Research Article New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

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1 Hidawi Publishig Corporatio e Scietific World Joural Volume 04, Article ID 45650, pages Research Article New Formulae for the High-Order Derivatives of Some Jacobi Polyomials: A Applicatio to Some High-Order Boudary Value Problems W. M. Abd-Elhameed, Departmet of Mathematics, Faculty of Sciece, Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia Departmet of Mathematics, Faculty of Sciece, Cairo Uiversity, Giza 6, Egypt Correspodece should be addressed to W. M. Abd-Elhameed; walee 9@yahoo.com Received 9 April 04; Accepted 7 August 04; Published 4 October 04 Academic Editor: Fazlollah Soleymai Copyright 04 W. M. Abd-Elhameed. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This paper is cocered with derivig some ew formulae expressig explicitly the high-order derivatives of Jacobi polyomials whose parameters differece is oe or two of ay degree ad of ay order i terms of their correspodig Jacobi polyomials. The derivatives formulae for Chebyshev polyomials of third ad fourth kids of ay degree ad of ay order i terms of their correspodig Chebyshev polyomials are deduced as special cases. Some ew reductio formulae for summig some termiatig hypergeometric fuctios of uit argumet are also deduced. As a applicatio, ad with the aid of the ew itroduced derivatives formulae, a algorithm for solvig special sixth-order boudary value problems are implemeted with the aid of applyig Galerki method. A umerical example is preseted hopig to ascertai the validity ad the applicability of the proposed algorithms.. Itroductio Classical orthogoal polyomials are successfully employed for solvig ordiary ad partial differetial equatios i spectral ad pseudospectral methods (see, for istace, [ ]). I particular, the class of Jacobi polyomials P (α,β) plays promiet roles i the applicatios of mathematical aalysis. For example, Doha et al. i [4] used these polyomials for solvig some odd-order boudary value problems (BVPs). The suggested algorithms i this paper are based o selectig certai combiatios of Jacobi polyomials satisfyig the boudary coditios of the give differetial equatio.therearesomeotherarticlesiliteraturethat have extesive theoretical ad umerical studies about Jacobi polyomials (see, e.g., [5, 6]). It is well kow that the classical Jacobi polyomials have two parameters. Of course, special choices of their parameters give special kids of these polyomials. It is worthy to metio here that the classical Jacobi polyomials have six special well-kow kids of orthogoal polyomials; they are Legedre, ultraspherical, ad Chebyshev polyomials of the four kids. All six special kidsofjacobipolyomialshavetheirrolesadimportace from both theoretical ad practical poits of view (see, e.g., [7 ]). The spectral methods have promiet roles i various applicatios such as fluid dyamics. These methods are global methods. There are three popular methods of spectral methods; they are tau, collocatio, ad Galerki methods (see, e.g., [, ]).Thechoiceofthesuitableusedspectral method suggested for solvig the give equatio depeds certailyothetypeofthedifferetialequatioadalsoo the type of the boudary coditios govered by it. The explicit formulae for the high-order derivatives of various orthogoal polyomials i terms of their origial polyomials are useful whe spectral ad pseudospectral methods are used for obtaiig umerical solutios of various types of differetial equatios. Formula for the highorder derivatives for Chebyshev polyomials i terms of

