Abstract
In this paper, Galerkin method has been introduced using Legendre polynomials as basis functions over the interval \([1, 1]\) to solve the eighthorder linear boundary value problems with twopoint 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 eighthorder linear differential equation of the form
subject to the following boundary conditions
where u(x), f(x) are continuous functions in the space \(\ell ^{2}]1, 1[\) and \(a_{i}(x)= x^{i}\).
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]. Higherorder boundary value problems arise in many fields. For instance, sixth and eighthorder 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 Atype stars may be modelled by sixthorder boundary value problems [4]. Dynamo actions in some stars may be modelled by such equations [5]. Shen [6] derived an eighthorder 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 eighthorder differential equations as shown by Paliwal and Pande [7]. Bishop et al. [8] showed that an eighthorder differential equation arises in torsional vibration of uniform beams. Existence and uniqueness of solutions of 2nth 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 [13–15], 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 eighthorder boundary value problems is presented here. Boutayeb and Twizell [18] used finite difference methods, Akram and Siddiqi [19, 20] used nonic and nonpolynomial spline functions, respectively, Akram and Rehman [21] developed reproducing kernel space, Viswanadham and Ballem [22] used Galerkin method with quintic Bspline, 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 Bsplines, whereas Abbasbandy and Shirzadi [27] developed variational iteration method.
In this paper, the Legendre Galerkin method has been elaborated for the solution of linear eighthorder boundary value problem with twopoint 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 analysis". The transformation of nonhomogeneous boundary conditions and change of interval are discussed in "Handling of boundary conditions and solution domain". The practical usefulness and applicability of the method have been discussed via examples in "Numerical examples".
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 \([1,1]\) is defined as
where
and satisfy the following recurrence relations
Legendre polynomials are orthogonal on \([1,1]\) with respect to the weight function 1, i.e.
and
Lemma 2.1
Let n and m be any two integers such that \(nm\le N\) and \(m>0\), then
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 \(n\ge m\), then
Proof
The proof is divided into two parts.
Case I
For \(n=m\), we have
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 \(n,m \le N\) , then
where \(i=1,3,5,\ldots ,2k+1\le Nm\).
Proof

(1)
Integrating \(\int _{1}^{1}L'_{n}(x)L_{m}(x){\rm d}x\) by parts gives
$$\begin{aligned} \int _{1}^{1}L'_{n}(x)L_{m}(x){\rm d}x&= \left[ L_{n}(x)L_{m}(x)\right] _{1}^{1}\int _{1}^{1}L_{n}(x)L'_{m}(x){\rm d}x \nonumber \\ &= \left[ 1+(1)^{n+m+1}\right] \int _{1}^{1}L_{n}(x)L'_{m}(x){\rm d}x. \end{aligned}$$(7)For \(n=m+i, i=1,3,5,\ldots ,2k+1\le Nm\) and using Lemma 2.2 lead to
$$\begin{aligned} \int _{1}^{1}L'_{n}(x)L_{m}(x){\rm d}x &= 2. \end{aligned}$$For \(n=m+i, i=0,2,4,\ldots ,2k\le Nm\), Eq. (7) yields
$$\begin{aligned} \int _{1}^{1}L'_{n}(x)L_{m}(x){\rm d}x &= 0. \end{aligned}$$For \(n\le m\) and considering the previous cases with Lemma 2.2 yield \(\int _{1}^{1}L'_{n}(x)L_{m}(x){\rm d}x=0.\)

(2)
The proof is divided into four parts.

(a)
For \(n=m+i, \quad i=2,4,6,\ldots ,2k\le Nm\).
$$\begin{aligned} \int _{1}^{1}L''_{n}(x)L_{m}(x){\rm d}x &= \left[ L'_{n}(x)L_{ni}(x)\right] _{1}^{1}\int _{1}^{1}L'_{n}(x)L'_{ni}(x){\rm d}x\\ &= n(n+1)\left[ L_{n}(x)L'_{ni}(x)\right] _{1}^{1}\\&\quad +\int _{1}^{1}L_{n}(x)L''_{ni}(x){\rm d}x\\ &= n(n+1)\left[ L_{n}(x)L'_{ni}(x)\right] _{1}^{1},\\ & \quad \left[ {\text{using Lemma 2.1}}\right] \\ &= n(n+1)m(m+1). \end{aligned}$$ 
(b)
For \(n=m+i, i=1,3,5,\ldots ,2k+1\le Nm\), then \(\int _{1}^{1}L''_{n}(x)L_{m}(x){\rm d}x=0\).

