p )) 2 4( b w (α),k tk α = 400(( a T (t p ) 1) µ(2 p = 200(( b T (t p ))+2µ(( a T (t p )) 2 a T (t p )) 2( (t p ))+12). a T (t p ) a T (t p )) 2 a T (t p )) 2 4( (t p )) (t p ))+12), (29) a T (t p )) (30) For sutable collocato pots we use the roots of shfted Chebyshev polyoal T +1 α (t). Also, by substtutg Eq.(26) the tal codtos u(0)=v(0)=0, we ca fd ( 1) a = 0, ( 1) b = 0. (31) Equatos (29) ad (30), together the equatos of the tal codtos (31), gve (2 + 2) of o-lear algebrac equatos whch ca be solved usg the Newto terato ethod, for the ukows a ad b,,1,...,. 1.II: Ipleetato of fractoal FDM I ths secto, the fractoal fte dfferece ethod wth the dscrete forula ( [27], [28]) s used to estate the te α-order fractoal dervatve to solve uercally the syste of FDEs (25). Usg ( [27], [28]) the restrcto of the exact soluto to the grd pots cetered at x = k,=1,2,...,n, Eqs.(25) σ α,k ω (α) (u +1 u )+O(k)= 400(u 2 v )u 2(u 1) µ(2u 4).(u 2 4u 2v + 12), σ α,k σ α,k ω (α) (v +1 v )+O(k) = 200(u 2 v )+2µ(u 2 4u 2v + 12), ω (α) (u +1 u )= 400(u 2 v )u (32) (33) 2(u 1) µ(2u 4).(u 2 4u 2v + 12)+TE 1 (t), (34) σ α,k ω (α) (v +1 v )=200(u 2 v ) + 2µ(u 2 4u 2v + 12)+T E 2 (t), (35) where T E 1 (t) ad T E 2 (t) are the trucato ters. Thus, accordg to Eqs.(34) ad (35), the uercal schee s cosstet, frst order correct te. The resultg fte dfferece equatos are defed by σ α,k ω (α) (u +1 u )= 400(u 2 v )u 2(u 1) µ(2u 4).(u 2 4u 2v + 12), (36) σ α,k ω (α) (v +1 v )=200(u 2 v ) + 2µ(u 2 4u 2v + 12), =1,2,...,N. (37) Ths schee presets a o-lear syste of algebrac equatos. I our calculato, we used the Newto terato ethod to solve ths syste.

6 2016 M. M. Khader et al: Nuercal Study for the Fractoal Dfferetal Equatos... Fg. 1: The behavor of the Chebyshev collocato soluto wth =4, FDM soluto wth k=0.002 ad RK4 soluto at α = 1. Fg. 2: The behavor of the Chebyshev collocato soluto wth =4 ad FDM soluto wth k=0.002 at α = I fgures 1 ad 2, we preseted a coparso betwee the approxate soluto (u(t), v(t)) usg the Chebyshev collocato ethod wth = 4, uercal soluto usg the fractoal fte dfferece ethod wth k = ad the soluto usg Ruge-Kutta ethod for α = 1 ad α = 0.85, respectvely. Fro these fgures, we ca coclude that the obtaed uercal solutos of the proposed ethods are excellet agreeet wth those obtaed fro Ruge-Kutta ethod. Table 1: The uercal soluto of the syste (40) usg the Chebyshev collocato ethod at α = Optzato proble 2: Cosder the equalty costraed optzato proble [26] ze f(x)=(x 1 1) 2 +(x 1 x 2 ) 2 +(x 2 x 3 ) 2 +(x 3 x 4 ) 4 +(x 4 x 5 ) 4, subect to h 1 (x)=x 1 + x 2 2+ x =0, h 2 (x)=x 2 x 2 3+ x =0, h 3 (x)=x 1 x 5 2=0. (38) The soluto of (38) s x =(1.19,1.362,1.47,1.64,1.68) ad ths s ot a exact soluto. For solvg the above proble, we covert t to a ucostraed optzato proble wth quadratc pealty fucto (4) for θ = 2, the we have F(x, µ)= f(x)+ 1 2 µ 3 l=1(h l (x)) 2, (39) where µ R + s a auxlary pealty varable. The correspodg o-lear syste of FDEs fro (6) s defed as D α x(t)= f(x) µ h(x)h(x), 0<α 1, (40) wth the followg tal codtos x(0)=(2,2,2,2,2) T that s ot feasble. The obtaed uercal results of the proble (40) usg the proposed ethods are preseted tables 1-5, where table 1, we preseted the uercal soluto x(t)=(x 1 (t),x 2 (t),...,x 5 (t)) usg Chebyshev collocato ethod wth =5 at α = 1 ad table 2, we preseted the uercal soluto usg the fractoal FDM wth