2 The Scietific World Joural their origial polyomials is give i Karageorghis [4]. The correspodig formulae for Legedre, ultraspherical, ad Jacobi polyomials are give, respectively, i Phillips [5] ad Doha [6]. Due to their great importace i various applicatios, high eve-order BVPs have bee ivestigated by a large umber of authors. For example, sixth-order BVPs are kow to ariseiastrophysics;thearrowcovectiglayersbouded by stable layers, which are believed to surroud A-type stars, may be modeled by sixth-order BVPs. Sixth-order BVPs were hadled by umerous umerical techiques. For example, Doha ad Abd-Elhameed i [7] have developed efficiet solutios of multidimesioal sixth-order BVPs based o employig symmetric geeralized Jacobi-Galerki method. Also, Bhrawy et al. i [8] have developed a extesio of the Legedre-Galerki method for hadlig sixth-order BVPs with variable polyomial coefficiets. There are other cotributios cocerig sixth-order BVPs; amog the techiques used for hadlig this kid of BVPs are sic-galerki method [9], opolyomial splie techique i [0], splie collocatio method i [], Adomia decompositio method with Gree s fuctio i [], parametric quitic splie solutio i [], homotopy perturbatio method i [4], ad fourth order fiite differece method i [5]. For extesive studies about the existece ad uiqueess of solutios of such problems, the iterested reader ca be refered to the importat book of Agarwal [6]. For other studies i high eve- ad high odd-order BVPs, see, for example, [7 ]. The mai obective of this paper is twofold: (i) developig ew formulae for the high-order derivatives of some Jacobi polyomials with certai parameters, (ii) employig the ew itroduced formulae i solvig some sixth-order BVPs based o applyig a spectral Galerki method. The rest of the paper is as follows. I the ext sectio, some useful properties of Jacobi polyomials are preseted. I Sectio, we derive two ew formulae which give explicitly the high-order derivatives of Jacobi polyomials whose parameters differece is oe i terms of their origial Jacobi polyomials. I Sectio4, we give other two ew formulae which give explicitly the high-order derivatives of Jacobi polyomials whose parameters differece is two i terms of their origial Jacobi polyomials. Some ew reductio formulae for summig some termiatig hypergeometric fuctios of the type F () are deduced i Sectio 5. Sectio 6 is devoted to implemetig ad presetig a Galerki algorithm for umerically solvig cerati sixth-order BVPs icludig a umerical example aimig to illustrate the accuracy ad the efficiecy of the proposed algorithm. Fially, coclusios are give i Sectio 7. w ( x) α ( x) β (see, e.g., [, ]) are a sequece of polyomials P (α,β) ( 0,,,...), each, respectively, of degree. These polyomials have the followig Gauss hypergeometric form: P (α,β) (α)! F x (, λ; α ) It is clear that P (α,β) ( ) (β )! λαβ, () (α)! F (, λ; β x ), () (z) k Γ (zk). () Γ (z) P (β,α) ( x) ( ) P (α,β), (), P (α,β) ( ) ( ) (β ). (4)! The Jacobi polyomials may be geerated usig the recurrece relatio ( ) ( λ) ( λ ) P (α,β) (λ ) xp (α,β) (α β )(λ)p (α,β) (α)(β)(λ)p (α,β), startig from P (α,β) 0 ad P (α,β) (/)[α β (λ )x], or obtaied alteratively from Rodrigue s formula P (α,β) ( )! ( x) α β d dx [( x)α β ]. The polyomials P (α,β) satisfy the orthogoality relatio ( x) α β P (α,β) i 0, i, h (α,β) i, i, P (α,β) dx (5) (6) (7). Some Properties of Jacobi Polyomials The classical Jacobi polyomials associated with the real parameters (α >, β > ) ad the weight fuctio h (α,β) i αβ Γ (iα) Γ(iβ) (iαβ)i!γ(iαβ). (8)

3 The Scietific World Joural The followig structure formula is useful i the sequel (see Raiville [4]): P (α,β) (λ )(λ ) [(λ ) ( λ ) DP (α,β) (α β)(λ )(λ)dp (α,β) (α)(β)(λ)dp (α,β) ],. (9) Also, the followig theorem is of importat use hereafter. Theorem. For all q,theqth derivative of the Jacobi polyomial P (α,β) is give explicitly by D q P (α,β) q i0 θ i,,α,β,q P (α,β) i, (0) θ i,,α,β,q ( λ) q (qλ) i (iqα) i q Γ (iλ) q ( i q)!γ(i λ) F ( qi,iqλ, i α ; i q α, i λ ). () (For a proof of Theorem,seeDoha[6].) Remark. Although the F () i formula () istermiatig, it caot be summed i a closed form except for certai special choices of the two parameters α ad β. Icaseof αβ,this F () cabesummediaclosedformwiththe aid of Watso s idetity (see Doha [6]).. High-Order Derivatives of P (α,α) P (α,α) ad The mai obective of this sectio is to state ad prove two theorems i which the derivatives of the Jacobi polyomials P (α,α) ad P (α,α) are give i terms of their correspodig Jacobi polyomials. We first state ad prove a lemma i which the first derivative of P (α,α) is expressed i terms of their origial polyomials. Lemma. For all,oehas DP (α,α) ( )/ / d,r P (α,α) r e,r P (α,α) r, () ( r α ) Γ (α) Γ ( rα) d,r, Γ (α) Γ ( rα) (r) Γ (α) Γ ( rα) e,r, Γ (α) Γ ( rα ) () ad z deotes the largest iteger less tha or equal to z. Proof. Let us deote S m m 0 a P (α,α), (4) I S m dx. (5) The itegratio of both sides of the structure formula (9)(for the case βα)yields P (α,α) dxα P (α,α) α β γ which i tur implies that β P (α,α) γ P (α,α),, α (α)( α ), ( α )( α ), (α) (α)( α ), I α 0 P (α,α) 0 dx m a [α P (α,α) γ P (α,α) ]. β P (α,α) The last equatio may be writte alteratively i the form I 0 (6) (7) (8) A P (α,α), (9) a α a β γ a A,,m,...,, a a m 0. The differece equatio (0)ca be solved to give a g k, A k k (k) odd k (k) eve f k, A k, 0,,...,m, (0) ()

4 4 The Scietific World Joural g k, f k, (kα) Γ (kα) Γ (α), () Γ (kα) Γ (α) (k ) Γ (kα) Γ (α). () Γ (kα) Γ (α) If we substitute ()ito(4), we get S m m 0 k (k) odd g k, A k k (k) eve f k, A k, P (α,α). (4) O the other had, if we differetiate (9) with respect to x, we have S m 0 A DP (α,α). (5) Equatig the two right-had sides of the two relatios (4) ad (5)yields m 0 k (k) odd 0 g k, A k A DP (α,a). k (k) eve f k, A k, P (α,α) (6) Expadig the left-had side of (6) ad collectig the similar terms, (6) may be writte after some rather maipulatio i the form 0 B A B ( )/ / 0 A DP (α,a), (7) g, r P (α,α) r f, r P (α,α) r, (8) ad g k,, f k, are give by () ad(), respectively. This immediately yields DP (α,α) ( )/ / d,r P (α,α) r e,r P (α,α) r, (9) d,r e,r ( r α ) Γ (α) Γ ( rα), Γ (α) Γ ( rα) (r) Γ (α) Γ ( rα), Γ (α) Γ ( rα ) ad this completes the proof of of Lemma. (0) Now, we exted the result of Lemma to give the qthderivative of the Jacobi polyomials P (α,α) i terms of their origial polyomials. The geeral formula is stated i the followig theorem. Theorem 4. For all q,theqth-derivative of the classical Jacobi polyomials P (α,α) of ay degree ad for ay order i terms of their correspodig Jacobi polyomials is give explicitly by ( q)/ D q P (α,α) ( q )/ A,r,q P (α,α) r q B,r,q P (α,α) r q, A,r,q ( q (qr )!