Research Article A Jacobi Dual-Petrov-Galerkin Method for Solving Some Odd-Order Ordinary Differential Equations

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1 Abstract and Appled Analyss Volume 011, Artcle ID 94730, 1 pages do: /011/94730 Research Artcle A Jacob Dual-Petrov-Galerkn Method for Solvng Some Odd-Order Ordnary Dfferental Equatons E. H. Doha, 1 A. H. Bhrawy,, 3 and R. M. Hafez 4 1 Department of Mathematcs, Faculty of Scence, Caro Unversty, Gza 1613, Egypt Department of Mathematcs, Faculty of Scence, Kng Abdulazz Unversty, Jeddah 1589, Saud Araba 3 Department of Mathematcs, Faculty of Scence, Ben-Suef Unversty, Ben-Suef 6511, Egypt 4 Department of Basc Scence, Insttute of Informaton Technology, Modern Academy, Caro 11931, Egypt Correspondence should be addressed to A. H. Bhrawy, albhrawy@yahoo.co.uk Receved 31 October 010; Revsed 13 February 011; Accepted 16 February 011 Academc Edtor: Smeon Rech Copyrght q 011 E. H. Doha 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. A Jacob dual-petrov-galerkn JDPG method s ntroduced and used for solvng fully ntegrated reformulatons of thrd- and ffth-order ordnary dfferental equatons ODEs wth constant coeffcents. The reformulated equaton for the Jth order ODE nvolves n-fold ndefnte ntegrals for n 1,...,J. Extenson of the JDPG for ODEs wth polynomal coeffcents s treated usng the Jacob-Gauss-Lobatto quadrature. Numercal results wth comparsons are gven to confrm the relablty of the proposed method for some constant and polynomal coeffcents ODEs. 1. Introducton A well-known advantage of spectral methods s hgh accuracy wth relatvely fewer unknowns when compared wth low-order fnte-dfference methods 1,. On the other hand, spectral methods typcally gve rse to full matrces, partally negatng the gan n effcency due to the fewer degrees of freedom. In general, the use of the Jacob polynomals wth α, β 1, and n s the polynomal degree has the advantage of obtanng solutons of ordnary dfferental equatons ODEs n terms of the Jacob ndces see for nstance, 3 5. Several such pars α, β have been used for approxmate solutons of ODEs see We avod developng approxmaton results for each partcular par of ndces and nstead carryouta study wth generalndces. Wth ths motvaton, we ntroduce n ths paper a famly of the Jacob polynomals wth general ndces. Thrd-order dfferental equatons have applcatons n many engneerng models, see for nstance Ffth-order dfferental equatons generally arse n the mathematcal P α,β n

2 Abstract and Appled Analyss modelng of vscoelastc flows and other branches of mathematcal physcs and engneerng scences, see Exstence and unqueness of solutons of such boundary value problems are dscussed, for nstance, n 18. In ths paper, the proposed dfferental equatons are ntegrated q tmes, where q s the order of the equaton, and we make use of the formulae relatng the expanson coeffcents of ntegraton appearng n these ntegrated forms of the proposed dfferental equatons to the Jacob polynomals themselves see, Doha 19. An advantage of ths approach s that the general equaton n the algebrac system then contans a fnte number of terms. We, therefore, motvated our nterest n ntegrated forms of these dfferental equatons. The nterested reader s referred to Doha and Bhrawy 7, 0. The man am of ths paper s to propose a sutable way to approxmate some ntegrated forms of thrd- and ffth-order ODEs wth constant coeffcents usng a spectral method, based on the Jacob polynomals such that t can be mplemented effcently and at the same tme has a good convergence property. It s worthy to note here that oddorder problems lack the symmetry of even-order ones, so we propose a Jacob dual-petrov- Galerkn JDPG method. The method leads to systems wth specally structured matrces that can be effcently nverted. We apply the method for solvng the ntegrated forms of thrd- and ffth-order ODEs by usng compact combnatons of the Jacob polynomals, whch satsfy essentally all the underlyng homogeneous boundary condtons. To be more precse, for the JDPG we choose the tral functons to satsfy the underlyng boundary condtons of the dfferental equatons, and we choose the test functons to satsfy the dual boundary condtons. Extenson of the JDPG for polynomal coeffcentodes sobtanedby approxmatng the weghted nner products n the JDPG by usng the Jacob-Gauss-Lobatto quadrature. Fnally, examples are gven to llustrate the effcency and mplementaton of the method. Comparsons are made to confrm the relablty of the method. The remander of ths paper s organzed as follows. In Secton we gve an overvew of the Jacob polynomals and ther relevant propertes needed hereafter. Secton 3 s devoted to the theoretcal dervaton of the JDPG method for thrd-order dfferental equatons wth constant and varable polynomal coeffcents. Secton 4 gves the correspondng results for those obtaned n Secton 3, but for the ffth-order dfferental equatons wth constant coeffcent and two choces of boundary condtons. In Secton 5, we present some numercal results exhbtng the accuracy and effcency of our numercal algorthms. Some concludng remarks are gven n the fnal secton.. Prelmnares Let S N I be the space of polynomals of degree at most N on the nterval I 1, 1.Weset W N { u S N : u ±1 u }, } W N {u S N : u ±1 u 1 1 0,.1 and let P α,β n x n 0, 1,,... be the Jacob polynomals orthogonal wth the weght functons x 1 x α 1 x β,whereα, β > 1.

