2 edition of **Numerical methods for integrating oscillatory functions.** found in the catalog.

Numerical methods for integrating oscillatory functions.

M. Newby

- 69 Want to read
- 28 Currently reading

Published
**1978** in Bradford .

Written in English

**Edition Notes**

M.Sc. dissertation. Typescript.

Series | Dissertations |

The Physical Object | |
---|---|

Pagination | 121p. |

Number of Pages | 121 |

ID Numbers | |

Open Library | OL13693946M |

Mathematica 8 adds a number of dramatic improvements to its core algorithms, with a new generation of methods for globally solving equations and inequalities, either symbolically or numerically. New symbolic-numeric methods allow you to numerically integrate a wide class of highly oscillatory functions automatically. Mathematica 8 reaches a new high-water mark in exact linear algebra.

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Functions Gradimir V. Milovanovic and Marija P. Stani´ ´c Abstract 1 Introduction By a highly–oscillating function we mean one with large number of local maxima and minima over some interval.

The computation of integrals of highly–oscillating functions is one of the most important issues in numerical analysis since such inte-File Size: KB. Some specific nonstandard methods for numerical integration of highly oscillating functions, mainly based on some contour integration methods and applications of some kinds of Gaussian quadratures.

NIntegrate of a highly oscillatory integral double exponential oscillatory. Ask Question Asked 3 years, 8 I see the problem from another point of view. Basically, the integral is a Gaussian multiplying a highly oscillatory function (two functions indeed, a Sin and a Cos). Thanks for contributing an answer to Mathematica Stack Exchange.

Abstract. Some specific nonstandard methods for numerical integration of highly oscillating functions, mainly based on some contour integration methods and applications of some kinds of Gaussian quadratures, including complex oscillatory weights, are presented in this by: 5.

Many effective methods have been proposed for the oscillatory integrals in order to overcome the difficulty caused by the high oscillation such as Filon-type methods [10,11,27], Levin methods [15 Author: Sheehan Olver.

4. Buyst and L. Schotsmans,A method of Gaussian type for the numerical integration of oscillating functions, ICC Bull. 3 (), – Google ScholarCited by: A Method for the Numerical Evaluation of Finite Integrals of Oscillatory Functions By I.

Longman 1. Introduction. In two previous publications [1, 2] the author has demon-strated a method, based on Euler's transformation of slowly convergent alternat. The integration of systems containing highly oscillatory functions is a central point in many practical problems in physics, chemistry and engineering.

Highly oscillatory integrals are allegedly difficult to calculate by the standard classic integration formulae when the frequency is significantly larger than the number of quadrature by: Method for numerical integration of difficult oscillatory integral.

Ask Question Asked 7 years, 3 months ago. i.e. integrating g from 0 to 1 should be the same as integrating f from 0 to $\infty$. Levin-type methods are the best established methods for these kinds of problems. The term "numerical integration" first appears in in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.

Quadrature is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean.

A Comparison of Some Methods for the Evaluation of Highly Oscillatory Integrals (by G.A. Evans and r) describes the weighted Clenshaw-Curtis approach to oscillatory integrals. Numerical Approximation of Highly Oscillatory Integrals (PDF) by Sheehan Olver.

Read “Lecture 37” in the textbook Numerical Linear Algebra. satisfies w'(x) =- A(x)w(x) with A(x) an m x m matrix of non-rapidly oscillatory functions, m = kl. Numerical examples In this section we demonstrate the application of the collocation method to three type of oscillatory functions.

The first is the well-studied case S(x) = Jv(rx), also treated in []. TheFile Size: KB. !is an oscillatory kernel which satis es a di erential equation. The aim of this thesis is the numerical approximation of such oscillatory integrals. Perhaps surprisingly, high oscillations make numerical quadrature easier: we will develop methods which actually File Size: 3MB.