Γ(α) Γ( rα ) Γ( q rα)) ((q )!r!γ (α) Γ( q rα) Γ( q rα )), B,r,q ( q (q r)!γ (α) Γ( rα ) Γ( q rα)) ((q )!r!γ (α) Γ( q rα) Γ( q rα )). () () () Proof. We proceed by iductio o q. For q, Theorem 4 reduces idetically to Lemma. Assume that relatio () holds,adwehavetoshowthat ( q )/ D q P (α,α) ( q)/ A,r,q P (α,α) r q B,r,q P (α,α) r q. (4)

5 The Scietific World Joural 5 If we differetiate relatio (), the we get ( q)/ D q P (α,α) ( q )/ A,r,q DP (α,α) r q adivirtueoflemma,oecawrite D q P (α,α) B,r,q DP (α,α) r q, (5), (6) r 0 (((q )!( 4 qα) Γ( α )) (!Γ( qα )) ), B,r,q ( q (qr)γ(α) Γ( q rα)) ((q )!Γ( α )Γ( q r α )) r 0 (((q )!( 4 qα) ( q)/ ( q )/ r A,r,q s0 ( q )/ ( q )/ r B,r,q s0 d q r,s P (α,α) q r s e q r,s P (α,α) q r s, Γ( α )) (!Γ( qα )) ). (40) ( q)/ ( q)/ r A,r,q s0 ( q )/ ( q)/ r B,r,q s0 e q r,s P (α,α) q r s d q r,s P (α,α) q r s. (7) If we expad ad ad collectig similar terms ad after some legthy maipulatios, we get ( q )/ D q P (α,α) A,r,q B,r,q r 0 r 0 ( q)/ A,r,q P (α,α) r q B,r,q P (α,α) r q, A,,q d q,r B,,q e q,r, A,,q e q,r B,,q d q,r. It is ot difficult to show that A,r,q ( q ( r α ) Γ (α) Γ( q rα)) ((q )!Γ (α) Γ( q rα)) (8) (9) To complete the proof of Theorem 4, the followig lemma which ca be easily proved by iductio is required. Lemma 5. For every oegative iteger r,oe has r 0 (q )!( 4 qα)γ( α/)!γ( qα/) (qr)!γ( rα/) qr!γ( q rα/). (4) Now, the applicatio of Lemma 5 i (40)yields A,r,q ( q (q r)!γ (α) Γ( rα ) Γ( q rα)) (q!r!γ (α) Γ( q rα) A,r,q, Γ( q rα )) B,r,q ( q (qr)!γ(α) Γ( rα ) Γ( q rα))

6 6 The Scietific World Joural (q!r!γ (α) Γ( q rα ) B r,,q. Γ( q rα )) (4) This proves formula(4) ad hece completes the proof of Theorem 4. Remark 6. It is worthy to ote here that relatio ()maybe writtealterativelyitheequivaletform E,i,q q D q P (α,α) i0 q Γ (α) Γ (iα) (q )!Γ (iα) Γ (α) E,i,q P (α,α) i, (4) (iq) eve, (( i q ) /)!Γ (( i q α ) /) (( i q ) /)!Γ (( i q α ) /), (( i q) /)!Γ (( i q α ) /) (( i q ) /)!Γ (( i q α 4) /), (iq) odd. (44) Remark 7. As a direct cosequece of relatio (), the qthderivative of P (α,α) is give by ( q)/ D q P (α,α) ( q )/ A,r,q P (α,α) r q B,r,q P (α,α) r q, (45) A,r,q ad B,r,q are give by ()ad(), respectively, or alteratively i the form q D q P (α,α) i0 ( ) iq E,i,q P (α,α) i, (46) Corollary 8. For all q,theqth-derivative of V of ay degree ad for ay order i terms of their correspodig polyomials is give explicitly by D q V q (q )! q (q )! ( q )/ ( q)/ ( r)!(rq )! V r!( r q)! r q ( r )!(rq)! V r! ( r q)! r q. (47) Corollary 9. For all q,theqth-order derivative of W of ay degree ad for ay order i terms of their correspodig polyomials is give explicitly by D q W q (q )! q (q )! ( q )/ ( q)/ ( r)!(rq )! W r!( r q)! r q ( r )!