Solutions of Seventh Order Boundary Value Problems Using Ninth Degree Spline Functions and Comparison with Eighth Degree Spline Solutions

Parcha Kalyani^*, Mihretu Nigatu Lemma

School of Mathematical & Statistical Sciences, Hawassa University, Hawassa, Ethiopia.

DOI: 10.4236/jamp.2016.42032                

Abstract

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.

Keywords

Seventh Order Boundary Value Problem, Boundary Value Problem, Ninth Degree Spline Function, Absolute Error

Received 10 June 2015; accepted 19 February 2016; published 22 February 2016

1. Introduction

Consider the linear seventh order differential equation

with the boundary conditions

Generally, this problem is difficult to solve analytically. Several numerical and semi-analytical methods have been developed for solving high order boundary value problems. For instance, a different approach of solving linear two-point boundary value problem has first been suggested by Bickley in 1968 [1] . He used cubic spline interpolation to model the solution curve and applied the differential equation as well as the boundary conditions to solve for the unknown constants. As a result, a set of equations can be produced approximating the analytical solution. Numerical methods based on spline functions generate solutions of ordinary and partial differential equations of high accuracy. The first place that the word “spline” is used in connection with smooth, piecewise polynomial approximation with mathematical reference has been made in the year 1946 by Schoenberg [2] . In late 1960’s, there were no handful of articles mentioning spline functions. Maclaren [3] , Rubin and Khosla [4] , Sastry [5] , Schoenberg [6] made great contributions in the development of splines. Convergence properties of the cubic spline method have been discussed by Ahlberg and Nilson [7] . Univariate splines have been studied intensely in 60’s. By the mid 70’s, splines were well understood to permit a fairly comprehensive treatment in the form of books. Some of the books which discuss splines include Ahlberg et al. [8] , deBoor [9] , Prenter [10] , Schumaker [11] , Shikin and Plis [12] , Spath [13] . The earlier studies [14] - [16] employed spline functions for the smooth approximate solution of ordinary and partial differential equations. Spline functions of various degrees have been demonstrated by them using approximate methods of solving second, third, fourth and fifth order linear boundary value problems. There are number of research articles published on this subject, yet it remains an active research area. Techniques such as quadratic, cubic, quartic, quintic, sextic, septic and higher degree splines are used to discuss the numerical solution of linear and nonlinear BVPs. Kumar and Srivastava [17] have given a survey on recent spline techniques for solving boundary value problems in ordinary differential equations using cubic, quintic and sextic polynomial and non-polynomial splines. Thomas [18] presents extensive use of splines for Boeing.

In the present paper, the seventh order boundary value problems are solved using ninth degree spline approximation and compared with the solution obtained by eighth degree spline solution [19] .

2. Construction of Ninth Degree Spline

We divide the interval into n subintervals with grid points, starting at x₀, the function y(x) in the interval is represented by ninth degree spline S(x), which is an approximate solution of y(x).

proceeding to the next interval, we add a term, proceeding in to the next interval we add another term and so on until we reach x_n. Thus the function y(x) is represented in the form

(1)

It can be shown that S(x) and its first six derivatives are continuous across nodes.

Method of Obtaining the Solution of Seventh Order Boundary Value Problems Using Ninth Degree Spline Function

Consider the linear seventh order differential equation

(2)

with the boundary conditions

(3)

From (3), and taking spline approximation in (2) at for, we get (n + 8) equations in (n + 9) unknowns,. To have the solution for the unknowns one more equation is required. So we assume that, after determining these unknowns we substitute them in (1) and

thus we get ninth degree spline approximation of. Putting in the spline function thus determined we get the solution at the grid points. The system of equations to be satisfied by the coefficients is derived below. From Equation (1) we get

(4)

Substituting (1) and (4) in the differential Equation (2) at we get

(5)

where and.