3 Abstract and Appled Analyss 3 Let x α,β N,j,0 j N, bethezerosof 1 x x P α,β N.Denotebyϖ α,β N,j,0 j N, the weghts of the correspondng Gauss-Lobatto quadrature formula, whch are arranged n decreasng order. We defne the dscrete nner product and norm of weghted space L I as follows: N u, v,n u k 0 x α,β N,k v x α,β N,k ϖ α,β N,k, u,n u, u,n.. Obvously see, e.g., formula.5 of 1, u, v,n u, v u, v S N 1..3 Thus, for any u S N, the norms u,n and u wα,β concde. For any u C I, the Jacob-Gauss-Lobatto nterpolaton operator I P α,β N u x S N, satsfyng I P α,β N u x α,β N,k u x α,β N,k, 0 k N..4 We also denote by I c N IP 1/, 1/ N and I l N IP 0,0 N the Chebyshev-Gauss-Lobatto and Legendre- Gauss-Lobatto nterpolaton operators, respectvely. For any real numbers α, β > 1, the set of the Jacob polynomals forms a complete L I -orthogonal system, and P α,β k,p α,β j w α,β h kδ k,j,.5 where δ k,j s the Kronecker functon and h k α β 1 Γ k α 1 Γ k β 1 k α β 1 Γ k 1 Γ k α β 1..6 The followng specal values wll be of fundamental mportance n what follows see, 4 P α,β n 1 α 1 n n!, P α,β n 1 1 n β 1 n, n! D q P α,β q 1 Γ n α 1 n λ n 1 q n q!γ q α 1, Dq P α,β n 1 1 n q D q P β,α n 1,.7 where a k Γ a k /Γ a and λ 1 β α.

4 4 Abstract and Appled Analyss If we defne the q tmes repeated dfferentaton and ntegraton of P α,β n x by D q P α,β n x and I q,α,β n x, respectvely, then cf. Doha 19, D q P α,β n n q x q α,β n λ q C n q, q, α, β P x,.8 I q,α,β n x q n q λ n q q q C n q, q, α, β P α,β x π q 1 x,.9 q 0, n q 1 for α β 1, q 0, n q for α / 1 or β / 1, where l q λ C l, q, α, β q α 1 l!γ λ l Γ λ l l q λ α 1 3F, q α 1 λ wth π q 1 x beng a polynomal of degree at most q 1.ItstobenotedthatI q,α,β n x may be obtaned from D q P α,β n x by replacng q wth negatve q. In general, the hypergeometrc seres 3F cannot be summed n explct form, but t can be summed by Watsons dentty 5, f α β. The followng two lemmas wll be of fundamental mportance n what follows. Lemma.1 see, 19, 6. One has x l l j 0 D γ,δ lj P γ,δ j x,.11 where D γ,δ lj j l! 1 γ δ j l j 1 γ j! l j! 1 γ δ j..1 1 γ δ j j Lemma.. If one wrtes I q,α,β k q α,β k x S q k,, α, β P x π q 1 x,.13 q

5 Abstract and Appled Analyss 5 then S q k,, α, β q k q λ Proof. It s mmedately obtaned from relaton.9. q C k q, q, α, β, q 1,, Thrd-Order Dfferental Equaton We are nterested n usng the JDPG method to solve the thrd-order dfferental equaton u 3 x γ 1 u x γ u 1 x γ 3 u x g x, n I, 3.1 subject to u ±1 u 1 1 0, 3. where γ 1, γ,andγ 3 are constants and g s a gven source functon. In ths paper, we consder the fully ntegrated form of the ODE, gven by u x γ 1 1 u x dx 1 γ u x dx γ 3 3 u x dx 3 f x d P α,β x, u ±1 u 1 0, 3.3 where q tmes {}}{ q tmes q {}}{ 3 u x dx q u x dx dx dx, f x g x dx In ths work we assume that f, the three-fold ndefnte ntegral form of g, can be evaluated analytcally. We set { S N span P α,β 0 x,p α,β 1 x,...,p α,β N }, x 3.5 then the dual-petrov-galerkn approxmaton to 3.3 s to fnd u N W N such that 1 u N,v γ 1 u N dx 1,v γ u N dx,v 3 γ 3 u N dx 3,v f d P α,β,v v W N, 3.6

