Asymptotic expansions for the Bessel transformations with high oscillations by using the homotopy technique

Abstract

In this paper, we establish the asymptotic expansions for the integration containing the Bessel functions with high oscillations by using the homotopy technique. The proposed method is constructed by approximating the non-oscillatory or low-oscillatory factor in practice. Two cases are considered, that is, the case 0 [a, b] and the case 0∈[a, b], where [a, b] denotes the integration interval. The efficiency of the technique and the correctness of the theory are shown by several examples.

Keywords:

• homotopy perturbation method
• asymptotic expansion
• the Bessel transformation
• oscillatory integrals
• numerical quadrature

2000 AMS Subject Classifications :

• 65D32
• 65D30

Acknowledgements

The authors thank the associated editor and referees for their valuable comments and suggestions for great improvement of this paper. The first author acknowledges the financial supports from Postdoctoral Foundation of Central South University and Start-up Projects of Guangdong Ocean University.