MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation89657510.1155/2012/896575896575Research ArticleAn Extension of the Legendre-Galerkin Method for Solving Sixth-Order Differential Equations with Variable Polynomial CoefficientsBhrawyA. H.1,2AlofiA. S.1El-SoubhyS. I.3MailybaevAlexei1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589Saudi Arabiakau.edu.sa2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511Egyptbsu.edu.eg3Department of Mathematics, Faculty of Science, Taibah University, Madinah 20012Saudi Arabiataibahu.edu.sa20122222012201226042011121220112012Copyright © 2012 A. H. Bhrawy et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomial coefficients. The complexities of the algorithm are O(N) operations for a one-dimensional domain with (N5) unknowns. An efficient and accurate direct solution for algorithms based on the Legendre-Galerkin approximations developed for the two-dimensional sixth-order elliptic equations with variable coefficients relies upon a tensor product process. The proposed Legendre-Galerkin method for solving variable coefficients problem is more efficient than pseudospectral method. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques.

1. Introduction

Spectral methods are preferable in numerical solutions of ordinary and partial differential equations due to its high-order accuracy whenever it works [1, 2]. Recently, renewed interest in the Galerkin technique has been prompted by the decisive work of Shen , where new Legendre polynomial bases for which the matrices systems are sparse are introduced. We introduce a generalization of Shen's basis to numerically solve the sixth-order differential equations with variable polynomial coefficients.

Sixth-order boundary-value problems arise in astrophysics; the narrow convecting layers bounded by stable layers, which are believed to surround A-type stars, may be modeled by sixth-order boundary-value problems [4, 5]. Further discussion of the sixth-order boundary-value problems is given in . The literature of numerical analysis contains little work on the solution of the sixth-order boundary-value problems [4, 5, 7, 8]. Theorems that list conditions for the existence and uniqueness of solutions of such problems are thoroughly discussed in , but no numerical methods are contained therein.

From the numerical point of view, Shen , Doha and Bhrawy , and Doha et al.  have constructed efficient spectral-Galerkin algorithms using compact combinations of orthogonal polynomials for solving elliptic equations of the second and fourth order with constant coefficients in various situations. Recently, the authors in [14, 15] and  have developed efficient Jacobi dual-Petrov-Galerkin and Jacobi-Gauss collocation methods for solving some odd-order differential equations. Moreover, the Bernstein polynomials have been applied for the numerical solution of high even-order differential equations (see, [17, 18]).

For sixth-order differential equations, Twizell and Boutayeb  developed finite-difference methods of order two, four, six, and eight for solving such problems. Siddiqi and Twizell  used sixth-degree splines, where spline values at the mid knots of the interpolation interval and the corresponding values of the even order derivatives were related through consistency relations. A sixth-degree B-spline functions is used to construct an approximate solution for sixth-order boundary-value problems (see ). Moreover, Septic spline solutions of sixth-order boundary value problems are introduced in . El-Gamel et al.  proposed Sinc-Galerkin method for the solutions of sixth-order boundary-value problems. In fact, the decomposition and modified domain decomposition methods to investigate solution of the sixth-order boundary-value problems are introduced in . Recently, Bhrawy  developed a spectral Legendre-Galerkin method for solving sixth-order boundary-value problems with constant coefficients. In this work, we introduce an efficient direct solution algorithm to generalize the work in [3, 22].

The main aim of this paper is to extend the application of Legendre-Galerkin method (LGM) to solve sixth-order elliptic differential equations with variable coefficients by using the expansion coefficients of the moments of the Legendre polynomials and their high-order derivatives. We present appropriate basis functions for the Legendre-Galerkin method applied to these equations. This leads to discrete systems with sparse matrices that can be efficiently inverted. The complexities of the algorithm is O(N) operations for a one-dimensional domain with (N-5) unknowns. The direct solution algorithms developed for the homogeneous problem in two-dimensions with constant and variable coefficients rely upon a tensor product process. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.