(rq)! W r! ( r q)! r q. (48) 4. High-Order Derivatives of P (α,α) P (α,α) ad I this sectio, ad followig similar procedures to those followed i Sectio, we ca obtai ew derivatives formulae for the high-order derivatives of the Jacobi polyomials P (α,α) ad P (α,α) i terms of their correspodig Jacobi polyomials. The details of the required computatios arelegthyadwillotbegivehere. Theorem 0. For all q,theqth-derivative of the classical Jacobi polyomials P (α,α) of ay degree ad for ay order i terms of their correspodig Jacobi polyomials is give explicitly by E,i,q is give by (44). As a special case of the two formulae ()ad(45)adif we set α /, the derivatives of Chebyshev polyomials of third kid (V ) ad of fourth kid (W ) cabeeasily deduced. These two results are give i the followig two corollaries. ( q)/ D q P (α,α) ( q )/ G,r,q P (α,α) r q H,r,q P (α,α) r q, (49)

7 The Scietific World Joural 7 G,r,q H,r,q ( q r (qr )!Γ(α) Γ ( r α )) ((q )!r!γ (α)( q rα) Γ( rα)) Γ ( q r α ) Γ ( q r α 5/) Γ ( q 4r α ) Γ ( q r α 5/) ( (qr α ) α(qr) r (q r) q r α α), ( q r (q r)!γ (α) Γ ( ( rα))) ((q )!r!γ (α)( q rα) Γ( rα)) Γ( q rα/)γ( q rα) Γ ( q r α /) Γ ( q 4r α ). (50) (5) Remark. As a direct cosequece of relatio (49), the qthderivative of P (α,α) is give by ( q)/ D q P (α,α) ( q )/ G,r,q P (α,α) r q H,r,q P (α,α) r q, (5) G,r,q ad H,r,q are give by (50)ad(5), respectively. 5. Reductio Formulae for Some Termiatig Hypergeometric Fuctios of the Type F () I this sectio, four ew reductio formulae for the F () that appears i relatio () are deduced for certai choices of the two parameters α ad β.theseformulaearegiveithe followig four corollaries. Corollary. For all i,, q Z 0 ad i q,oehas F ( iq,iqα, i α ; i q α, i α ) Γ (iα/) Γ(iqα) π(q )! (( iq )/)!Γ(( i q)/) Γ((iqα)/)Γ((i qα)/), (iq)eve, (( iq )/)!Γ(( i q)/) Γ((iqα)/)Γ((i qα4)/), (iq)odd, F ( iq,iqα, i α ; i q α, i α ) Γ (iα/) Γ(iqα) π(q )!(α) (( i q ) /)!Γ (( i q ) /) Γ (( i q α)/)γ(( i q α )/), (iq)eve, (( iq )/)!Γ(( i q)/) Γ((iqα)/)Γ((i qα)/), (iq)odd. (5) Corollary. For all i,, q Z 0 ad i q,oehas F ( iq,iqα, i α ; i q α, i α 4 ) (iα) Γ (i α 4) Γ(iqα) (q )!Γ (iα) (( i q ) /)!Γ (( i q ) /) Γ (( i q α 5)/)Γ(( i q α 4)/) ξ i,,α, (iq)eve, 4(( iq )/)!Γ(( i q)/) Γ (( i q α 4)/)Γ(( i q α )/), (iq)odd, F ( iq,iqα, i α ; i q α, i α ) Γ (iα/) Γ(iqα) π(q )!(iα)(α )(α) Γ (( i q α )/)Γ(( i q α)/) η i,,α, (iq)eve, 4 (( i q ) /)!Γ (( i q ) /) (( i q ) /)!Γ (( i q ) /) Γ (( i q α)/)γ(( i q α )/), (iq)odd, (54)

8 8 The Scietific World Joural ξ i,,α i α(i) i q qα 4, η,i,α i α(i ) i q qα. (55) Proof. The proofs of the four reductio formulae i Corollaries ad cabededucedimmediately,bycomparigthe four relatios (4), (46), (49), ad (5)withthecorrespodig results obtaied from the geeral relatio i (0). 6. Jacobi Galerki Algorithms for Sixth-Order Two-Poit Boudary Value Problems I this sectio, ad as a applicatio, we are iterested i applyig the itroduced derivatives formulae which were obtaied i Sectios ad 4, for the sake of solvig the followig sixth-order two-poit BVPs: y (6) γy f, x (, ), (56) govered by the ohomogeeous boudary coditios y (r) (±) ±α r, r 0,,, (57) γ is a real costat. We draw the reader s attetio that (56) goveredbythe ohomogeeous boudary coditios (57) ca be easily trasformed to the equatio (see [7]) u (6) γu g, x (, ), (58) govered by the homogeeous boudary coditios u (r) (±) 0,,,, (59) g f 5 i0 ad η i, 0 i, are some costats. Now, we defie the followig spaces: η i x i, (60) S N spa P (α,β) 0,P (α,β),p (α,β),...,p (α,β) V N V S N :D V (±) 0,0,,. N, (6) The Jacobi-Galerki procedure for solvig (56)subect to the boudary coditios (57)istofidu N V N such that ( D 6 u N, V) w γ(u N, V) w (g, V) w, V V N, (6) w ( x) α ( x) β,ad(u, V) w wuv dx is the scalar ier product i the weighted space L w (, ). 