A numerical method for the integration of oscillatory functions

A new method for the calculation of the integrals $$I_1 (m) = \int\limits_a^b {f(x)\sin mxdx} andI_2 (m) = \int\limits_a^b {f(x)\cos mxdx}$$ is presented. The functionf(x) is approximated by a sum of Chebyshev polynomials. The Chebyshev coefficients are then used to calculate a Neumann series approximation forI ₁(m) andI ₂(m). The numerical examples demonstrate that this method is very accurate and efficient.     

A method for indefinite integration of oscillatory and singular functions

We propose a general method for computing indefinite integrals of the form

Numerical analysis of a fast integration method for highly oscillatory functions

The integration of systems containing Bessel functions is a central point in many practical problems in physics, chemistry and engineering. This paper presents a new numerical analysis for the collocation method presented by Levin for and gives more accurate error analysis about the integration of systems containing Bessel functions. The effectiveness and accuracy of the quadrature is tested for Bessel functions with large arguments. AMS subject classification (2000)  65D32, 65D30     

On the convergence of the method for indefinite integration of oscillatory and singular functions

We consider the problem of convergence and error estimation of the method for computing indefinite integrals proposed in Keller [8]. To this end, we have analysed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. Properties of this operator were never investigated before. The obtained results lead us to the conclusion that the studied method is always convergent. We also give a rigorous proof of the error estimates.     

Moment-free numerical integration of highly oscillatory functions

^** Email: s.olver{at}damtp.cam.ac.uk The aim of this paper is to derive new methods for numericallyapproximating the integral of a highly oscillatory function.We begin with a review of the asymptotic and Filon-type methodsdeveloped by Iserles and Nørsett. Using a method developedby Levin as a point of departure, we construct a new methodthat utilizes the same information as a Filon-type method, andobtains the same asymptotic order, while not requiring the computationof moments. We also show that a special case of this methodhas the property that the asymptotic order increases with theaddition of sample points within the interval of integration,unlike all the preceding methods whose orders depend only onthe endpoints.     

Analysis of a collocation method for integrating rapidly oscillatory functions

A collocation method for approximating integrals of rapidly oscillatory functions is analyzed. The method is efficient for integrals involving Bessel functions J_v(rx) with a large oscillation frequency parameter r, as well as for many other one- and multi-dimensional integrals of functions with rapid irregular oscillations. The analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases.     

A new method for exact integration of some perturbed stiff linear systems of oscillatory type

This article presents a new family of real functions with values within the ring of M(m,R) matrices, Φ-functions for perturbed linear systems and a numerical method adapted for integration of this type of problem. This method permits the system solution to be expressed as a series of Φ-functions. The coefficients of this series are obtained through recurrences in which the perturbation intervenes.The Φ-functions series method has the advantage of being exactly integrated in the perturbed problem. For this purpose an appropriate B matrix is selected and used to construct the operator described in this article, thus annihilating the disturbance terms, transforming the system into a homogenous second-order system, which is exactly integrated with the two first Φ-functions.The article ends with a detailed study of four perturbed systems which illustrate how the method is used in stiff problems or in highly oscillatory problems, contrasting its behaviour by studying its accuracy in comparison with other well-known codes.     

Reducing factorization of a semiprime number to the integration of highly oscillatory functions

We reduce the problem of factoring a semiprime integer to the problem of (numerically) integrating a certain highly oscillatory function. We provide two algorithms for addressing this problem, one based on the residue theorem and the other on the (extended) Cauchy argument principle. We show that in the former algorithm, computing the residue of the function at a certain pole leads to us obtaining the factors of the semiprime integer. In the latter, we consider a contour integral for which we are able to obtain an analytical solution with several branches. The computational difficulty reduces to that of discovering the branch of the solution which gives the precise integral. We address this problem by numerically computing an upper and a lower bound of the integral and then considering the branch that fits these bounds. The time complexity of the algorithms is left as an open problem.     

A parabolic acceleration time integration method for structural dynamics using quartic B-spline functions

In this paper, an explicit time integration method is proposed for structural dynamics using periodic quartic B-spline interpolation polynomial functions. In this way, at first, by use of quartic B-splines, the authors have proceeded to solve the differential equation of motion governing SDOF systems and later the proposed method has been generalized for MDOF systems. In the proposed approach, a straightforward formulation was derived in a fluent manner from the approximation of response of the system with B-spline basis. Because of using a quartic function, the system acceleration is approximated with a parabolic function. For the aforesaid method, a simple step-by-step algorithm was implemented and presented to calculate dynamic response of MDOF systems. The proposed method has appropriate convergence, accuracy and low time consumption. Accuracy and stability analyses have been done perfectly in this paper. The proposed method benefits from an extraordinary accuracy compared to the existing methods such as central difference, Runge–Kutta and even Duhamel integration method. The validity and effectiveness of the proposed method is demonstrated with four examples and the results of this method are compared with those from some of the existent numerical methods. The high accuracy and less time consumption are only two advantages of this method.     

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.     

A Gautschi-type method for oscillatory second-order differential equations

Summary. We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of García-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation. Received January 21, 1998 / Published online: June 29, 1999     

A fast numerical algorithm for the integration of rational functions

A new iterative method for high-precision numerical integration of rational functions on the real line is presented. The algorithm transforms the rational integrand into a new rational function preserving the integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the degree of the input and the desired order of the method. Both parameters are arbitrary. The formulas can be precomputed. Iteration yields an approximation of the desired integral with mth order convergence. Examples illustrating the automatic generation of these formulas and the numerical behaviour of this method are given.     

Efficient integration for a class of highly oscillatory integrals

This paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polynomial and can be evaluated efficiently in O(N log N) operations, where N + 1 is the number of Clenshaw-Curtis points in the interval of integration. In addition, we derive the corresponding error bound in inverse powers of the frequency ω for the Clenshaw-Curtis-Filon-type method for the class of highly oscillatory integrals. The efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results.     

A note on the Gautschi-type method for oscillatory second-order differential equations

The Gautschi-type method has been proposed by Hochbruck and Lubich for oscillatory second-order differential equations. They conjecture that this method allows for a uniform error bound independent of the size of the system. The conjecture is proved in this note.     

Asymptotic expansions for oscillatory integrals using inverse functions

We treat finite oscillatory integrals of the form ∫ _a ^b F(x)e ^ikG(x) dx in which both F and G are real on the real line, are analytic over the open integration interval, and may have algebraic singularities at either or both interval end points. For many of these, we establish asymptotic expansions in inverse powers of k. No appeal to the theories of stationary phase or steepest descent is involved. We simply apply theory involving inverse functions and expansions for a Fourier coefficient ∫ _a ^b φ(t)e ^ikt dt. To this end, we have assembled several results involving inverse functions. Moreover, we have derived a new asymptotic expansion for this integral, valid when , −1<σ ₁<σ ₂<⋅⋅⋅. The authors were supported by the Office of Advanced Scientific Computing Research, Office of Science, US Department of Energy, under Contract DE-AC02-06CH11357.     

A note on the numerical evaluation of finite integrals of oscillatory functions

A method is described for the numerical evaluation of integrals of the form ∫ _?1 ¹ f(x)K(m,x)dx, wheref(x) is smooth in [?1,1], whileK(m,x) is highly oscillatory for large values ofm.