This paper is organized as follows. In the next section, we discuss an algorithm for solving the one-dimensional sixth-order elliptic equations with variable polynomial coefficients. In Section 3, we extend our results of Sections 2 to the two-dimensional sixth-order equations with variable polynomial coefficients. In Section 4, we present two numerical examples to exhibit the accuracy and efficiency of the proposed numerical algorithms. Also a conclusion is given in Section 5.

2. One-Dimensional Sixth-Order Equations with Polynomial Coefficients

We first introduce some basic notation which will be used in the sequel. We denote by Ln(x) the nth degree Legendre polynomial, and we set SN=span{L0(x),L1(x),,LN(x)},WN={vSN  :v(j)(±1)=0,j=0,1,2}, where v(j)(x) denotes jth-order differentiation of v(x) with respect to x.

We recall that the {Ln(x)} satisfy the orthogonality relation-11Lm(x)Ln(x)dx={0,mn,hn,m=n,hn=22n+1,m,n0. We recall also that Ln(x) is a polynomial of degree n and therefore Ln(q)(x)Sn-q. The following relation (the qth derivative of Lk(x)) will be needed for our main results (see Doha )DqLk(x)=i=0(k+i)  evenk-qCq(k,i)Li(x), whereCq(k,i)=2q-1(2i+1)Γ((1/2)  (q+k-i))Γ((1/2)(q+k+i+1))Γ(q)Γ((1/2)(2-q+k-i))Γ((1/2)(3-q+k+i)).

Some other useful relations areLn(±1)=(±1)n,Ln(q)(±1)=(±1)n+q(n+q)!2q(n-q)!q!.

In this section, we are interested in using the Legendre-Galerkin method to solve the variable polynomial coefficients sixth-order differential equation in the form:-χ3(x)u(6)(x)+i=02(-1)iαiχi(x)u(2i)(x)=f(x),  x[-1,1], subject tou(j)(±1)=0,j=0,1,2, where αi, i=0,1,2 are constants and χi(x), i=0,1,2,3 are given polynomials. Moreover, f(x) is a given source function. Without loss of generality, we suppose that χ3(x)=xμ,χi(x)=xν, and χ0(x)=xσ where μ, ν  , and σ are positive integers.

2.1. Basis of Functions

The problem of approximating solutions of ordinary or partial differential equations by Galerkin approximation involves the projection onto the span of some appropriate set of basis functions, typically arising as the eigenfunctions of a singular Sturm-Liouville problem. The members of the basis may satisfy automatically the boundary conditions imposed on the problem. As suggested in [3, 1012], one should choose compact combinations of orthogonal polynomials as basis functions to minimize the bandwidth and condition number of the resulting system. As a general rule, for one-dimensional sixth-order differential equations with six boundary conditions, one can choose the basis functions of expansion ϕk(x) of the formϕk(x)=j=03η  j(k)Lk+2j(x);η  0(k)=1,k=0,1,,N-6. We will choose the coefficients {ηj(k)} such that ϕk(x) verifies the boundary conditions (2.7). Making use of (2.5) and (2.8), hence {ηj(k)} can be uniquely determined to obtainηj(k)=(-1)j  3(2k+4j+1)(k+3/2)3(3-j)!j!(k+j+1/2)4,        j=1,2,3, where (a)b=Γ(a+b)/Γ(a). Now, substitution of (2.9) into (2.8) yields ϕk(x)=Lk(x)+j=13(-1)j3(2k+4j+1)(k+3/2)3(3-j)!j!(k+j+1/2)4Lk+2j(x),k=0,1,2,,N  -  6. It is obvious that {ϕk(x)} are linearly independent. Therefore by dimension argument we haveWN=span{ϕk(x):k=0,1,2,,N-6}.

