APPROXIMATE INTEGRATION OF HIGHLY OSCILLATING FUNCTIONS

Numerical integration of highly–oscillating functions

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 integrals abound in applications in many branches of mathematics as well as in other sciences, e.g., quantum physics, fluid mechanics, electromagnetics, etc....

Highly Oscillating Conductivities with Jumps

We study the regularized, oscillatory, multiscale, quasilinear, elliptic problem in divergence form − div(Qη(x, x , u)∇u) = f, in Ω, (0.1) u = g, on ∂Ω, (0.2) assuming that the conductivity has the form Qη(x, y, r) = a−(x) + (a+(x)− a−(x))βη(r − ψ(x, y)), (0.3) for some domain Ω ⊂ Rn, n ≥ 1, a regularization βη of the Heaviside step function and a locally periodic function ψ(x, y), Y -periodic ...

Integration of Highly Oscillatory Problems Through G-Functions

Perturbed harmonic oscillators describe many models in physics and engineering. A method for solving this type of problem is based on the utilization of Scheifele’s functions, consisting of a refinement of the Taylor series method. One disadvantage of the method is that it is difficult to determine, in each case, the recurrence relations necessary for arriving to the solutions. In this paper we...

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 by Iserles and Nørsett. Using a method developed by Levin as a point of departure, we construct a new method that utilizes the same information as the Filon-type method, and obtains the same asymptot...

A Numerical Integration Method by Using Generalized Series of Functions