The asymptotic behaviour of the θ-methods with constant stepsize for the generalized pantograph equation

Abstract

This paper is concerned with the long-time behaviour of the numerical solutions to a class of linear non-autonomous neutral delay differential equation with proportional delays. Our purpose is to give some asymptotic estimates of the θ-methods with constant stepsize discretization and formulate their upper bounds. Asymptotic estimate not only describes more accurate than asymptotic stability, but also gives an upper bound estimate of the solution for the long-time behaviour. We also compare the known results and show that our formulae improve and generalize these results. Some numerical examples are given in the end of this paper to confirm the theoretical results.

Keywords:

• generalized pantograph equation
• θ-methods
• neutral functional differential equations
• asymptotic behaviour
• asymptotic stability

2010 AMS Subject Classifications:

• 65L05
• 65L20
• 34K28
• 34K40

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 11371074, 11271311), the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ1020), the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 13A108), and The Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities.