High-order continuous third derivative formulas with block extensions for y″=f(x, y, y′)

Abstract

A class of high-order continuous third derivative formulas (CTDFs) for second-order ordinary differential equations (ODEs) is proposed. The CTDFs are used to generate main and additional methods which are combined to give the block third derivative formulas (BTDFs). The BTDFs are applied as single-block matrix equations to simultaneously provide approximate solutions for second-order ODEs. The stability properties of the BTDFs are discussed. Numerical examples are given to illustrate the accuracy and efficiency of the block extensions of the methods.

Keywords:

• continuous method
• block extension
• third derivative
• second order
• initial value problems

2010 AMS Subject Classifications :

• 65L05
• 65L06
• 65L12