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Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator

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Abstract

In this work, we perform a complete asymptotic analysis and the construction of affordable quadrature rules for a class of oscillatory infinite Bessel transform with a general oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points, we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter \(\omega \). The resulting asymptotic expansions clarify the large \(\omega \) behavior of the transform and provide powerful tools for designing quadrature rules and conducting error analysis. As a consequence, efficient and affordable new modified Filon-type methods for computing the transform numerically are proposed. Particularly, we carry out the rigorous error analysis and obtain asymptotic error estimates in inverse powers of \(\omega \). Numerical examples can confirm our analysis. The accuracy can be improved greatly by either adding more derivatives interpolation at endpoints or adding more interior nodes. Moreover, only using a small number of nodes and multiplicities, we can obtain the required accuracy level. For fixed number of nodes and multiplicities, the higher accuracy can be achieved with the larger values of \(\omega \).

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Acknowledgements

This research was supported in part by Zhejiang Provincial Natural Science Foundation of China under Grant Nos. LY18A010009, LY17A010029, National Natural Science Foundation of China (Grant Nos. 11301125, 11971138, 11571087, 11701133), Research Foundation of Hangzhou Dianzi University (Grant No. KYS075613017). The authors would like to express their most sincere thanks to the referees and editors for their very helpful comments and suggestions, which greatly improved the quality of this paper.

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  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, Zhejiang, People’s Republic of China

    Hongchao Kang & Hong Wang

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  1. Hongchao Kang
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  2. Hong Wang
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Correspondence to Hongchao Kang.

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This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant Nos. LY18A010009, LY17A010029, National Natural Science Foundation of China (Grant Nos. 11301125, 11971138, 11571087, 11701133), Research Foundation of Hangzhou Dianzi University (Grant No. KYS075613017).

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Kang, H., Wang, H. Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator. J Sci Comput 82, 29 (2020). https://doi.org/10.1007/s10915-020-01132-0

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  • DOI: https://doi.org/10.1007/s10915-020-01132-0

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