New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient.

Journal of Advanced Research (2015) 6, 673–686 Cairo University Journal of Advanced Research ORIGINAL ARTICLE New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method E.H Doha a, W.M Abd-Elhameed a b a,b , Y.H Youssri a,* Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia A R T I C L E I N F O Article history: Received 19 September 2013 Received in revised form March 2014 Accepted 10 March 2014 Available online 17 March 2014 Keywords: Dual-Petrov–Galerkin method Generalized Jacobi polynomials Nonhomogeneous Dirichlet conditions Convergence analysis A B S T R A C T Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University Introduction Techniques for finding approximate solutions for differential equations, based on classical orthogonal polynomials are popularly known as spectral methods The term ‘‘spectral’’ was probably originated from the fact that the trigonometric functions fei k x g are the eigenfunctions of the Laplace * Corresponding author Tel.: +20 1001875669; fax: +23 35676509 E-mail address: youssri@sci.cu.edu.eg (Y.H Youssri) Peer review under responsibility of Cairo University Production and hosting by Elsevier operator with the periodic boundary conditions This fact and the availability of Fast Fourier Transform (FFT) are the main advantages of the Fourier spectral method Thus, using Fourier series to solve partial differential equations, with principal differential operator being the Laplace operator (or its power) with periodic boundary conditions, results in very alternative numerical algorithms [1–6] The spectral methods aim to approximate functions (solutions of differential equations) by means of truncated series of orthogonal polynomials There are three well-known methods of spectral methods, namely, tau, collocation and Galerkin methods [2,4] The choice of the suitable spectral method suggested for solving the given equation depends certainly on the type of the differential equation and the type of the boundary conditions governed by it The spectral 2090-1232 ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University http://dx.doi.org/10.1016/j.jare.2014.03.003 674 methods take various orthogonal polynomials as trial functions The use of the Chebyshev, Legendre, ultraspherical and classical Jacobi polynomials is suitable for non-periodic problems, while the use of Laguerre and Hermite polynomials is suitable for handling the problems defined respectively on the half line, and on the whole line [4,6,7] Standard spectral methods are capable of providing very accurate approximations to well-behaved smooth functions with significantly less degrees of freedom when compared with finite difference or finite element methods [1,2,4,8] Classical orthogonal polynomials are used successfully and extensively for the numerical solution of differential equations in spectral and pseudospectral methods [2,8–13] The classical Jacobi polynomials Pða;bÞ ðxÞ have great imporn tance in analysis from both theoretical and practical points of view [14] The six special polynomials of the classical Jacobi polynomials namely, ultraspherical, Chebyshev polynomials of the four kinds, and Legendre polynomials have been extensively employed in numerical analysis and in particular in spectral methods [15] It is known that the Jacobi polynomials are precisely the only polynomials arising as eigenfunctions of a singular Sturm–Liouville problem ([2, Section 9.2]) The construction of the generalized Jacobi polynomials was first introduced by Guo et al [16] They extended the definition of the classical Jacobi polynomials Pða;bÞ ðxÞ to allow their n parameters a; b to take negative integers In Guo et al [16], it has been shown that the generalized Jacobi polynomials, with parameters corresponding to the number of boundary conditions in a given differential equation, are considered as the natural basis functions for the spectral approximation of this problem Moreover, it has been shown that the use of generalized Jacobi polynomials simplifies the numerical analysis for the spectral approximations of boundary value problems (BVPs) and also leads to very efficient numerical algorithms Recently, Abd-Elhameed et al [17] and Doha et al [18] have analyzed in detail some numerical algorithms for solving the differentiated and integrated forms of third and fifth-order boundary value problems based on the application of the spectral method namely Petrov–Galerkin method In these two articles, the authors have employed two new families of general parameters generalized Jacobi polynomials A large number of books and research articles dealing with the theory of ordinary differential equations, or their practical applications in various fields, contain mainly results from the theory of second-order linear differential equations, and some results from the theory of some special linear differential equations of higher even order However, there are few studies for handling third- and fifth- order BVPs This is due to that the application of the collocation method on suck kinds of BVPs leads to high condition numbers More precisely, it leads to condition numbers of order N6 for third-order BVPs and N10 for fifth-order BVPs, respectively, where N is the number of retained modes These high condition