The Numerical Solution of Linear Sixth Order Boundary Value Problems with Quartic B-Splines

Abstract

A quartic B-spline 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 B-spline 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 B-spline 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: 𝑦⁽⁶⁾(𝑥)+𝑓(𝑥)𝑦(𝑥)=𝑔(𝑥), (1) subject to the conditions 𝑦(𝑎)=𝐴₀,𝑦^′(𝑎)=𝐴₁,𝑦^′′(𝑎)=𝐴₂,𝑦(𝑏)=𝐵₀,𝑦^′(𝑏)=𝐵₁,𝑦^′′(𝑏)=𝐵₂, (2) where 𝑓(𝑥) and 𝑔(𝑥) are continuous functions on [𝑎,𝑏], and 𝐴_𝑖(𝑖=0,1,2) and 𝐵_𝑖(𝑖=0,1,2) 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]. Sinc-Galerkin 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 B-spline 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 B-spline has been widely applied for the approximation solution of boundary value problems. Caglar used quartic B-spline to solve the linear cubic order boundary value problem [23]. Besides, quartic B-spline was also used to solve fifth order boundary value problems and the obstacle problems [24, 25]. Our method is based on quartic B-spline interpolation. It is second order convergent and with lower computational cost. Moreover, we also can get the approximate derivative values of 𝑦^(𝑘)(𝑥)(𝑘=1,2,3,4,5) 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 B-splines has been described and some preliminary results of quartic B-spline interpolation have been presented. In Section 3, we mainly give the quartic B-spline solution of linear sixth-order 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 B-Spline

2.1. Definition of Quartic B-Spline

For an interval [𝑎,𝑏]⊂𝑅, we introduce a set of equally spaced knots of partition Ω={𝑥₀,𝑥₁,…,𝑥_𝑛}, and we assume that 𝑛≥5, 𝑥_𝑖=𝑎+𝑖ℎ(𝑖=0,1,…,𝑛), 𝑥₀=𝑎,𝑥_𝑛=𝑏.

Let 𝑆₄[𝜋] be the space of continuously differentiable, piecewise, quartic-degree polynomials on 𝜋. Consider the B-splines basis in 𝑆₄[𝜋]. A detailed description of B-spline functions generated by subdivision can be founded in [26].

The zero degree B-spline is defined as 𝑁_𝑖,0(𝑥)=1,𝑥∈𝑥_𝑖,𝑥_𝑖+1,0,otherwise, (3) and for positive constant 𝑝, it is defined in the following recursive form: 𝑁_𝑖,𝑝(𝑥)=𝑥−𝑥_𝑖𝑥_𝑖+𝑝−𝑥_𝑖𝑁_{𝑖,𝑝−1}(𝑥)+𝑥_𝑖+𝑝+1−𝑥𝑥_𝑖+𝑝+1−𝑥_𝑖+1𝑁_{𝑖+1,𝑝−1},𝑝≥2. (4)

