Efficient numerical methods of integrals with products of two Bessel functions and their error analysis
In this paper, we propose and analyze three efficient methods for numerical approximation of oscillatory integrals with products of two Bessel functions. Firstly, the explicit formulas and asymptotic estimates of the generalized moments are derived by using the Meijer G function. Next, we design a F...
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Veröffentlicht in: | BIT 2025, Vol.65 (1) |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | In this paper, we propose and analyze three efficient methods for numerical approximation of oscillatory integrals with products of two Bessel functions. Firstly, the explicit formulas and asymptotic estimates of the generalized moments are derived by using the Meijer G function. Next, we design a Filon-type method by utilizing ordinary Hermite interpolation polynomials. On this basis, we propose a modified Filon-type method based on Taylor interpolation polynomials with two points. In particular, based on special Hermite interpolation polynomials at Clenshaw–Curtis points, we also give a more efficient Clenshaw–Curtis–Filon-type method that can produce more accurate numerical results. Moreover, the recursive relations of the required modified moments are derived. Importantly, we perform rigorous error analysis of the proposed numerical methods in inverse powers of the oscillation frequency by large amount of theoretical analysis. With the increase of the oscillation frequency, the accuracy improves rapidly when both the number of nodes and the multiplicities are fixed. For the fixed oscillation frequency, the accuracy of the obtained approximate values also increases greatly as either the multiplicities or the number of nodes becomes large. Finally, we compare two of these methods at the same computational cost and find that the Clenshaw–Curtis–Filon-type method gives more accurate results. Some preliminary numerical experiments validate our theoretical analysis and verify the efficiency and accuracy of the proposed methods. |
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ISSN: | 0006-3835 1572-9125 |
DOI: | 10.1007/s10543-024-01045-6 |