2.2. Treatment of Variable Polynomial Coefficients

A more general situation which often arises in the numerical solution of differential equations with polynomial coefficients by using the Legendre Galerkin method is the evaluation of the expansion coefficients of the moments of high-order derivatives of infinitely differentiable functions. The formula of Legendre coefficients of the moments of one single Legendre polynomials of any degree isxmLj(x)=n=02mΘm,n(j)Lj+m-n(x),m,j0, with L-r(x)=0,  r1, whereΘm,n(j)=(-1)n2j+m-nm!(2j+2m-2n+1)Γ(j+1)Γ(j+m-n+1)×k=max(0,j-n)min(j+m-n,j)(j+m-nk)Γ(j+k+1)2k(n+k-j)!Γ(3j+2m-2n-k+2)×l=0j-k(-1)lΓ(2j+m-n-k-l+1)Γ(j+m+l-n+1)l!(j-k-l)!Γ(j-l+1)Γ(k+l+1)×2F1(j-k-n,j+m+l-n+1;3j+2m-2n-k+2;2). For more details about the above formula, the reader is referred to Doha . This formula can be used to facilitate greatly the setting up of the algebraic systems to be obtained by applying the LGM for solving differential equations with polynomial coefficients of any order. The following lemma is very important and needed in what follows.

Lemma 2.1.

We have, for arbitrary constants ,j0, xlϕj(q)(x)=i=03[n=02lΘl,n(j+2i)Lj+2i+l-n(x)]ηi(j),xlϕj(q)(x)=i=03[s=0j+2i-q[n=02lΘl,n(s)Ls+l-n(x)]Cq(j+2i,s)]ηi(j), where ϕj(x), ηi(j), and Θ,n(j) are as defined in (2.8), (2.9), and (2.13), respectively.

Proof.

Immediately obtained from relations (2.3), (2.8), and (2.12), the Legendre-Galerkin approximation to (2.6)-(2.7) is, to find uNWN such that (-xμ  uN(6)(x),v(x))+i=12αi((-1)ixν    uN(2i)(x),v(x))+α0(xσuN(x),v(x))=(f(x),v(x))N,vWN, where (u,v)=-11uvdx is the scalar product in L2(-1,1) and (·,·)N is the inner product associated with the Legendre-Gauss-Lobatto quadrature. It is clear that if we take ϕk(x) as defined in (2.8) and v(x)=ϕk(x), then we find that (2.15) is equivalent to (-xμuN(6)(x),ϕk(x))+i=12αi((-1)ixνuN(2i)(x),ϕk(x))+α0(xσuN(x),ϕk(x))=(f(x),ϕk(x))N,k=0,1,,N-6.

Hence, by setting fk=(f,ϕk(x))N,f=(f0,f1,,fN-6)T,uN(x)=n=0N-6anϕn(x),a=(a0,a1,,aN-6)T,Zil=(zkji,l)0k,jN-6;0i3,l  is  a  positive  integer, where zkjr,l=((-1)rxlϕj(2r)(x),ϕk(x)),r=0,1,2,3,   then the matrix system associated with (2.16) becomes (Z3μ+i=12αiZiν+α0Z0σ)a=f, where the nonzero elements of the matrices Z3μ, Z2ν, Z1ν, and Z0σ are given explicitly in the following theorem.

Theorem 2.2.

If we take ϕk(x) as defined in (2.8), and if we denote zkj3,μ=(-xμϕj(6)(x),ϕk(x)), zkjr,ν=((-1)rxνϕj(2r)(x),ϕk(x)), r=1,2 and zkj0,σ=(xσϕj(x),ϕk(x)) then the nonzero elements (zkj3,μ), (zkjr,ν), (zkj0,σ) for 0k, jN-6 are given by zkj3,μ=-l=03[i=03(s=02μ(C6(j+2i,k+2l-μ+s)  Θμ,s(k+2l-μ+s))ηi(j))ηl(k)hk+2l],j=k+2p-μ,p=0,1,,μ,zkjr,v=(-1)rl=03[i=03(s=02v(C2r(j+2i,k+2l-v+s)Θv,s(k+2l-v+s))ηi(j))ηl(k)hk+2l],j=k+2p-v+2r-6,  p=0,1,,v+6-2r,r=1,2,zkj0,σ=l=03[i=03(Θσ,j+2i+σ-(k+2l)(j+2i)ηi(j))ηl(k)hk+2l],j=k+2p-σ-6,p=0,1,,σ+6.

Proof.

