Erratum to: Numerical Solution of Riemann–Hilbert Problems: Painlevé II
TL;DR: A new, spectrally accurate method for solving matrixvalued Riemann–Hilbert problems numerically is described and can be used to relate initial conditions with asymptotic behaviour.
Abstract: We describe a new, spectrally accurate method for solving matrixvalued Riemann–Hilbert problems numerically. The effectiveness of this approach is demonstrated by computing solutions to the homogeneous Painlevé II equation. This can be used to relate initial conditions with asymptotic behaviour.
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TL;DR: In this article, the small dispersion limit for the Korteweg-de Vries (KdV) equation is studied numerically and a quantitative comparison of the numerical solution with various asymptotic formulae for small ϵ in the whole ( x, t ) -plane is given.
44 citations
Cites methods from "Erratum to: Numerical Solution of R..."
...Solutions to the Painlevé II equation have been computed as the solution of a Riemann-Hilbert problem in [53]....
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TL;DR: Higher Transcendental Functions Based on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project as discussed by the authors, are presented in Table 1.
Abstract: Higher Transcendental Functions Based, in part, on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project. Vol. 1. Pp. xxvi + 302. 52s. Vol. 2. Pp. xvii + 396. 60s. (London: McGraw-Hill Publishing Company, Ltd., 1953.)
4,428 citations
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...Using these maps, we obtain the following formulæ: Theorem 3.2 Define µm(z) = bm+12 c∑ j=1 z2j−1 2j − 1 , ψ0(z) = 2 iπ arctanh z, ψm(z) = z m [ ψ0(z)− 2 iπ { µ−m−1(z) for m < 0 µm(1/z) for m > 0 ] ....
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TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.
Abstract: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.
4,083 citations
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01 Oct 2000TL;DR: In this paper, the authors present an asymptotics for orthogonal polynomials in Riemann-Hilbert problems and Jacobi operators for continued fractions.
Abstract: Riemann-Hilbert problems Jacobi operators Orthogonal polynomials Continued fractions Random matrix theory Equilibrium measures Asymptotics for orthogonal polynomials Universality Bibliography.
1,572 citations
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...Keywords Riemann–Hilbert problems, spectral methods, collocation methods, Painlevé transcendents....
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TL;DR: Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable and deserves to be known as the standard method of polynometric interpolation.
Abstract: Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.
1,177 citations
"Erratum to: Numerical Solution of R..." refers background in this paper
...In practice, f will vanish at t = +1, as this will correspond to z =∞....
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TL;DR: In this paper, a method for numerical integration of a well-behaved function over a finite range of argument is described, which consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term.
Abstract: A new method for the numerical integration of a "well-behaved" function over a finite range of argument is described. It consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term. Illustrative examples are given, and the method is compared with the most commonly-used alternatives, namelySimpson's rule and the method ofGauss.
919 citations