We apply this recursion to get the quartic B-spline 𝑁_𝑖,4(𝑥)(𝑖=−2,−1,…,𝑛+1); it is defined in 𝑆₄[𝜋] as follows: 𝑁_𝑖,4(𝑥)=124ℎ⁴𝑥−𝑥_𝑖−2⁴,𝑥∈𝑥_𝑖−2,𝑥_𝑖−1,𝑥−𝑥_𝑖−2⁴−5𝑥−𝑥⁴_𝑖−1,𝑥∈𝑥_𝑖−1,𝑥_𝑖,𝑥−𝑥_𝑖−2⁴−5𝑥−𝑥_𝑖−1⁴+10𝑥−𝑥_𝑖⁴,𝑥∈𝑥_𝑖,𝑥_𝑖+1,𝑥−𝑥_𝑖+3⁴−5𝑥−𝑥_𝑖+2⁴,𝑥∈𝑥_𝑖+1,𝑥_𝑖+2,𝑥−𝑥_𝑖+3⁴,𝑥∈𝑥_𝑖+2,𝑥_𝑖+3,0,otherwise. (5) The properties of quartic B-spline functions:(1)compact supported: 𝑁_𝑖,4(𝑥)=≥0,𝑥∈𝑥_𝑖−2,𝑥_𝑖+3,=0,otherwise. (6)(2)Normalization: ∑^𝑛+1_𝑖=−2𝑁_𝑖,4(𝑥)=1.(3)Translation invariance: 𝑁_𝑖,4(𝑥)=𝑁_0,4(𝑥−(𝑖−1)ℎ)(𝑖=−2,−1,…,𝑛+1).(4)Derivation formula: 𝑁^(𝑘)_𝑖,4(𝑥)=(4!/(4−𝑘)!)∑^𝑛_𝑗=1𝛼_𝑘,𝑗𝑁_{𝑖+𝑗,4−𝑘}, where 𝛼_0,0𝛼=1,_𝑘,0=𝛼_𝑘−1,0𝑥_{𝑖+3−𝑘}−𝑥_𝑖,𝛼_𝑘,𝑘=−𝛼_{𝑘−1,𝑘−1}𝑥_𝑖+5−𝑥_𝑖+𝑘,𝛼_𝑘,𝑗=𝛼_{𝑘−𝑗,𝑗}−𝛼_{𝑘−1,𝑗−1}𝑥_{𝑖+𝑗+5−𝑘}−𝑥_𝑖+𝑗. (7)By some trivial computations, we can obtain the value of 𝑁^(𝑘)_𝑖,4(𝑥)(𝑖=−2,−1,…,𝑛+1, 𝑘=0,1,2,3) at the knots, which are listed in Table 1.

Table 1 

The values of 𝑁^(𝑘)_𝑖,4(𝑥) at the knots.