7 Appl. Math. If. Sc. 7, No. 5, (2013) / Table 2: The uercal soluto of the syste (40) usg the fractoal FDM at α = Table 3: The uercal soluto of the syste (40) usg the RK4 ethod at α = geerated fro the NLP proble. The fractoal dervatve s cosdered the Caputo sese. The propertes of the Chebyshev polyoals are used to reduce the syste of fractoal dfferetal equatos to the soluto of syste of algebrac equatos. It s evdet that the overall errors ca be ade saller by addg ew ters fro the seres (26). The covergece aalyss of the proposed ethod ad dervato a upper boud of the error are troduced. Fro llustratve exaples, t ca be see that the proposed uercal approach ca obta very accurate ad satsfactory results. The uercal coparso aog the fourth order Ruge-Kutta (α = 1) ad the soluto obtaed usg fte dfferece ethod wth the proposed ethods shows that our techque perfor rapd covergece to the optal solutos of the optzato probles. Also, fro the obtaed uercal results we ca coclude that our results are excellet agreeet wth the exact soluto ad those fro the RK4 ethod. All uercal results are obtaed usg Matlab. Table 4: The uercal soluto of the syste (40) usg the Chebyshev collocato ethod at α = Table 5: The uercal soluto of the syste (40) usg the fractoal FDM at α = k = at α = 1 ad table 3, we preseted the uercal soluto usg the fourth order Ruge-Kutta ethod. But tables 4 ad 5, we preseted the uercal soluto of the sae syste (40) wth α = 0.85 usg the two proposed ethods, respectvely. Fro these tables, we ca coclude that our solutos of the proposed ethods are excellet agreeet wth the soluto usg RK4 ethod. 5 Cocluso ad rearks I ths artcle, we pleeted a effcet uercal ethod for solvg the syste of FDEs whch s Ackowledgeet The authors are very grateful to the aoyous referees for carefully readg the paper ad for ther coets ad suggestos that proved the paper. Refereces [1] K. B. Oldha ad J. Spaer, The Fractoal Calculus, Acadec Press, New York, (1974). [2] I. Podluby, Fractoal Dfferetal Equatos, Acadec Press, New York, (1999). [3] R. L. Bagley ad P. J. Torvk, J. Appl. Mech., 51, (1984). [4] M. M. Meerschaert ad C. Tadera, Appl. Nuer. Math., 56, (2006). [5] N. H. Swela, M. M. Khader ad R. F. Al-Bar, Physcs Letters A, 371, (2007). [6] I. Hash, O. Abdulazz ad S. Moa, Coucatos Nolear Scece ad Nuercal Sulatos, 14, (2009). [7] M. M. Khader, Coucatos Nolear Scece ad Nuercal Sulatos, 16, (2011). [8] M. M. Khader, N. H. Swela ad A. M. S. Mahdy, J. of Appled Matheatcs ad Boforatcs, 1, 1-12 (2011). [9] M. M. Khader ad A. S. Hedy, Iteratoal Joural of Pure ad Appled Matheatcs, 74, (2012). [10] M. M. Khader, Arab Joural of Matheatcal Sceces, 18, (2012). [11] N. H. Swela, M. M. Khader ad A. M. S. Mahdy, Joural of Appled Matheatcs, 1-14 (2012). [12] F. Evrge ad N. Özder, J. Coputatoal Nolear Dyacs, 6, (2010). [13] F. Evrge ad N. Özder, Fractoal Dyacs ad Cotrol, (2012). [14] N. H. Swela, M. M. Khader ad A. M. S. Mahdy, Joural of Fractoal Calculus ad Applcatos, 15, 1-12 (2012).

