Numerical integration of the Cauchy problem with non-singular special points
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, s...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Peoples’ Friendship University of Russia (RUDN University)
2023-09-01
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| Series: | Discrete and Continuous Models and Applied Computational Science |
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| Online Access: | https://journals.rudn.ru/miph/article/viewFile/35918/22460 |
| _version_ | 1797682191370551296 |
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| author | Aleksandr A. Belov Igor V. Gorbov |
| author_facet | Aleksandr A. Belov Igor V. Gorbov |
| author_sort | Aleksandr A. Belov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T23:55:59Z |
| format | Article |
| id | doaj.art-631d40337473467e92536d50056d27db |
| institution | Directory Open Access Journal |
| issn | 2658-4670 2658-7149 |
| language | English |
| last_indexed | 2024-03-11T23:55:59Z |
| publishDate | 2023-09-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | Discrete and Continuous Models and Applied Computational Science |
| spelling | doaj.art-631d40337473467e92536d50056d27db2023-09-18T12:29:16ZengPeoples’ Friendship University of Russia (RUDN University)Discrete and Continuous Models and Applied Computational Science2658-46702658-71492023-09-0131321822710.22363/2658-4670-2023-31-3-218-22721020Numerical integration of the Cauchy problem with non-singular special pointsAleksandr A. Belov0https://orcid.org/0000-0002-0918-9263Igor V. Gorbov1https://orcid.org/0009-0005-5335-6179M.V. Lomonosov Moscow State UniversityM.V. Lomonosov Moscow State UniversitySolutions 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.https://journals.rudn.ru/miph/article/viewFile/35918/22460ordinary differential equationscauchy problemmultiple zerosolution transformation |
| spellingShingle | Aleksandr A. Belov Igor V. Gorbov Numerical integration of the Cauchy problem with non-singular special points Discrete and Continuous Models and Applied Computational Science ordinary differential equations cauchy problem multiple zero solution transformation |
| title | Numerical integration of the Cauchy problem with non-singular special points |
| title_full | Numerical integration of the Cauchy problem with non-singular special points |
| title_fullStr | Numerical integration of the Cauchy problem with non-singular special points |
| title_full_unstemmed | Numerical integration of the Cauchy problem with non-singular special points |
| title_short | Numerical integration of the Cauchy problem with non-singular special points |
| title_sort | numerical integration of the cauchy problem with non singular special points |
| topic | ordinary differential equations cauchy problem multiple zero solution transformation |
| url | https://journals.rudn.ru/miph/article/viewFile/35918/22460 |
| work_keys_str_mv | AT aleksandrabelov numericalintegrationofthecauchyproblemwithnonsingularspecialpoints AT igorvgorbov numericalintegrationofthecauchyproblemwithnonsingularspecialpoints |