6 6 Abstract and Appled Analyss where x 1 x α 1 x β and u, v w uvw dx s the nner product n the weghted I space L I. The norm n L I wll be denoted by The Jacob Dual-Petrov-Galerkn Method We choose compact combnatons of the Jacob polynomals as bass functons to mnmze the bandwdth hopng to mprove the condton number of the coeffcent matrx correspondng to 3.6. We choose the test bass and tral functons of expansons φ k x and ψ k x to be of the form φ k x P α,β k x ɛ k P α,β k 1 x ε kp α,β k x ζ kp α,β k 3 x, 3.7 ψ k x P α,β k x ρ k P α,β k 1 x ϱ kp α,β k x σ kp α,β k 3 x, 3.8 where ɛ k, ε k, ζ k, ρ k, ϱ k,andσ k are the unque constants such that φ k x W N and ψ k x W N. From the boundary condtons, φ k ±1 φ 1 k 1 0and.7,henceɛ k, ε k and ζ k can be unquely determned by usng mathematca to gve see, 7 ɛ k k 1 k λ k α β 1 k α 1 k β 1 k λ 4, ε k k 1 k λ 1 k β α 3 k α 1 k β 1 k λ 5, 3.9 ζ k k 1 3 k λ 1 k α 1 k β 1 k λ 4. Usng.7,andψ k ±1 ψ 1 1 0, one verfes readly that k ρ k k 1 k λ k β α 1 k α 1 k β 1 k λ 4, ϱ k k 1 k λ 1 k α β 3, k β 1 k α 1 k λ σ k k 1 3 k λ 1 k β 1 k α 1 k λ 4.

7 Abstract and Appled Analyss 7 Now t s clear that 3.6 s equvalent to un,ψ k x γ 1 1 u N dx 1,ψ k x γ 3 3 u N dx 3,ψ k x f x γ u N dx,ψ k x d P α,β x,ψ k x,n, k 0, 1,...,N, 3.11 where,,n s the dscrete nner product assocated wth the Jacob-Gauss-Lobatto quadrature. The constants d 0, d 1,andd would not appear f we take k 3n 3.11, therefore we get un,ψ k x γ 1 1 u N dx 1,ψ k x γ 3 3 u N dx 3,ψ k x γ u N dx,ψ k x f x,ψ k x, k 3, 4,...,N.,N 3.1 If we take φ k x and ψ k x as defned n 3.7 and 3.8, respectvely, and f we denote f k f, ψ k x,n, f f 3,f 4,...,f N T, N 3 u N x v n φ n x, v v 0,v 1,...,v N 3 T, n 0 a kj φ j 3 x,ψ k x, b kj 1 φ j 3 x dx 1,ψ k x, 3.13 c kj φ j 3 x dx,ψ k x, d kj 3 φ j 3 x dx 3,ψ k x, then W N span { φ 0 x,φ 1 x,...,φ N 3 x }, W N span{ ψ 3 x,ψ 4 x,...,ψ N x }, 3.14

8 8 Abstract and Appled Analyss and the nonzero elements a kj, b kj, c kj,and d kj for 3 k, j N are gven as follows: a kk ζ k 3 h k, a k,k 1 ε k h k ζ k ρ k h k 1, a k,k ɛ k 1 h k ε k 1 ρ k h k 1 ζ k 1 ϱ k h k, a k,k 3 h k ɛ k ρ k h k 1 ε k ϱ k h k ζ k σ k h k 3, a k,k 4 ρ k h k 1 ɛ k 1 ϱ k h k ε k 1 σ k h k 3, a k,k 5 ϱ k h k ɛ k σ k h k 3, a k,k 6 σ k h k 3, 3.15 b k,j [ R 1 j, k, α, β hk R 1 j, k 1,α,β ρk h k 1 R 1 j, k,α,β ϱk h k R 1 j, k 3,α,β σk h k 3 ], j k l 1,l 0, 1,...,8, c k,j [ R j, k, α, β hk R j, k 1,α,β ρk h k 1 R j, k,α,β ϱk h k R j, k 3,α,β σk h k 3 ], j k l,l 0, 1,...,10, d k,j R 3 j, k, α, β hk R 3 j, k 1,α,β ρk h k 1 R 3 j, k,α,β ϱk h k R 3 j, k 3,α,β σk h k 3, j k l 3, l 0, 1,...,1, where R j, k, α, β S j 3,k,α,β ɛj 3 S j,k,α,β ε j 3 S j 1,k,α,β ζj 3 S j, k, α, β Hence by settng A a kj, B bkj, C ckj, D dkj, 3 k, j N, 3.17 then 3.1 s equvalent to the followng matrx equaton: A γ1 B γ C γ 3 D v f All the analytcal formulae of the nonzero elements of matrces A, B, C and D can be obtaned by drect computatons usng the propertes of the Jacob polynomals for detals see, 7, 8. In the case of α, β / 0, α, β 1,, we can form explctly the LU factorzaton, that s, A γ 1 B γ C γ 3 D LU. In general, the expense of calculatng LU factorzaton of an N N dense matrx M s O N 3 operatons, and the expense of solvng Ax b, provded

