Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)3591810.22363/2658-4670-2023-31-3-218-227YENIDIArticlesСтатьиResearch ArticleNumerical integration of the Cauchy problem with non-singular special pointsЧисленное интегрирование задач Коши с несингулярными особыми точкамиhttps://orcid.org/0000-0002-0918-9263BelovAleksandr A.БеловА. А.

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

aa.belov@physics.msu.ruhttps://orcid.org/0009-0005-5335-6179GorbovIgor V.ГорбовИ. В.

Master’s degree student of Faculty of Physics

garri-g@bk.ruM.V. Lomonosov Moscow State UniversityМосковский государственный университет им. М.В. ЛомоносоваPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народов12092023313VOL 31, NO3 (2023)ТОМ 31, №3 (2023)21822712092023Copyright ©; 2023, Belov A.A., Gorbov I.V.Copyright ©; 2023, Белов А.А., Горбов И.В.2023Belov A.A., Gorbov I.V.Белов А.А., Горбов И.В.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/35918

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.

Решения многих прикладных задач Коши для обыкновенных дифференциальных уравнений имеют один или несколько кратных нулей на отрезке интегрирования. Примерами являются уравнения специальных функций математической физики. Наличие кратных нулей существенно затрудняет численный расчёт, поскольку такие задачи являются плохо обусловленными. Из-за ошибок округления в решении может не остаться ни одного верного знака. Поэтому кратные нули следует отнести к особым точкам ОДУ. В данной работе предложена локальная замена искомой функции, которая преобразует кратный нуль решения в простой. Расчёт последнего не представляет трудностей. Это позволяет кардинально повысить точность и надёжность расчёта. Проведены иллюстративные примеры, которые подтверждают преимущества предлагаемого метода.

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