Eigenvalue approximations for linear periodic differential equations with a singularity
We consider the second order, linear differential equation \begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*} where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Szeged
1999-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=16 |
| _version_ | 1797830796864651264 |
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| author | B. J. Harris F. Marzano |
| author_facet | B. J. Harris F. Marzano |
| author_sort | B. J. Harris |
| collection | DOAJ |
| description | We consider the second order, linear differential equation
\begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*}
where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$. |
| first_indexed | 2024-04-09T13:42:54Z |
| format | Article |
| id | doaj.art-9e6f6a3776854122b916118e9995dc0e |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:42:54Z |
| publishDate | 1999-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-9e6f6a3776854122b916118e9995dc0e2023-05-09T07:52:56ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38751999-01-011999711810.14232/ejqtde.1999.1.716Eigenvalue approximations for linear periodic differential equations with a singularityB. J. Harris0F. Marzano1Northern Illinois University, DeKalb, Illinois, U.S.A.Edinboro University of Pennsylvania, Edinboro, Pennsylvania, U.S.A.We consider the second order, linear differential equation \begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*} where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=16 |
| spellingShingle | B. J. Harris F. Marzano Eigenvalue approximations for linear periodic differential equations with a singularity Electronic Journal of Qualitative Theory of Differential Equations |
| title | Eigenvalue approximations for linear periodic differential equations with a singularity |
| title_full | Eigenvalue approximations for linear periodic differential equations with a singularity |
| title_fullStr | Eigenvalue approximations for linear periodic differential equations with a singularity |
| title_full_unstemmed | Eigenvalue approximations for linear periodic differential equations with a singularity |
| title_short | Eigenvalue approximations for linear periodic differential equations with a singularity |
| title_sort | eigenvalue approximations for linear periodic differential equations with a singularity |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=16 |
| work_keys_str_mv | AT bjharris eigenvalueapproximationsforlinearperiodicdifferentialequationswithasingularity AT fmarzano eigenvalueapproximationsforlinearperiodicdifferentialequationswithasingularity |