9 Abstract and Appled Analyss 9 that the factorzaton s known, s O N see, 7. However, n the case of banded matrx A of bandwdth r, we need just O r N operatons to factorze and O rn operatons to solve a lnear system. In the case of γ / 0, 0, 1,, the square matrx A γ 1 B γ C γ 3 D has bandwdth of 13. So we need just O 13N operatons to solve the lnear system If r N, ths represents a very substantal savng. It s worthy to note that, for α, β / 0, the algebrac system 3.18 resultng from fully ntegrated reformulaton of 3.1 s sparse and s therefore cheaper to solve than those obtaned from the dfferentated form see 7, Theorem3.1. Moreover, the savngs n computatonal effort ncrease as the sze of the systems grow. Thus, we have demonstrated the advantage of usng the ntegrated forms over the dfferentated ones for constant coeffcents ODEs. 3.. A Quadrature JDPG Method The JDPG can be extended for ODEs wth polynomal coeffcents because of analytcal form of a product of an algebrac polynomal, and the Jacob polynomals are known. Now the formula of the Jacob coeffcents of the moments of one sngle Jacob polynomal of any degree see, Doha s x m P α,β j x m n 0 Θ m,n j P α,β j m n x m, j 0, 3.19 wth P α,β r x 0, r 1, where Θ m,n j 1 n j m n m! j m n λ Γ j m n λ Γ j α 1 Γ j β 1 Γ j m n α 1 Γ j m n β 1 Γ j λ mn j m n,j k max 0,j n j m n Γ j k λ k k n k j!γ 3j m n k λ 1 j k 1 l Γ j m n k l α 1 Γ j m l n β 1 l! j k l!γ j l α 1 Γ k l β 1 l F1 j k n, j m l n β 1; 3j m n k λ 1;. Ths formula can be used to facltate greatly the settng up of the algebrac systems to be obtaned by applyng the spectral methods for solvng dfferental equatons wth polynomal coeffcents of any order.

10 10 Abstract and Appled Analyss Let us consder the followng ntegrated form of the thrd-order dfferental equaton: 1 3 u x γ 1 x u x dx 1 γ x u x dx γ 3 x u x dx 3 f x d P α,β x, n I 1, 1, u ±1 u 1 1 0, 3.1 where γ 1 x, γ x, andγ 3 x are the varable polynomal coeffcents of the dfferental equaton. The quadrature dual-petrov-galerkn method for 3.1 s to fnd u N W N such that u N,v N,N 1 γ 1 x u N dx 1,v N,N γ x u N dx,v N,N 3 γ 3 x u N dx 3,v N f d P α,β,v N,N,N v N W N, 3. where u, v,n s the dscrete nner product of u and v assocated wth the Jacob-Gauss- Lobatto quadrature. Let us consder N 3 u N ã k φ k, a ã 0, ã 1,...,ã N 3 T, fk f, ψ k,n, f f3, f 1,..., f T, N 3.3 k 0 and usng Lemma. and formula 3.19,wecanobtan ã j 1 φ j 3,ψ,N, bj γ 1 x φ j 3 x dx 1,ψ,,N 3 c j γ x φ j 3 x dx,ψ, dj γ 3 x φ j 3 x dx 3,ψ.,N,N 3.4 And by settng à ã kj, B bkj, C c kj, D dkj, 3 k, j N, 3.5 then the lnear system of 3. becomes à B C D a f. 3.6

11 Abstract and Appled Analyss Ffth-Order Dfferental Equaton In ths secton, we consder the ffth-order dfferental equaton of the form u 5 γ 1 u 3 γ u 1 γ 3 u g x, x I, 4.1 but by consderng ts ntegrated form, namely, u x γ 1 u x dx γ 4 u x dx 4 γ 3 5 u x dx 5 f x 4 d P α,β x Frst Choce of Boundary Condtons Here, we apply the dual-petrov-galerkn approxmaton to 4. subject to the boundary condtons u ±1 u 1 ±1 u We set { } V N v S N : v ±1 v 1 ±1 v 1 0, } V N {v S N : v ±1 v 1 ±1 v 1 0 ; 4.4 then the Jacob dual-petrov-galerkn approxmaton to 4. s to fnd u N V N such that 4 u N,v γ 1 u N dx,v γ u N dx,v γ 3 u N dx 5,v f d P α,β,v v V N.,N 4.5 We consder the followng the Jacob dual-petrov-galerkn procedure for 4.1 : fndu N V N such that u N,v γ 1 u N dx,v γ 3 5 u N dx 5,v γ f 4 u N dx 4,v 4 d P α,β,v v V N. 4.6