numbers will lead to instabilities caused by rounding errors [9,19,20] In this paper, we introduce some efficient spectral algorithms for reducing these condition numbers to be of OðN2 Þ and OðN4 Þ for third- and fifth-order BVPs, respectively, based on certain nonsymmetric generalized Jacobi Petrov–Galerkin method The study of odd-order equations is of mathematical and physical interest As an example, third-order equation contains a type of operator which appears in many physical applications such as the Kortweg–de Vries equation The oscillation E.H Doha et al properties of third-order differential equations can be found in the monographs of Mckelvey [21] For more applications of odd-order differential equations, see the monograph by Gregus [22], in which many physical and engineering applications of third-order differential equations are discussed [22] In the sequence of papers of Abd-Elhameed [23], Doha and Abd-Elhameed [24,25], Doha and Bhrawy [26] and Doha et al [27], the authors handled second-, fourth-, 2nth- and 2n ỵ 1ịth-order two point boundary value problems In these articles, they suggested some numerical algorithms based on constructing combinations of various orthogonal polynomials together with the application of the Galerkin method Recently, Doha and Abd-Elhameed [28] have introduced and used a family of orthogonal polynomials called ‘‘symmetric generalized Jacobi polynomials’’ for handling multidimensional sixth-order two point boundary value problems by the Galerkin method For other studies on third- and fifth-order BVPs, one can be referred for example to [29,30] Since, the main differential operator in odd-order differential equations is not symmetric, it is convenient to use a Petrov–Galerkin method The main difference between the two spectral mthods namely, Galerkin and Petrov–Galerkin methods, is that in case of Galerkin method, the test functions coincide with the trial functions, while in Petrov–Galerkin method, the trial and test functions are chosen in a way such that they satisfy respectively, the boundary conditions and their dual conditions of the differential equation The main objective in this article is to introduce new algorithms for handling third- and fifth-order BVPs, based on applying the nonsymmetric generalized Jacobi Petrov–Galerkin method (GJPGM) The linear systems resulted from the application of GJPGM are band and hence they can be efficiently inverted The structure of the paper is as follows In ‘‘Some peoperties of classical and generalized Jacobi polynomials’’ Section, some properties of classical and generalized Jacobi polynomials are given In ‘‘Dual Petrov-Galerkin algorithms for third-order elliptic linear differential equations’’ and ‘‘Dual Petrov-Galerkin algorithms for fifth-order elliptic linear differential equations’’ Sections, GJPGM is applied for the sake of solving third- and fifth-order linear BVPs with constant coefficients governed by homogenous boundary conditions In ‘‘Structure of the coefficients matrices in the linear systems (23) and (32)’’ Section, the linear systems resulting from the application of GJPGM are investigated In ‘‘Condition number of the resulting matrices’’ Section, we discuss the condition numbers of the obtained systems In ‘‘Convergence analysis’’ Section, we state and prove two theorems for the convergence of the proposed algorithms In ‘‘Numerical results’’ Section, some numerical results accompanied by some comparisons with the other available algorithms appeared in literature are given Conclusions are given in ‘‘Concluding remarks’’ Section Some properties of classical and generalized Jacobi polynomials Classical Jacobi polynomials The classical Jacobi polynomials associated with the real parameters ða > À1; b > À1Þ [14,31,32], are a sequence of Generalized Jacobi Solutions for odd-order BVPs 675 polynomials Pa;bị xị; x 1; 1ị; n ẳ 0; 1; 2; Þ, each n respectively of degree n Now, and for the sake of simplicity in the upcoming computations, it is useful to define the following normalized classical Jacobi polynomials by Ra;bị xị ẳ n a;bị xị a;bị 1ị Pn Pn n!Caỵ1ị This means that, Ra;bị xị ẳ Cnỵaỵ1ị Pa;bị xị In such n n a1;a1ị aị case Rn 2 xị ẳ Caị n xị, where Cn ðxÞ is the ultraspherical ða;bÞ polynomial Moreover, Rn ðxÞ may be generated with the aid of the following three term recurrence relation: h i 1n Caỵ12ị 1 2a n aỵnỵ2 ; Caị ị D ð1 À x Þ ð1 À x n ðxÞ ẳ Cnỵaỵ2ị Caị n 1ị ẳ 1; n ¼ 0; 1; 2; : This definition has the desirable properties that Cð0Þ n ðxÞ is identical with the Chebyshev polynomials of the first kind Tn ðxÞ, Cðn2Þ ðxÞ is the Legendre polynomials Pn ðxÞ, and C1ị n xị is equal to nỵ1 Un xị is the Chebyshev polynomials of the second kind (see, [33]) Now, the following theorem is useful in what follows ða;bÞ 2n ỵ kịn ỵ a ỵ 1ị2n ỵ k 1ịRnỵ1 xị xị ỵ a2 b2 ị2n ỵ kịRa;bị xị ẳ 2n ỵ k 1ị3 xRa;bị n n 2nn ỵ bị2n ỵ k ỵ a;bị 1ịRn1 xị; n ẳ 1; 2; ; a;bị R0 xị a;bị starting from ẳ1 and R1 xị ẳ 2aỵ1ị ẵa b ỵ k ỵ 1ịx, or obtained from Rodrigues formula  n Ca ỵ 1ị xịa ỵ xịb Dn Ra;bị xị ẳ n Cn ỵ a ỵ 1ị ẵ1 xịaỵn ỵ xịbỵn ; where k ẳ a ỵ b ỵ 1; zịk ẳ Cz ỵ kị ; Czị D d ; dx The orthogonality relation of Rða;bÞ ðxÞ is n & Z 0; mn; a;bị ỵ xịa ỵ xịb Ra;bị xịR xịdx ẳ m n ha;b ; m ẳ n; n 1ị 2k n! Cn ỵ b ỵ 1ịCa ỵ 1ịị2 ẳ : 2n ỵ kịCn ỵ kịCn ỵ a ỵ 1ị The polynomials Ra;bị ðxÞ are eigenfunctions of the singular n Sturm–Liouville equation: ð1 x2 ị/00 xị ỵ ẵb a k ỵ 1ịx/0 xị ỵ nn ỵ kị/xị ẳ 0: The following relations are useful in the sequel i h a;bị a;b1ị a1;bị Rk xị ẳ k ỵ a ỵ 1ịRkỵ1 xị aRkỵ1 xị ; kỵ1 h i a;bị a;b1ị a1;bị Rk xị ẳ k ỵ bịRk xị þ aRk ðxÞ ; kþaþb h i 2ða þ 1Þ aỵ1;bị a;bị a;bị xịRk xị ẳ Rk xị Rkỵ1 xị ; 2k ỵ a ỵ b ỵ aỵ1;bỵ1ị x2 ị Rk1 a;bị Dq R k xị ẳ xị ẳ 2ị 3ị 4ị 4a ỵ 1ị 2k ỵ k 1ị3 h a;bị k þ bÞð2k þ k þ 1ÞRkÀ1 ðxÞ  ða;bÞ À k ỵ a ỵ 1ị2k ỵ k 1ịRkỵ1 xị  i a;bị ỵ a bị2k ỵ kịRk xị ; 5ị k q ỵ 1ịq k ỵ kịq aỵq;bỵqị Rkq xị: 2q a ỵ 1ịq iẳ0 where Cnq;i a ỵ q; b ỵ q; a; bị n ỵ q ỵ kịi i ỵ q ỵ a ỵ 1ịniq Ci ỵ kị ẳ n i qị! C2i þ kÞi!