Cubic B-Spline Collocation and Adomian Decomposition Methods on 4th Order Multi-point Boundary Value Problems

Abstract

In this paper, Cubic B-Spline collocation method (CBSCM) and Adomian Decomposition Method (ADM) are applied to obtain numerical solutions to fourth-order linear and nonlinear differential equations. The CBSCM was based on finite element method involving collocation method with cubic B-spline as a basis function. While ADM was based on multistage decomposition method. We discovered in the illustrative examples considered, that result by ADM were compatible with the closed form solutions well over twenty, in some cases over thirty, decimal places and with extremely minimal absolute errors. Results by CBSCM gave correct solutions to atmost six decimal places with sizeable absolute errors. These has further revealed the importance and superiority of ADM over CBSCM in providing semi-analytic solution to this class of differential equations.

A note on the convergence of the Adomian decomposition method

“…The Adomian decomposition method (ADM) was developped in the 1980's by the American physicist G. Adomian (1923Adomian ( − 1996 ( [1]- [4]) as a powerful method for solving functional equations. The main idea of this method lies in the decomposition of the solution u of a vector nonlinear equation u = f+N (u) where N is an analytic operator and f a given vector, into a series u = +∞ n=0 u n such that the term u n+1 is determined from the terms u 0 , u 1 , ..., u n by a reccurence relation involving a polynomial A n generated by the Taylor expansion of the operator N. Many papers on the applications of the ADM to the problems arising from different areas of pure and applied sciences have been published ( [5]- [7], [9], [15], [16], [23]). Many works have been also devoted to the convergence of the ADM ( [8], [10]- [14]).…”

Correlations between Forging Tool Lifetime and the Continuous Curvature Transitions

“…This requirement is practically equivalent to usage of cubic B-spline curves [5]. The Pro/E CAD system, similarly to other market leader CAD systems, uses the NURBS (Non Uniform Rational B-Spline) curves for complex tasks [6].…”