The proof of this theorem is rather lengthy, but it is not difficult once Lemma 2.1 is applied.

From Theorem 2.2, we see that Z3μ is a band matrix with an upper bandwidth of μ, lower bandwidth of μ, and an overall bandwidth 2μ+1. The sparse matrices Z2v, Z1v, and Z0σ have bandwidths of 2v+5, 2v+9, and 2σ+13, respectively.

In general, the expense of calculating an LU factorization of an N×N dense matrix A is O(N3) operations, and the expense of solving Ax=b, provided that the factorization is known, is O(N2). However, in the case that a banded A has bandwidth of r, we need just O(r2N) operations to factorize and O(rN) operations to solve a linear system. In the case of αi=0, i=0,1,2, the square matrix Z3μ has bandwidth of 2μ+1. We need just O((2μ+1)2N) operations to factorize and O((2μ+1)N) operations to solve the linear system (2.19). If μ    N this represents a very substantial saving. Notice also that the system (2.19) reduces to a diagonal system for μ=0 and αi=0, i=0,1,2.

2.3. Constant Coefficients

In the special case, (μ=ν=σ=0, i.e., the sixth-order differential equation with constant coefficients), the corresponding matrix system becomes (Z30+i=13αiZi0)a=f, where Zi0=(zkji,0)0k,jN-6; i=0,1,2,3.

Corollary 2.3.

If μ=ν=σ=0 then the nonzero elements (zkj3,0), (zkj2,0), (zkj1,0), (zkj0,0) for 0k, jN-6 are given as follows: zkk3,0=26(k+32)2(k+32)3,zkj2,0=l=03[i=03ηi(j)C4(j+2i,k+2l)]ηl(k)hk+2l,j=k+2p-2,p=0,1,2,zkj1,0=-l=03[i=03ηi(j)C2(j+2i,k+2l)]ηl(k)hk+2l,j=k+2p-4,p=0,1,2,3,4,zk+2p,k0,0=zk,k+2p0,0=i=03-p(-1)2i+p(3i)(3i+p)(2k+4p+4i+1)(k+3/2)3(k+2p+3/2)32(k+p+i+1/2)4(k+2p+i+1/2)4,p=0,1,2,3.

Note that the results of Corollary 2.3 can be obtained immediately as a special case from Theorem 2.2. For more details see .

It is worthy to note here that if αi=0, i=0,1,2, then the nonzero elements of the matrix Z30 are given by (2.21) and the solution of the linear system is given explicitly by ak=(f,ϕk)N/zkk3,0.

Obviously Z20, Z10, and Z00 are symmetric positive definite matrices. Furthermore, Z30 is a diagonal matrix, Z20 can be split into two tridiagonal submatrices, Z10 can be split into two pentadiagonal submatrices, and A0 can be split into two sparse submatrices with bandwidth of 7. Therefore, the system can be efficiently solved. More precisely for k+j odd, zkjr,0=0, r=0,1,2,3. Hence system (Z30+i=13αiZi0)a=f of order N-5 can be decoupled into two separate systems of order (N/2-2) and (N/2-3), respectively. In this way one needs to solve two systems of order n instead of one of order 2n, which leads to substantial savings. Moreover, in the case of αi0, i=0,1,2, we can form explicitly the LU factorization, that is, Z30+i=13αiZi0=LU. The special structure of L and U allows us to obtain the solution in O(N) operations.

Remark 2.4.

If the boundary conditions are nonhomogeneous, one can split the solution u(x) into the sum of a low-degree polynomial which satisfies the nonhomogeneous boundary conditions plus a sum over the basis functions ϕ(x) that satisfy the equivalent homogeneous boundary conditions.

