Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles
In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infini...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-02-01
|
| Series: | Mathematical and Computational Applications |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2297-8747/29/1/11 |
| _version_ | 1797297568193970176 |
|---|---|
| author | Gilbert Kerr Nehemiah Lopez Gilberto González-Parra |
| author_facet | Gilbert Kerr Nehemiah Lopez Gilberto González-Parra |
| author_sort | Gilbert Kerr |
| collection | DOAJ |
| description | In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases. |
| first_indexed | 2024-03-07T22:23:13Z |
| format | Article |
| id | doaj.art-d6b0fe73c80a45d2902674ea009af839 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-07T22:23:13Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-d6b0fe73c80a45d2902674ea009af8392024-02-23T15:26:21ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472024-02-012911110.3390/mca29010011Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit CyclesGilbert Kerr0Nehemiah Lopez1Gilberto González-Parra2Department of Mathematics, New Mexico Tech, Socorro, NM 87801, USADepartment of Mathematics, New Mexico Tech, Socorro, NM 87801, USADepartment of Mathematics, New Mexico Tech, Socorro, NM 87801, USAIn this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases.https://www.mdpi.com/2297-8747/29/1/11systems of linear delay differential equationsretardedneutralLaplace transformnon-harmonic Fourier serieslimit cycles |
| spellingShingle | Gilbert Kerr Nehemiah Lopez Gilberto González-Parra Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles Mathematical and Computational Applications systems of linear delay differential equations retarded neutral Laplace transform non-harmonic Fourier series limit cycles |
| title | Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles |
| title_full | Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles |
| title_fullStr | Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles |
| title_full_unstemmed | Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles |
| title_short | Analytical Solutions of Systems of Linear Delay Differential Equations by the Laplace Transform: Featuring Limit Cycles |
| title_sort | analytical solutions of systems of linear delay differential equations by the laplace transform featuring limit cycles |
| topic | systems of linear delay differential equations retarded neutral Laplace transform non-harmonic Fourier series limit cycles |
| url | https://www.mdpi.com/2297-8747/29/1/11 |
| work_keys_str_mv | AT gilbertkerr analyticalsolutionsofsystemsoflineardelaydifferentialequationsbythelaplacetransformfeaturinglimitcycles AT nehemiahlopez analyticalsolutionsofsystemsoflineardelaydifferentialequationsbythelaplacetransformfeaturinglimitcycles AT gilbertogonzalezparra analyticalsolutionsofsystemsoflineardelaydifferentialequationsbythelaplacetransformfeaturinglimitcycles |