2.2. Quartic B-Spline Interpolation

For a given function 𝑦(𝑥) (assuming to be sufficiently smooth), there exists a unique quartic B-spline 𝑠(𝑥)=∑^𝑖=𝑛+1_𝑖=−2𝑐_𝑖𝑁_𝑖,4(𝑥) satisfying the interpolation conditions 𝑠𝑥_𝑖=𝑦𝑥_𝑖,(),𝑠𝑖=0,1,…,𝑛_′(𝑎)=𝑦^′(𝑎),𝑠^′′(𝑎)=𝑦^′′(𝑎),𝑠^′(𝑏)=𝑦^′(𝑏),𝑠^′′(𝑏)=𝑦^′′(𝑏). (8) For 𝑗=0,1,…,𝑛, let 𝑦_𝑗=𝑠(𝑥_𝑗)=𝑦(𝑥_𝑗), 𝑃_𝑗=𝑠^′(𝑥_𝑗), 𝑄_𝑗=𝑠^′′(𝑥_𝑗), and 𝑅_𝑗=𝑠⁽³⁾(𝑥_𝑗) for short. Through a simple calculation by Table 1, we have 𝑦_𝑗=^𝑛+1_𝑖=−2𝑐_𝑖𝑁_𝑖,4𝑥_𝑗=1𝑐24^𝑗−2+11𝑐_𝑗−1+11𝑐_𝑗+𝑐_𝑗+1,𝑃_𝑗=^𝑛+1_𝑖=−2𝑐_𝑖𝑁^′_𝑖,4𝑥_𝑗=16ℎ−𝑐_𝑗−2−3𝑐_𝑗−1+3𝑐_𝑗+𝑐_𝑗+1,𝑄_𝑗=^𝑛+1_𝑖=−2𝑐_𝑖𝑁^′′_𝑖,4𝑥_𝑗=12ℎ²𝑐_𝑗−2−𝑐_𝑗−1−𝑐_𝑗+𝑐_𝑗+1,𝑅_𝑗=^𝑛+1_𝑖=−2𝑐_𝑖𝑁^′′′_𝑖,4𝑥_𝑗=1ℎ³−𝑐_𝑗−2+3𝑐_𝑗−1−3𝑐_𝑗+𝑐_𝑗+1. (9)(10)(11)(12) So, we get ℎ4𝑃_𝑗−2+11𝑃_𝑗−1+11𝑃_𝑗+𝑃_𝑗+1=−𝑦_𝑗−2−3𝑦_𝑗−1+3𝑦_𝑗+𝑦_𝑗+1,ℎ²𝑄12^𝑗−1+10𝑄_𝑗+𝑄_𝑗+1=𝑦_𝑗−1−2𝑦_𝑗+𝑦_𝑗+1,ℎ³𝑅24^𝑗−2+11𝑅_𝑗−1+11𝑅_𝑗+𝑅_𝑗+1=−𝑦_𝑗−2+3𝑦_𝑗−1−3𝑦_𝑗+𝑦_𝑗+1. (13) Using operator notation in [27, 28], we obtain 𝑃_𝑗=4ℎ−𝐸⁻²−3𝐸⁻¹+3𝐼+𝐸𝐸⁻²+11𝐸⁻¹𝑦+11𝐼+𝐸^𝑗,𝑄_𝑗=12ℎ²𝐸⁻¹−2𝐼+𝐸𝐸⁻¹𝑦+10𝐼+𝐸^𝑗,𝑅_𝑗=24ℎ³−𝐸⁻²+3𝐸⁻¹−3𝐼+𝐸𝐸⁻²+11𝐸⁻¹𝑦+11𝐼+𝐸^𝑗, (14) where the operators are defined 𝐸𝑦(𝑥)=𝑦(𝑥+ℎ), 𝐷𝑦(𝑥)=𝑦^′(𝑥), and 𝐼𝑦(𝑥)=𝑦(𝑥). Let 𝐸=exp(ℎ𝐷) and expand them in powers of ℎ𝐷; we get 𝑦^′𝑥_𝑗=𝑃_𝑗−1ℎ720⁴𝑦⁽⁵⁾𝑥_𝑗+𝑂ℎ⁶,𝑦^′′𝑥_𝑗=𝑄_𝑗+1ℎ240⁴𝑦⁽⁶⁾𝑥_𝑗+𝑂ℎ⁶,𝑦^′′′𝑥_𝑗=𝑅_𝑗−1ℎ12²𝑦⁽⁵⁾𝑥_𝑗+1ℎ240⁴𝑦⁽⁷⁾𝑥_𝑗+𝑂ℎ⁶. (15)

We can use 𝑅_𝑗 to construct numerical difference formula for 𝑦⁽⁴⁾(𝑥_𝑗), 𝑦⁽⁵⁾(𝑥_𝑗), and 𝑦⁽⁶⁾(𝑥_𝑗)(𝑗=1,2,…,𝑛−1) by the Taylor series expansion as follows: 𝑦⁽⁴⁾𝑥_𝑗=𝑅_𝑗+1−𝑅_𝑗−1ℎ2ℎ+𝑂²,𝑦⁽⁵⁾𝑥_𝑗=𝑅_𝑗+1−2𝑅_𝑗+𝑅_𝑗−1ℎ²+𝑂ℎ²,𝑦⁽⁶⁾𝑥_𝑗=𝑅_𝑗+2−𝑅_𝑗𝑅2ℎ−2^𝑗+1−𝑅_𝑗−1+𝑅2ℎ_𝑗−𝑅_𝑗−2×ℎ2ℎ²⁻¹+𝑂ℎ²,=𝑅_𝑗+2−2𝑅_𝑗+1+2𝑅_𝑗−1−𝑅_𝑗−22ℎ³+𝑂ℎ². (16)(17)(18) Substituting (9)–(12) into (15)–(18) yields 𝑦^′𝑥_𝑗=16ℎ−𝑐_𝑗−2−3𝑐_𝑗−1+3𝑐_𝑗+𝑐_𝑗+1+𝑂ℎ⁴,𝑦^′′𝑥_𝑗=12ℎ²𝑐_𝑗−2−𝑐_𝑗−1−𝑐_𝑗+𝑐_𝑗+1+𝑂ℎ⁴,𝑦^′′′𝑥_𝑗=1ℎ³−𝑐_𝑗−2+3𝑐_𝑗−1−3𝑐_𝑗+𝑐_𝑗+1+𝑂ℎ²,𝑦⁽⁴⁾𝑥_𝑗=12ℎ⁴𝑐_𝑗−3−3𝑐_𝑗−2+2𝑐_𝑗−1+2𝑐_𝑗−3𝑐_𝑗+1+𝑐_𝑗+2+𝑂ℎ²,𝑦⁽⁵⁾𝑥_𝑗=1ℎ⁵−𝑐_𝑗−3+5𝑐_𝑗−2−10𝑐_𝑗−1+10𝑐_𝑗−5𝑐_𝑗+1+𝑐_𝑗+2+𝑂ℎ⁴,𝑦⁽⁶⁾𝑥_𝑗=12ℎ⁶𝑐_𝑗−4−5𝑐_𝑗−3+9𝑐_𝑗−2−5𝑐_𝑗−1−5𝑐_𝑗+9𝑐_𝑗+1−5𝑐_𝑗+2+𝑐_𝑗+3+𝑂ℎ². (19)