Free Online Library: Integrating oscillatory functions in Matlab, II.(Report) by "Electronic Transactions on Numerical methods for integrating oscillatory functions.

book Analysis"; Numerical methods for integrating oscillatory functions. book and Internet Mathematics Approximation Research Approximation theory Functional equations Functions Functions (Mathematics) Mathematical optimization Optimization theory. Integrate a Highly Oscillating Function Use hybrid symbolic-numeric methods to immediately solve problem 1 of the SIAM challenge problems, a difficult, highly.

other function by a polynomial, and use the fruits of the formula for numerical approximation. It appears that Mathematica version includes numerical methods for oscillatory integrands including trigonometric and Bessel functions, though the methods are not speci ed.

New Runge–Kutta methods specially adapted to the numerical integration of IVPs with oscillatory solutions are obtained. The coefficients of these methods are frequency-dependent such that certain particular oscillatory solutions are computed exactly (without truncation errors).Cited by: In this paper we give a short account of the most important methods for the evalua-tion of integrals of oscillatory functions (Sections 2 and 3), and an uniﬁed approach for such a purpose in Section 4.

FILON’S RULE, GAUSSIANFORMULAE ANDINTEGRATION BETWEEN ZEROS The earliest formulas for numerical integration of rapidly oscillatory.

Stack Exchange network consists of Q&A communities including Stack Overflow, Numerical intergration of a complex, oscillatory function (Bessel function, Singularities) Ask Question Browse other questions tagged numerical-methods bessel-functions. Current research made contribution to the numerical analysis of highly oscillatory ordinary differential equations.

Highly oscillatory functions appear to be at the forefront of the research in numerical analysis. In this work we developed efficient numerical algorithms for solving highly oscillatory differential equations.

The main important achievements are: to the contrary of classical Author: Marianna Khanamiryan. on haar wavelets and hybrid functions, Comput Math Applpp. – [9] Siraj-ul-Islam,Aziz, I, KhanW, Numerical integration of multi-dimensional highly oscillatory, gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions, Engineering Analysis with Boundary Elements,pp.

– pose of the research is to nd appropriate integration methods for the oscillatory integrals, we made use of di erent in-built Mathematica functions. We calculated the oscillatory integrals by the use of those various functions.

We then compared the result from di erent integration methods to choose the best method that does the calculation faster. This book provides an up-to-date overview of methods for computing special functions and discusses when to use them in standard parameter domains, as well as in large and complex domains.

The first part of the book covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Moment-free numerical integration of highly oscillatory functions Sheehan Olver ∗ Septem Abstract The aim of this paper is to derive new methods for numerically approximating the integral of a highly oscillatory function.

We begin with a review of the asymptotic and Filon-type methods developed by Iserles and Nørsett. Moment-free numerical integration of highly oscillatory functions The aim of this paper is to derive new methods for numerically approximating the integral of a highly oscillatory function.

We begin with a review of the asymptotic and Filon-type methods developed byCited by: NIntegrate symbolically analyzes its input to transform oscillatory and other integrands, subdivide piecewise functions, and select optimal algorithms.

The method suboption "SymbolicProcessing" specifies the maximum number of seconds for which to attempt performing symbolic analysis of. Numerical experiments are carried out for the algorithms discussed in Section 3 and Section 4 for linear hyperbolic equations with oscillatory coefficients and oscillatory initial values.

In the numerical experiments, we take the time step size At equal to the space grid size h (the CFL condition is satisfied) and always com-Cited by: 6. An introduction to highly oscillatory problemsThe wonderful world of asymptotic expansionsOscillatory integrals Design goal of hybrid numerical-asymptotic methods Say f (n) is the solution to Lf = 0.

Say Q[f ] is the numerical solution to the approximate equation L hf = 0. Then if f (n) ˘ X1 k=1 a kn k; n ˛1 and g(n) ˘ X1 k=1 b kn k; n ˛1. formula (DIFSUB) of Gear [13], [14] and the blended linear multistep methods of Skeel and Kong [24], and the symmetric multistep methods of Lambert and Watson [17].

