Effective collocation methods to solve Volterra integral equations with weakly singular highly oscillatory Fourier or Airy kernels

Abstract

In this paper two collocation methods, the direct linear interpolation collocation method and the direct high order interpolation collocation method, are proposed to solve the second kind of Volterra integral equations with weakly singular highly oscillatory Fourier or Airy kernels. The purpose is to solve the equations by Kummer hypergeometric and Gamma functions for solving the modified moments, where the modified moments are transformed from the integration interval [0,x] to the interval [−1,1]. The detailed collocation procedure is provided and the effectiveness of these algorithms are proved by convergence analysis and numerical experiments. Further, numerical examples are used to prove that the direct linear interpolation collocation method and the direct high order interpolation collocation method are also effective for solving highly oscillatory equations without weakly singularities.

Keywords:

• Weakly singular kernel
• Highly oscillatory kernel
• Volterra integral equation
• Fourier transforms
• Airy function

2010 AMS Subject Classification:

• 65R20

Acknowledgments

We would like to thank Dr. Joseph Elliot at the University of Kansas for her assistance with English language and grammatical editing of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was funded by Hunan Provincial Natural Science Foundation of China (grant numbers 2022JJ30276, 2021JJ30297) and the Science and Technology Program of Hunan Province (grant number 2019TP1014).