A smooth approximation for the solution of a fourth-order boundary value problem based on nonpolynomial splines
type: Item
instance of: scholarly article
title: A smooth approximation for the solution of a fourth-order boundary value problem based on nonpolynomial splines
zbMATH Open document ID: 0810.65079
DOI: 10.1016/0377-0427(92)00112-M
published in: Journal of Computational and Applied Mathematics
publication date: 1995-04-09T00:00:00Z
review text: The authors consider smooth approximations of the solution of the fourth- order linear ordinary differential equation \(y^{(4)} + f(x) y = g(x)\), \(a \leq x \leq b\), subject to the boundary conditions \(y(a) - A_ 1 = 0 = y(b) - A_ 2\), \(y'(a) - B_ 1 = 0 = y'(b) - B_ 2\) whereby \(f(x)\) and \(g(x)\) are continuous functions on \([a, b]\) and \(A_ i\), \(B_ i\), \(i = 1,2,\) are finite real arbitrary constants. The aim of this paper is to construct a new spline method that is based on a nonpolynomial spline space. Indeed, since \(\{1, x, \dots, x^ m, \sin kx, \cos kx\}\) generates an extended complete Chebyshev space on \([a, b]\), it is possible to construct spline approximations that have a polynomial and a trigonometric part. The spline functions proposed in this paper have the form \[ a \sin kx + b \cos kx + c x^ 3 + d x^ 2 + e x + f. \] It is shown that the parameter \(k\) can be used to raise the order of accuracy of the method. Two numerical examples are given to sustain the autonomous parametric spline method and to compare it with a quintic polynomial spline method of second order.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 699108
zbMATH Keywords: smooth approximations
zbMATH Keywords: fourth-order linear ordinary differential equation
zbMATH Keywords: nonpolynomial spline space
zbMATH Keywords: numerical examples
zbMATH Keywords: parametric spline method
author: M. Van Daele
author: G. Vanden Berghe
author: H. De Meyer
reviewed by: Pavol Chocholatý
describes a project that uses: MACSYMA
MaRDI profile type: Publication
cites work: Q5719651
cites work: Uniqueness of best \(L_ 1\)-approximations from periodic spline spaces
cites work: Q3262547
cites work: Q3895980
cites work: Discrete Variable Methods for a Boundary Value Problem with Engineering Applications
cites work: Smooth spline approximations for the solution of a boundary value problem with engineering applications
description: scientific article; zbMATH DE number 699108