12 1 Abstract and Appled Analyss Now, we choose the bass and the dual bass functons Φ k x and Ψ k x to be of the form Φ k x P α,β k x ɛ k P α,β k 1 x ε kp α,β k x ζ k P α,β k 3 x μ k P α,β k 4 x υ kp α,β k 5 x, k 0, 1,...,N 5, Ψ k x P α,β k x ρ k P α,β k 1 x ϱ kp α,β k x σ kp α,β k 3 x 4.7 ς k P α,β k 4 x τ kp α,β k 5 x, k 0, 1,...,N 5. It s not dffcult to show that the bass functons Φ k x V k 5 and the dual bass functons Ψ k x V k 5. We choose the coeffcents ɛ k, ε k, ζ k, μ k,and υ k such that Φ k x verfes the boundary condtons 4.3. Makng use of.7, then the boundary condtons 4.3 lead to lnear system for these coeffcents. The computaton of the exact soluton of such lnear system for the unknown coeffcents s extremely tedous by hand, and we have resorted to the symbolc computaton software mathematca verson 6, hence ɛ k, ε k, ζ k, μ k,and υ k can be unquely determned to gve ɛ k k 1 k λ k α 3β 1 k α 1 k β 1 k λ 6, ε k k 1 k λ 1 k λ 4 k α 1 k β 1 k λ 6 [ ] k 4 α 3 k α 9α 3β 3 α 1 β 16, ζ k k 1 3 k λ 1 k α 1 3 k β 1 k λ 7 [ k 4 β 3 ] k 3α 3α β 3 α 5 β 16, 4.8 μ k k 1 4 k 3α β 5 k λ 1 3 k α 1 3 k β 1 k λ 6 k λ 9, k 1 υ k 5 k λ 1 4 k α 1 3 k β 1 k λ 6. 4

13 Abstract and Appled Analyss 13 Snce the dual bass functons Ψ k x satsfy the dual boundary condtons, and makng use of.7 then the unknown coeffcents ρ k, ϱ k, σ k, ς k,and τ k are determned by usng Mathematca to gve ρ k k 1 k λ k 3α β 1 k α 1 k β 1 k λ 6, ϱ k k 1 k λ 1 k λ 4 k α 1 k β 1 k λ 6 [ k 4 β 3 ] k 3α 3α β 3 α 3 β 16, k 1 σ k 3 k λ 1 k α 1 k β 1 3 k λ [ ] k 4 α 3 k α 15α 3β 3 α 1 β 16, k 1 4 k 3β α 5 k λ 1 3 ς k k α 1 k β 1 3 k λ 6 k λ 9, k 1 τ k 5 k λ 1 4 k α 1 k β 1 3 k λ 6. 4 It s clear that 4.6 s equvalent to 4 u N, Ψ k x γ 1 u N dx, Ψ k x γ u N dx 4, Ψ k x γ 3 5 u N dx 5, Ψ k x f x 4 d P α,β x, Ψ k x, k 0, 1,...,N The constants d 0, d 1, d, d 3,andd 4 would not appear f we take k 5n 4.10, therefore we get 4 u N, Ψ k x γ 1 u N dx, Ψ k x γ u N dx 4, Ψ k x 5 γ 3 u N dx 5, Ψ k x f x, Ψ k x, k 5, 6,...,N. 4.11

