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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. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by J m (z) = 1 2 H m (1) (z) + H m (2) (z) , where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy's residue theorem to convert these Fourier-type integrals into the infinite integrals on [ 0 , + ∞) . Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we study both the fast computation and the related error analysis of many different finite and infinite oscillatory Bessel transforms. By transforming the Bessel function into Confluent hypergeometric function and using its asymptotic expansion, we develop an efficient approach by the contours in the complex plane, corresponding to the paths of steepest descent. Furthermore, we conduct their error analyses in inverse powers of the frequency ω. The related error analyses show that the precision can be improved by either adding more Gauss-Laguerre nodes or increasing the frequency ω. In particular, the method exhibits the high error order. Meanwhile, we apply the proposed method to some other kinds of oscillatory integrals through some simple changes of variables. Numerical experiments can verify our theoretical analysis. It is also shown that the new algorithms are more efficient than the existing Filon-type method and can achieve the same level of accuracy with the method (the XMX method) proposed in Xu, Milovanović and Xiang [46] by some numerical experiments under the given conditions. A combination of the n 1 -point Gauss-Laguerre quadrature rule and the n 2 -point generalized Gauss-Laguerre quadrature rule are used in the XMX method. However, we only use the Gauss-Laguerre quadrature rule for one time in the established method, which makes the theoretical and error analysis simpler than the XMX method. [ABSTRACT FROM AUTHOR]

Filon collocation methods are well known for their good performances in solving the second-kind highly oscillatory Volterra integral equations with weak singularities. As the frequency increases, these methods can provide much more accurate solutions for oscillatory integral equations. This paper is devoted to investigating frequency-related properties of collocation solutions of highly oscillatory Volterra equations with weakly singular kernels. Optimal convergence rates are obtained by analyzing the error equations and studying asymptotic properties of oscillatory integrals involving Bessel functions. Moreover, numerical experiments are conducted to verify the given theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we present and analyze efficient methods for computing the Cauchy principal value integral of the oscillatory Bessel function ∫ a b f (x) x − c J m (ω x) d x , where 0 ≤ a < c < b. For the two cases of a > 0 and a = 0 , we establish new steepest descent integration paths, and employ the complex line integration technique to obtain affordable quadrature rules. According to the relation formula between the Meijer G-function and the Bessel function, a crucial explicit expression of the required integral ∫ 0 + ∞ x j x − c J m (ω x) d x can be derived in terms of the Meijer G-function. Furthermore, for the proposed quadrature formulae, we perform rigorous error analysis in inverse powers of ω. Particularly, compared with a combination method given in the work (Kang et al., 2022), the proposed method takes less time and produces more accurate results. We verify error analysis of the proposed method by experimental results. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed numerical methods. [ABSTRACT FROM AUTHOR]

This work introduces and analyzes both the complex integration method and the modified complex integration method for two classes of highly oscillatory Bessel transforms with algebraic or algebraic-logarithmic singularities. We first transform the original integrals into the infinite integrals on 1 , + ∞) by the change of variable t = x − β , where β > 0. With the help of the integral representation of the Bessel function, the integral representation and the asymptotic series of the Whittaker function, we then use the Cauchy's residue theorem and convert the above infinite integrals into the infinite integrals on 0 , + ∞). The integrands of the resulting infinite integrals are not oscillatory and decay exponentially fast. Fortunately, they can be efficiently evaluated by using the Gauss-Laguerre quadrature rule. Furthermore, we conduct their error analysis in inverse powers of the frequency ω. Finally, numerical experiments are provided to confirm our theoretical analysis. We can see from theoretical analysis and numerical experiments that the accuracy of the quadrature increases both for the case of growing the number of nodes n and fixed ω , and for the case of an increasing ω and fixed the number of nodes n. The comparison of the proposed two methods and the Filon-type method presented in Kang and Ma (2017), is briefly discussed as well. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, based on the asymptotic property of the solution, we derive the corresponding convergence rates in terms of the frequency for the direct-Filon and linear continuous collocation methods, which solves an open problem in Brunner (2010) [1]. Numerical tests verify that the asymptotic orders obtained are optimal. [Copyright &y& Elsevier]