Since S(x) approximates, from (1) and from the boundary conditions (3) we obtain

(6)

(7)

(8)

(9)

(10)

(11)

(12)

From (5)-(12) we have (n + 8) equations, if these equations are taken in the order (7), (9), and (11) with, (12), (10), (8) and (6) the coefficient matrix of the unknowns, , k, j, h, g, e, d, c,

b, a will be an upper triangular matrix with two lower sub diagonals. The forward elimination is then simple with only two multipliers at each step, and back substitution is correspondingly easy.

3. Numerical Illustrations

In this section we consider three linear boundary value problems. Their numerical solution and absolute errors are given at different step lengths. The approximate solution, exact solutions and absolute errors at the grid points are summarized in tabular form. Further the approximate solution and exact solution have been shown graphically. The comparison of maximum absolute errors at different step lengths has been presented in tabular form.

3.1. Example 1

Consider the linear non homogeneous seventh order boundary value problem with constant coefficients.

(13)

With the boundary conditions

(14)

The exact solution is

We find the solution of (13)-(14) by taking step lengths h = 0.2 and h = 0.1 at equal subintervals.

Solution with h = 0.2

The ninth degree spline S(x) which approximates y(x) is given by

(15)

where, , , , ,.

We have 13 unknowns and the conditions to be satisfied by these unknowns are

(16)

Since, , , , it follows that a = 0, b = 1, c = 0 and d = −0.5, hence the spline S(x) reduces to the form

(17)

Differentiating (17)

(18)

(19)

(20)

and the seventh derivative is

(21)

Solving set of equations obtained from (16) we get the following values,

, ,

, ,

,

,

Substituting these values in Equation (15) we get the spline approximation of.

The values of, and the corresponding absolute errors at have been given in the Table 1 and the comparison has been shown in Figure 1.

Solution with h = 0.1

Since h = 0.1 we suppose the grid points x₀, x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀, where, x₀ = 0, x₁ = 0.1, x₂ = 0.2, x₃ = 0.3, x₄ = 0.4, x₅ = 0.5, x₆ = 0.6, x₇ = 0.7, x₈ = 0.8, x₉ = 0.9, x₁₀ = 1.

From Equation (1) ninth degree spline S(x) which approximate s u(x) becomes

(22)

From Equation (22) and the boundary conditions we get the following values

, ,

, ,

, ,

Figure 1. Comparison of approximate solution and exact solution for example 1 with h = 0.2.

Table 1. Numerical solution, exact solution and absolute error of example 1with h = 0.2.

, ,

, ,

Substituting these values in Equation (22) we get the spline approximation of. The values of, and the corresponding absolute errors at x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀ have been given in the Table 2 and the comparison has been shown in Figure 2.

3.2. Example 2

Consider non-homogeneous linear seventh order boundary value problem with variable coefficients

(23)

Figure 2. Comparison of and for example1 with h = 0.1.

Table 2. Numerical solution, exact solution and absolute error of example 1 with h = 0.1.

Subject to the boundary conditions

(24)

The exact solution is

We find the solution of (23)-(24) by taking the step lengths h = 0.2 and h = 0.1 at equal sub intervals.

Solution when h = 0.2

Since h = 0.2 we suppose the grid points, , , , ,

From Equation (1) ninth degree spline S(x) which approximates u(x).

(25)

From S(x) and boundary conditions we get the following values.

a = 1, b = 0, c = −0.5, d = −0.333333 Equation (25) reduces to the form

(26)

From equation

we get j = −0.00119047619047 and from the remaining conditions we have the following values

, , ,

, ,

, ,

Substituting these values in Equation (25) we get the spline approximation S(x) of u(x). The values of S(x), u(x) and the corresponding absolute errors at x₁, x₂, x₃ and x₄ has been given in Table 3 and the comparison has been shown in Figure 3.

Figure 3. Comparison of approximate solution and exact solution of example 2 with h = 0.2.

Table 3. Numerical solution S(x), exact solution u(x) and absolute error of example 2 with h = 0.2.

