The spectral problem for a class of highly oscillatory Fredholm integral operators

@article{Brunner2010TheSP,
  title={The spectral problem for a class of highly oscillatory Fredholm integral operators},
  author={Hermann Brunner and Arieh Iserles and Syvert P. N{\o}rsett},
  journal={Ima Journal of Numerical Analysis},
  year={2010},
  volume={30},
  pages={108-130}
}
Let ℱ ω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K o (x, y)e iω|x―y| . We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f (x; ω) to the associated Fredholm integral equation f = μℱ ω f + a as ω → ∞, thus refining a… 

Figures from this paper

The computation of the spectra of highly oscillatory Fredholm integral operators
where ω ≫ 1. Our main tool is the finite section method: an eigenfunction is expanded in an orthonormal basis of the underlying space, resulting in an algebraic eigenvalue problem. We consider two
On Volterra integral operators with highly oscillatory kernels
We study the high-oscillation properties of solutions to integral equations associated with two classes of Volterra integral operators: compact operators with highly oscillatory kernels that are
On the Spectrum Computation of Non-oscillatory and Highly Oscillatory Kernel with Weak Singularity
TLDR
A spectral Galerkin method with modified Fourier expansion is developed to compute the spectra of highly oscillatory kernel and if the bilinear form associated with the kernel is positive definite, the convergence rate is doubled.
Oscillation Preserving Galerkin Methods for Fredholm Integral Equations of the Second Kind with Oscillatory Kernels
Solutions of Fredholm integral equations of the second kind with oscillatory kernels likely exhibit oscillation. Standard numerical methods applied to solving equations of this type have poor
On the singular values and eigenvalues of the Fox–Li and related operators
The Fox–Li operator is a convolution operator over a finite interval with a special highly oscillatory kernel. It plays an important role in laser engineering. However, the mathematical analysis of
Spectral element approximation of Fredholm integral eigenvalue problems
Spectral theory of large Wiener–Hopf operators with complex-symmetric kernels and rational symbols
Abstract This paper is devoted to the asymptotic behaviour of individual eigenvalues of truncated Wiener–Hopf integral operators over increasing intervals. The kernel of the operators is
Modifid Interpolatory Projection Method for Weakly Singular Integral Equation Eigenvalue Problems
  • Xin ZhangYun He
  • Mathematics
    Acta Mathematicae Applicatae Sinica, English Series
  • 2019
This paper deals with eigenvalue problems for linear Fredholm integral equations of the second kind with weakly singular kernels. A new discrete method is proposed for the approximation of
...
...

References

SHOWING 1-10 OF 18 REFERENCES
Efficient quadrature of highly oscillatory integrals using derivatives
  • A. IserlesS. P. Nørsett
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2005
In this paper, we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the
On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation
TLDR
Based on analytic continuation, rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand, which is compared with the oscillatory integration techniques recently developed by Iserles and Norsett.
On the Quadrature of Multivariate Highly Oscillatory Integrals Over Non-polytope Domains
  • S. Olver
  • Mathematics, Computer Science
    Numerische Mathematik
  • 2006
TLDR
A Levin-type method for approximating multivariate highly oscillatory integrals, subject to a non-resonance condition, which does not require the knowledge of moments to derive an approximation when the oscillator is complicated and when the domain is neither a simplex nor a polytope.
Quadrature methods for multivariate highly oscillatory integrals using derivatives
TLDR
It is demonstrated that, in the absence of critical points and subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices, a direct counterpart of the univariate case.
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
The Construction of cubature rules for multivariate highly oscillatory integrals
TLDR
An efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains by developing Cubature rules that only require the evaluation of the integrand and its derivatives in a limited set of points.
The notion of approximate eigenvalues applied to an integral equation of laser theory
The integral operator with kernel (iri/ir)1'2 exp [—it)(x — yf] on the interval \x\, y < 1 serves to model the behavior of a class of lasers. Although the kernel is simple, it is not Hermitian; this
Eigensystems Associated with the Complex-Symmetric Kernels of Laser Theory
TLDR
Both analytical and numerical techniques and results are presented, some of which are more broadly applicable than others, and promising areas for further investigation are suggested.
The Numerical Solution of Integral Equations of the Second Kind
Preface 1. A brief discussion of integral equations 2. Degenerate kernel methods 3. Projection methods 4. The Nystrom method 5. Solving multivariable integral equations 6. Iteration methods 7.
...
...