3. Description of Numerical Method

In the section, we give the quartic B-spline method for the linear sixth order boundary value problem. Let 𝑠(𝑥)=∑^𝑛+1_𝑖=−2𝑐_𝑖𝑁_𝑖,4(𝑥) be the approximate solution of 6BVP (1)-(2) and 𝑠(𝑥)=∑^𝑛+1_𝑖=−2𝑐_𝑖𝑁_𝑖,4(𝑥) the approximate spline of 𝑠(𝑥). Discretize (1) at the knots 𝑥_𝑖(𝑖=2,3,…,𝑛−2), we get 𝑦⁽⁶⁾𝑥_𝑖+𝑓𝑥_𝑖𝑦𝑥_𝑖=𝑔𝑥_𝑖. (20) By (9) and (18), we turn (20) into 12ℎ⁶𝑐_𝑖−4−5𝑐_𝑖−3+9𝑐_𝑖−2−5𝑐_𝑖−1−5𝑐_𝑖+9𝑐_𝑖+1−5𝑐_𝑖+2+𝑐_𝑖+3+𝑓𝑥_𝑖×𝑐24_𝑖−2+11𝑐_𝑖−1+11𝑐_𝑖+𝑐_𝑖+1=𝑔𝑥_𝑖+𝑂ℎ², (21) where 𝑓(𝑥_𝑖) and 𝑔(𝑥_𝑖)are the value of 𝑓(𝑥) and 𝑔(𝑥) at the knots 𝑥_𝑖(𝑖=2,3,…,𝑛−2) for short. Change (21) equivalently, we yield 12𝑐_𝑖−4−5𝑐_𝑖−3+9𝑐_𝑖−2−5𝑐_𝑖−1−5𝑐_𝑖+9𝑐_𝑖+1−5𝑐_𝑖+2+𝑐_𝑖+3+𝑓𝑥_𝑖×𝑐_𝑖−2+11𝑐_𝑖−1+11𝑐_𝑖+𝑐_𝑖+1ℎ⁶=24ℎ⁶𝑔𝑥_𝑖+𝑂ℎ⁸. (22)

Dropping the term 𝑂(ℎ⁸) from (22), we get a linear system with 𝑛−3 linear equations in 𝑛+4  unknowns 𝑐_𝑖(𝑖=−2,−1,…,𝑛+1)12𝑐_𝑖−4−5𝑐_𝑖−3+9𝑐_𝑖−2−5𝑐_𝑖−1−5𝑐_𝑖+9𝑐_𝑖+1−5𝑐_𝑖+2+𝑐_𝑖+3+𝑓𝑥_𝑖×𝑐_𝑖−2+11𝑐_𝑖−1+11𝑐_𝑖+𝑐_𝑖+1ℎ⁶=24ℎ⁶𝑔𝑥_𝑖(𝑖=2,3,…,𝑛−2), (23) so seven more equations are needed.

