Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations
In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change...
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2017-03-01
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| Series: | Mathematical Modelling and Analysis |
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| Online Access: | https://journals.vgtu.lt/index.php/MMA/article/view/881 |
| _version_ | 1819158591113789440 |
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| author | Serge Bruno Yamgoue Olivier Tiokeng Lekeufack Timoleon Crepin Kofane |
| author_facet | Serge Bruno Yamgoue Olivier Tiokeng Lekeufack Timoleon Crepin Kofane |
| author_sort | Serge Bruno Yamgoue |
| collection | DOAJ |
| description | In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach. |
| first_indexed | 2024-12-22T16:26:57Z |
| format | Article |
| id | doaj.art-a890c1b84f214de6a1f70c658e5cf079 |
| institution | Directory Open Access Journal |
| issn | 1392-6292 1648-3510 |
| language | English |
| last_indexed | 2024-12-22T16:26:57Z |
| publishDate | 2017-03-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj.art-a890c1b84f214de6a1f70c658e5cf0792022-12-21T18:20:08ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102017-03-0122210.3846/13926292.2017.1276983Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential EquationsSerge Bruno Yamgoue0Olivier Tiokeng Lekeufack1Timoleon Crepin Kofane2Department of Physics, Higher Teachers Training College Bambili, The University of Bamenda, Po. Box 39 Bamenda, CameroonLaboratoire de Mecanique, Departement de Physique, Faculte de Sciences, Universite de Yaounde I, B.P. 812 Yaound´e – CameroonLaboratoire de Mecanique, Departement de Physique, Faculte de Sciences, Universite de Yaounde I, B.P. 812 Yaounde – Cameroon; Centre d’Excellence Africain des Technologies de l’Information et de la Communication (CETIC), Universite de Yaounde I, Yaounde, CameroonIn this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach.https://journals.vgtu.lt/index.php/MMA/article/view/881harmonic balancelinearizationexplicit approximationssolitonshyperbolic solutions |
| spellingShingle | Serge Bruno Yamgoue Olivier Tiokeng Lekeufack Timoleon Crepin Kofane Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations Mathematical Modelling and Analysis harmonic balance linearization explicit approximations solitons hyperbolic solutions |
| title | Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations |
| title_full | Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations |
| title_fullStr | Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations |
| title_full_unstemmed | Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations |
| title_short | Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations |
| title_sort | harmonic balance for non periodic hyperbolic solutions of nonlinear ordinary differential equations |
| topic | harmonic balance linearization explicit approximations solitons hyperbolic solutions |
| url | https://journals.vgtu.lt/index.php/MMA/article/view/881 |
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