8 2018 M. M. Khader et al: Nuercal Study for the Fractoal Dfferetal Equatos... [15] M. Abraowtz ad I.A. Stegu, Hadbook of atheatcal fuctos, Dover, New York, (1964). [16] N. H. Swela ad M. M. Khader, ANZIAM, 51, (2010). [17] D. G. Lueberger, Itroducto to Lear ad Nolear Prograg. Addso-Wesley, Calfora, (1973). [18] W. Su ad Y. Yua, Optzato Theory ad Methods: Nolear Prograg, Sprger-Verlag, New York, (2006). [19] K. J. Arrow, L. Hurwcz ad H. Uzawa, Studes Lear ad No-Lear Prograg, Staford Uversty Press, Calfora, (1958). [20] A. V. Facco ad G. P. Mccorck, Nolear prograg: sequetal ucostraed zato techques, Joh Wley, New York, (1968). [21] H. Yaashta, Math. Progra, 1, (1976). [22] G. D. Sth, Nuercal Soluto of Partal Dfferetal Equatos, Oxford Uversty Press, (1965). [23] N. H. Swela, M. M. Khader ad A.M. Nagy, Joural of Coputoal ad Appled Matheatcs, 235, (2011). [24] S. Sako, A. Klbas ad O. Marchev, Fractoal Itegrals ad Dervatves: Theory ad Applcatos, Gordo ad Breach, Lodo, (1993). [25] M. A. Syder, Chebyshev Methods Nuercal Approxato, Pretce-Hall, Ic. Eglewood Clffs, N. J. (1966). [26] K. Schttkowsk, More Test Exaples For Nolear Prograg Codes, Sprger-Verlag, Berl, (1987). [27] A. Dego Muro, Coputers ad Matheatcs wth Applcatos, 56, (2008). [28] N. H. Swela, M. M. Khader ad A. M. S. Mahdy, Joural of Fractoal Calculus ad Applcatos, 2, 1-9 (2012). Da Proble. He s the Head of the Departet of Matheatcs, Faculty of Scece, Caro Uversty, scece May He s referee ad edtor of several teratoal ourals, the frae of pure ad appled Matheatcs. Hs a research terests are uercal aalyss, optal cotrol of dfferetal equatos, fractoal ad varable order calculus, bo-foratcs ad cluster coputg, ll-posed probles. Ar M. S. Mahdy receved the PhD degree Matheatcs Departet-Faculty of Scece, Zagazg Uversty (2013). Hs research terests are the areas of Pure Matheatcs cludg the atheatcal ethods ad uercal techques for solvg fractoal dfferetal equatos. He has publshed research artcles reputed teratoal ourals of atheatcal. Mohaed M. Khader receved the PhD degree Matheatcs Departet- Faculty of Scece-Beha Uversty-Egypt (2009). Hs research terests are the areas of Pure Matheatcs cludg the atheatcal ethods ad uercal techques for solvg FDEs. He has publshed research artcles reputed teratoal ourals of atheatcal. He s referee of atheatcal ourals. Nasser H. Swela s professor of uercal aalyss at the Departet of Matheatcs, Faculty of Scece, Caro Uversty. He was a chael syste Ph.D. studet betwee Caro Uversty, Egypt, ad TU-Much, Geray. He receved hs Ph.D. Optal Cotrol of Varatoal Iequaltes, the