Numerical solution for solving special eighth-order linear boundary value problems using Legendre Galerkin method

In this paper, Galerkin method has been introduced using Legendre polynomials as basis functions over the interval $$[-1, 1]$$[-1,1] to solve the eighth-order linear boundary value problems with two-point boundary conditions. Legendre Galerkin method is an effective tool in numerically solving such problems. The performance and applicability of the method is illustrated through some examples that reveal the method presents much better results. The obtained numerical results are convincing and very close to the analytical ones.


Introduction
Consider the general eighth-order linear differential equation of the form where u(x), f(x) are continuous functions in the space ' 2 À 1; 1½ and a i ðxÞ ¼ x i . The boundary value problems of higher order have been investigated due to their mathematical importance and the potential for the applications in hydrodynamic and hydromagnetic stability [1,2]. Higher-order boundary value problems arise in many fields. For instance, sixth-and eighth-order differential equations are modelled by thermal instability as ordinary convection and overstability in horizontal layer of fluid heated from below subject to the action of rotation [2,3]. Generally, such problems are known to arise in astrophysics. The narrow convecting layers bounded by stable layers which are believed to surround A-type stars may be modelled by sixth-order boundary value problems [4]. Dynamo actions in some stars may be modelled by such equations [5]. Shen [6] derived an eighth-order differential equation by governing bending and axial vibrations. Equations for the equilibrium in terms of components for an orthotropic thin circular cylindrical shell subjected to a load that is symmetric about the shell result in an eighth-order differential equations as shown by Paliwal and Pande [7]. Bishop et al. [8] showed that an eighth-order differential equation arises in torsional vibration of uniform beams. Existence and uniqueness of solutions of 2n-th order boundary value problems are discussed by Agarwal [9,10]. The analytical solutions of such problems cannot be found easily. Therefore, the authors suggested different approximate algorithms using Legendre, Hermite and Laguerre [11,12] polynomials. Among them, Legendre polynomials have extensively been particularly used in the area of physics and engineering. For instance, Legendre and assocaited Legendre polynomials are also widely used in [13][14][15], to solve the fractional problems. Spectral methods also have gained a good reputation among numerical analysts as a robust numerical tool for a wide variety of problems in applied mathematics and scientific computing. Many researchers used the spectral approach [16,17] to solve the ordinary and partial differential equations, respectively. A selective review for getting the numerical solution of the eighth-order boundary value problems is presented here. Boutayeb and Twizell [18] used finite difference methods, Akram and Siddiqi [19,20] used nonic and non-polynomial spline functions, respectively, Akram and Rehman [21] developed reproducing kernel space, Viswanadham and Ballem [22] used Galerkin method with quintic B-spline, Inc and Evans [23] constructed Adomian decomposition method, Wazwaz [24] developed modified Adomian decomposition method, Siddiqi and Iftikhar [25] used homotopy analysis method, and Ballem and Viswanadham [26] presented the Galerkin method with septic B-splines, whereas Abbasbandy and Shirzadi [27] developed variational iteration method.
In this paper, the Legendre Galerkin method has been elaborated for the solution of linear eighth-order boundary value problem with two-point boundary conditions defined in Eq. (1) with Eq. (2).
In ''Preliminaries'', some important definitions, lemmas and theorems regarding Legendre polynomials are discussed. Legendre Galerkin method is explained in ''Description of the method''. Convergence and error analysis of the method are discussed in ''Convergence and error analy-

Preliminaries
Legendre polynomials are widely used as a mathematical tool in applied sciences as well as in engineering field. These polynomials are defined precisely and easily differentiated and integrated as well.
Legendre polynomials of degree n over the interval ½À1; 1 is defined as and satisfy the following recurrence relations ð2n þ 1ÞL n ðxÞ ¼ L 0 nþ1 ðxÞ À L 0 nÀ1 ðxÞ; ð3Þ nL n ðxÞ ¼ xL 0 n ðxÞ À L 0 nÀ1 ðxÞ: ð4Þ Legendre polynomials are orthogonal on ½À1; 1 with respect to the weight function 1, i.e. and Lemma 2.1 Let n and m be any two integers such that n À m N and m [ 0, then Z 1

À1
L n ðxÞL 00 nÀm ðxÞdx ¼ 0: Proof Integrating the left hand side by parts and using Eq. (6) yield the result.
Lemma 2.2 Let n and m be any two integers such that n ! m, then Z 1

À1
L n ðxÞL 0 m ðxÞdx ¼ 0: Proof The proof is divided into two parts.
Case II For n [ m, the integral on the left, using Eqs. (3) and (6), can be written as Theorem 2.1 Let n and m be any two integers such that n; m N , then For n m and considering the previous cases with For n\m, then integrating R 1 À1 L 00 n ðxÞL m ðxÞdx by parts and using Eq.

Description of the method
To solve the linear eighth-order boundary value problem (1) by the Galerkin method along with Legendre basis, u(x) is approximated as where a 0 j s, j ¼ 0; 1; 2; . . .; n are the Legendre coefficients. To determine these coefficients a j , orthogonalizing the residual with respect to the basis functions, i.e.