Introduction. The development of numerical integration formulas for stiff as well as highly oscillatory systems of differential equations has attracted considerable. How to integrate a highly oscillatory function. Follow 16 views (last 30 days) Mila Z on 22 Nov Vote.

0 ⋮ You can't prove divergence of this integral using numerical methods. Try analytical methods instead (e.g.

show that the function behaves like constant*cos(2*y) for large y). DRAFT: Integration of oscillatory integrals, a computer-algebra approach Richard Fateman Computer Science University of California Berkeley, CA, USA Novem Abstract The numerical integration of oscillatory integrals is an important and well-studied area of mathemat-ical inquiry.

on developing numerical methods eﬃcient for solving linear and non-linear systems of highly oscillatory diﬀerential equations.

The key problem of our discussion throughout the thesis is a family of highly oscillatory vector-valued integrals of the form, I[f] = Z b a Xωfdt, X′ ω = AωXω, Xω(0) = I.

4 NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS 0 1 2 −1 − − − − 0 1 time y y=e−t dy/dt Fig. Graphical output from running program in MATLAB. The plot shows the function. NUMERICAL SOLUTION OF HIGHLY OSCILLATORY DIFFERENTIAL EQUATIONS BY MAGNUS SERIES METHOD In this study, the differential equation known as Lie-type equation where the solutions of the equation stay in the Lie-Group is considered.

The solution of this equation can be represented as an innite series whose terms consist of integrals and commutators,File Size: KB. () Multistep numerical methods for the integration of oscillatory problems in several frequencies.

Advances in Engineering Software() Semi-analytical integration algorithms based on the use of several kinds of anomalies as temporal by: terms motivated a combination of asymptotic and numerical approaches taken in [29,15,28,16,17,25].

A numerical approach to the evaluation of oscillatory integrals impacts many problems in other applications as may be seen in [19]. In this paper we introduce a new semi-analytic method for the numerical eval-uation of oscillatory integrals.

() New methods for oscillatory systems based on ARKN methods. Applied Numerical Mathematics() Geometric integrators for multiple time-scale by: Davis, P., and Rabinowitz, P.Methods of Numerical Integration,2nded.(Orlando,FL: Academic Press).

Classical Formulas for Equally Spaced Abscissas Where would any book on numerical analysis be without Mr. Simpson and his “rule”. The classical formulas for integrating a function whose value is known at.

$\begingroup$ @RonGordon Thanks for the link. What is the intuition for it though. Is any of what I wrote in the last to paragraphs of my OP correct at all. Does one simply note that since the function is oscillating sinusoidally, the contributions from its peaks and troughs will cancel each other out if it has a large frequency, since then the neighbouring peaks and troughs will be so close.

(Schatz, Thomee & Wendland ). Another important source of highly oscillatory integrals is geometric numerical integration and methods for highly oscillatory diﬀer-ential equations that expand the solution in multivariate integrals (Degani & SchiﬀIserlesIserles a).

NUMERICAL SOLUTIONS TO TWO-DIMENSIONAL INTEGRATION PROBLEMS by Alexander D. Carstairs Under the Direction of Valerie Miller, PhD ABSTRACT This paper presents numerical solutions to integration problems with bivariate integrands.

Using equally spaced nodes in Adaptive Simpson’s Rule as a base case, two ways of sampling the domain.In this work, we present an adaptive Levin-type method for high-precision computation of highly oscillatory integrals with integrands of the form f(x)exp(i!g(x)).

If g0has no real zero in the integration in-terval and the integrand is su ciently smooth, the method can attain arbitrarily high asymptotic orders without computation of by: 1.Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review David Cohen1, Tobias Jahnke2, Katina Lorenz1, and Christian Lubich1 1 Mathematisches Institut, Univiversit¨at Tu¨bingen, Tu¨bingen.

{Cohen, Lorenz, Lubich}@ 2 Freie Universit¨at Berlin, Institut fu¨r Mathematik II, BioComputing Group, Arnimallee 2–6, Berlin.