Three algorithms for Hadamard finite-part integrals and fractional derivatives
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TLDR
Three algorithms for the evaluation of the Hadamard finite-part integral of the form f(t) (1−t) 1+α) , where α is a positive non-integer, are described.About:
This article is published in Journal of Computational and Applied Mathematics.The article was published on 1995-09-22 and is currently open access. It has received 15 citations till now. The article focuses on the topics: Chebyshev nodes & Minimal polynomial (field theory).read more
Citations
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Finite difference methods with non-uniform meshes for nonlinear fractional differential equations
Changpin Li,Qian Yi,An Chen +2 more
TL;DR: In this article, finite difference methods with non-uniform meshes for solving nonlinear fractional differential equations are presented, where the non-equidistant stepsize is non-decreasing and the rectangle formula and trapezoid formula are proposed based on theNon- uniform meshes.
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Uniform approximations to finite Hilbert transform and its derivative
TL;DR: In this article, interpolating a smooth function f(t) at abscissae in the interval of integration is shown to be essential for uniformly approximating Cauchy principal value integrals and Hadamard finite part integrals.
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On the issue of imposing boundary conditions on quantum fields
TL;DR: In this article, Hadamard regularization is invoked in order to fill the gap between the infinities appearing in the QFT renormalized results and the finite values obtained in the literature with other procedures.
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The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions
TL;DR: A threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand.
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Dynamical Casimir Effect with Semi-Transparent Mirrors, and Cosmology
TL;DR: In this article, the authors consider the dynamical Casimir effect (or Fulling-Davis theory), where related regularization problems appear, with a view to an experimental verification of this theory.
References
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Handbook of Mathematical Functions
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Higher Transcendental Functions
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.
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The special functions and their approximations
C. W. Clenshaw,Yudell L. Luke +1 more