Convergence and error analysis
In this section, the convergence and error analysis of the Legendre Galerkin method have been studied in detail.
Convergence of the method Lemma 4.1 Let xðtÞ 2 H k À 1; 1½ (a Sobolev space) and let x n ðtÞ ¼ P n i¼0 c i L n ðtÞ be the best approximation polynomial of x(t) in the ' 2 -norm, then kxðtÞ À x n ðtÞk ' 2 ½À1;1 c 0 n Àk kxðtÞk H k À1;1½ ; and c 0 is a non-negative constant which depends on the selected norm and is free from x(t) and n.
Theorem 4.1 Assume j : X ! X is bounded, with X a Banach space, and k À j : X ! X is bijective. Further, assume kj À jL n k ! 0 as n ! 1; then for all sufficiently large n, say n ! N, the operator ðk À jL n Þ À1 exists as a bounded operator from X to X. Moreover, it is uniformly bounded such that sup n ! N kðk À jL n Þ À1 k\1: For the solution of ðk À jL n Þx m ¼ L n y, x m 2 X and ðk À jÞx ¼ y, x À x m ¼ kðk À L n jÞ À1 ðx À L n ðxÞÞ; jkj kk À jL n k kx À L n ðxÞk kx À x n k jkjkðk À jL n Þ À1 kkx À L n ðxÞk: Proof [31].
Consequently, the approximation rate of Legendre polynomials is n Àk with respect to Lemma 4.1, and also from Theorem 4.1, kx À x n k converge to zero as soon as kx À L n k.

Error analysis of the method
In this subsection, an error estimator for eighth-order boundary value problems using Legendre Galerkin approximation has been discussed.
Consider e n ðxÞ ¼ uðxÞ À u n ðxÞ as the error function of Legendre approximation u n ðxÞ to u(x), where u(x) is the exact solution of Eq. (1) with boundary conditions defined in Eq. (2). So, u n ðxÞ satisfies the following problem: where P n ðxÞ is a perturbation term linked with u n ðxÞ obtained as follows P n ðxÞ ¼ u ð8Þ n ðxÞ þ X 7 i¼0 a i ðxÞu ðiÞ n ðxÞ À f ðxÞ; i ¼ 0; 1; 2; 3: We find an approximation e n;N ðxÞ to e n ðxÞ in the same way as in description of the method, for the solution of Eq. (1) with Eq. (2). Subtracting Eqs. (22) and (23) from Eqs. (1) and (2), respectively, yields the error function of the form P n ðxÞ ¼ Àe ð8Þ n ðxÞ À We solve this problem using the Legendre Galerkin method to get the approximation e n;N ðxÞ.

Handling of boundary conditions and solution domain
If the boundary conditions are nonhomogeneous or the solution domain is [a, b], then these conditions are converted to homogeneous conditions and the domain of the solution must be converted to ½À1; 1. Consider subject to the following boundary conditions u ðjÞ ðaÞ ¼ h j ; u ðjÞ ðbÞ ¼ / j ; Using the linear transformation t ¼ bÀa 2 x þ bþa 2 , then Eq. (27) takes the form where subject to the following boundary conditions To transform the nonhomogeneous boundary conditions in Eq. (30) to homogeneous boundary conditions, we replace where WðxÞ is the interpolating polynomial such that W ðjÞ ðÀ1Þ ¼ H j and W ðjÞ ð1Þ ¼ U j , j ¼ 0; 1; 2; 3. Also, g j x j The problem takes the form:

Numerical examples
Some examples have been constructed to measure the accuracy of the proposed method. Numerical results obtained by the method show the betterment of the method also.
The proposed method is implemented to the problem for n ¼ 10. The comparison between the absolute errors of the proposed method and that developed by Viswanadham and Ballem [22] is shown in Table 1 and Fig. 1, respectively.
The proposed method is implemented to the problem for n ¼ 10. The comparison between the absolute errors of the proposed method and that developed by Ballem and Viswanadham [26] is shown in Table 2 and Fig. 2, respectively.    subject to the boundary conditions uð0Þ ¼ 0; uð1Þ ¼ 0; u 0 ð0Þ ¼ À1; u 0 ð1Þ ¼ 2sin1; u 00 ð0Þ ¼ 0; u 00 ð1Þ ¼ 4 cos 1 þ 2 sin 1; u 000 ð0Þ ¼ 7; u 000 ð1Þ ¼ 6 cos 1 À 6 sin 1: The exact solution of the problem is uðxÞ ¼ ðx 2 À 1Þ sin x. The proposed method is implemented to the problem for n ¼ 10. The comparison between the absolute errors of the proposed method and that developed by Viswanadham and Ballem [22] is shown in Table 3 and Fig. 3, respectively.

Conclusion
In this paper, Galerkin method using Legendre polynomials as basis function has been developed to approximate the linear eighth-order boundary value problems. In this method, the nonhomogeneous boundary conditions are transformed to the homogeneous boundary conditions and the solution domain is converted to the interval ½À1; 1. By comparing the results of the proposed method with other existing methods, it is found that the results are improved and become remarkable. Consequently, the solution may converge efficiently to the analytical one by increasing the order of the problem.
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