14 14 Abstract and Appled Analyss If we take Φ k x and Ψ k x as defned n 4.7 and f we denote f k f, Ψ k x, f f 5,f 6,...,f N T, N 5 u N x v n Φ n x, v v 0,v 1,...,v N 5 T, n 0 p kj Φ j 5 x, Ψ k x, q kj Φ j 5 x dx, Ψ k x 4 5 s kj Φ j 5 x dx 4, Ψ k x t kj Φ j 5 x dx 5, Ψ k x,,, 4.1 then V N span{φ 0 x, Φ 1 x,...,φ N 5 x }, V N span{ψ 5 x, Ψ 6 x,...,ψ N x }, 4.13 and the nonzero elements p kj, q kj, s kj,and t kj for 5 k, j N are gven as follows: p kk υ k 5 h k, p k,k 1 μ k 4 h k υ k 4 ρ k h k 1, p k,k ζ k 3 h k μ k 3 ρ k h k 1 υ k 3 ϱ k h k, p k,k 3 ε k h k ζ k ρ k h k 1 μ k ϱ k h k υ k σ k h k 3, p k,k 4 ɛ k 1 h k ε k 1 ρ k h k 1 ζ k 1 ϱ k h k μ k 1 σ k h k 3 υ k 1 ς k h k 4, p k,k 5 h k ɛ k ρ k h k 1 ε k ϱ k h k ζ k σ k h k 3 μ k ς k h k 4 υ k τ k h k 5, 4.14 p k,k 6 ρ k h k 1 ɛ k 1 ϱ k h k ε k 1 σ k h k 3 ζ k 1 ς k h k 4 μ k 1 τ k h k 5, p k,k 7 ϱ k h k ɛ k σ k h k 3 ε k ς k h k 4 ζ k τ k h k 5, p k,k 8 σ k h k 3 ɛ k 3 ς k h k 4 ε k 3 τ k h k 5, p k,k 9 ς k h k 4 ɛ k 4 τ k h k 5, p k,k 10 τ k h k 5, q k,j R j, k, α, β hk R j, k 1,α,β ρk h k 1 R j, k,α,β ϱk h k R j, k 3,α,β σk h k 3 R j, k 4,α,β ςk h k 4 R j, k 5,α,β τk h k 5, j k l, l 0, 1,...,14,

15 Abstract and Appled Analyss 15 s k,j R 4 j, k, α, β hk R 4 j, k 1,α,β ρk h k 1 R 4 j, k,α,β ϱk h k R 4 j, k 3,α,β σk h k 3 R 4 j, k 4,α,β ςk h k 4 R 4 j, k 5,α,β τk h k 5, j k l 4, l 0, 1,...,18, t k,j R 5 j, k, α, β hk R 5 j, k 1,α,β ρk h k 1 R 5 j, k,α,β ϱk h k R 5 j, k 3,α,β σk h k 3 R 5 j, k 4,α,β ςk h k 4 R 5 j, k 5,α,β τk h k 5, j k l 5, l 0, 1,...,0, 4.15 where R j, k, α, β S j 5,k,α,β ɛj 5 S j 4,k,α,β εj 5 S j 3,k,α,β ζ j 5 S j,k,α,β μj 5 S j 1,k,α,β υj 5 S j, k, α, β Then 4.11 s equvalent to the followng matrx equaton: P γ1 Q γ S γ 3 T v f Second Choce of Boundary Condtons In ths subsecton, we consder the ffth-order dfferental equaton 4.1 wth the followng boundary condtons: u ±1 u 1 ±1 u Equaton 4.1 subject to the boundary condtons 4.18 has been consdered n 9, 30. Let us denote Z N Ẑ N { } v S N : v ±1 v 1 ±1 v 3 1 0, { } v S N : v ±1 v 1 ±1 v ; 4.19 then the Jacob dual-petrov-galerkn approxmaton of 4.1 subject to 4.18 conssts of fndng u N Z N such that 4 u N,v γ 1 u N dx,v γ u N dx,v γ 3 u N dx 5,v f d P α,β,v v Ẑ N. 4.0

16 16 Abstract and Appled Analyss We consder the followng choce of bass functons: ϕ k x P α,β k x ξ 1,k P α,β k 1 x ξ,kp α,β k x ξ 3,kP α,β k 3 x ξ 4,kP α,β k 4 x ξ 5,kP α,β k 5 x, 4.1 and dual bass functons: ϕ k x P α,β k x δ 1,k P α,β k 1 x δ,kp α,β k x δ 3,kP α,β k 3 x δ 4,kP α,β k 4 x δ 5,kP α,β k 5 x, 4. where ξ 1,k, ξ,k, ξ 3,k, ξ 4,k, ξ 5,k, δ 1,k, δ,k, δ 3,k, δ 4,k,andδ 5,k are chosen to be the unque constants such that ϕ k x Z N and ϕ k x Ẑ N,forallk 0, 1,...,N 5. Equaton 4.0 s equvalent to the followng matrx equaton: P γ 1 Q γ Ŝ γ 3 T v f, 4.3 where the elements of the matrces P, Q, Ŝ,and T can be obtaned smlarly as n the prevous sectons, but detals are not gven here. 5. Numercal Results In ths secton some examples are consdered amng to llustrate how one can apply the proposed algorthms presented n the prevous sectons. Comparsons between JDPG method and other methods proposed n 9 3 are made. Example 5.1. Consder the one-dmensonal thrd-order equaton u 3 x πu x πu 1 x 3πu x f x, x I, u ±1 ± 1, u 1 1 π 4, 5.1 wth an exact smooth soluton πx u x sn Table 1 lsts the maxmum pontwse error of u u N, usng the JDPG wth varous choces of α, β, andn. Example 5.. Consder the one-dmensonal ffth-order dfferental problem u 5 γ 1 u 3 γ u 1 γ 3 u f x, x I, u ±1 0, u 1 ±1 π, u 1, 5.3 wth the exact soluton u x x cos π/ x.