ði þ a þ 1ÞnÀi   Àn þ q ỵ i; n ỵ i ỵ q ỵ k; i ỵ a ỵ F2 ;1 : i þ q þ a þ 1; 2i þ k þ (For the proof of Theorem 1, Doha [34]) where ha;b n Theorem The qth derivative of the normalized Jacobi polynomial Rða;bÞ ðxÞ is given explicitly by n nÀq X a;bị Dq Ra;bị xị ẳ n ỵ kịq 2q n! Cnq;i a ỵ q; b ỵ q; a; bịRi ðxÞ; n Nonsymmetric generalized Jacobi polynomials Following [16], a family of generalized Jacobi polynomials/ functions with indexes a, b R can be dened Let wa;b xị ẳ xịa ỵ xịb We denote by L2wa;b 1; 1Þ the weighted L2 space with inner product: Z ðu; vịwa;b xị :ẳ uxịvxịwa;b xịdx; I and the associated norm jjujjwa;b ẳ u; uị2wa;b Now, we aim to define Jacobi polynomials with parameters a and/or b À1, which will be called ‘‘nonsymmetric generalized Jacobi polynomials (GJPs)’’ These polynomials will satisfy some selected properties that are essentially relevant to spectral approximations In this work, the values of the two parameters a and b are restricted to take negative integers Now, and if we assume that ‘; m are two integers, then we can define GJPS by ðÀ‘;ÀmÞ xị ỵ xịm Rkk0 xị; k0 ẳ ỵ mị;;m 1; > > > > > ;mị < xị Rkk0 xị; k0 ẳ ; 1;m > 1; ;mị Jk xị ẳ m ;mị > > k0 ¼ Àm;‘ > À1;m À1; ð1 þ xÞ RkÀk0 ðxÞ; > > > : ð‘;mÞ RkÀk0 xị; k0 ẳ 0;;m > 1: It should be noted here that the GJPs have the characterization that for ‘; m Z and ‘; m P 1, ðÀ‘;ÀmÞ ð6Þ ðaÀ1;aÀ1Þ Note It is worth noting that Rn 2 ðxÞ is identical to the ultraspherical polynomials, CðaÞ n xị, which is explicitly dened by Di Jk 1ị ẳ 0; ;mị 1ị Dj Jk ẳ 0; i ẳ 0; 1; ; ‘ À 1; j ¼ 0; 1; ; m À 1: It is not difficult to verify that 2k À 2;1ị Jk xị ẳ Lk3 xị Lk2 xị k À 1Þð2k À 3Þ 2k À ! 2k À Lk ðxÞ ; k P 3; À LkÀ1 ðxÞ þ 2k À 676 ðÀ1;À2Þ Jk ðÀ3;À2Þ Jk ðÀ2;À3Þ Jk E.H Doha et al xị ẳ D3 uN xị; vðxÞÞ À a1 ðD2 uN ðxÞ; vðxÞÞ À b1 ðDuN xị; vxịị ỵ c1 uN xị; vxịị ẳ fxị; vxịị; 8v VÃN : 2k À LkÀ3 ðxÞ þ LkÀ2 ðxÞ 2k À 2k À ! 2k À Lk ðxÞ ; k P 3; À LkÀ1 ðxÞ À 2k À The choice of basis functions 24 ð2k À 7Þ LkÀ5 ðxÞ À LkÀ4 ðxÞ ð2k À 5Þð2k À 7Þðk À 2Þ 2k À 22k 5ị 22k 7ị Lk3 xị ỵ LkÀ2 ðxÞ 2k À 2k À ! 2k À ð2k À 5Þð2k À 7Þ LkÀ1 ðxÞ À Lk xị ; k P 5; ỵ 2k 2k 1ị2k 3ị xị ẳ xị ẳ ỵ We can choose suitable basis functions and their dual basis by setting 2;1ị 2;1ị xị ẳ x2 ị1 xịRk xị; 1;2ị 1;2ị wk xị ẳ Jkỵ3 xị ẳ x2 ị1 ỵ xịRk xị; uk xị ẳ Jkỵ3 k ẳ 0;1; ;N 3; k ẳ 0;1; ;N À 3: It is worthy noting here that the basis fuk xịg are orthogonal on ẵ1; in the sense that & Z 0; k–j; uj ðxÞuk xị dx ẳ ; k ẳ j: h2;1 1 xị ỵ xị k 2k Lk5 xị ỵ Lk4 xị 2k 5ị2k 7Þ 2k À À ð9Þ 2ð2k À 5Þ 2ð2k À 7Þ LkÀ3 ðxÞ À LkÀ2 ðxÞ 2k À 2k À ! 2k À ð2k À 5Þð2k 7ị Lk1 xị ỵ Lk xị ; k P 5; 2k À ð2k À 1Þð2k À 3Þ where Lk ðxÞ is the Legendre polynomial of the kth degree ðÀ‘;ÀmÞ fJk ðxÞg are natural candidates as basis functions for PDFs with the following boundary conditions: Di u1ị ẳ ; i ¼ 0; 1; ; ‘ 1; Dj u1ị ẳ bj ; j ẳ 0; 1; ; m À 1: Dual Petrov–Galerkin algorithm for third-order elliptic linear differential equations The idea behind this choice is to use trial and test functions to guarantee the satisfaction of the underlying boundary and dual boundary conditions of the third-order differential equations under investigation In contrast to other bases [1,2,24], these choices lead to linear systems with specially structured matrices that are well-conditioned, i.e have bounded condition numbers and therefore, can be efficiently inverted These and other items will be discussed in the section entitled ‘‘Condition numbers of the resulting matrices’’ It is clear that the two sets of orthogonal polynomials fuk ðxÞg and fwk ðxÞg are linearly independent, and therefore we have VN ¼ spanfuk xị : k ẳ 0; 1; 2; ; N À 3g; This section is concerned with using GJPGM for solving the following third-order elliptic linear differential equation uð3Þ ðxÞ À a1 uð2Þ ðxÞ À b1 uð1Þ ðxÞ ỵ c1 uxị ẳ fxị; x 1; 1ị; and VN ẳ spanfwk xị : k ẳ 0; 1; 2; ; N À 3g: ð7Þ Now the following two important lemmas will be stated and proved ð8Þ Lemma governed by the homogeneous boundary conditions uặ1ị ẳ u1ị 1ị ẳ 0: We dene the space 2ị 2;1ị D3 Jkỵ3 1;2ị xị ẳ 2k ỵ 1ịk ỵ 3ịRk xị: 10ị 1ị V ẳ fu H Iị : uặ1ị ẳ u 1ị ẳ 0g; and its dual space V ẳ fu H2ị Iị : uặ1ị ẳ u1ị 1ị ẳ 0g; Proof By using Leibnitzs rule, we have 2;1ị D3 Jkỵ3 2;1ị xị ẳ x2 ị1 xịD3 Rk where ỵ 33x H2ị Iị ẳ fu : kuk2;wa;b < 1g; !12 X k kdx ukwaỵk;bỵk : kuk2;wa;b ẳ ỵ 63x kẳ0 Let PN be the space of all polynomials of degree less than or equal to N Setting VN ¼ V \ PN and VÃN ¼ Và \ PN We observe that: n o 2;1ị 2;1ị 2;1ị VN ẳ span J3 xị; J4 ðxÞ; ; JN ðxÞ ; n o ðÀ1;À2Þ ðÀ1;À2Þ ðÀ1;À2Þ ðxÞ; J4 ðxÞ; ; JN xị : VN ẳ span J3 The dual PetrovGalerkin approximation of (7) and (8) is to find uN VN such that ðxÞ ð2;1Þ À 2x À 1ÞD