Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem
In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative...
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| Format: | Article |
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MDPI AG
2020-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/2/213 |
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| author | Assiya Zhumanazarova Young Im Cho |
| author_facet | Assiya Zhumanazarova Young Im Cho |
| author_sort | Assiya Zhumanazarova |
| collection | DOAJ |
| description | In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution. |
| first_indexed | 2024-12-13T07:01:38Z |
| format | Article |
| id | doaj.art-41f3e3fe56494f67a5a9490b5bb411c7 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-13T07:01:38Z |
| publishDate | 2020-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-41f3e3fe56494f67a5a9490b5bb411c72022-12-21T23:55:55ZengMDPI AGMathematics2227-73902020-02-018221310.3390/math8020213math8020213Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value ProblemAssiya Zhumanazarova0Young Im Cho1Department of Computer Engineering, Gachon University, Gyeonggi-do 461-701, KoreaDepartment of Computer Engineering, Gachon University, Gyeonggi-do 461-701, KoreaIn this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.https://www.mdpi.com/2227-7390/8/2/213small parametersingular perturbationboundary functionsasymptotic estimates |
| spellingShingle | Assiya Zhumanazarova Young Im Cho Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem Mathematics small parameter singular perturbation boundary functions asymptotic estimates |
| title | Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem |
| title_full | Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem |
| title_fullStr | Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem |
| title_full_unstemmed | Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem |
| title_short | Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem |
| title_sort | asymptotic convergence of the solution of a singularly perturbed integro differential boundary value problem |
| topic | small parameter singular perturbation boundary functions asymptotic estimates |
| url | https://www.mdpi.com/2227-7390/8/2/213 |
| work_keys_str_mv | AT assiyazhumanazarova asymptoticconvergenceofthesolutionofasingularlyperturbedintegrodifferentialboundaryvalueproblem AT youngimcho asymptoticconvergenceofthesolutionofasingularlyperturbedintegrodifferentialboundaryvalueproblem |