A high-order numerical scheme for the impulsive fractional ordinary differential equations

ABSTRACT

In this paper, we use a good technique to construct a high-order numerical scheme for the impulsive fractional ordinary differential equations (IFODEs). This technique is based on the so-called block-by-block method, which is a common method for the integral equations. In our approach, the classical block-by-block method is improved so as to avoid the coupling of the unknown solutions at each block step with an exception in the first two steps between two adjacent pulse points. The convergence and stability analysis of the scheme are given. It proves that the numerical solution converges to the exact solution with order 3+q for $0<q\le 1$, where q is the order of the fractional derivative. A series of numerical examples are provided to support the theoretical results.

KEYWORDS:

• Impulsive fractional ordinary differential equations
• Volterra integral equation
• high-order numerical scheme
• convergence analysis
• stability analysis

2010 AMS SUBJECT CLASSIFICATION:

• 65L12

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper was supported by National Natural Science Foundation of China (grant nos. 11501140, 11671166, and U1530401), Foundation of Guizhou Science and Technology Department (grant nos. [2017]1086 and [2014]2098), Postdoctoral SFC (grant no. 2015M580038) and Innovation Group Major Program of Guizhou Province (grant no. KY[2016]029).