Research Article  Open Access
Mingzhu Li, Lijuan Chen, Qiang Ma, "The Numerical Solution of Linear Sixth Order Boundary Value Problems with Quartic BSplines", Journal of Applied Mathematics, vol. 2013, Article ID 962165, 7 pages, 2013. https://doi.org/10.1155/2013/962165
The Numerical Solution of Linear Sixth Order Boundary Value Problems with Quartic BSplines
Abstract
A quartic Bspline method is proposed for solving the linear sixth order boundary value problems. The method converts the boundary problem to solve a system of linear equations and obtains coefficients of the corresponding Bspline functions. The method has the convergence of two order. It develops not only the quartic spline approximate solution but also the higher order approximate derivatives. Two numerical examples are presented to verify the theoretical analysis and show the validity and applicability of the method. Compared with other existing recent methods, the quartic Bspline method is a more efficient and effective tool.
1. Introduction
In the paper, we consider the linear sixth order boundary value problems (6BVP) of the form: subject to the conditions where and are continuous functions on , and and are given finite real constants. Many mathematical models arising in various applications can be written as boundary value problems. One such problem is the sixth boundary value problem which plays an important role in astrophysics and the narrow convecting layers bounded by stable layers [1–4]. Further discussion of sixth order boundary value problems is given in [5, 6] and in a book by Chandrasekhar [7]. Theorems that list conditions for the existence and uniqueness of solution of such type of boundary value problems can be found in the book written by Agarwal [8]. However, it is difficult to obtain the analytic solutions of (1)(2). Therefore, the availability of numerical method for this problem is of practical importance.
Over the years, there are several authors who worked on this type of boundary value problems by using different methods. For example, finite difference method was developed by Boutayeb and Twizell [2, 3, 9]. A modified form of the decomposition method was established by Wazwaz [10] and used to solve such BVPs [11]. SincGalerkin method, variational iteration method, and homotopy perturbation method were developed to study the same problem [12–14]. Spline functions have been also used to construct efficient and accurate numerical methods for solving boundary value problems. For example, Siddiqi and Akram solved the same boundary value problems by using different splines such as quintic splines, septic splines, and nonpolynomial splines [15–17]. Loghmani and Viswanadham used sixth and septic Bspline functions to solve sixth order boundary value problems [18, 19]. Instead of the above two ways, other differential spline collocation methods can also be used [20–22].
It is well known that the quartic Bspline has been widely applied for the approximation solution of boundary value problems. Caglar used quartic Bspline to solve the linear cubic order boundary value problem [23]. Besides, quartic Bspline was also used to solve fifth order boundary value problems and the obstacle problems [24, 25]. Our method is based on quartic Bspline interpolation. It is second order convergent and with lower computational cost. Moreover, we also can get the approximate derivative values of at the knots. This is another advantage of our method, since some methods cannot obtain those results.
This paper is arranged as follows. In Section 2, the definition of quintic Bsplines has been described and some preliminary results of quartic Bspline interpolation have been presented. In Section 3, we mainly give the quartic Bspline solution of linear sixthorder boundary value problems based on the results. In Section 4, the convergence of the method has been demonstrated. In Section 5, numerical examples of linear boundary value problems are presented, which illustrate the performance of this method.
2. Quartic BSpline
2.1. Definition of Quartic BSpline
For an interval , we introduce a set of equally spaced knots of partition , and we assume that , , .
Let be the space of continuously differentiable, piecewise, quarticdegree polynomials on . Consider the Bsplines basis in . A detailed description of Bspline functions generated by subdivision can be founded in [26].
The zero degree Bspline is defined as and for positive constant , it is defined in the following recursive form:
We apply this recursion to get the quartic Bspline ; it is defined in as follows: The properties of quartic Bspline functions:(1)compact supported: (2)Normalization: .(3)Translation invariance: .(4)Derivation formula: , where By some trivial computations, we can obtain the value of , at the knots, which are listed in Table 1.

2.2. Quartic BSpline Interpolation
For a given function (assuming to be sufficiently smooth), there exists a unique quartic Bspline satisfying the interpolation conditions For , let , , , and for short. Through a simple calculation by Table 1, we have So, we get Using operator notation in [27, 28], we obtain where the operators are defined , , and . Let and expand them in powers of ; we get
We can use to construct numerical difference formula for , , and by the Taylor series expansion as follows: Substituting (9)–(12) into (15)–(18) yields
3. Description of Numerical Method
In the section, we give the quartic Bspline method for the linear sixth order boundary value problem. Let be the approximate solution of 6BVP (1)(2) and the approximate spline of . Discretize (1) at the knots , we get By (9) and (18), we turn (20) into where and are the value of and at the knots for short. Change (21) equivalently, we yield
Dropping the term from (22), we get a linear system with linear equations in unknowns so seven more equations are needed.
By the boundary conditions at we have
By the boundary conditions at we get
We can construct an approximate formula as follows: where the coefficients are determined by maximizing the error order.
Substituting (12) into (28), we have
Dropping the term from (29), we can get
Take (22), (25), (27), and (29) together, we get linear equation with as unknowns. The linear system can be written in matrix notations
Take (23), (25), (27), and (30) together, we get linear equation with as unknowns. The linear system can be written in matrix notations where
After solving the linear system (32), we obtain the quartic spline approximate solution . Furthermore, we can take as the approximation of .
4. Convergence Analysis
By (31) and (32), we have is invertible, and if we assume that then is also invertible.
Hence, we get
By (35) and (36) and note , we have
Hence,
Generally, we get
5. Numerical Results
In the section, we give some computational results of numerical experiments with method based on previous sections to support our theoretical discussion. We use double precision arithmetic in order to reduce the roundoff errors to a minimum.
Example 1. We consider the following equation: The exact solution is given by .
The results of maximum absolute errors for this problem are tabulated in Table 2.

Next, we compare our method with the other spline method. Consider another sixth order boundary value problem.
Example 2. Consider the boundary value problem which has the exact solution .
The example has been solved by the collocation method based on the sixth Bspline [18], and the numerical results are stated in Table 3. Also, the system of differential equation along with the given boundary conditions was solved by Wazwaz using Adomian decomposition method [10] and Noor using variational iteration method [13]. The respective maximum absolute errors are given in Table 3. Obviously, the results of our method are very encouraging.
6. Conclusion
In the section, we employ the quartic Bspline for solving the sixth order boundary value problems. Properties of the Bspline function are utilized to reduce the computation of this problem to some algebraic equations. The method is computationally attractive and applications are demonstrated through illustrative examples. The obtained results showed that this approach can solve the problem effectively, and the comparison shows that the proposed technique is in good agreement with the existing results in the literature.
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Copyright © 2013 Mingzhu Li 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.