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Abstract This paper explores the fast multipole methods (FMMs) to accelerate the approximation for weakly singular Volterra integral equations with highly oscillatory trigonometric kernels. By constructing the fast translation path, the FMM is utilized to speed up the iterative method, which reduces the complexity from O (N 2) to O (N). Especially, we use the collocation method to discretize the Volterra integral equation with constants and linear elements respectively, then apply the GMRES to solve the dense and non-symmetric linear system. In addition, the highly oscillatory integrals derived from the algorithm are calculated effectively by the steepest descent method. The proposed method shows that the numerical solutions become more accurate as the frequency increases. Both of the optimal convergence rates of truncation and the error bounds analysis are represented in the end. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the numerical evaluation of hypersingular finite-part integrals with two kinds of highly oscillatory integrands. Suppose that f is analytic in the first quadrant of the complex plane, based on complex integration theory, both of them are transformed into the problem of integrating two integrals on [ 0 , + ∞) , such that the integrands do not oscillate and decay exponentially and thus can be computed efficiently by constructing the corresponding Gaussian quadrature rule for them. Moreover, error analyses are made for the proposed methods. Finally, several numerical examples are given to verify the theoretical results and illustrate the accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: In this work we propose and analyse a numerical method for computing a family of highly oscillatory integrals with logarithmic singularities. For these quadrature rules we derive error estimates in terms of , the number of nodes, the rate of oscillations and a Sobolev-like regularity of the function. We prove that the method is not only robust but the error even decreases, for fixed , as increases. Practical issues about the implementation of the rule are also covered in this paper by: (a) writing down ready-to-implement algorithms; (b) analysing the numerical stability of the computations and (c) estimating the overall computational cost. We finish by showing some numerical experiments which illustrate the theoretical results presented in this paper. [Copyright &y& Elsevier]

Abstract: We analyze the convergence behavior of Filon-type quadrature rules by making explicit the dependence on both , the parameter that controls the oscillatory behavior of the integrand, and , the number of function evaluations. We provide explicit conditions on the domain of analyticity of the integrand to ensure convergence for . [Copyright &y& Elsevier]

Abstract: With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form where is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case . In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program. [Copyright &y& Elsevier]

This paper explores numerical techniques for evaluating a class of oscillatory infinite integrals of the form ∫ 0 ∞ f (k) J m (k) J n (h k) sin (t k) d k. These oscillatory integrals arise from the transient super Green function associated with cylindrical surfaces in a multi-domain method for three-dimensional hydrodynamic problems in the time domain. The original integral is decomposed into two sub-integrals which are studied in the real k -axis and the complex plane, respectively. The second sub-integral is reformulated by contour integrals whose integrands are exponentially decreasing and well suited for numerical evaluation. The techniques proposed in this paper are shown to be very efficient and accurate by comparing with results from Mathematica. [ABSTRACT FROM AUTHOR]

This paper investigates the implementation of Clenshaw–Curtis algorithms on singular and highly oscillatory integrals for efficient evaluation of the finite Fourier-type transform of integrands with endpoint singularities. In these methods, integrands are truncated by orthogonal polynomials and special function series term by term. Then their singularity types are computed using third and fourth-order homogeneous recurrence relations. The first approach reveals its efficiency for low, moderate and very high frequencies, whereas the second one, is more efficient for small values of frequencies. Moreover, all the results were found quite satisfactory. Algorithms and programming code in MATHEMATICA® 9.0 are provided for the implementation of methods for automatic computation on a computer. Lastly, illustrative numerical experiments and comparison of the proposed Clenshaw–Curtis algorithms to the steepest descent method are mentioned in support of our theoretical analysis in the examples section. [ABSTRACT FROM AUTHOR]