6.. Basis Fuctios i Terms of Certai Parameters Jacobi Polyomials. I this sectio, we cosider four kids of basis fuctios satisfyig the boudary coditios (59). Assume that these basis fuctios ca be expressed i the followig forms: φ,k P (α,α) k d,m,k P (α,α) km, 6 m φ,k P (α,α) k d,m,k P (α,α) km, 6 m φ,k P (α,α) k d,m,k P (α,α) km, 6 m φ 4,k P (α,α) k d 4,m,k P (α,α) km, 6 m (6) ad k0,,,...,n 6. The coefficiets d i,m,k, i 4, are uiquely determied such that each member of φ i,k, i 4, satisfies the boudary coditios (57). For example, the coefficiets d,m,k are give explicitly by d,m,k ( ) m/ Γ (α)( m/ ) (km)!(kα/) m/, Γ (kmα)(kα9/) m/ m, 4, 6, (( ) (m )/ (m 7) Γ (α)( (m )/ ) Γ(k)(kα/) (m )/ ) (Γ(kmα)(kα9/) ()/ ), m,, 5. (64) Now, the four variatioal formulatios correspodig to the four choices of the basis fuctios ca be writte as ( D 6 u i,n,φ i,k ) wi γ(u i,n,φ i,k ) wi (g, φ i,k ) wi, i 4, (65) w ( x) α α, w ( x) α α, w ( x) α α, w 4 ( x) α α, u i,n N 6 c i,r φ i,r. (66)

9 The Scietific World Joural 9 For i 4, let us deote g i,k (g,φ i,k ) wi, g i (g i,0,g i,,...,g i,n 6 ) T, A i (a i k ) 0 k, N 6, B i (b i k ) 0 k, N 6, c i (c i,0,c i,,...,c i,n 6 ) T, (67) adthe(65) isequivalettothefollowigfourmatrix systems: (A i γb i ) c i g i, (68) the ozero elemets of the matrices A i ad B i, i 4, ca be give explicitly. Now, we give a umerical example to show the applicability ad the efficiecy of the proposed algorithm. Example 4. Cosider the followig sixth-order liear boudary value problem (see El-Gamel et al. [9], Akram ad Siddiqi [0], ad Lamii et al. []): y (6) (t) y(t) 6e t, t [0, ], y (0), y (0) 0, y (0), y () 0, y () e, y () e. The aalytic solutio to (69)is( t)e t. The trasformatio t ( x)/ turs (69)ito u (6) u 6e /, x [, ], u ( ), u ( ) 0, u ( ), u () 0, u () e, u () e, (69) (70) with the aalytic solutio u (/)( x)e /. I Table, we deote E u u N by the absolute errors resultig from the applicatio of the Jacobi Galeki method (JGM) for problem (70) forvariousvaluesoftheparameters α, β, adn. Moreover,Table displays a compariso betwee the best errors resultig from the applicatio of JGM with the results obtaied by applyig the followig three methods: (i) sic-galerki method developed i [9], (ii) opolyomial splie techique developed i [0], (iii) splie collocatio method developed i []. The results i Table idicate that our method is more accurate if compared with all of the above-metioed methods. 7. Coclusios This paper is cocered with itroducig four ew aalytical formulae for the qth-derivative of certai parameters Jacobi polyomials i terms of their correspodig Jacobi polyomials. Moreover, some ew reductio formulae for Table : Maximum absolute error E for N8,, 6. N α β E Table : Compariso betwee the best errors of differet methods. Error JGM Method i [9] Method i[0] Methodi[] E summig some termiatig hypergeometric fuctios of uit argumet are deduced. As a applicatio, ad with the aid of the ew itroduced derivatives formulae, some spectral solutios of a special sixth-order boudary value problem are preseted. To the best of our kowledge, all the preseted theoretical formulae i this paper are completely

