On efficient computation of highly oscillatory retarded potential integral equations

ABSTRACT

This paper presents an efficient numerical method for retarded potential integral equations with highly oscillatory spatially time-harmonic incident waves, which is based on inverse Fourier transforms and efficient algorithms for the highly oscillatory Volterra integral equation of the first kind. From the integral equation, it leads to an efficient approximation by applying the Clenshaw–Curtis-type method which costs the same operations independent of large values of frequencies. Applying inverse Fourier transforms yields numerical results on solving the retarded potential integral equations. Preliminary numerical results show the efficiency and accuracy of the approximations.

KEYWORDS:

• Retarded potential integral equations
• spatially time-harmonic incident wave
• Bessel function
• highly oscillatory Volterra integral equations
• Clenshaw–Curtis-type method

2010 AMS SUBJECT CLASSIFICATIONS:

• 65D32
• 65R20

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Fast evaluation of the n nodes and weights with $\mathcal{O}\left(n\right)$ operations for the Gauss–Jacobi quadrature was given by Glaser et al. [11], Bogaert et al. [3], and Hale and Townsend [14]. A Matlab file for computation of these nodes and weights can be found in Chebfun system [31].

Additional information

Funding

This work is supported partly by NSF of China [No. 11371376, 11771454], the Innovation-Driven Project, the Mathematics and Interdisciplinary Sciences Project of Central South University and the Fundamental Research Funds for the Central Universities of Central South University [No. 2017zzts060] and scientific research project of Department of Education of Hunan Province [No. 17C0677].