hp-Legendre–Gauss collocation method for impulsive differential equations

ABSTRACT

The original Legendre–Gauss collocation method is derived for impulsive differential equations, and the convergence is analysed. Then a new hp-Legendre–Gauss collocation method is presented for impulsive differential equations, and the convergence for the hp-version method is also studied. The results obtained in this paper show that the convergence condition for the original Legendre–Gauss collocation method depends on the impulsive differential equation, and it cannot be improved, however, the convergence condition for the hp-Legendre–Gauss collocation method depends both on the impulsive differential equation and the meshsize, and we always can choose a sufficient small meshsize to satisfy it, which show that the hp-Legendre–Gauss collocation method is superior to the original version. Our theoretical results are confirmed in two test problems.

KEYWORDS:

• hp-Version
• Legendre–Gauss collocation methods
• impulsive differential equations

2010 AMS SUBJECT CLASSIFICATIONS:

• 65L05
• 65L60

Disclosure statement

No potential conflict of interest was reported by the author.

Funding

The work was supported by the National Nature Science Foundation of China (No. 11101130), the Research Fund of the Heilongjiang Provincial Education Department for the academic backbone of the excellent young people (No. 1254G044), Science and Technology Innovation Team in Higher Education Institutions of Heilongjiang Province (No. 2014TD005) and the Heilongjiang University Science Funds for Distinguished Young Scholars (No. JCL201303).