Rk xị 2;1ị 2;1ị 1ịDRk xị ỵ 6Rk xị: 2 Making use of the relation ð2;1Þ ð1 À x2 ị1 xịD3 Rk 2;1ị xị ẳ ỵ 6x 7x2 ịD2 Rk xị 2;1ị ỵ k 1ịk þ 5Þðx À 1ÞDRk ðxÞ; we obtain ðÀ2;À1Þ D3 Jkþ3 2;1ị xị ẳ 2x2 1ịD2 Rk ỵ 63x xị ỵ ẵk 1ịk ỵ 5ịx 1ị 2;1ị 1ịDRk xị 2;1ị ỵ 6Rk xị; which in turn with Eq (2), and after some rather lengthy manipulation, yields Generalized Jacobi Solutions for odd-order BVPs 677 h i ðÀ2;À1Þ ð2;1Þ 2;1ị D3 Jkỵ3 xị ẳ k ỵ 1ịk ỵ 3ị ð1 À xÞDRk ðxÞ À 2Rk ðxÞ : " # N3 N2 X X 2;1ị 1;2ị D ak Jkỵ3 xị ẳ ek;2 Rk xị; kẳ0 Making use of the two relations (4) and (6), we have 2;1ị D3 Jkỵ3 then h i 3;2ị 2;1ị xị ẳ k ỵ 1ịk ỵ 3ị kk ỵ 4ịx 1ịRk1 xị ỵ 12Rk ðxÞ : Finally, and in virtue of (2) and (3), and after some manipulation, we get ð2Þ ð1;2Þ xị ẳ 2k ỵ 1ịk ỵ 3ịRk xị: By recalling the denition of Pochhammers symbol, , we have zịn ẳ Czỵnị Czị 2ị 14ị where 2ị ak ẳ 2k ỵ 3ị2 ; 2k ỵ 5ị ck ẳ 2ị ek;2 ẳ ak1 ak1 ỵ ak bk ỵ akỵ1 ckỵ1 ; 2ị 2;1ị D3 Jkỵ3 13ị kẳ0 2ị bk ẳ k ỵ 1ịk ỵ 3ị ; k þ 32 2ðkÞ2 : ð2k þ 3Þ Also, if " # NÀ3 NÀ1 X X ðÀ2;À1Þ ð1;2Þ D ak Jkỵ3 xị ẳ ek;1 Rk xị; kẳ0 Lemma 15ị kẳ0 then 2;1ị D2 Jkỵ3 xị 2k ỵ 3ị2 1;2ị k ỵ 1ịk ỵ 3ị 1;2ị ẳ R xị Rk xị 2k ỵ 5ị kỵ1 k ỵ 32 2;1ị DJkỵ3 xị ẳ 1ị 1ị 1ị 1ị ek;1 ẳ ak2 ak2 ỵ ak1 bk1 ỵ ak ck ỵ akỵ1 dkỵ1 1ị ỵ akỵ2 lkỵ2 ; 2kị2 1;2ị R xị; 2k ỵ 3ị k1 16ị where k þ 3Þ3 ðk þ 3Þ2 ð1;2Þ À Á Rð1;2Þ ðxÞ R xị 2k ỵ 2ị k ỵ 52 kỵ2 k ỵ 32 kỵ1 1ị ak ẳ k ỵ 3ị3 ; 2k ỵ 2ị k ỵ 52 1ị k ỵ 1ịk ỵ 3Þ ð1;2Þ ðkÞ2 À Á Á Rð1;2Þ ðxÞ Rk ðxÞ þ À k þ 32 k þ 12 k1 ck ẳ ỵ k 1ị3 R1;2ị xị; 2k ỵ 2ị k ỵ 12 k2 lk ẳ 1ị k ỵ 3ị2 1ị ; bk ẳ k ỵ 32 k ỵ 1ịk þ 3Þ À Á ; k þ 32 ðkÞ2 1ị ; dk ẳ k ỵ 12 k 1ị3 : 2k ỵ 2ị k þ 12 Finally, if ðÀ2;À1Þ Jkþ3 ðxÞ ðk þ 4ị3 R1;2ị xị ẳ 4k ỵ 2ị k þ 52 kþ3 NÀ3 N X X ðÀ2;À1Þ ð1;2Þ ak Jkỵ3 xị ẳ ek;0 Rk xị; 3k ỵ 3ị3 3k ỵ 3ị2 1;2ị R1;2ị xị R xị 4k ỵ 2ị k ỵ 32 kỵ2 k ỵ 32 kỵ1 ỵ 3k þ 1Þðk þ 3Þ ð1;2Þ 3ðkÞ2 À Á Á Rð1;2Þ xị Rk xị ỵ k ỵ 12 4 k ỵ 12 k1 3k 1ị3 R1;2ị xị 4k ỵ 2ị k 12 k2 0ị 0ị 0ị ỵ akỵ2 gkỵ2 ỵ akỵ3 fkỵ3 ; k ỵ 4ị3 ; 4k ỵ 2ị k ỵ 52 3kị2 0ị ; lk ẳ k ỵ 12 0ị fk ẳ 0ị 18ị 0ị bk ẳ 0ị dk ẳ 0ị gk ẳ 3k ỵ 3ị3 ; 4k ỵ 2ị k ỵ 32 3k ỵ 1ịk ỵ 3ị ; k ỵ 12 3k 1ị3 ; 4k ỵ 2ị k À 12 ðk À 2Þ3 À Á : 4k ỵ 2ị k 12 Now, the application of PetrovGalerkin method on Eq (7), yields 11ị ẳ fxị; wk xịị; 19ị where kẳ0 where Moreover, if 0ị D3 uN xị a1 D2 uN b1 DuN ỵ c1 uN ; wk xịị # bk ẳ 2k ỵ 1ịk ỵ 3ịak : 0ị 3k ỵ 3ị2 0ị ; ck ẳ k ỵ 32 Theorem For arbitrary constants ak , one has k¼0 0ị ek;0 ẳ ak3 ak3 ỵ ak2 bk2 ỵ ak1 ck1 ỵ ak dk ỵ akỵ1 lkỵ1 0ị Now, based on the two Lemmas and 2, the following theorem can be obtained N3 X X 2;1ị 1;2ị ak Jkỵ3 xị ẳ N 3bk Rk xị; then ak ẳ Proof The proof of Lemma is rather lengthy and it can be accomplished by following the same procedure used in the proof of Lemma h D3 17ị kẳ0 where ðk À 2Þ3 À Á Rð1;2Þ ðxÞ: À 4ðk ỵ 2ị k 12 k3 " kẳ0 12ị uN xị ẳ N3 X ak /k xị; 2;1ị /k xị ẳ Jkỵ3 xị; kẳ0 1;2ị wk xị ẳ Jkỵ3 xị; k ẳ 0; 1; ; N À 3: 678 E.H Doha et al Substitution of formulae (11), (13), (15) and (17) into (19) yields NÀ3 NÀ2 NÀ1 X X X ð1;2Þ ð1;2Þ ð1;2Þ bj Rj ðxÞ À a1 ej;2 Rj ðxÞ À b1 ej;1 Rj ðxÞ jẳ0 jẳ0 ỵ c1 N X 1;2ị 1;2ị ej;0 Rj xị; Jkỵ3 xị ! In this section we aim to apply the GJPGM for solving the following fifth-order elliptic linear equation jẳ0   1;2ị ẳ f; Jkỵ3 xị ; 20ị jẳ0 jẳ0 jẳ0 N X 1;2ị ej;0 Rj ! 1;2ị xị; Rk xị jẳ0   1;2ị ẳ f; Rk xị ; w where w ẳ x ị1 ỵ xị Making use of the orthogonality relation (1), it is not difficult to show that Eq (20) is equivalent to ð21Þ w This linear system may be put in the form À Á bk À a1 ek;2 b1 ek;1 ỵ c1 ek;0 ẳ fk ; k ¼ 0; 1; N À 3; ð22Þ ; k ỵ 1ịk ỵ 2ịk ỵ 3ị 23ị À ÁT fà ¼ fÃ0 ; fÃ1 ; ; fÃNÀ3 ; and the nonzero elements of the matrices B1 ; E2 ; E1 and E0 are given explicitly in the following theorem Theorem The nonzero elements of the matrices B1 ẳ b1kj ị and Ei ẳ ðeikj Þ; i 2, for k; j N À 3, are given as follows: ẳ 2k ỵ 1ịk ỵ 3ị; ẳ 2kỵ3ịkỵ4ị ; 2kỵ5 e2kk ẳ e2kỵ1;k ẳ 2kỵ1ịkỵ2ị ; 2kỵ5 4kỵ1ịkỵ3ị ; 2kỵ3ị2kỵ5ị 4kỵ1ịkỵ3ị e1kk ẳ 2kỵ3ị2kỵ5ị ; 8kỵ1ịkỵ2ị e1k;kỵ1 ẳ 2kỵ3ị2kỵ5ị2kỵ7ị ; e1k;kỵ2 ẳ 2kỵ1ịkỵ2ịkỵ3ị ; kỵ4ị2kỵ5ị2kỵ7ị 8kỵ3ịkỵ4ị e1kỵ1;k ẳ 2kỵ3ị2kỵ5ị2kỵ7ị ; e1kỵ2;k ẳ 2kỵ3ịkỵ4ịkỵ5ị ; kỵ2ị2kỵ5ị2kỵ7ị e0kk ẳ 3kỵ1ịkỵ3ị ; 2kỵ12ị e0k;kỵ1 ẳ 43kỵ1ị ; kỵ32ị3 e0k;kỵ2 ẳ 4kỵ4ị kỵ33 ; ị4 3kỵ1ị kỵ1ị3 kỵ52ị e0k;kỵ3 ẳ 4kỵ5ị 3kỵ3ị5 kỵ32ị e0kỵ2;k ẳ 4k ; ; e0kỵ1;k ẳ 3kỵ3ị ; 4kỵ32ị kỵ4ị e0kỵ3;k ẳ 4kỵ2ị kỵ3 : ị3 kuk3;wa;b ẳ !12 : k¼0 Now, setting VN ¼ V \ PN and VÃN ¼ Và \ PN We observe that: n o 3;2ị 3;2ị 3;2ị VN ẳ span J5 xị; J6 ðxÞ; ; JN ðxÞ ; n o 2;3ị 2;3ị 2;3ị VN ẳ span J5 xị; J6 xị; ; JN xị : ỵ b2 D3 uN ðxÞ; vðxÞÞ À c2 ðD2 uN ðxÞ; vðxÞÞ where e2k;kỵ1 H Iị ẳ fu : kuk3;wa;b < 1g; X k@ kx uk2waỵk;bỵk D5 uN xị; vxịị ỵ a2 ðD4 uN ðxÞ; vðxÞÞ which may be written simply in the matrix form b1kk where The dual Petrov–Galerkin approximation of (24) and (25) is to find uN VN such that where B1 ỵ a1 E2 ỵ b1 E1 ỵ c1 E0 ịa ẳ f ; 25ị V ẳ fu H3ị Iị : uặ1ị ẳ u1ị ặ1ị ẳ u2ị 1ị ẳ 0g; 3ị where   1;2ị fk ẳ f; Rk xị : a ẳ a0 ; a1 ; ; aNÀ3 ÞT ; governed by the homogeneous boundary conditions and w h1;2 k ẳ 24ị V ẳ fu H3ị Iị : uặ1ị ẳ u1ị ặ1ị ¼ uð2Þ ð1Þ ¼ 0g; fk ; h1;2 k x ðÀ1; 1Þ; We define the following two spaces j¼0 fk ¼ ðbk À a1 ek;2 À b1 ek;1 ỵ c1 ek;0 ịh1;2 k ; k ẳ 0; 1; N 3; ẳ fxị; uặ1ị ẳ u1ị ặ1ị ẳ u2ị 1ị ẳ 0: N3 N2 N1 X X X ð1;2Þ ð1;2Þ ð1;2Þ bj Rj ðxÞ À a1 ej;2 Rj ðxÞ À b1 ej;1 Rj ðxÞ fÃk ẳ u5ị xị ỵ a2 u4ị xị ỵ b2 u3ị xị c2 u2ị xị d2 u1ị xị ỵ l2 uðxÞ where bk and ek;2Àq ; q are given by (12), (14), (16) and (18), respectively Eq (20) is equivalent to ỵ c1 Dual PetrovGalerkin algorithm for fifth-order elliptic differential equations À d2 ðDuN ðxÞ; vxịị ỵ l2 uN xị; vxịị ẳ fxị; vxịị; 8v VÃN : ð26Þ The choice of basis functions We can choose suitable basis functions and their dual basis – in the same way as in the previous case and for the same reasons – by setting ðÀ3;À2Þ uk ðxÞ ẳ Jkỵ5 3;2ị xị ẳ x2 ị xịRk xị; k ẳ 0;1; .; N 5; 2;3ị 2;3ị wk xị ẳ Jkỵ5 xị ẳ x2 ị ỵ xịRk xị; k ¼ 0;1; .; N À 5: It is worthy noting here that the basis f/k ðxÞg are orthogonal on ½À1; 1Š in the sense that & Z 0; kj; uj xịuk xị dx ẳ 3;2 hk ; k ẳ j: 1 xị ỵ xÞ It is clear that the two sets of orthogonal polynomials f/k ðxÞg and fwk ðxÞg are linearly independent, and therefore we have VN ẳ spanfuk xị : k ẳ 0; 1; 2; ; N À 5g; and VN ẳ spanfwk xị : k ẳ 0; 1; 2; ; N À 5g: Generalized Jacobi Solutions for odd-order BVPs 679 The following two lemmas are needed and Lemma Jkỵ5 3;2ị 3;2ị D5 Jkỵ5 2;3ị xị ẳ 3k ỵ 1ịk ỵ 2ịk ỵ 4ịk ỵ 5ịRk xị ẳ ỵ xị: 3;2ị xị ẳ 12 h 2;1ị k ỵ 2ị2k ỵ 7ịRk xị 2k ỵ 5ị3 ỵ i 2;1ị 2;1ị ỵ 2k ỵ 3ịRkỵ1 xị k ỵ 4ị2k ỵ 5ịRkỵ2 xị : 3;2ị 3;4ị 3;4ị ỵ 2kị7 Rk1 xị 2k ỵ 5ịk ỵ 1Þ7 Rk i ðxÞ : Finally, from the two relations (2) and (3), and after some lengthy manipulation, we get 3;2ị D5 Jkỵ5 2;3ị xị ẳ 3k ỵ 1ịk ỵ 2ịk ỵ 4ịk ỵ 5ịRk ỵ 15k 3ị5 3k 4ị5 R2;3ị xị ỵ R2;3ị xị: 32k ỵ 3ị k 32 k4 32k þ 3Þ k À 32 kÀ5 ð31Þ h 3;4ị 2k ỵ 7ịk 1ị7 Rk2 xị 2k ỵ 5ị3 xị ẳ 45k ỵ 1ị2 k ỵ 4ị2 2;3ị 15kị3 k ỵ 4ị 2;3ị Rk1 xị Rk xị ỵ 16 k ỵ 12 16 k ỵ 12 15k 1ị4 2;3ị 15k 2Þ5 Á R ðxÞ À À Á Rð2;3Þ ðxÞ À k 12 k2 32k ỵ 3ị k À 12 kÀ3 à Making use of this relation and with the aid of the two relations (6) (for q ẳ 2) and (10), we obtain D5 Jkỵ5 15k ỵ 5ị5 15k ỵ 4ị5 R2;3ị xị þ À Á Rð2;3Þ ðxÞ 32ðk þ 3Þ k þ 52 kỵ4 32k ỵ 3ị k ỵ 52 kỵ3 15k ỵ 4ị4 2;3ị 15k ỵ 2ịk ỵ 4ị3 2;3ị R xị Rkỵ1 xị k ỵ 32 kỵ2 16 k ỵ 32 Proof Setting a ¼ 2; b ¼ in relation (5), we get ð1 À x2 ÞRk 3ðk þ 6Þ5 À Á Rð2;3Þ ðxÞ 32ðk þ 3Þ k þ 72 kþ5 Applying Petrov–Galerkin method to (24) and (25) and if we make use of the two Lemmas and 4, and after performing some lengthy manipulation, then the numerical solution of (24) and (25) can be obtained This solution is given in the following Theorem P ðÀ3;À2Þ Theorem If uN xị ẳ N5 ak Jkỵ5 xị is the Petrov– Galerkin approximation to (24) and (25), then the expansion coefficients fak : k ¼ 0; 1; ; N À 5g satisfy the matrix system xị: Lemma B2 ỵ a2 G4 ỵ b2 G3 ỵ c2 G2 ỵ d2 G1 ỵ l2 G0 ịa ẳ f ; 3k ỵ 2ịk ỵ 4ị3 2;3ị 3k ỵ 1ị2 k ỵ 4ị2 2;3ị 3;2ị D4 Jkỵ5 xị ẳ Rkỵ1 xị ỵ Rk xị 2k ỵ k ỵ 52 3kị3 k þ 4Þ ð2;3Þ þ RkÀ1 ðxÞ; ð2k þ 5Þ ð27Þ 3k ỵ 1ị2 k ỵ4ị2 2;3ị 3kị k ỵ 4ị ð2;3Þ À Á Rk ðxÞ À À 3Á RkÀ1 xị kỵ2 k ỵ 52 3k À 1Þ4 ð2;3Þ Á R ðxÞ; À À k ỵ 32 k2 3;2ị D2 Jkỵ5 xị ẳ ỵ 3;2ị DJkỵ5 xị ẳ 28ị 3k ỵ 4ị5 9k ỵ 4ị4 2;3ị R2;3ị xị ỵ R xị 8k ỵ 3ị k ỵ 72 kỵ3 k ỵ 52 kỵ2 9k ỵ 2ịk ỵ 4ị3 2;3ị 9k ỵ 1ị2 k ỵ 4ị2 2;3ị Rkỵ1 xị Rk xị k ỵ 52 k ỵ 32 9kị3 k ỵ 4ị 2;3ị 9k 1ị4 2;3ị Rk1 xị ỵ R xị k þ 12 kÀ2 k þ 32 3ðk 2ị5 R2;3ị xị; ỵ 8k ỵ 3ị k ỵ 12 k3 29ị 3k ỵ 5ị5 R2;3ị xị 16k ỵ 3ị k ỵ 72 kỵ4 b2kk ẳ rk ; rk ; g4kk ẳ k ỵ 52 g4kỵ1;k ẳ 3k ỵ 2ị5 ; k ỵ 3ị2k ỵ 7ị g3k;kỵ1 ẳ 3k þ 1Þ3 ðk þ 5Þ À Á ; k ỵ 52 g3k;kỵ2 ẳ 3k ỵ 1ị4 ; k ỵ 72 g3kỵ2;k ẳ 3k ỵ 4ị4 ; k ỵ 72 3rk ; g2kk ẳ k ỵ 32 g3kỵ1;k ẳ g2k;kỵ1 ẳ g2k;kỵ2 ẳ 9k ỵ 1ị4 ; k ỵ 52 3k ỵ 1ị3 k ỵ 5ị ; 2k ỵ 3k ỵ 2ịk þ 4Þ3 À Á ; k þ 52 9k ỵ 1ị3 k ỵ 5ị ; k ỵ 52 g2k;kỵ3 ẳ 3k ỵ 1ị5 ; 8k ỵ 6ị k ỵ 72 9k ỵ 2ịk ỵ 4ị3 2;3ị 9k ỵ 1ị2 k þ 4Þ2 ð2;3Þ À Á À Á Rkþ1 ðxÞ À Rk xị k ỵ 32 k ỵ 32 g2kỵ2;k ẳ 9k ỵ 4ị4 ; k ỵ 52 ỵ 9kị3 k ỵ 4ị 2;3ị 3k 1ị4 2;3ị Rk1 xị ỵ R xị k ỵ 12 k2 k ỵ 12 g2kỵ3;k ẳ 3k ỵ 4ị5 ; 8k ỵ 3ị k ỵ 72 g1k;kỵ1 ẳ 9k ỵ 1ị3 k ỵ 5ị ; k ỵ 32 30ị g4k;kỵ1 ẳ rk ; g3kk ẳ k ỵ 52 3k ỵ 4ị5 3k ỵ 4ị4 2;3ị R2;3ị xị ỵ R xị 4k ỵ 3ị k þ 52 kþ3 k þ 52 kþ2 3ðk À 2Þ5 3ðk À 3Þ5 ð2;3Þ À Á Rð2;3Þ xị R xị; 16k ỵ 3ịk 12ị4 k4 4k ỵ 3ị k 12 k3  and where the nonzero elements of the matrices B2 ¼   i Gi ¼ gk;j ; ð0 i 4Þ, for k; j N À 5, are given as ỵ b2k;j follows: 3kỵ 4ị4 2;3ị 3k ỵ 2ịkỵ 4ị3 2;3ị 3;2ị R xịỵ D3 Jkỵ5 xị ẳ Rkỵ1 xị k ỵ 72 kỵ2 k ỵ 52 ỵ 32ị  g2kỵ1;k ẳ 9k ỵ 2ịk ỵ 4ị3 ; k ỵ 52 3rk ; g1kk ẳ k ỵ 32 680 E.H Doha et al g1k;kỵ2 ẳ g1k;kỵ4 3k ỵ 1ị4 ; k ỵ 52 g1k;kỵ3 ẳ 3k ỵ 1ị5 ; 4k ỵ 6ị k ỵ 52 3k ỵ 1ị5 ; ẳ 16k ỵ 7ị k ỵ 72 g1kỵ1;k ẳ 9k þ 2Þðk þ 4Þ3 À Á ; k þ 32 g1kỵ3;k ẳ 3k ỵ 4ị5 ; 4k ỵ 3ị k ỵ 52 g1kỵ4;k 3k ỵ 5ị5 ; ẳ 16k ỵ 3ị k ỵ 72 g0k;kỵ1 15k ỵ 1ị3 k ỵ 5ị ẳ ; 16 k ỵ 32 g0k;kỵ2 ẳ 15k ỵ 1ị4 ; k ỵ 32 g0k;kỵ4 ẳ 15k ỵ 1ị5 ; 32k ỵ 7ị k ỵ 52 g0k;kỵ5 ẳ 3k ỵ 1ị5 ; 32k ỵ 8ị k ỵ 72 g0kỵ2;k ẳ 15k ỵ 4ị4 ; k ỵ 32 g0kỵ3;k ẳ 15k ỵ 4ị5 ; 32k ỵ 3ị k ỵ 52 g0kỵ5;k ẳ 3k ỵ 6ị5 ; 32k ỵ 3ị k ỵ 72 g1kỵ2;k ẳ 3k ỵ 4ị4 ; k ỵ 52 where fk ẳ g0k;kỵ3 ẳ 15rk ; ẳ 16 k ỵ 12 15k ỵ 1ị5 ; 32k ỵ 6ị k ỵ 52 g0kỵ1;k ẳ 15k ỵ 2ịk þ 4Þ3 À Á ; 16 k þ 32 g0kỵ4;k ẳ 15k ỵ 5ị5 ; 32k ỵ 3ị k ỵ 52 1;2ị x2 ị1 ỵ xịfxịRk xịdx P 3;2ị Corollary If uN xị ẳ N5 xị is the PetrovGalerkẳ0 ak Jkỵ5 kin approximation to problem (24) and (25), for a2 ¼ b2 ¼ c2 ¼ d2 ¼ l2 ¼ 0, then the expansion coefficients fak : k ¼ 0; 1; ; N À 5g are given explicitly by ak ẳ g0kk R1 kỵ3 fk ; 384 k ẳ 0; 1; ; N À 5; where Z 2;3ị fk ẳ x2 ị1 ỵ xịfxịRk ðxÞdx: À1 Now, each of the matrices E3Àq ð1 q 3Þ and G5Àq ð1 q 5Þ is a band matrix and the total number of nonzero diagonals upper or lower the main diagonal for each matrix is q Thus the coefficient matrices D1 and D2 are at most fourband and six-band