(c)
For \(n=m\), then
$$\begin{aligned} \int _{1}^{1}L''_{n}(x)L_{m}(x){\rm d}x &= \left[ L'_{n}(x)L_{ni}(x)\right] _{1}^{1}\int _{1}^{1}L'_{n}(x)L'_{ni}(x){\rm d}x\\ &= n(n+1)m(m+1)\\ &= 0. \end{aligned}$$ 
(d)
For \(n<m\), then integrating \(\int _{1}^{1}L''_{n}(x)L_{m}(x) {\rm d}x\) by parts and using Eq. (6) leads to \(\int _{1}^{1}L''_{n}(x)L_{m}(x) {\rm d}x=0\).
Description of the method
To solve the linear eighthorder boundary value problem (1) by the Galerkin method along with Legendre basis, u(x) is approximated as
where \(\alpha _{j}'s\), \(j=0,1,2,\ldots ,n\) are the Legendre coefficients. To determine these coefficients \(\alpha _{j}\), orthogonalizing the residual with respect to the basis functions, i.e.
where
We approximate the integrals in Eq. (9) by integrating by parts such that all derivatives transfer from u to \(L_{r}\). For convenience, few of the inner products of Eq. (9) can be calculated, as
Lemma 3.1
The following relations hold:
Proof

1.
As
$$\begin{aligned} \langle u^{(8)}(x),L_{r}(x)\rangle &= \int _{1}^{1}u^{(8)}(x)L_{r}(x){\rm d}x. \end{aligned}$$Integrating the right hand terms of the above equation by parts leads to
$$\begin{aligned} \langle u^{(8)}(x),L_{r}(x)\rangle &= B_{T,8}+\sum _{k=4}^{7}(1)^{k+1}\left[ u^{(k)}(x)L_{r}^{(7k)}(x)\right] _{1}^{1}\\&\quad+\int _{1}^{1}u(x)L_{r}^{(8)}(x){\rm d}x, \end{aligned}$$where the boundary term
$$\begin{aligned} B_{T,8} &= \sum _{k=0}^{3}(1)^{k+1}\left[ u^{(k)}(x)L_{r}^{(7k)}(x)\right] _{1}^{1} \end{aligned}$$is zero using the boundary conditions defined in Eq. (2) yielding the relation.

