Stabilized cubic c¹-spline collocation method for solving first-order ordinary initial value problems

Abstract

This paper is concerned with a construction of a “variable” one-parameter cubic C ¹-spline collocation method for solving the initial value problem (IVP)viz

where the collocation point . The presented method will be shown to be strongly unstable if β ≤ 1/2. It turns out that the proposed method is a continuous extension of the A-stabilized version of Simpson's rule introduced in [1], if β = 1. Moreover, the method is of order three, for β ∊ (1/2, 1) and is of order four, if β = 1 and y ∊ C ⁵[0,b].

Keywords:

• First order initial value problem
• A-stabilized Simpson's rule
• collocation methods
• absolute stability
• A-stability

AMS Subject clasification:

• 65L06

C.R. Categories:

• G.1.7