3. Two-Dimensional Sixth-Order Equations with Polynomial Coefficients

In this section, we extend the results of Section 2 to deal with the two-dimensional sixth-order differential equations with variable polynomial coefficients:-X3(x)Y3(y)(Δ)3u(x,y)+r=12αrXr(x)Yr(y)(-Δ)ru(x,y)+α0X0(x)Y0(y)u(x,y)=f(x,y),in  Ω, subject to the boundary conditionsixiu(±1,y)=0,i=0,1,2,iyiu(x,±1)=0,i=0,1,2, where Ω=(-1,1)×(-1,1), the differential operator Δ is the well-known Laplacian defined by Δ=(2/x2)+(2/y2), (0/x0)u(±1,y)=u(±1,y), (0/x0)u(x,±1)=u(x,±1) and f(x,y) is a given source function. Moreover, Xi(x) and Yi(y), i=0,1,2,3 are given polynomials. Without loss of generality, we suppose that X3(x)=xμ, Y3(y)=yν, Xi(x)=xρ, Yi(y)=yσ, i=1,2, X0(x)=xδ, and Y0(y)=yɛ where μ, ν, ρ, σ, δ, and ɛ are positive integers.

The Legendre-Galerkin approximation to (3.1)-(3.2) is, to find uNWN2 such that(xμyν(-Δ)3uN,v)+r=12αr(xρyσ(-Δ)ruN,v)+α0(xδyɛuN,v)=(f,v)NvWN2. It is clear that if we take ϕk(x) as defined in (2.8), thenWN2=span{ϕi(x)ϕj(y),i,j=0,1,,N-6}.

We denoteuN=k=0N-6j=0N-6ukjϕk(x)ϕj(y),fkj=(f,ϕk(x)ϕj(y))N,U=(ukj),F=(fkj),k,j=0,1,,N-6. Taking v(x,y)=ϕ(x)ϕm(y) in (3.3) for ,m=0,1,,N-6, then one can observe that (3.3) is equivalent to the following equation:l,m=0N-6{{zil3,μulmzjm0,ν+3zil2,μulmzjm1,ν+3zil1,μulmzjm2,ν+zil0,μulmzjm3,ν}+α2{zil2,ρulmzjm0,σ+2zil1,ρulmzjm1,σ+zil0,ρulmzjm2,σ}+α1{zil1,ρulmzjm0,σ+zil0,ρulmzjm1,σ}+α0{zil0,δulmzjm0,ɛ}}=fij,i,j=0,1,,N-6, which may be written in the matrix formZ3μUZ0ν+3Z2μUZ1ν+3Z1μUZ2ν+Z0μUZ3ν+α2(Z2ρUZ0σ+2Z1ρUZ1σ+Z0ρUZ2σ)+α1(Z1ρUZ0σ+Z0ρUZ1σ)+α0Z0δUZ0ɛ=F, where {Ziϱ  for  i=0,1,2,3  and  ϱ  is  apositive  integer} are the matrices defined in Theorem 2.2.

The direct solution algorithm here developed for the sixth-order elliptic differential equation in two dimensions relies upon a tensor product process, which is defined as follows. Let P and R be two matrices of size n×n and m×m, respectively. Their tensor productPR=(P11RP1nRPn1RPnnR), is a matrix of size mn×mn.

We can also rewrite (3.7) in the following form using the Kronecker matrix algebra (See, Graham ):Lv[Z3μZ0ν+3Z2μZ1ν+3Z1μZ2ν+Z0μZ3ν+α2(Z2ρZ0σ+2Z1ρZ1σ+Z0ρZ2σ)+α1(Z1ρZ0σ+Z0ρZ1σ)+α0Z0δZ0ɛ]v=f, where f and v are F and U written in a column vector, that is,f=(f00,f10,,fN-6,0;f01,f11,,fN-6,1;;f0,N-6,,fN-6,N-6)T,v=(u00,u10,,uN-6,0;u01,u11,,uN-6,1;;u0,N-6,,uN-6,N-6)T, and denotes the tensor product of matrices, that is, Z3μZ0ν=(Z3μzij0,ν)i,j=0,1,,N-6. In brief, the solution of (3.1) subject to (3.2) can be summarized in Algorithm 1.

<bold>Algorithm 1</bold>

Compute the matrices {Zjμ,Zjν  for=0,1,2,3}, {Zjρ,Zjσ  for  j=0,1,2} and {Z0δ,Z0ɛ}

from Theorem 2.2.