10 0 The Scietific World Joural ew. Moreover, we do believe that these ew itroduced formulae ca be applied for solvig other kids of high-order boudary value problems. Coflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets This Proect was fuded by the Deaship of Scietific Research (DSR), Kig Abdulaziz Uiversity, Jeddah, uder Grat o The author, therefore, ackowledges with thaks DSR techical ad fiacial support. Refereces [] W.M.Abd-Elhameed,E.H.Doha,adY.H.Youssri, Efficiet spectral-petrov-galerki methods for third- ad fifth-order differetial equatios usig geeral parameters geeralized Jacobi polyomials, Quaestioes Mathematicae, vol. 6, o., pp.5 8,0. []E.H.DohaadW.M.Abd-Elhameed, Efficietspectral- Galerki algorithms for direct solutio of secod-order equatios usig ultraspherical polyomials, SIAM Joural o Scietific Computig,vol.4,o.,pp ,00. [] E. H. Doha ad W. M. Abd-Elhameed, Efficiet spectral ultraspherical-dual-petrov-galerki algorithms for the direct solutio of ( )th-order liear differetial equatios, Mathematics ad Computers i Simulatio, vol.79,o.,pp. 4, 009. [4] E. H. Doha, W. M. Abd-Elhameed, ad Y. H. Youssri, Efficiet spectral-petrov Galerki methods for the itegrated forms of third- ad fifth-order elliptic differetial equatios usig geeral parameters geeralized Jacobi polyomials, Applied Mathematics ad Computatio, vol.8,o.5,pp , 0. [5] S. Kazem, A itegral operatioal matrix based o Jacobi polyomials for solvig fractioal-order differetial equatios, Applied Mathematical Modellig, vol.7,o.,pp.6 6, 0. [6] E. H. Doha, W. M. Abd-Elhameed, ad H. M. Ahmed, The coefficiets of differetiated expasios of double ad triple Jacobi polyomials, Bulleti of the Iraia Mathematical Society, vol. 8, o., pp , 0. [7] M. R. Eslahchi, M. Dehgha, ad S. Amai, The third ad fourth kids of Chebyshev polyomials ad best uiform approximatio, Mathematical ad Computer Modellig, vol. 55, o. 5-6, pp , 0. [8] K. Julie ad M. Watso, Efficiet multi-dimesioal solutio of PDEs usig Chebyshev spectral methods, Joural of Computatioal Physics,vol.8,o.5,pp ,009. [9]J.C.MasoadD.C.Hadscomb,Chebyshev Polyomials, Chapma & Hall, New York, NY, USA, CRC, Boca Rato, Fla, USA, 00. [0] K. T. Elgidy ad K. A. Smith-Miles, O the optimizatio of Gegebauer operatioal matrix of itegratio, Advaces i Computatioal Mathematics,vol.9,o.-4,pp.5 54,0. [] L. Zhu ad Q. Fa, Solvig fractioal oliear Fredholm itegro-differetial equatios by the secod kid Chebyshev wavelet, Commuicatios i Noliear Sciece ad Numerical Simulatio,vol.7,o.6,pp. 4,0. [] J. P. Boyd, Chebyshev ad Fourier Spectral Methods, Dover, Mieola, NY, USA, d editio, 00. []C.Cauto,M.Y.Hussaii,A.Quarteroi,adT.A.Zag, Spectral Methods i Fluid Dyamics, Spriger, New York, NY, USA, 989. [4] A. Karageorghis, A ote o the Chebyshev coefficiets of the geeral order derivative of a ifiitely differetiable fuctio, Joural of Computatioal ad Applied Mathematics,vol.,o., pp. 9, 988. [5] T. N. Phillips, O the Legedre coefficiets of a geeral-order derivative of a