matrices, respectively These special structures of D1 and D2 simplify greatly the solution of the two linear systems (23) and (32) These two systems can be decomposed by LU-factorization Moreover, the operations required for constructing these factorizations are of order 21ðN À 2Þ and 55ðN À 4Þ, respectively Also, the number of operations required for solving the two decomposable triangular systems are of order 13ðN À 2Þ and 21ðN À 4Þ respectively Note The total number of operations mentioned in the previous discussion includes the number of all subtractions, additions, divisions and multiplications [35] Treatment of nonhomogeneous boundary conditions where rk ẳ 3k ỵ 1ịk ỵ 2ịk ỵ 4ịk ỵ 5ị Structure of the coefficient matrices in the linear systems (23) and (32) This section is concerned with discussing the structure of the coefficient matrices B1 and E3Àq ð1 q 3Þ which appear in the linear system (23), and the coefficient matrices B2 and G5Àq ð1 q 5Þ in the linear system (32) Hence, we will discuss the structure of the two combined matrices D1 ẳ B1 ỵ a1 E2 þ b1 E1 þ c1 E0 ; and D2 ¼ B2 ỵ a2 G4 ỵ b2 G3 ỵ c2 G2 þ d2 G1 þ l2 G0 : Also, the influence of these structures on the efficiency for solving the two systems (23) and (32) will be discussed Since the two matrices B1 and B2 are diagonal, the two cases correspond to a1 ¼ b1 ¼ c1 ¼ in (23) and a2 ¼ b2 ¼ c2 ¼ d2 ¼ l2 ¼ in (32) lead to two diagonal systems The results for these two cases are summarized in the following two important corollaries P ðÀ2;À1Þ Corollary If uN ðxÞ ẳ N3 xị is the Galerkin kẳ0 ak Jkỵ3 approximation to problem (7) and (8), for a1 ¼ b1 ¼ c1 ¼ 0, then the expansion coefficients fak : k ¼ 0; 1; ; N À 3g are given explicitly by: ak ẳ kỵ2 fk ; 16 k ¼ 0; 1; ; N À 3; This section is devoted to describe the way of how third- and fifth-order BVPs governed by nonhomogeneous boundary conditions can be converted to BVPs governed by homogeneous boundary conditions Now, Let us consider the one-dimensional third-order equation uð3Þ ðxÞ a1 u2ị xị b1 u1ị xị ỵ c1 uxị ẳ fxị; x I ẳ 1; 1ị; governed by the nonhomogeneous boundary conditions: uặ1ị ẳ aặ ; u1ị 1ị ẳ a1 : 33ị Now, and if we make use of the transformation Vxị ẳ uxị ỵ a0 ỵ a1 x ỵ a2 x2 ; 34ị where a 3aỵ ỵ 2a1 ; a ỵ aỵ 2a1 a2 ¼ ; a0 ¼ a1 ¼ aÀ À aỵ ; then, the transformation (34) turns the nonhomogeneous boundary conditions (33) into the homogeneous boundary conditions: Vặ1ị ẳ V1ị 1ị ẳ 0: 35ị Generalized Jacobi Solutions for odd-order BVPs 681 Hence, it is sufficient to solve the following modied onedimensional third-order equation: 3ị 2ị 1ị f xị ẳ l2 a0 d2 a1 2c2 a2 ỵ 6ba3 þ 24a2 a4 Þ þ ðl2 a1 À 2d2 a2 6c2 a3 ỵ 24b2 a4 ịx ỵ l2 a2 À 3d2 a3 À 12c2 a4 Þx2 à V ðxÞ a1 V xị b1 V xị ỵ c1 Vxị ẳ f xị; x I ẳ 1; 1ị; þ ðl2 a3 À 4d2 a4 Þx3 þ l2 a4 x4 ỵ fxị: 36ị governed by the homogeneous boundary conditions (35), where VðxÞ is given by (34), and fà ðxÞ ẳ fxị ỵ 2a1 a2 b1 a1 ỵ c1 a0 ị ỵ 2b1 a2 ỵ c1 a0 ịx ỵ c1 a2 x2 : Now, the application of the GJPGM to the modified Eq (36), leads to the following equivalent system of equations B1 ỵ a1 E2 ỵ b1 E1 ỵ c1 E0 ịa ẳ f ; B1 ; E2 ; E1 and E0 are Á the matrices defined in Theorem 3, and fà ¼ fÃ0 ; fÃ1 ; ; fÃNÀ3 , where À2a1 a2 À b1 a1 ỵ c1 a0 ; k ẳ 0; > > > > > > < 65 2b1 a2 ỵ c1 a0 ị; k ẳ 1; fk ẳ > 10 > ca; k ¼ 2; > > > > : k P 3; fk ; R1 ð1;2Þ and fk ẳ 1 ỵ x ị1 ỵ xÞRk ðxÞfðxÞdx The same procedure can be applied to solve the following fifth-order BVP: If the GJPGM is applied to Eq (39), then the following equivalent system of equations is obtained B2 ỵ aG4 ỵ bG3 ỵ cG2 ỵ dG1 ỵ lG0 ịa ẳ f ; where B2 ; Gi ð0 i 4Þ are the matrices defined in Theorem 4, and À Á fà ¼ fÃ0 ; fÃ1 ; ; fÃNÀ5 ; l2 a0 d2 a1 2c2 a2 ỵ 6b2 a3 ỵ 24a2 a4 ; k ¼ 0; > > > > > k ¼ 1; > ðl2 a1 À 2d2 a2 6c2 a3 ỵ 24b2 a4 ị; > > < ðl a À 3d a À 12c a ị; k ẳ 2; 2 fÃk ¼ 350 > ðl a À 4d a Þ; k ¼ 3; > 33 > > > 238 > l a ; k ¼ 4; > > : 143 fk ; k P 5; R1 2;3ị and fk ẳ 1 þ x Þð1 þ xÞRk ðxÞfðxÞdx Condition numbers of the resulting matrices In such case, (37) and (38) can be turned into In the direct collocation method, the condition numbers behave like OðN6 Þ and OðN10 Þ for third- and fifth-order BVPs, respectively, (N: maximal degree of polynomials) In this article, improved condition numbers with OðN4 Þ and OðN6 Þ are obtained, respectively, for third- and fifth-order BVPs The advantage with respect to propagation of rounding errors is demonstrated For GJPGM, the resulting systems obtained for the two differential equations u3ị xị ẳ fxị and u5ị xị ẳ fxị are B1 a1 ¼ fà and B2 a2 ¼ fà , where B1 and B2 are two diagonal matrices their elements are given by b1kk and b2kk , where ÀVð5Þ ðxÞ þ a2 Vð4Þ ðxÞ þ b2 Vð3Þ ðxÞ À c2 V2ị xị d2 V1ị xị b1kk ẳ 2k ỵ 1ịk ỵ 3ị; u5ị xị ỵ a2 u4ị xị ỵ b2 uð3Þ ðxÞ À c2 uð2Þ ðxÞ À d2 uð1Þ xị ỵ l2 uxị ẳ fxị; x 1; 1ị; 37ị governed by the nonhomogeneous boundary conditions uặ1ị ẳ aặ ; u1ị ặ1ị ẳ a1ặ ; u2ị 1ị ẳ b: 38ị ỵ l2 Vxị b2kk ẳ 3k ỵ 1ịk ỵ 2ịk ỵ 4ịk ỵ 5ị: where Thus we note that the condition numbers of the matrices B1 and B2 behave like Oðk2 Þ and Oðk4 Þ for large values of k, respectively The evaluation of the condition numbers for the matrices B1 and B2 are easy because of their special structures, since B1 and B2 are diagonal matrices, so their eigenvalues are their diagonal elements In such case, the condition number can be dened as: Vxị