Find the tensor product of the matrices in the previous step.

Compute F and write it in a column vector f.

Obtain a column vector v by solving (3.9).

4. Numerical Results

We report on two numerical examples by using the algorithms presented in the previous sections. It is worthy to mention that the pure spectral-Galerkin technique is rarely used in practice, since for a general right-hand side function f we are not able to compute exactly its representation by Legendre polynomials. In fact, the so-called pseudospectral method is used to treat the right-hand side; that is, we replace f by INf (polynomial interpolation over the set of Gauss-Lobatto points); see for instance; Funaro .

Example 4.1.

Here we present some numerical results for a one-dimensional sixth-order equation with polynomial coefficients. We only consider the case χ3(x)=(x4+x+1), χ2(x)=x2+2x+3, χ1(x)=x2+1 and χ0(x)=1; that is, we consider the -(x4+1)u(6)(x)+γ2(x2+2x+3)u(4)(x)-γ1(x2+1)u′′(x)+γ0u(x)=f(x), subject to u(±1)=u(±1)=u(±1)=0,        xI, where f(x) is chosen such that the exact solution of (4.1) is u(x)=(1-x2)sin2(2πx).

For LGM, we have uN(x)=k=0N-6akϕk(x), the vectors of unknowns a is the solution of the following system((Z34+Z30)+γ2(Z22+2Z21+3Z20)+γ1(Z12+Z10)+γ0Z00)a=f, where the nonzero elements of the matrices Z34, Z30, Z22, Z21, Z20, Z12, Z10 and Z00 can be evaluated explicitly from Theorem 2.2. Table 1 lists the maximum pointwise error of u-uN, using the LGM with various choices of γ0,  γ1,  γ2 and N. Numerical results of this problem show that the Legendre Galerkin method converge exponentially.

Maximum pointwise error of u-uN for N=20,30,40.

N     γ0        γ1        γ2    LGM            γ0        γ1        γ2            LGM
2030118.66·10-41359.22·10-4
301.67·10-91.67·10-9
406.10·10-166.10·10-16

In order to examine the algorithm proposed in Section 3, we will consider a problem for a two-dimensional sixth-order elliptic differential equation with constant coefficients.

Example 4.2.

Consider the two-dimensional sixth-order equation -Δ3u+γ2Δ2u-γ1Δu+γ0u=f(x,y), subject to the boundary conditions u(±1,y)=u(x,±1)=0,ux(±1,y)=uy(x,±1)=0,2ux2(±1,y)=2uy2(x,±1)=0, where f(x,y) is chosen such that the exact solution of (4.4)-(4.5) is u(x,y)=(1-x2)(1-y2)sin2(2πx)sin2(2πy).

In Table 2, we list the maximum pointwise error of u-uN by the LGM with two choices of γ0, γ1, γ2 and various choices of N. The results indicate that the spectral accuracy is achieved and that the effect of roundoff errors is very limited.

Maximum pointwise error of u-uN for N=20,30,40.

N     γ0        γ1        γ2            LGM            γ0        γ1        γ2            LGM
203 1 11.18·10-300 01.18·10-3
302.39·10-91.18·10-9
402.11·10-142.08·10-14
5. Concluding Remarks

We have presented stable and efficient spectral Galerkin method using Legendre polynomials as basis functions for sixth-order elliptic differential equations in one and two dimensions. We concentrated on applying our algorithms to solve variable polynomials coefficients differential equations by using the expansion coefficients of the moments of the Legendre polynomials and their high-order derivatives. Numerical results are presented which exhibit the high accuracy of the proposed algorithms.

Acknowledgments

The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have improved the paper. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.

