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In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral ∫ 0 1 f (x) x - τ J m (ω x γ) d x with the Cauchy type singular point, where 0 < τ < 1 , m ≥ 0 , 2 γ ∈ N +. The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based on the special transformation of the integrand and the additivity of the integration interval, we convert the integral into three integrals. The explicit formula of the first one is expressed in terms of the Meijer G function. The second is computed by using the complex integration method and the Gauss– Laguerre quadrature rule. For the third, we adopt the Clenshaw– Curtis– Filon– type method to obtain the quadrature formula. In particular, the important recursive relationship of the required modified moments is derived by utilizing the Bessel equation and the properties of Chebyshev polynomials. Importantly, the strict error analysis is performed by a large amount of theoretical analysis. Our proposed methods only require a few nodes and interpolation multiplicities to achieve very high accuracy. Finally, numerical examples are provided to verify the validity of our theoretical analysis and the accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to explore the convolution quadrature based on a class of two-point Hermite collocation methods. Incorporating derivatives into the numerical scheme enhances the accuracy while preserving stability, which is confirmed by the convergence analysis for the discretization of the initial value problem. Moreover, we employ the resulting quadrature to evaluate a class of highly oscillatory integrals. The frequency-explicit convergence analysis demonstrates that the proposed convolution quadrature surpasses existing convolution quadratures, achieving the highest convergence rate with respect to the oscillation among them. Numerical experiments involving convolution integrals with smooth, weakly singular, and highly oscillatory Bessel kernels illustrate the reliability and efficiency of the proposed convolution quadrature. [ABSTRACT FROM AUTHOR]

Our recent work (Kang and Wang in J Sci Comput 82:1–33, 2020) performed a complete asymptotic analysis and proposed a modified Filon-type method for a class of oscillatory infinite Bessel transform with a general oscillator. In this paper, we present and analyze a different method by converting the integration path to the complex plane for this class of oscillatory infinite Bessel transform. In particular, we establish a series of new quadrature rules for this transform and carry out rigorous analysis, including the cases that the oscillator g(x) has either zeros or stationary points. The error analysis shows the advantages that this approach exhibits high asymptotic order, and the accuracy improves significantly as either the frequency ω or the number of nodes n increases. Furthermore, the constructed method shows higher accuracy and error order by comparing with the existing modified Filon-type method in our recent work (Kang and Wang 2020) at the same computational cost. Some numerical experiments are provided to verify the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ p {\ell^{p}} -quasi-norm minimization. Compared to ℓ 1 {\ell^{1}} -norm minimization, the choice of 0 < p < 1 {0

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 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 ω . The resulting asymptotic expansions clarify the large ω 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 ω . 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 ω . [ABSTRACT FROM AUTHOR]

A class of Fredholm integral equations of the second kind, with respect to the exponential weight function $$w(x)=\exp (-(x^{-\alpha }+x^\beta ))$$ , $$\alpha >0$$ , $$\beta >1$$ , on $$(0,+\infty )$$ , is considered. The kernel k( x, y) and the function g( x) in such kind of equations, can grow exponentially with respect to their arguments, when they approach to $$0^+$$ and/or $$+\infty $$ . We propose a simple and suitable Nyström-type method for solving these equations. The study of the stability and the convergence of this numerical method in based on our results on weighted polynomial approximation and 'truncated' Gaussian rules, recently published in Mastroianni and Notarangelo (Acta Math Hung, 142:167-198, 2014), and Mastroianni, Milovanović and Notarangelo (IMA J Numer Anal 34:1654-1685, 2014) respectively. Moreover, we prove a priori error estimates and give some numerical examples. A comparison with other Nyström methods is also included. [ABSTRACT FROM AUTHOR]

In this paper, we introduce efficient methods for the approximation of solutions to weakly singular Volterra integral equations of the second kind with highly oscillatory Bessel kernels. Based on the asymptotic analysis of the solution, we derive corresponding convergence rates in terms of the frequency for the Filon method, and for piecewise constant and linear collocation methods. We also present asymptotic schemes for large values of the frequency. These schemes possess the property that the numerical solutions become more accurate as the frequency increases. [ABSTRACT FROM AUTHOR]