高振荡Bessel变换的新型Hermite卷积积分公式及其应用
Novel Hermite Convolution Quadrature for a Class of Highly Oscillatory Bessel Transforms and Its Application
摘要: 高振荡问题已广泛出现于许多科学工程应用领域，例如电磁、声波散射、量子化学和图像分析等问题。高振荡数值积分是高振荡问题数值解中的重要研究方向，由于其被积函数具有高振荡性，传统数值积分解法面临许多挑战。基于Lubich的卷积求积，本文针对一类高振荡的Bessel变换提出了一种新型的Hermite卷积积分公式，并研究了其在高振荡Volterra积分方程的数值解中的应用。通过理论与数值实验表明，该方法在计算含高振荡Bessel核的卷积积分以及积分方程时，计算精度不受振荡频率的影响，是一种高效的计算方法。
Abstract: Highly oscillatory problems frequently arise in many science and engineering fields, such as electromagnetic, acoustic scattering, quantum chemistry and image analysis problems. Numerical calculation of highly oscillatory integration is an important research interest of numerical studies on highly oscillatory problems. Due to the high oscillation of the integrand, classical numerical integration encounters great challenges. Based on Lubich’s convolution quadrature, this paper proposes novel Hermite convolution quadrature for a class of highly oscillatory Bessel transforms and studies its application in the numerical solutions to highly oscillatory Volterra integral equations. Both theoretical and numerical evidences indicate that the proposed method is particularly efficient in computing highly oscillatory integrals and solving highly oscillatory integral equations, and its accuracy is not affected by the oscillation parameter.
文章引用：任浩, 曾港, 李恒杰. 高振荡Bessel变换的新型Hermite卷积积分公式及其应用[J]. 应用数学进展, 2022, 11(4): 2352-2361. https://doi.org/10.12677/AAM.2022.114247

参考文献

[1] Asheim, A. and Huybrechs, D. (2009) Local Solutions to High Frequency 2D Scattering Problems. Journal of Computational Physics, 229, 5357-5372. [Google Scholar] [CrossRef] 
[2] Cakoni, F. and Colton, D. (2006) Qualitative Methods in Inverse Scattering Theory. Springer, Berlin. [Google Scholar] [CrossRef] 
[3] Colton, D. and Kress, R. (1983) Integral Equation Methods in Scattering Theory. Wiley, Hoboken.
[4] Langdon, S. and Chandler-Wilde, S.N. (2006) A Wavenumber Independent Boundary Element Method for an Acoustic Scattering Problem. SIAM Journal on Numerical Analysis, 43, 2450-2477. [Google Scholar] [CrossRef] 
[5] Nédélec, J. (2001) Acoustic and Electromagnetic Equations. Springer, Berlin, 144. [Google Scholar] [CrossRef] 
[6] 向淑晃. 一些高振荡积分、高振荡积分方程的高性能计算[J]. 中国科学: 数学, 2012, 42(7): 651-670.
[7] Filon, L. (1930) On a Quadrature Formula for Trigonometric Integrals. Proceedings of the Royal Society of Edinburgh, 49, 38-47. [Google Scholar] [CrossRef] 
[8] Nørsett, S. and Iserles, A. (2005) Efficient Quadrature of Highly Oscillatory Integrals Using Derivatives. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461, 1383-1399. [Google Scholar] [CrossRef] 
[9] Domínguez, V., Graham, I. and Smyshlyaev, V. (2011) Stability and Error Estimates for Filon-Clenshaw-Curtis Rules for Highly-Oscillatory Integrals. IMA Journal of Numerical Analysis, 31, 1253-1280. [Google Scholar] [CrossRef] 
[10] Xiang, S., Cho, Y., Wang, H., et al. (2011) Clenshaw-Curtis-Filon-Type Methods for Highly Oscillatory Bessel Transforms and Applications. IMA Journal of Numerical Analysis, 31, 1281-1314. [Google Scholar] [CrossRef] 
[11] Majidian, H., Firouzi, M. and Alipanah, A. (2022) On the Stability of Filon-Clenshaw-Curtis Rules. Bulletin of the Iranian Mathematical Society, 1-22. [Google Scholar] [CrossRef] 
[12] Wu, Q. and Sun, M. (2021) Numerical Steepest Descent Method for Hankel Type of Hypersingular Oscillatory Integrals in Electromagnetic Scattering Problems. Advances in Mathematical Physics, 2021, Article ID: 8021050. [Google Scholar] [CrossRef] 
[13] Levin, D. (1982) Procedures for Computing One- and Two-Dimensional Integrals of Functions with Rapid Irregular Oscillations. Mathematics of Computation, 38, 531-538. [Google Scholar] [CrossRef] 
[14] Li, J., Wang, X., Wang, T., et al. (2010) An Improved Levin Quadrature Method for Highly Oscillatory Integrals. Applied Numerical Mathematics, 60, 833-842. [Google Scholar] [CrossRef] 
[15] Khan, S., Zaman, S., Arshad, M., et al. (2021) A Well-Conditioned and Efficient Levin Method for Highly Oscillatory Integrals with Compactly Supported Radial Basis Functions. Engineering Analysis with Boundary Elements, 131, 51-63. [Google Scholar] [CrossRef] 
[16] He, J. and El-Dib, Y. (2021) Homotopy Perturbation Method with Three Expansions for Helmholtz-Fangzhu Oscillator. International Journal of Modern Physics B, 35, Article ID: 2150244. [Google Scholar] [CrossRef] 
[17] Kang, H., Xiang, C., Xu, Z., et al. (2021) Efficient Quadrature Rules for the Singularly Oscillatory Bessel Transforms and Their Error Analysis. Numerical Algorithms, 88, 1493-1521. [Google Scholar] [CrossRef] 
[18] Ke, X., Ying, S. and Wang, Y. (2021) One-Step Extrapolation Method Based on the Multiple-Angle Formula. Chinese Journal of Geophysics, 64, 2480-2493.
[19] Lubich, C. (1988) Convolution Quadrature and Discretized Operational Calculus. I. Numerische Mathematik, 52, 129-145. [Google Scholar] [CrossRef] 
[20] Lubich, C. (1988) Convolution Quadrature and Discretized Operational Calculus. II. Numerische Mathematik, 52, 413-425. [Google Scholar] [CrossRef] 
[21] Lubich, C. and Ostermann, A. (1993) Runge-Kutta Methods for Parabolic Equations and Convolution Quadrature. Mathematics of Computation, 60, 105-131. [Google Scholar] [CrossRef] 
[22] Xiang, S. and Brunner, H. (2013) Efficient Methods for Volterra Integral Equations with Highly Oscillatory Bessel Kernels. BIT Numerical Mathematics, 53, 241-263. [Google Scholar] [CrossRef] 
[23] Ma, J. and Liu, H. (2018) On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels. Symmetry, 10, 239. [Google Scholar] [CrossRef] 

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