ifiitely differetiable fuctio, IMA Joural of Numerical Aalysis, vol. 8, o. 4, pp , 988. [6] E. H. Doha, O the coefficiets of differetiated expasios ad derivatives of Jacobi polyomials, Joural of Physics A: Mathematical ad Geeral, vol. 5, o. 5, pp , 00. [7] E. H. Doha ad W. M. Abd-Elhameed, Efficiet solutios of multidimesioal sixth-order boudary value problems usig symmetric geeralized Jacobi-Galerki method, Abstract ad Applied Aalysis,vol.0,ArticleID74970,9pages,0. [8]A.H.Bhrawy,A.S.Alofi,adS.I.El-Soubhy, Aextesio of the Legedre-Galerki method for solvig sixth-order differetial equatios with variable polyomial coefficiets, Mathematical Problems i Egieerig, vol.0,articleid , pages, 0. [9] M. El-Gamel, J. R. Cao, ad A. I. Zayed, Sic-Galerki method for solvig liear sixth-order boudary-value problems, Mathematics of Computatio, vol. 7, o. 47, pp. 5 4, 004. [0] G. Akram ad S. S. Siddiqi, Solutio of sixth order boudary value problems usig o-polyomial splie techique, Applied Mathematics ad Computatio,vol.8,o.,pp , 006. [] A. Lamii, H. Mraoui, D. Sbibih, A. Tiii, ad A. Zida, Splie collocatio method for solvig liear sixth-order boudaryvalue problems, Iteratioal Joural of Computer Mathematics,vol.85,o.,pp ,008. [] W. Al-Hayai, Adomia decompositio method with Gree s fuctio for sixth-order boudary value problems, Computers & Mathematics with Applicatios, vol.6,o.6,pp , 0. [] A. Kha ad T. Sultaa, Parametric quitic splie solutio for sixth order two poit boudary value problems, Filomat, vol. 6,o.6,pp. 45,0. [4] M. A. Noor ad S. T. Mohyud-Di, Homotopy perturbatio method for solvig sixth-order boudary value problems, Computers & Mathematics with Applicatios,vol.55,o.,pp , 008. [5] P. K. Padey, Fourth order fiite differece method for sixth order boudary value problems, Computatioal Mathematics ad Mathematical Physics,vol.5,pp.57 6,0. [6] R. P. Agarwal, Boudary Value Problems for High Ordiary Differetial Equatios, World Scietific, Sigapore, 986. [7] W. M. Abd-Elhameed, E. H. Doha, ad M. A. Bassuoy, Two legedre-dual-petrov-galerki algorithms for Solvig the itegrated forms of high odd-order boudary value problems, The Scietific World Joural, vol.04,articleid0964, pages, 04.

11 The Scietific World Joural [8]M.A.Noor,K.I.Noor,adS.T.Mohyud-Di, Variatioal iteratio method for solvig sixth-order boudary value problems, Commuicatios i Noliear Sciece ad Numerical Simulatio,vol.4,o.6,pp ,009. [9] A. Kha ad T. Sultaa, Parametric quitic splie solutio of third-order boudary value problems, Iteratioal Joural of Computer Mathematics, vol. 89, o., pp , 0. [0] A. Aslaov, A geeral formula for the series solutio of high-order liear ad oliear boudary value problems, Mathematical ad Computer Modellig,vol.55,o.-4,pp , 0. [] E. H. Doha ad W. M. Abd-Elhameed, O the coefficiets of itegrated expasios ad itegrals of Chebyshev polyomials of third ad fourth kids, Bulleti of the Malaysia Mathematical Scieces Society,vol.7,o.,pp.8 98,04. [] G. E. Adrews, R. Askey, ad R. Roy, Special Fuctios, Cambridge Uiversity Press, Cambridge, UK, 999. [] M. Abramowitz ad I. A. Stegu, Eds., Hadbook of Mathematical Fuctios, vol.55ofapplied Mathematical Series, Natioal Bureau of Stadards, New York, NY, USA, 970. [4] E. D. Raiville, Special Fuctios, Macmilla,NewYork,NY, USA, 960.