Solution when h = 0.1

Since h = 0.1 we suppose the grid points x₀, x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀ where

x₀ = 0, x₁ = 0.1, x₂ = 0.2, x₃ = 0.3, x₄ = 0.4, x₅=0.5, x₆ = 0.6, x₇ = 0.7, x₈ = 0.8, x₉ = 0.9, x₁₀ = 1

From Equation (1) ninth degree spline S(x) which approximates u(x) becomes

(27)

From S(x) and boundary conditions we get the following values.

a = 1, b = 0, c= −0.5, d = −0.333333, with these values (27) reduces to the form

(28)

proceeding as in the above, we get the following values

, ,

, ,

, ,

, ,

, ,

The values of S(x), u(x) and the corresponding absolute errors at x₁, x₂, ₃, x₄, x₅, x₆, x₇, x₈, x₉, x₁₀ has been given in Table 4 and the comparison has been shown in Figure 4.

3.3. Example 3

Consider the linear non-homogeneous seventh order boundary value problem with constant coefficients.

subject to the boundary conditions

(29)

The exact solution is

We find the solution of (29) by taking the step lengths h = 0.2 and h = 0.1 at equal sub intervals.

The values of S(x), u(x) and the corresponding absolute errors at x₁, x₂, x₃ and x₄ have been given in Table 5 and the comparison has been shown in Figure 5. The values at x₁, x₂, ₃, x₄, x₅, x₆, x₇, x₈, x₉ and x₁₀ have been given in Table 6 and the comparison has been shown in Figure 6.

Figure 4. Comparison of approximate solution and exact solution of example 2 with h = 0.1.

Figure 5. Comparison of approximate solution and exact solution of example 3 with h = 0.2.

Figure 6. Comparison of approximate solution and exact solution of example 3with h = 0.1.

Table 4. Numerical solution S(x), exact solution u(x) and absolute error of example 2 with h = 0.2.

Table 5. Numerical solution S(x), exact solution u(x), absolute error of example 3 with h = 0.2.

Table 6. Numerical solution S(x), exact solution u(x) and absolute error of example 3 with h = 0.1.

4. Comparative Study of Eighth Degree Spline and Ninth Degree Spline Approximation

The numerical results obtained by ninth degree spline approximation are compared with the numerical results obtained by eighth degree spline approximation [11] at different step lengths. Comparison is given in tabular form and shown graphically (Tables 7-9, Figures 7-9).

Figure 7. Comparison of the absolute error [11] and ninth degree spline approximation for example 1 at h = 0.1.

Figure 8. Comparison for absolute error of [11] and ninth degree spline approximation for example 2 at h = 0.1.

Figure 9. Comparison for absolute error of [11] and ninth degree spline approximation for example 3 at h = 0.1.

Table 7. Comparison of absolute errors [11] and absolute errors obtained by our method for example 1 at h = 0.1.

Table 8. Comparison of absolute error [11] and absolute error obtained by our method for example 2 at h = 0.1.

Table 9. Comparison of absolute error [11] and absolute error obtained by our method for example 3 at h= 0.1.

5. Conclusion

A ninth degree spline solution has been employed of example 1, 2 and 3 at step lengths h = 0.2 and h = 0.1. Numerical solutions are summarized in the tables and the comparison has been shown in figures. The maximum absolute errors at the given step length are −2.90100 × 10⁻⁹ and 7.2899 × 10⁻¹⁰, 8.6950 × 10⁻⁹, 9.00000 × 10⁻¹⁰, 2.2970 × 10⁻⁹ and 9.0000 × 10⁻¹¹ respectively. These values show that the agreement between approximate solution and exact solution is good. It is observed that the solution is more accurate when step length is small. We also compare our results with the results obtained using eighth degree spline solution [11] . From the tables and graphs, we conclude that the ninth degree spline solutions are more accurate (10 - 11) than the solutions obtained by using eighth degree spline functions.

Conflicts of Interest

The authors declare no conflicts of interest.

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