Numerical integration of the Cauchy problem with non-singular special points

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Abstract

Solutions of many applied Cauchy problems for ordinary differential equations have one or more multiple zeros on the integration segment. Examples are the equations of special functions of mathematical physics. The presence of multiples of zeros significantly complicates the numerical calculation, since such problems are ill-conditioned. Round-off errors may corrupt all decimal digits of the solution. Therefore, multiple zeros should be treated as special points of the differential equations. In the present paper, a local solution transformation is proposed, which converts the multiple zero into a simple one. The calculation of the latter is not difficult. This makes it possible to dramatically improve the accuracy and reliability of the calculation. Illustrative examples have been carried out, which confirm the advantages of the proposed method.

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1. Introduction Consider the Cauchy problem for an ordinary differential equation (ODE)
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About the authors

Aleksandr A. Belov

M.V. Lomonosov Moscow State University; Peoples’ Friendship University of Russia (RUDN University)

Email: aa.belov@physics.msu.ru
ORCID iD: 0000-0002-0918-9263

Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Computational Mathematics and Artificial Intelligence of Peoples’ Friendship University of Russia (RUDN University); Researcher of Faculty of Physics, M.V. Lomonosov Moscow State University

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

Igor V. Gorbov

M.V. Lomonosov Moscow State University

Author for correspondence.
Email: garri-g@bk.ru
ORCID iD: 0009-0005-5335-6179

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., Gorbov I.V.

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