Author
Alfredo Deaño
Other affiliations: University of Kent, University of Copenhagen Faculty of Science, Carlos III Health Institute ...read more
Bio: Alfredo Deaño is an academic researcher from Charles III University of Madrid. The author has contributed to research in topics: Orthogonal polynomials & Asymptotic expansion. The author has an hindex of 15, co-authored 54 publications receiving 619 citations. Previous affiliations of Alfredo Deaño include University of Kent & University of Copenhagen Faculty of Science.
Papers published on a yearly basis
Papers
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TL;DR: Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.
Abstract: We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand.
61 citations
TL;DR: In this article, the authors studied the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane.
48 citations
TL;DR: In this article, the authors present a method to compute efficiently solutions of systems of ODEs that possess highly oscillatory forcing terms, based on asymptotic expansions in inverse powers of the oscillatory parameter, and feature two fundamental advantages with respect to standard numerical ODE solvers.
Abstract: We present a method to compute efficiently solutions of systems of ordinary differential equations (ODEs) that possess highly oscillatory forcing terms. This approach is based on asymptotic expansions in inverse powers of the oscillatory parameter, and features two fundamental advantages with respect to standard numerical ODE solvers: first, the construction of the numerical solution is more efficient when the system is highly oscillatory, and, second, the cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided, featuring the Van der Pol and Duffing oscillators and motivated by problems in electronic engineering.
34 citations
TL;DR: In this article, the authors consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation, and they use a Bessel function identity to expand the oscillator into asymptotic series, and this allows them to extend Filon-type approach to this setting.
Abstract: In this paper, we consider linear ordinary differential equations originating in electronic engineering, which exhibit exceedingly rapid oscillation. Moreover, the oscillation model is completely different from the familiar framework of asymptotic analysis of highly oscillatory integrals. Using a Bessel-function identity, we expand the oscillator into asymptotic series, and this allows us to extend Filon-type approach to this setting. The outcome is a time-stepping method that guarantees high accuracy regardless of the rate of oscillation.
34 citations
Cited by
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01 Mar 1987
TL;DR: The variable-order Adams method (SIVA/DIVA) package as discussed by the authors is a collection of subroutines for solution of non-stiff ODEs.
Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
1,955 citations
01 Jan 2007
TL;DR: Two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weightfunction are known or can be calculated.
Abstract: Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.
1,007 citations
Book•
01 Jan 1966
TL;DR: Boundary value problems in physics and engineering were studied in this article, where Chorlton et al. considered boundary value problems with respect to physics, engineering, and computer vision.
Abstract: Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s
733 citations