ẳ uxị ỵ a0 þ a1 x þ a2 x2 þ a3 x3 þ a4 x4 ; Condition number of the matrix ¼ fà xị; x I ẳ 1; 1ị; governed by the homogenous boundary conditions Vặ1ị ẳ V1ị ặ1ị ẳ V2ị 1ị ẳ 0; with a0 ẳ 2a1 ỵ 8a1ỵ 2b 5a 11aỵ ; 16 a1 ẳ a1 ỵ a1ỵ ỵ 3a 3aỵ ; a2 ẳ 6a1ỵ ỵ 2b 3a ỵ 3aỵ ; a3 ẳ a1 a1ỵ a ỵ aỵ ; a4 ẳ and 4aỵ ỵ 3a 3aỵ ; 16 39ị ẳ Max eigenvalue of the matrixị : Min ðeigenvalue of the matrixÞ In Table 1, we list the values of the conditions numbers of the matrices B1 and B2 , respectively, for different values of N P3 P5 Remark If we add q¼1 E3Àq ð1 q 3ị and qẳ1 G5q q 5Þ, where the matrices E3Àq and G5Àq are the matrices their nonzero elements are given explicitly in Theorems and 4, to the matrices B1 and B2 , respectively, then we find that the P eigenvalues of the matrices D1 ẳ B1 ỵ 3qẳ1 E3q ; D2 ẳ B2 ỵ P5 q¼1 G5Àq are all real and positive Moreover, the effect of these additions does not significantly change the values of the condition numbers for the systems This means that matrices 682 E.H Doha et al Table Condition number for the matrices Bn ; n ¼ 1; n N amin amax Cond ðBn Þ Cond ðBn Þ=N2n 16 20 24 28 32 36 40 448 720 1056 1456 1920 2448 3040 74.667 120 176 242.667 320 408 506.667 2:917  10À1 3:000  10À1 3:056  10À1 3:095  10À1 3:125  10À1 3:148  10À1 3:167  10À1 16 20 24 28 32 36 40 120 112,320 310,080 695,520 1:361  106 2:416  106 3:992  106 6:234  106 936 120 5796 11,340 20137.6 33,264 51,984 1:428  10À2 1:615  10À2 1:747  10À2 1:845  10À2 1:920  10À2 1:981  10À2 2:029  10À2 Table N 16 20 24 28 32 36 40 jak j < Condition number for the matrices Dn ; n ¼ 1; Cond ðD1 Þ 55.287 88.679 129.929 179.037 236.003 300.826 373.507 Cond ðD1 Þ N2 Cond ðD2 Þ N4 Cond ðD2 Þ À1 2:159  10 2:217  10À1 2:256  10À1 2:284  10À1 2:305  10À1 2:321  10À1 2:334  10À1 À2 1:262  10 1:424  10À2 1:539  10À2 1:624  10À2 1:689  10À2 1:742  10À2 1:784  10À2 827.262 2278.4 5104.45 9980.18 17715.3 2925.4 45677.4 9L n o 2;1ị Proof Since Jkỵ3 xị : k ¼ 0; 1; 2; basis of H2ị 1; 1ị, then ak ẳ h2;1 k 2;1ị Convergence analysis Z 1 2;1ị Jkỵ3 xịuxị xị2 ỵ xị are orthogonal dx; where h2;1 k is as defined in (1) With the aid of the relation Jk B1 and B2 , which resulted from the highest derivatives of the differential equations under investigation, play the most important role in the propagation of the roundoff errors The numerical results of Table illustrate this remark 8k P 0: ; pk2 xị ẳ 2k LkÀ3 ðxÞ À LkÀ2 ðxÞ ðk À 1Þð2k À 3Þ 2k À ! 2k À Lk ðxÞ ; Lk1 xị ỵ 2k and after integration by parts two times, we get, Z k ỵ 1ịk ỵ 3ị ak ẳ Ik xịf2ị xịdx; 22k þ 3Þ À1 where In this section, we state and prove two theorems to ascertain that the nonsymmetric generalized Jacobi polynomials expansion of a function uðxÞ Hð2Þ ðÀ1; 1Þ, converges uniformly to uðxÞ For k ) 1; ak : bk and ak bk mean that limk!1 abkk ¼ and limk!1 abkk < 1, respectively The following theorem is needed in the sequel LkÀ2 ðxÞ LkÀ1 ðxÞ 3Lk xị Ik xị ẳ k 12 2k ỵ 1ị2k ỵ 5ị 2k 1ị2k ỵ 5ị ỵ 3Lkỵ3 xị Lkỵ4 xị 2k þ 5Þð2k þ 9Þ ð2k þ 5Þð2k þ 7Þ ð2k þ 3ÞLkþ5 ðxÞ À Á þ k þ 52 À Theorem (Bernstein type Inequality [36]) The well-known Legendre polynomials Lk xị; k ẳ 0; 1; 2; , satisfy the following inequality rffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi sin hLk cos hị < ; pk ẳ Theorem A function uxị ẳ xị ỵ xịfxị Hð2Þ ðÀ1; 1Þ, with jfð2Þ ðxÞj L, can be expanded as an infinite sum of nonsymmetric ogeneralized Jacobi polynomials n 2;1ị X bm;k Lkỵm2 xị; say: mẳ0 < h < p: Jkỵ3 32k ỵ 3ịLkỵ1 xị 3Lkỵ2 xị ỵ 42k ỵ 1ị k ỵ 52 2k ỵ 1ị2k ỵ 7ị xị : k ¼ 0; 1; 2; , and the series converges uni- formly to uðxÞ Explicitly, the expansion coefficients in P 2;1ị uxị ẳ xị, satisfy the following inequality: kẳ0 ak Jkỵ3 Now, making use of the substitution x ¼ cos h, yields Z p  ðk ỵ 1ịk ỵ 3ị X ak ẳ bm;k Lkỵm2 coshịf2ị coshịsinhdh ; 22k ỵ 3ị mẳ0 Therefore, we have jak j k ỵ 1ịk ỵ 3ịL X jbm;k j 22k ỵ 3ị mẳ0 Z p  jLkỵm2 cos hịj sin h dh : Generalized Jacobi Solutions for odd-order BVPs 683 PN5 k ỵ 1ịk ỵ 3ÞL X jbm;k j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jak j < pffiffiffiffiffiffi 2pð2k ỵ 3ị mẳ0 k ỵ m Z p p  sin h dh B2 ỵ a2 G4 ỵ b2 G3 ỵ c2 G2 ỵ d2 G1 ỵ l2 G0 ịa ẳ f ; where the nonzero elements of the matrices B2 and Gi ð0 i 4Þ are given explicitly in Theorem Table lists the maximum pointwise error E for u À uN to (41), using GJPGM for various values of m and the coefficients a2 ; b2 , c2 ; d2 and l2 4k ỵ 1ịk ỵ 3ịL k ỵ 2ị4k ỵ 16k ỵ 3ị : < p k 12 k ỵ 52 k 2ị2k ỵ 3ịp Now it can be easily shown that k ỵ 1ị3 4k2 ỵ 16k ỵ 3ị < 4k ỵ 2ị and  k  Example Consider the one dimensional third-order nonhomogeneous equation > k ỵ 1ị ; and accordingly, ( 8Lk ỵ 2ị 9L 9L 9L ẳ jak j < p < pffiffiffiffiffiffiffiffiffiffiffi $ : p k À 2ðk þ 1Þ6 p k À 2ðk À 2Þ pðk À 2Þ2 pk2 This completes the proof of the theorem h Theorem A function uxịẳ1xị3 1ỵxị2 fxị2H3ị 1;1ị, with jf3ị ðxÞj L, can be expanded as an infinite sum n of nonsymgeneralized Jacobi polynomials 3;2ị Jkỵ5 Lp ; k2 8k P 0: Proof The proof is similar to the proof of Theorem h ( the one dimensional ð5Þ ð4Þ ( uð3Þ ðxÞ À a1 uð2Þ ðxÞ À b1 u1ị xị ỵ c1 uxị ẳ fxị; uặ1ị ẳ u1ị 1ị ẳ 0: Table lists the maximum pointwise error E for u À uN to (42), using GJPGM for various values of m and the coefficients a1 ; b1 and c1 In the following, and for the sake of comparison, we give two other numerical examples to show the effectiveness of