A fast algorithm for solving the generalized airfoil equation

In this chapter we present a survey of many of the polynomial-based projection and quadrature methods that have been developed in the past 15 years for the numerical solution of Cauchy singular integral equations with constant coefficients on (−1, 1) Emphasis is placed on equations of the first kind, where an elementary proof of the inversion formula for the airfoil equation due to Peters enables one to develop the theory and subsequent numerical analysis using techniques which are well known for Fredholm equations of the second kind Convergence proofs are given for many of the algorithms, and numerical examples are drawn from the literature to illustrate the efficiency of these methods when the data are smooth However, in many practical problems where either the kernel and/or the right-hand side may have discontinuities, this is not the case, and further work needs to be done Some possibilities for doing this are examined as well

Introduction to the Numerical Solution of Cauchy Singular Integral Equations

A fast and well-conditioned spectral method for singular integral equations

In this paper, we propose piecewise-polynomial methods for the approximation of Cauchy principal value integrals and develop a simple, efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise-polynomial quadratures. We present two examples to illustrate the advantages of these quadratures versus the Gauss–Jacobi quadratures.

Piecewise-polynomial quadratures for Cauchy singular integrals

When a function is expanded as $$f(x) \approx \sum \limits _{n=0}^{N-1}\, a_{n}\phi_{n}(x)$$
 for some set of basis functions $${\phi_{j}(x)}$$
 , its spectral coefficients a
 n
 generally have an asymptotic approximation, as n→∞, in the form of an inverse power series plus terms that decrease exponentially with n. If f(x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of “beyond-all-orders” or “exponential” asymptotics. The method of steepest descent for integrals and other complex-path integration techniques can successfully connect the exponentially small behavior of the spectral coefficients to the singularities of f(x) off the expansion interval. Many examples are given in both one and two dimensions.

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

The Chebyshev polynomials of both the first and second kind are of fundamental importance when considering the particular case of the singular integral equation $a(t)\phi (t) + ({{b(t)} / \pi })\lambda \int_{ - 1}^1 {({{\phi (\tau )} / {(\tau - t)}})} d\tau = f(t), - 1 < t < 1$, in which $a \equiv 0$ and $b \equiv - 1$ on $[ - 1,1]$. We identify the two sets of orthogonal polynomials which play a corresponding role for the singular integral equation with general a, b and consider some of the relationships between these two sets of polynomials.

Orthogonal Polynomials Associated with Singular Integral Equations Having a Cauchy Kernel

Complex variable techniques are used to estimate the Fourier coefficients of functions expanded in series of Jacobi, Laguerre and Hermite polynomials.

Asymptotic Estimates of Fourier Coefficients

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Orthogonal Polynomials Associated with Singular Integral Equations Having a Cauchy Kernel

Asymptotic Estimates of Fourier Coefficients