Quadratures with super power convergence

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Abstract

The calculation of quadratures arises in many physical and technical applications. The replacement of integration variables is proposed, which dramatically increases the accuracy of the formula of averages. For infinitely smooth integrand functions, the convergence law becomes super power. It is significantly faster than the power law and is close to exponential one. For integrals with bounded smoothness, power convergence is realized with the maximum achievable order of accuracy.

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1. Introduction Applied tasks. In many physical problems, it is required to approximate integrals that are not taken in elementary functions. Here are some examples: 2. Calculation of special functions of mathematical physics: Fermi-Dirac functions equal to the moments of the Fermi distribution, gamma function, cylindrical functions and a number of others. 3. Calculation of Fourier coefficients of a given function, Fourier and Laplace transforms. 4. Numerical solution of integral equations, both correctly posed and incorrect. 5. Solving boundary value problems for partial differential equations (including eigenvalue problems) written in integral form, etc. Such integrals must be calculated with high accuracy up to computer round-off errors. Calculation of quadratures. Commonly, trapezoid, mean and Simpson methods on a uniform grid are used for grid calculation of quadratures. The majorant error estimation is well known for these methods. For trapezoid and mean formulas it is
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About the authors

Aleksandr A. Belov

M. V. Lomonosov Moscow State University; RUDN University

Author for correspondence.
Email: aa.belov@physics.msu.ru
ORCID iD: 0000-0002-0918-9263
Scopus Author ID: 57191950560
ResearcherId: Q-5064-2016

Candidate of Physical and Mathematical Sciences, Researcher of Faculty of Physics, M. V. Lomonosov Moscow State University; Assistant Professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation; 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Maxim A. Tintul

M. V. Lomonosov Moscow State University

Email: maksim.tintul@mail.ru
ORCID iD: 0000-0002-5466-1221

Master’s Degree Student of Faculty of Physics

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation

Valentin S. Khokhlachev

M. V. Lomonosov Moscow State University

Email: valentin.mycroft@yandex.ru
ORCID iD: 0000-0002-6590-5914

Master’s Degree Student of Faculty of Physics

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation

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Copyright (c) 2023 Belov A.A., Tintul M.A., Khokhlachev V.S.

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