2.
The inner product of \(\{a_{7}(x)u^{(7)}(x)\}\) with \(L_{r}(x)\) is obtained, as
$$\begin{aligned} \langle a_{7}(x)u^{(7)}(x),L_{r}(x)\rangle &= B_{T,7}+\sum _{k=4}^{6}(1)^{k}\left[ u^{(k)}(x)\{a_{7}(x)L_{r}(x)\}^{(6k)}\right] _{1}^{1}\\&\quad\int _{1}^{1}u(x)\{a_{7}(x)L_{r}(x)\}^{(7)}{\rm d}x, \end{aligned}$$where the boundary term
$$\begin{aligned} B_{T,7} &= \sum _{k=0}^{3}(1)^{k}\left[ u^{(k)}(x)\{a_{7}(x)L_{r}(x)\}^{(6k)}\right] _{1}^{1}=0 \end{aligned}$$
gives the relation. The other relations can be obtained similarly.
Theorem 3.1
If Eq. (8) is the assumed approximate solution of the boundary value problem (1)–(2), then the discrete system for determining the coefficients \(\{\alpha _{j}\}_{j=0}^{n}\) is given by
It can be written, in matrix form, as
where
and
The term \(\mu _{j,r}\) can be calculated using the results given in Preliminaries, while the boundary term \(\nu _{j,r}\) can be calculated as
After solving the linear system (21) having \((n+1)\) equations with \((n+1)\) unknowns yield, the column vector \(X=\left( \alpha _{0}, \alpha _{1}, \alpha _{2},\ldots , \alpha _{n}\right) ^{\rm T}\). Thus, u(x) can now be approximated by Eq. (8).
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 \(x(t)\in H^{k}]1, 1[\) (a Sobolev space) and let \(x_{n}(t)=\sum _{i=0}^{n}c_{i}L_{n}(t)\) be the best approximation polynomial of x(t) in the \(\ell ^{2}\) norm, then
and \(c_{0}\) is a nonnegative constant which depends on the selected norm and is free from x(t) and n.
Proof
Theorem 4.1
Assume \(\kappa :X\rightarrow X\) is bounded, with X a Banach space, and \(\lambda \kappa :X\rightarrow X\) is bijective. Further, assume
then for all sufficiently large n , say \(n\ge N\), the operator \((\lambda \kappa L_{n})^{1}\) exists as a bounded operator from X to X. Moreover, it is uniformly bounded such that
For the solution of \((\lambda \kappa L_{n})x_{m}=L_{n}y\), \(x_{m}\in X\) and \((\lambda \kappa )x=y\),
Proof
[31].
Consequently, the approximation rate of Legendre polynomials is \(n^{k}\) with respect to Lemma 4.1, and also from Theorem 4.1, \(\xx_{n}\\) converge to zero as soon as \(\xL_{n}\\).
Error analysis of the method
In this subsection, an error estimator for eighthorder boundary value problems using Legendre Galerkin approximation has been discussed.
Consider \(e_{n}(x)=u(x)u_{n}(x)\) as the error function of Legendre approximation \(u_{n}(x)\) to u(x), where u(x) is the exact solution of Eq. (1) with boundary conditions defined in Eq. (2). So, \(u_{n}(x)\) satisfies the following problem:
with boundary conditions
where \(P_{n}(x)\) is a perturbation term linked with \(u_{n}(x)\) obtained as follows
We find an approximation \(e_{n,N}(x)\) to \(e_{n}(x)\) 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
and
We solve this problem using the Legendre Galerkin method to get the approximation \(e_{n,N}(x)\).
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 \([1,1]\). Consider
subject to the following boundary conditions
Using the linear transformation \(t=\frac{ba}{2}x+\frac{b+a}{2}\), 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 \(\Psi (x)\) is the interpolating polynomial such that \(\Psi ^{(j)}(1)=\Theta _{j}\) and \(\Psi ^{(j)}(1)=\Phi _{j}\), \(j=0,1,2,3\). Also,
and
The problem takes the form:
subject to the following boundary conditions
where
Let
be an approximate solution of Eq. (32). Then,
be the approximate solution of Eq. (31). Using the inverse linear transformation \(x=\frac{2}{ba}t\frac{b+a}{ba}\) in Eq. (35) yields the approximate solution u(t) of Eq. (27).
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.
Example 1
Consider the following differential equation:
subject to the boundary conditions
The exact solution of the problem is \(u(x)=x(1x)e^{x}\).
The proposed method is implemented to the problem for \(n=10\). 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.
Example 2
Consider the following differential equation:
subject to the boundary conditions
The exact solution of the problem is \(u(x)=(x^{2}1)\sin x\).
The proposed method is implemented to the problem for \(n=10\). 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.
Example 3
Consider the following differential equation:
subject to the boundary conditions
The exact solution of the problem is \(u(x)=(x^{2}1)\sin x\).
The proposed method is implemented to the problem for \(n=10\). 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 eighthorder 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 \([1,1]\). 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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Elahi, Z., Akram, G. & Siddiqi, S.S. Numerical solution for solving special eighthorder linear boundary value problems using Legendre Galerkin method. Math Sci 10, 201–209 (2016). https://doi.org/10.1007/s4009601601949
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DOI: https://doi.org/10.1007/s4009601601949
Keywords
 Galerkin method
 Legendre polynomials
 Eighth order
 Numerical solutions