By the boundary conditions at 𝑥=𝑎𝑦(𝑎)=𝐴₀,𝑦^′(𝑎)=𝐴₁,𝑦^′′(𝑎)=𝐴₂, (24) we have 𝑐₋₂+11𝑐₋₁+11𝑐₀+𝑐₁=24𝐴₀,−𝑐₋₂−3𝑐₋₁+3𝑐₀+𝑐₁=6ℎ𝐴₁,𝑐₋₂−𝑐₋₁−𝑐₀+𝑐₁=2ℎ²𝐴₂. (25)

By the boundary conditions at 𝑥=𝑏𝑦(𝑏)=𝐵₀,𝑦^′(𝑏)=𝐵₁,𝑦^′′(𝑏)=𝐵₂, (26) we get 𝑐_𝑛−2+11𝑐_𝑛−1+11𝑐_𝑛+𝑐_𝑛+1=24𝐵₀,−𝑐_𝑛−2−3𝑐_𝑛−1+3𝑐_𝑛+𝑐_𝑛+1=6ℎ𝐵₁,𝑐_𝑛−2−𝑐_𝑛−1−𝑐_𝑛+𝑐_𝑛+1=2ℎ²𝐵₂. (27)

We can construct an approximate formula as follows: 𝑦⁽⁶⁾(𝑎)=−5𝑅₀+18𝑅₁−24𝑅₂+14𝑅₃−3𝑅₄24ℎ³+𝑂ℎ², (28) where the coefficients are determined by maximizing the error order.

Substituting (12) into (28), we have 5𝑐₋₂−33𝑐₋₁+93𝑐₀−145𝑐₁+135𝑐₂−75𝑐₃+23𝑐₄−3𝑐₅=24ℎ⁶𝑦⁽⁶⁾(𝑎)+𝑂ℎ⁵. (29)

Dropping the term 𝑂(ℎ⁵) from (29), we can get 5𝑐₋₂−33𝑐₋₁+93𝑐₀−145𝑐₁+135𝑐₂−75𝑐₃+23𝑐₄−3𝑐₅=24ℎ⁶𝑦⁽⁶⁾(𝑎). (30)

Take (22), (25), (27), and (29) together, we get 𝑛+4 linear equation with 𝑐_𝑖(𝑖=−2,−1,…,𝑛+1) as unknowns. The linear system can be written in matrix notations 𝐴+ℎ⁶𝐹𝐵𝐶=𝐷+𝐸. (31)

Take (23), (25), (27), and (30) together, we get 𝑛+4 linear equation with 𝑐_𝑖̃(𝑖=−2,−1,…,𝑛+1) as unknowns. The linear system can be written in matrix notations 𝐴+ℎ⁶𝐹𝐵𝐶=𝐷, (32) where 𝐶=𝑐₋₂,𝑐₋₁,𝑐₀,…,𝑐_𝑛+1^𝑇,𝐶=𝑐̃₋₂,𝑐̃₋₁,𝑐₀̃,…,𝑐̃_𝑛+1^𝑇,𝐷=24ℎ⁶𝑦⁽⁶⁾(𝑎),24𝐴₀,6ℎ𝐴₁,2ℎ²𝐴₂,24ℎ⁶𝑔𝑥₂,…,24ℎ⁶𝑔𝑥_𝑛−1,24𝐵₀,6ℎ𝐵₁,2ℎ²𝐵₂^𝑇,𝐸=𝑒₋₂,𝑒₋₁,…,𝑒_𝑛+1^𝑇,𝑒₋₁=𝑒₀=𝑒₁=𝑒_𝑛−1=𝑒_𝑛=𝑒_𝑛+1𝑒=0,₋₂=𝑂ℎ⁵,𝑒_𝑖=𝑂ℎ⁸(),𝐴=𝑖=2,…,𝑛−2_{5−3393−145135−7523−3111111−1−3311−1−1112−60108−60−60108−6012⋱⋱⋱⋱⋱⋱⋱⋱12−60108−60−60108−6012111111−1−3311−1−11},𝐹=000𝑓𝑥₂⋱𝑓𝑥_𝑛−1000,𝐵=.000000000111111000⋱⋱⋱⋱⋱⋱⋱000111111000000000 (33)

After solving the linear system (32), we obtain the quartic spline approximate solution 𝑦(𝑥)≈𝑠(̃𝑥)=∑^𝑛+1_𝑖=−2𝑐_𝑖̃𝑁_𝑖,4(𝑥). Furthermore, we can take 𝑠̃^(𝑘)(𝑥)(𝑘=1,2,3,4,5) as the approximation of 𝑦^(𝑘)(𝑥)(𝑘=1,2,3,4,5).

