A Legendre–Petrov–Galerkin method for solving Volterra integro-differential equations with proportional delays

ABSTRACT

In this paper, a fully discrete Legendre–Petrov–Galerkin method is presented for solving functional Volterra integro-differential equations with vanishing delays. This method produces a fully discrete linear system. We prove that this system has a unique solution for sufficiently large n, where n+1 denotes the order of the system. Moreover, we prove that the approximate solution and its corresponding derivative function arrive at an optimal convergence order $\mathcal{O}\left(n^{-m-1}\right)$ and $\mathcal{O}\left(n^{-m}\right)$ in $L^2$ norm, where m+1 is the regularity of the original solution. In addition, we prove that for sufficiently large n, the spectral condition number of the corresponding linear system is uniformly bounded. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

Keywords:

• Volterra integro-differential equation with proportional delays
• Legendre–Petrov–Galerkin method
• stability analysis
• convergence analysis
• spectral condition number

2010 AMS subject classifications:

• 65R20
• 45E05

Acknowledgements

The authors thank the referees for very helpful suggestions, which has helped us improve this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (91430104, 11671157,91430213), China Postdoctoral Science Foundation (2017M622590) and Shandong Natural Science Foundation (ZR2014JL003).