17 Abstract and Appled Analyss 17 Table 1: Maxmum pontwse error usng JDPG method for N 8, 16, 4, for Example 5.1. N α β JDPG α β JDPG Table : Maxmum pontwse error usng JDPG method for N 1, 4, 36, for Example 5.. N α β γ 1 γ γ 3 JDPG γ 1 γ γ 3 JDPG N N N N N N N N N Table lsts the maxmum pontwse error, usng the JDPG method wth varous choces of α, β, γ 1, γ, γ 3,andN. Numercal results of ths example show that the JDPG method converges exponentally. Example 5.3. Consder the followng ffth-order boundary value problem see 9 3 : u 5 x u x 15 10x e x, x 0, 1, u 0 0, u 1 0, u 1 0 1, u 1 1 e, u The analytc soluton of ths problem s u x x 1 x e x. Approxmate solutons u N x N 10, 13, 0, 6, 40 are obtaned by usng our proposed method. Table 3 exhbts a comparson between the error obtaned by usng JDPG method and the sxth-degree B-splne 31, sextc splne 3, nonpolynomal sextc splne 9, and the computatonal method n 30. The numercal results show that JDPG method s more accurate than the exstng methods.

18 18 Abstract and Appled Analyss Table 3: Maxmum pontwse error of u u N for N 15, 18, 5, 31, for Example 5.3. N 5 α β Our algorthm In 31 In 3 In 9 In Example 5.4. Consder the followng thrd-order ODE wth polynomal coeffcents: u 3 x x u x x 3 4x u 1 x 3x x u x g x, x 1, 1, 5.5 subject to u ±1 u 1 1 0, 5.6 where g s selected such that exact soluton s u x 1 x 1 x e x. Equaton 5.5 can be rearranged to take the form 1 u 3 x x u x x u x 3 xu x g x, 5.7 and accordngly ts fully ntegrated form s 1 3 u x x u x dx 1 x 3 u x dx xu x dx 3 f x d P α,β x. 5.8

19 Abstract and Appled Analyss 19 Table 4: Maxmum pontwse error usng quadrature JDPG method for N 1, 16, 0, 4, for Example 5.4. N α β Quadrature JDPG α β Quadrature JDPG Usng the quadrature dual-petrov-galerkn method descrbed n Secton 3., we evaluate the maxmum pontwse error of u u N wth varous choces of α, β, andn n Table 4. Numercal results show that there s a very good agreement between the approxmate soluton obtaned by the quadrature JDPG method and the exact soluton and at the same tme ascertan that the JDPG method converges exponentally. 6. Concludng Remarks In ths paper,wedescrbed a JDPG method forfully ntegrated forms of thrd- and ffth-order ODEs wth constant coeffcents. Because of the constant coeffcents, the matrx elements of the dscrete operators are provded explctly, and ths n turn greatly smplfes the steps and the computatonal effort for obtanng solutons. However, the ntegrated form of the source functon nvolvng severalfold ndefnte ntegrals should be known analytcally, and the rght hand sde vector requre quadrature approxmatons. Ths approach s also consdered for ODEs wth polynomal coeffcents. Numercal results exhbt the hgh accuracy of the proposed numercal methods of solutons. Acknowledgment The authors are very grateful to the referees for carefully readng the paper and for ther comments and suggestons whch have mproved the paper. References 1 J. P. Boyd, Chebyshev and Fourer Spectral Methods, Dover, Mneola, NY, USA, nd edton, 001. C. Canuto, M. Y. Hussan, A. Quarteron, and T. A. Zang, Spectral Methods n Flud Dynamcs,Sprnger Seres n Computatonal Physcs, Sprnger, New York, NY, USA, Y. Chen and T. Tang, Convergence analyss of the Jacob spectral-collocaton methods for Volterra ntegral equatons wth a weakly sngular kernel, Mathematcs of Computaton, vol. 79, no. 69, pp , E. H. Doha and A. H. Bhrawy, Effcent spectral-galerkn algorthms for drect soluton for secondorder dfferental equatons usng Jacob polynomals, Numercal Algorthms, vol. 4, no., pp , P. W. Lvermore and G. R. Ierley, Quas- norm orthogonal Galerkn expansons n sums of Jacob polynomals. Orthogonal expansons, Numercal Algorthms, vol. 54, no. 4, pp , K. Aghgh, M. Masjed-Jame, and M. Dehghan, A survey on thrd and fourth knd of Chebyshev polynomals and ther applcatons, Appled Mathematcs and Computaton, vol. 199, no. 1, pp. 1, 008.