4. Convergence Analysis

By (31) and (32), we have 𝐴+ℎ⁶𝐹𝐵𝐶−𝐶=𝐸. (34)𝐴 is invertible, and if we assume that ℎ⁶𝐴⁻¹_∞‖𝐹‖_∞‖𝐵‖_∞<1, (35) then (𝐼+ℎ⁸𝐴⁻¹𝐹𝐵) is also invertible.

Hence, we get 𝐶−𝐶=𝐼+ℎ⁶𝐴⁻¹𝐹𝐵⁻¹𝐴⁻¹𝐸. (36)

By (35) and (36) and note ‖𝐸‖_∞≤𝑘ℎ⁸, we have 𝐶−𝐶_∞≤𝐴⁻¹_∞‖𝐸‖_∞1−ℎ⁶𝐴⁻¹_∞‖𝐹‖_∞‖𝐵‖_∞≤𝑘ℎ⁶𝐴⁻¹_∞1−ℎ⁶𝐴⁻¹_∞‖𝐵‖_∞‖𝐹‖_∞ℎ²≤𝑘‖𝐵‖_∞‖𝐹‖_∞ℎ²=𝑂ℎ². (37)

Hence, ‖𝑠−𝑠‖̃_∞≤𝐶−𝐶_∞^𝑛_𝑖=1𝑁_𝑖,4(𝑥)=𝐶−𝐶_∞=𝑂ℎ². (38)

Generally, we get 𝑦−𝑠̃_∞≤𝑦−𝑠̃_∞+‖𝑠−𝑠‖̃_∞=𝑂ℎ²+𝑂ℎ²=𝑂ℎ². (39)

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 round-off errors to a minimum.

Example 1. We consider the following equation: 𝑦⁽⁶⁾(𝑥)(𝑥)𝑦(0)+𝑦=6cos𝑥,0≤𝑥≤1,=0,𝑦^′(0)=−1,𝑦^′′(0)𝑦(1)=2,=0,𝑦^′(1)=sin1,𝑦^′′(1)=2cos1. (40) The exact solution is given by 𝑦(𝑥)=(𝑥−1)sin𝑥.

The results of maximum absolute errors max[𝑦^(𝜈)(𝑥_𝑖)]=max_{1⩽𝑖⩽𝑛}|𝑦^(𝜈)(𝑥_𝑖)−𝑠^(𝜈)(𝑥_𝑖)|(𝜈=0,1,2,3,4,5) for this problem are tabulated in Table 2.

Table 2 

Max[𝑦^(𝜈)(𝑥_𝑖)] of quartic B-spline method for Example 1.

Next, we compare our method with the other spline method. Consider another sixth order boundary value problem.

Example 2. Consider the boundary value problem 𝑦⁽⁶⁾(𝑥)−𝑦(𝑥)=−6𝑒^𝑥𝑦(0),0≤𝑥≤1,=1,𝑦^′(0)=0,𝑦^′′(0)𝑦(1)=−1,=0,𝑦^′(1)=−𝑒,𝑦^′′(1)=−2𝑒, (41) which has the exact solution 𝑦(𝑥)=(1−𝑥)𝑒^𝑥.

The example has been solved by the collocation method based on the sixth B-spline [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.

Table 3 

Max[𝑦(𝑥_𝑖)] of different methods for Example 2.

6. Conclusion

In the section, we employ the quartic B-spline for solving the sixth order boundary value problems. Properties of the B-spline 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.