Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems

doi:10.1016/j.amc.2006.06.053

Applied Mathematics and Computation

Volume 184, Issue 2, 15 January 2007, Pages 476-484

https://doi.org/10.1016/j.amc.2006.06.053 Get rights and content

Abstract

In this paper, quadratic and cubic polynomial and nonpolynomial spline functions based methods are presented to find approximate solutions to second order boundary value problems. Using these spline functions we drive a few consistency relations which to be used for computing approximations to the solution for second order boundary value problems. The present approaches have less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of the proposed methods.

Introduction

Many problems in science and technology are formulated mathematically in boundary value problems for second order differential equations as in heat transfer and deflection in cables.We shall consider a numerical solution of the following linear second order two-point boundary value problem, see [6] $y^{(2)} + f (x) y = g (x), x \in [a, b],$ subjected to Neumann boundary conditions: $y^{(1)} (a) - A_{1} = y^{(1)} (b) - A_{2} = 0,$ where A_i, i = 1, 2 are finite real constants. The functions f(x) and g(x) are continuous on the interval [a, b]. The analytical solution of (1.1) subjected to (1.2) can not be obtained for arbitrary choices of f(x) and g(x).

The numerical analysis literature contains little on the solution of second order two-point boundary value problems (1.1) subjected to Neumann boundary conditions (1.2). Albasiny and Raghavarao [1], [4] solved linear second order two-point boundary problem (1.1) subjected to Dirichlet boundary conditions using cubic polynomial spline. Blue [5] solved this problem using quintic polynomial spline. While, Caglar et al. [3] solved this problem using cubic B-spline. Arshad Khan solved it using parametric cubic spline. Zahra [6], also solved this problem using quadratic polynomial spline at midknots.

In this paper, we use both polynomial and nonpolynomial spline functions to develop numerical methods for obtaining smooth approximations for the solution of the problem (1.1) subjected to Neumann boundary conditions (1.2).

Section snippets

Derivation of the methods

We introduce a finite set of grid points x_i by dividing the interval [a, b] into n equal parts. $\begin{matrix} x_{i} = a + ih, i = 0, 1, \dots, n, \\ x_{0} = a, x_{n} = b and h = \frac{b - a}{n} . \end{matrix}$

Let y(x) be the exact solution of the system (1.1) and S_i be an approximation to y_i = y(x_i) obtained by the spline function Q_i(x) passing through the points (x_i, S_i) and (x_i+1, S_i+1).

Spline solutions

The spline solution of (1.1) with the boundary condition (1.2) is based on the linear equations given by (2.9), (2.10), (2.11) for quadratic polynomial spline and (2.16), (2.17), (2.18) for cubic spline solution and (2.25), (2.26), (2.27) for nonpolynomial spline.

Let Y = (y_i+1/2), S = (S_i+1/2), C = (C_i), T = (t_i), E = (e_i+1/2) = Y − S be n-dimensional column vectors. Then we can write the standard matrix equations for nonpolynomial spline method in the form. $\begin{matrix} (i) NY = C + T, \\ (ii) NS = C, \\ (iii) NE = T \end{matrix}$ $We also have N = N_{0} + \frac{h^{2}}{24} BF,$

Convergence analysis

Our main purpose now is to derive a bound on ∥E∥_∞. We now turn back to the error equation (iii) in (3.1) and rewrite it in the form $E = N^{- 1} T = {(M_{0} + J_{0} + \frac{h^{2}}{24} BF)}^{- 1} T = {(I + M_{0}^{- 1} (J_{0} + \frac{h^{2}}{24} M_{0}^{- 1} BF))}^{- 1} M_{0}^{- 1} T$ we get $‖ E ‖_{\infty} ⩽ \frac{‖ M_{0}^{- 1} ‖_{\infty} ‖ T ‖_{\infty}}{1 - ‖ M_{0}^{- 1} ‖_{\infty} ‖ J_{0} + \frac{h^{2}}{24} BF ‖_{\infty}}$ provided $‖ M_{0}^{- 1} ‖_{\infty} ‖ (J_{0} + \frac{h^{2}}{24} BF) ‖_{\infty} < 1$ .

It was shown in [2] that, $‖ M_{0}^{- 1} ‖_{\infty} = \frac{h^{- 2}}{8} ((b - a)^{2} + h^{2}) = O (h^{- 2}),$

Lemma 4.1

The matrix N given by (3.2) is nonsingular, provided that $(2 h^{- 2} + (ω + 2 α) | f (x) |) β < 1,$ where $β = \frac{1}{8} [(b - a)^{2} + h^{2}]$ .

The proof of Lemma 4.1 follows from the following statement. If T is square matrix

Numerical examples and discussion

We now consider two numerical examples illustrating the comparative performance of nonpolynomial spline method (ii) in (3.1) over quadratic and cubic polynomials spline methods. All calculations are implemented by MATLAB 6.

Example 1

Consider the boundary value problem $\begin{matrix} y^{(2)} + y = - 1, \\ y^{(1)} (0) = (1 - \cos (1)) / \sin (1) = - y^{(1)} (1) . \end{matrix}$ The analytical solution of (5.1) is $y (x) = \cos (x) + \frac{1 - \cos (1)}{\sin (1)} \sin (x) - 1 .$

Example 2

Consider the boundary value problem $\begin{matrix} y^{(2)} + xy = (3 - x - x^{2} + x^{3}) \sin x + 4 x \cos x, \\ y^{(1)} (0) = - 1, y^{(1)} (1) = 2 \sin (1) . \end{matrix}$ The analytical solution of (5.3)

Conclusion

Table 5, Table 6 show that the accuracy of the nonpolynomial spline method is better than the two polynomial spline methods (quadratic and cubic splines) and is less than half the errors due to quadratic and cubic polynomial splines. Moreover, Nonpolynomial spline method has less computational cost over other polynomial spline methods and by arbitrary choices of α and ω we get the other two methods. So the results obtained by nonpolynomial spline method are very encouraging over other existing

References (6)

Arshad Khan
Parametric cubic spline solution of two-point boundary value problems
Applied Mathematics and Computations
(2004)
C.V. Raghavarao et al.
A note on application of cubic splines to two-point boundary value problems
Computers and Mathematics with Applications
(1994)
E.L. Albasiny et al.
Cubic spline solution to two-point boundary value problems
Computer Journal
(1968)

There are more references available in the full text version of this article.

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Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems

Abstract

Introduction

Section snippets

Derivation of the methods

Spline solutions

Convergence analysis

Numerical examples and discussion

Conclusion

Applied Mathematics and Computations

Computers and Mathematics with Applications

Cubic spline solution to two-point boundary value problems

Computer Journal