20 0 Abstract and Appled Analyss 7 E. H. Doha and A. H. Bhrawy, Effcent spectral-galerkn algorthms for drect soluton of the ntegrated forms of second-order equatons usng ultrasphercal polynomals, The ANZIAM Journal, vol. 48, no. 3, pp , M. Fernandno, C. A. Dorao, and H. A. Jakobsen, Jacob galerkn spectral method for cylndrcal and sphercal geometres, Chemcal Engneerng Scence, vol. 6, no. 3, pp , W. Henrchs, Spectral approxmaton of thrd-order problems, Journal of Scentfc Computng, vol. 14, no. 3, pp , P. W. Lvermore, Galerkn orthogonal polynomals, Journal of Computatonal Physcs, vol. 9, no. 6, pp , A. R. Aftabzadeh, C. P. Gupta, and J.-M. Xu, Exstence and unqueness theorems for three-pont boundary value problems, SIAM Journal on Mathematcal Analyss, vol. 0, no. 3, pp , A. R. Aftabzadeh and K. Demlng, A three-pont boundary value problem, Dfferental and Integral Equatons, vol. 4, no. 1, pp , F. Berns and L. A. Peleter, Two problems from dranng flows nvolvng thrd-order ordnary dfferental equatons, SIAM Journal on Mathematcal Analyss, vol. 7, no., pp , A. Boucherf, S. M. Bouguma, N. Al-Malk, and Z. Benbouzane, Thrd order dfferental equatons wth ntegral boundary condtons, Nonlnear Analyss: Theory, Methods & Applcatons, vol. 71, no. 1, pp. e1736 e1743, A. R. Daves, A. Karageorghs, and T. N. Phllps, Spectral Glarken methods for the prmary twopont boundary-value problems n modelng vscoelastc flows, Internat, Internatonal Journal for Numercal Methods n Engneerng, vol. 6, pp , A. Karageorghs, T. N. Phllps, and A. R. Daves, Spectral collocaton methods for the prmary twopont boundary-value problems n modelng vscoelastc flows, Internatonal Journal for Numercal Methods n Engneerng, vol. 6, no. 4, pp , G. L. Lu, New research drectons n sngular perturbaton theory: artfcal parameter approach and nverse-perturbaton technque, n Proceedngs of the 7th Conference on Modern Mathematcs and Mechancs, Shangha, Chna, R. P. Agarwal, Boundary Value Problems for Hgher Order Dfferental Equatons, World Scentfc, Teaneck, NJ, USA, E. H. Doha, Explct formulae for the coeffcents of ntegrated expansons of Jacob polynomals and ther ntegrals, Integral Transforms and Specal Functons, vol. 14, no. 1, pp , E. H. Doha and A. H. Bhrawy, A Jacob spectral Galerkn method for the ntegrated forms of fourthorder ellptc dfferental equatons, Numercal Methods for Partal Dfferental Equatons, vol. 5, no. 3, pp , B.-Y. Guo and L.-l. Wang, Jacob nterpolaton approxmatons and ther applcatons to sngular dfferental equatons, Advances n Computatonal Mathematcs, vol. 14, no. 3, pp. 7 76, 001. E. H. Doha, On the constructon of recurrence relatons for the expanson and connecton coeffcents n seres of Jacob polynomals, Journal of Physcs A, vol. 37, no. 3, pp , E. H. Doha and A. H. Bhrawy, Effcent spectral-galerkn algorthms for drect soluton of fourthorder dfferental equatons usng Jacob polynomals, Appled Numercal Mathematcs, vol.58,no.8, pp , B.-Y. Guo and L.-l. Wang, Jacob approxmatons n non-unformly Jacob-weghted Sobolev spaces, Journal of Approxmaton Theory, vol. 18, no. 1, pp. 1 41, G. N. Watson, A note on generalzed hypergeometrc seres, Proceedngs London Mathematcal Socety, vol. 3, pp , Y. L. Luke, Mathematcal Functons and Ther Approxmatons, Academc Press, New York, NY, USA, E. H. Doha, A. H. Bhrawy, and R. M. Hafez, A Jacob-Jacob dual-petrov-galerkn method for thrd- and ffth-order dfferental equatons, Mathematcal and Computer Modellng, vol. 53, no. 9-10, pp , A. H. Bhrawy and S. I. El-Soubhy, Jacob spectral Galerkn method for the ntegrated forms of second-order dfferental equatons, Appled Mathematcs and Computaton, vol. 17, no. 6, pp , S. S. Sddq, G. Akram, and S. A. Malk, Nonpolynomal sextc splne method for the soluton along wth convergence of lnear specal case ffth-order two-pont value problems, Appled Mathematcs and Computaton, vol. 190, no. 1, pp , 007.

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