CanutoC.HussainiM. Y.QuarteroniA.ZangT. A.Spectral Methods in Fluid Mechanics2006New York, NY, USASpringerxxii+5632223552BernardiC.MadayY.Approximations Spectrales de Problèmes aux Limites Elliptiques199210Paris, FranceSpringerii+2421208043ShenJ.Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomialsSIAM Journal on Scientific Computing19941561489150510.1137/09150891298626ZBL0811.65097BoutayebA.TwizellE.Numerical methods for the solution of special sixth-order boundary value problemsInternational Journal of Computer Mathematics199245207233TwizellE. H.BoutayebA.Numerical methods for the solution of special and general sixth-order boundary value problems, with applications to Bénard layer eigenvalue problemsProceedings of the Royal Society London Series A1990431188343345010.1098/rspa.1990.01421086351ZBL0722.65042BaldwinP.Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methodsPhilosophical Transactions of the Royal Society of London Series A1987322156628130590211310.1098/rsta.1987.0051ZBL0625.76043SiddiqiS. S.TwizellE. H.Spline solutions of linear sixth-order boundary-value problemsInternational Journal of Computer Mathematics1996603-4295304El-GamelM.CannonJ. R.ZayedA. I.Sinc-Galerkin method for solving linear sixth-order boundary-value problemsMathematics of Computation2004732471325134310.1090/S0025-5718-03-01587-42047089ZBL1054.65085AgarwalR. P.Boundary Value Problems for Higher Order Differential Equations1986SingaporeWorld Scientific Publishingxii+3071021979ZBL0681.76121DohaE. H.BhrawyA. H.Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomialsNumerical Algorithms200642213716410.1007/s11075-006-9034-62262512ZBL1103.65119DohaE. H.BhrawyA. H.Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomialsApplied Numerical Mathematics20085881224124410.1016/j.apnum.2007.07.0012428974ZBL1152.65112DohaE. H.BhrawyA. H.A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equationsNumerical Methods for Partial Differential Equations200925371273910.1002/num.203692510756ZBL1170.65099DohaE. H.BhrawyA. H.Abd-ElhameedW. M.Jacobi spectral Galerkin method for elliptic Neumann problemsNumerical Algorithms2009501679110.1007/s11075-008-9216-52487228ZBL1169.65111DohaE. H.BhrawyA. H.HafezR. M.A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equationsMathematical and Computer Modelling2011539-101820183210.1016/j.mcm.2011.01.0022782868ZBL1219.65077DohaE. H.BhrawyA. H.HafezR. M.A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equationsAbstract and Applied Analysis201120112194723010.1155/2011/9472302784393ZBL1216.65086BhrawyA. H.Abd-ElhameedW. M.New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation methodMathematical Problems in Engineering201120111483721810.1155/2011/8372182781569ZBL1217.65155DohaE. H.BhrawyA. H.SakerM. A.Integrals of Bernstein polynomials: an application for the solution of high even-order differential equationsApplied Mathematics Letters201124455956510.1016/j.aml.2010.11.0132749745DohaE. H.BhrawyA. H.SakerM. A.On the derivatives of Bernstein polynomials: an application for the solution of high even-order differential equationsBoundary Value Problems201120111682954310.1155/2011/8295432783106ZBL1220.33006LoghmaniG. B.AhmadiniaM.Numerical solution of sixth order boundary value problems with sixth degree B-spline functionsApplied Mathematics and Computation2007186299299910.1016/j.amc.2006.08.0682316723ZBL1171.65412SiddiqiS. S.AkramG.Septic spline solutions of sixth-order boundary value problemsJournal of Computational and Applied Mathematics2008215128830110.1016/j.cam.2007.04.0132400634ZBL1138.65062WazwazA.The numerical solution of sixth-order boundary value problems by the modified decomposition methodApplied Mathematics and Computation20011182-3311325181296110.1016/S0096-3003(99)00224-6ZBL1023.65074BhrawyA. H.Legendre-Galerkin method for sixth-order boundary value problemsJournal of the Egyptian Mathematical Society20091721731882598145ZBL1190.65177DohaE. H.On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomialsJournal of Physics A200437365767510.1088/0305-4470/37/3/0102065587ZBL1055.33007GrahamA.Kronecker Products and Matrix Calculus: With Applications1981England, UKEllis Horwood Ltd.640865FunaroD.Polynomial Approximation of Differential Equations19928Berlin, GermanySpringerx+305Lecture Notes in Physics1176949