A Complete Orthonormal System of Functions in $L^2 ( - \infty ,\infty )$ Space

Christov, C. I.

doi:10.1137/0142093

Cited by 125 publications

(118 citation statements)

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“…These estimates are not only applicable to the calculation of the Hubert transform. It may also be useful for computing Fourier transforms using Weber's method [21], for the inversion of the Laplace transform with Weeks' method [22], or for solving differential equations using expansions in the basis functions pn(x) [5,8,23].…”

Section: Discussionmentioning

confidence: 99%

“…The function u(x, t) was expanded in terms of the basis functions p"(x) with time-dependent coefficients. The .x-derivatives were computed by the methods described in [8,15,23], and the Hubert transform was computed by the method introduced here. This led to a nonlinear system of ordinary differential equations in the / variable, which we integrated with the explicit midpoint method.…”

Section: Discussionmentioning

confidence: 99%

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Computing the Hilbert Transform on the Real Line

Weideman¹

1995

Mathematics of Computation

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Abstract. We introduce a new method for computing the Hubert transform on the real line. It is a collocation method, based on an expansion in rational eigenfunctions of the Hubert transform operator, and implemented through the Fast Fourier Transform. An error analysis is given, and convergence rates for some simple classes of functions are established. Numerical tests indicate that the method compares favorably with existing methods.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Computing the Hilbert Transform on the Real Line

Weideman¹

1995

Mathematics of Computation

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“…Another e ective direct approach for problems in unbounded domains is based on rational approximations: Christov [11] and Boyd [12; 13] developed some spectral methods on inÿnite intervals by using mutually orthogonal systems of rational functions; most recently, Guo et al [14] developed a Legendre rational spectral method which is based on a weighted orthogonal system consisting of rational functions built from Legendre polynomials under a rational transformation. It is shown in Reference [14] that the Legendre rational method is an attractive alternative for problems in semi-inÿnite intervals.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev rational spectral and pseudospectral methods on a semi‐infinite interval

Guo

Shen

Wang

2001

Numerical Meth Engineering

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SUMMARYA weighted orthogonal system on the half-line based on the Chebyshev rational functions is introduced. Basic results on Chebyshev rational approximations of several orthogonal projections and interpolations are established. To illustrate the potential of the Chebyshev rational spectral method, a model problem is considered both theoretically and numerically: error estimates for the Chebyshev rational spectral and pseudospectral methods are established; preliminary numerical results agree well with the theoretical estimates and demonstrate the e ectiveness of this approach.

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“…(iii) Another class of spectral methods is based on rational approximations. For example, Christov [10] and Boyd [11] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of rational functions. Boyd [12] defined a new spectral basis, named rational Chebyshev functions, on the semi-infinite interval by mapping the Chebyshev polynomials.…”

Section: Introductionmentioning

confidence: 99%

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

Parand

Nikarya

Rad

et al. 2012

Zeitschrift Für Naturforschung A

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In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on R and is convergent for any x ∈ R. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

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A Complete Orthonormal System of Functions in $L^2 ( - \infty ,\infty )$ Space

Cited by 125 publications

References 5 publications

Computing the Hilbert Transform on the Real Line

Computing the Hilbert Transform on the Real Line

Chebyshev rational spectral and pseudospectral methods on a semi‐infinite interval

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

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