GJPGM third-order ( ð40Þ where fðxÞ is chosen such that the exact solution for (40) is uðxÞ ¼ ð1 À x2 Þxj sinðmpxÞ; j; m N We have uN xị ẳ PN3 2;1ị xị and the vector of unknowns kẳ0 ak x ị1 xịRk a ẳ a0 ; a1 ; ; aNÀ3 ÞT is the solution of the system B1 ỵ a1 E2 ỵ b1 E1 ỵ c1 E0 Þa ¼ fà , where the nonzero elements of the matrices B1 and Ei ð0 i 2Þ are given explicitly in Theorem In Table 3, the maximum pointwise error E for u À uN to Eq (40) is listed, using GJPGM for various values of j; m and the coefficients a1 ; b1 and c1 ( where fðxÞ is chosen such that the exact solution for (42) is uxị ẳ sinhmxị Setting Vxị ẳ uxị sinhmịxỵ ẵm coshmị sinhmị1 x2 ị, then the differential Eq (42) is equivalent to the differential equation Example ([37,38]) uð3Þ ðxÞ À a1 uð2Þ ðxÞ b1 u1ị xị ỵ c1 uxị ẳ fxị; uặ1ị ¼ uð1Þ ð1Þ ¼ 0; Example Consider equation ð42Þ  First, consider the following third-order boundary value problem Numerical results Example Consider equation uð3Þ ðxÞ À a1 uð2Þ xị b1 u1ị xị ỵ c1 uxị ẳ fxị; uặ1ị ẳ ặ sinhmị; u1ị 1ị ẳ m coshmị; m R; xị : k ẳ 0; 1; 2; g, and the series converges uniformly to uðxÞ Explicitly, the expansion coefcients in uxị ẳ P1 3;2ị xị, satisfy the following inequality: kẳ0 ak Jkỵ5 jak j < 3;2ị 4k ỵ 1ịk ỵ 3ịL X < p jbm;k j k 2ị2k ỵ 3ịp mẳ0 metric We have uN xị ẳ kẳ0 ak x2 Þ ð1 À xÞRk ðxÞ, and the vector of unknowns a ¼ ða0 ; a1 ; ; aNÀ5 ÞT is the solution of the system With the aid of Theorem 5, we have, the ð3Þ one dimensional 2ị 1ị u xị ỵ a2 u xị ỵ b2 u ðxÞ À c2 u ðxÞ À d2 u ðxÞ ỵl2 uxị ẳ fxị;uặ1ị ẳ u1ị ặ1ị ẳ u2ị 1ị ¼ 0; fifth-order ð41Þ where fðxÞ is chosen such that the exact solution for (41) is given by uðxÞ ¼ ð1 À x2 Þ ð1 À xÞ coshðmxÞ; m R y3ị tị ỵ ytị ẳ t2 ị cos t ỵ t2 6t 1ị sin t; t ẵ0; y0ị ẳ y1ị 0ị ỵ ẳ y1ị 1ị sin ẳ 0: 43ị The analytic solution to (43) is ytị ẳ t2 1ị sin t The transformation t ẳ 1ỵx turns Eq (43) into 3ị 1ỵx > < 8u xị ỵ uxị ẳ 27 2x x ị cos ; x ẵ1; 14 15 ỵ 10x x2 ị sin 1ỵx > : u1ị ẳ u1ị 1ị ỵ 12 ẳ u1ị 1ị sin ẳ 0: 44ị The analytic solution to (44) is uxị ẳ 14 x2 þ 2x À 3Þ sin 1þx PNÀ3 ð1;2Þ We set uN xị ẳ kẳ0 ak þ xÞ ð1 À xÞRk ðxÞþ ð1 þ xÞ ỵ x ỵ 21 ỵ xị sin 1ị In Table 6, the maximum pointwise error E for u À uN to Eq (44) are listed, using GJPGM The maximum absolute errors by our algorithm and by the second- and fourth-order quintic nonpolynomial spline [37], and by the quartic nonpolynomial spline method [38], for Example are presented in Table 684 Table E.H Doha et al Maximum pointwise error for u À uN , and N ¼ 8; 12; 16; 20; 24 N j 12 16 20 24 12 16 20 24 Table Maximum N ¼ 8; 12; 16; 20; 24 N 12 16 20 24 12 16 20 24 12 16 20 24 m 0 1 pointwise b2 a2 2 b1 1 12 16 20 24 12 16 20 24 12 16 20 24 a1 m c2 0 u À uN , c1 E a1 b1 c1 E 2:558  10À3 1:909  10À6 4:368  10À10 2:811  10À14 3:885  10À16 12 16 20 24 82 122 162 202 242 83 123 163 203 243 2:872:10À3 2:224  10À6 4:122  10À10 2:961  10À14 2:220  10À16 4:472  10À3 3:687  10À6 6:660  10À10 4:529  10À14 7:771  10À16 83 123 163 203 243 82 122 162 202 242 12 16 20 24 9:409  10À3 8:399  10À6 2:178  10À9 1:455  10À13 6:106  10À16 1:119  10À1 2:060  10À3 8:934  10À6 1:009  10À8 4:156  10À12 12 16 20 24 82 122 162 202 242 83 123 163 203 243 1:341  10À1 2:430  10À3 8:459  10À6 1:072  10À8 4:746  10À12 1:578  10À2 3:749  10À5 1:324  10À8 1:539  10À12 2:498  10À16 83 123 163 203 243 82 122 162 202 242 12 16 20 24 3:927  10À1 8:773  10À3 4:369  10À5 5:206  10À8 2:417  10À11 error for d2 l2 E 1:135  10À1 2:464  10À4 8:165  10À8 1:098  10À11 5:551  10À16 and À3 2 1 1 1 1 1:102  10 3:164  10À8 1:312  10À13 2:220  10À16 2:220  10À16 1:927  10À2 8:652  10À6 5:776  10À10 1:598  10À14 3:330  10À16 6:658  10À5 1:215  10À10 6:661  10À16 6:661  10À16 6:661  10À16 Table Maximum N ¼ 8; 12; 16 pointwise error for u À uN , and N m a1 b1 c1 E 12 16 0 2:804  10À8 9:536  10À14 1:110  10À16 12 16 1 1 2:819  10À8 9:736  10À14 1:110  10À16 12 16 1545  10À5 8:248  10À10 1:310  10À14 6:919  10À4 1:808  10À7 1:414  10À11 12 16 Table The maximum absolute error for Example N 11 14 E 1:077  10À3 1:131  10À7 3:567  10À12 2:579  10À16  Second, consider the fifth-order boundary value problem Example ([39–46]) 5ị t > < y tị ytị ẳ 15 ỵ 10tịe ; t ẵ0; 1; 1ị 2ị y0ị ¼ 0; y ð0Þ ¼ 1; y ð0Þ ¼ 0; > : y1ị ẳ 0; y1ị 1ị ẳ e: Table 45ị The analytic solution of (45) is ytị ẳ tð1 À tÞet The transforturns Eq (45) into mation t ẳ 1ỵx The best errors for Example Method Error Present method Quintic nonpolynomial spline method Lang [37] Quartic nonpolynomial spline method [38] 2:58  10À16 2:39  10À12 2:196  10À9 Generalized Jacobi Solutions for odd-order BVPs Table 685 Conflict of interest The maximum absolute error for Example N 12 15 E 2:178  10À3 1:776  10À8 4:601  10À13 6:351  10À17 The authors have declared no conflict of interest Compliance with Ethics Requirements Table This article does not contain any studies with human or animal subjects The best errors for Example Method Error Present method Cubic B-spline method Lang [39] Sixth-degree B-spline method [40] Sextic spline method [41] Finite difference method [42] Sextic spline method [43] Nonpolynomial sextic spline method [44] Sextic spline method [45] Quartic spline method [46] 6:35  10À17 1:14  10À5 2:08  10À2 2:01  10À5 1:15  10À2 4:84  10À7 1:61 1013 5:28 107 7:66 105 1ỵx 5ị > < 32u xị uxị ẳ 54 ỵ xịe ; 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