Spline solutions for nonlinear two point boundary value problems

Abstract

ABSTRACT. Necessary formulas are developed for obtaining cubic, quartic, quintic, and sextic spline solutions of nonlinear boundary value problems.These methods enable us to approximate the solution of the boundary value problems, as well as their successive derivatives smoothly. Numerical evidence is included to demonstrate the relative performance of these four techniques.

Multiderivative methods for non-linear second-order boundary value problems

“…Table 2 here It is noted from Table 2 that, for each of the three methods, || E || is reduced by the factor 2 p (approximately), where p is the order of the method, as h is successively halved. It is also noted that the novel sixth order method gives smaller error moduli than the corresponding methods outlined in Jain [12] and Usmani [6] where, for h =0.1, || E || =0.3E-04. Table 3 also contains, for m = 3 and 4, the values of ||E|| obtained by Usmani [6] using his second order cubic spline method, while Table 4 contains the values of ||E|| obtained using the fourth order methods of Usmani [6] based on quartic and quintic splines.…”

“…To test the effectiveness of the second, fourth and sixth order methods developed in the paper, each was tested on the following problems which were also used by Lal and Moffatt [10], Usmani [6,8,11] and Jain [12]. Table 2, together with the equivalent results of the sixth order method of Jain [12] which was based on Lobatto quadrature.…”

“…Commonly used finite difference methods are discussed by the authors of many texts (see, for instance [1,2,3]); the problem is treated using variational techniques in [3]; and shooting methods have received coverage in [4], for example. The possibility of using spline functions to obtain a smooth solution of (1) with (2) was first discussed in [5], and in [6] cubic, quartic, quintic and sextic splines are used to solve the boundary value problem. Attention was drawn in [7] to the connection between a cubic spline solution and the solution obtained using the well known Numerov method .…”

A smooth approximation for non-linear second order boundary value problems using composite non-polynomial spline functions

“…Spline functions were applied by many authors to establish the accurate and efficient numerical schemes for the solution of boundary value problems [4]. An exploration of the literature on a number of polynomial and non-polynomial spline techniques to solve the second order BVPs can be comprehended as quadratic spline method [8,26,32,42,49], cubic spline method [2-3, 5, 9-12, 15, 20-23, 27-28, 30-34, 36-38, 40-41, 50], quartic spline method [6, 13-14, 29, 47], quintic spline method [7,16,43,48] and others [39,46]. Voluminous research work have been contributed to this field but we are mainly concerned on those papers which have implemented non-polynomial splines for the solution of second order BVPs with various types of boundary conditions.…”

Some new linear relations for even degree polynomial splines on a uniform mesh

“…Recently, some relations which connect N -1 spline values with N -1 derivative values have been discovered for N = 4 and N = 6 ([6], [7]). The purpose of this note is to give the general form and the leading term of the truncation error for these new relations.…”