Solution of sixth order boundary value problems using non-polynomial spline technique

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Abstract

Non-polynomial spline is used for the numerical solutions of the sixth order linear special case boundary value problems. The method presented in this paper has also been proved to be second order convergent. Two examples are considered for the numerical illustrations of the method developed. The method is also compared with those developed by El-Gamel et al. [M. El-Gamel, J.R. Cannon, J. Latour, A.I. Zayed, Sinc-Galerkin method for solving linear sixth-order boundary-value problems, Math. Comput. 73 (247) (2003) 1325–1343] and Siddiqi and Twizell [S.S. Siddiqi, E.H. Twizell, Spline solutions of linear sixth-order boundary value problems, Int. J. Comput. Math. 60 (1996) 295–304], as well and is observed to be better.

Introduction

Sixth order boundary value problems arise in astrophysics, i.e., the narrow convecting layers bounded by stable layers which are believed to surround A-type stars may be modelled by sixth order boundary value problems [2].

Chandrasekhar [3] determined that when an infinite horizontal layer of fluid is heated from below and is under the action of rotation, instability sets in. When this instability is as ordinary convection, the ordinary differential equation is sixth order.

Agarwal [4] presented the theorems stating the conditions for the existence and uniqueness of solutions of sixth order boundary value problems, while no numerical methods are contained therein.

Boutayeb and Twizell [5] developed a family of numerical methods for the solution of special nonlinear sixth-order boundary value problems.

Siddiqi and Twizell [6] presented the solution of sixth order boundary value problem using the sextic spline. El-Gamel et al. [1] used Sinc-Galerkin method for the solutions of sixth order boundary value problems.

In this paper, non-polynomial spline function is used to develop a technique for the solution of sixth order boundary value problem, extending the method for non-polynomial spline solution of fifth order boundary value problem, developed by Siddiqi and Ghazala [7]. The method developed, is claimed to be better than those developed by El-Gamel et al. [1] and Siddiqi and Twizell [6], as discussed in Example 1, Example 2.

The method is of order two for arbitrary choices of α, β, γ and δ such that α+β+γ+δ2=12.

Consider the following boundary value problemy(6)(x)+f(x)y(x)=g(x),x[a,b],y(a)=α0,y(b)=α1,y(1)(a)=γ0,y(1)(b)=γ1,y(2)(a)=δ0,y(2)(b)=δ1,where αi, γi and δi; i = 0,1 are finite real constants while the functions f(x) and g(x) are continuous on [a, b].

The non-polynomial spline function, under consideration has the form Tn = Span{1, x, x2, x3, x4, x5, cos(kx), sin(kx)}, where k is taken to be the frequency of the trigonometric part of the spline function. It is to be noted that k can be real or pure imaginary which is used to raise the accuracy of the method.

Using derivative continuities at knots, the consistency relation between the values of spline and its sixth order derivatives at knots is determined in Section 2.

In Section 3, the non-polynomial spline solution approximating the analytic solution of the BVP (1.1) is determined, using the consistency relation involving the sixth order derivatives and the values of the spline alongwith the end conditions. The error bound of the solution is determined in Section 4. In Section 5, two examples are considered for the usefulness of the method developed.

Section snippets

Preliminary results

To develop the spline approximation to the problem (1.1), the interval [a, b] is divided into n equal subintervals, using the grid points xi = a + ih; i = 0,1,  , n, where h = (b  a)/n.

Consider the following restriction Si of S to each subinterval [xi, xi+1], i = 0,1,  , n  1,Si(x)=aicosk(x-xi)+bisink(x-xi)+ci(x-xi)5+di(x-xi)4+ei(x-xi)3+fi(x-xi)2+gi(x-xi)+pi.Letyi=Si(xi)Mi=Si(2)(xi),Ni=Si(4)(xi),Ti=Si(6)(xi),i=0,1,,n.Assuming y(x) to be the exact solution of the BVP (1.1) and yi be an approximation to y(xi),

Non-polynomial spline solution

The non-polynomial spline solution of the problem (1.1) is based on the system of linear equations given by Eqs. (2.8), (2.10), (2.11), (2.12), (2.13). If Y = [y1, y2,  , yn−1]T and Y=[y(x1),y(x2),,y(xn-1)]T, then the standard matrix equations for the method developed is in the form(i)MY=C+T,(ii)MY=C,(iii)ME=T,where M=(A+h6BF), C = [c1, c2,  , cn−1]T, T=[t˜1,t˜2,,t˜n-1]T, and A, B, F are (n  1) × (n  1) matrices.

AlsoA=-6424-64913799415-79094156186415-24914151-615-2015-611-615-2015-611-615-2015

Convergence of the method

To calculate the error bound, A−1 and ∥A−1 are determined in the following subsection:

Numerical examples

Example 1

Consider the following boundary value problemy(vi)(x)+y(x)=6(2xcos(x)+5sin(x)),x[-1,1],y(-1)=y(1)=0,y(1)(-1)=y(1)(1)=2sin(1),y(2)(-1)=-y(2)(1)=-4cos(1)-2sin(1).The analytic solution of the above problem isy(x)=(x2-1)sin(x).

The observed maximum errors (in absolute values) associated with yi, for the problem (5.1), corresponding to the different values of α, β, γ and δ, are summarized in Table 1.

Remark

The Table 2 shows that the errors in absolute values, are better than those shown by Siddiqi and

Conclusion

Non-polynomial spline method is developed for the approximate solution of sixth order linear special case BVP. The method is also proved to be second order convergent. It has been observed that the relative errors in absolute, are better than those shown by El-Gamel et al. [1] while the errors in absolute values, are better than those shown by Siddiqi and Twizell [6].

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    Boundary Value Problems for Higher-Order Differential Equations

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