Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs
We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This tre...
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| Format: | Article |
| Language: | English |
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Wiley
2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/59619 |
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| author | M. P. Markakis |
| author_facet | M. P. Markakis |
| author_sort | M. P. Markakis |
| collection | DOAJ |
| description | We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the
original equation is expressed. This treatment is applied to another nonlinear equation,
subjected to the same general class as that of Gambier, by constructing step by step an
appropriate analytical technique. The developed procedure yields a general exact closed
form solution of this equation, valid for specific values of the parameters involved and
containing two arbitrary (free) parameters evaluated by the relevant initial conditions.
We finally verify this technique by applying it to two specific sets of parameter values of
the equation under consideration. |
| format | Article |
| id | doaj-art-d9a4c7af14894fa5bb40153538f25f8d |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d9a4c7af14894fa5bb40153538f25f8d2025-02-03T05:52:33ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252007-01-01200710.1155/2007/5961959619Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEsM. P. Markakis0Department of Engineering Sciences, University of Patras, Patras 26500, GreeceWe establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.http://dx.doi.org/10.1155/2007/59619 |
| spellingShingle | M. P. Markakis Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs International Journal of Mathematics and Mathematical Sciences |
| title | Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs |
| title_full | Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs |
| title_fullStr | Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs |
| title_full_unstemmed | Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs |
| title_short | Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs |
| title_sort | use of a strongly nonlinear gambier equation for the construction of exact closed form solutions of nonlinear odes |
| url | http://dx.doi.org/10.1155/2007/59619 |
| work_keys_str_mv | AT mpmarkakis useofastronglynonlineargambierequationfortheconstructionofexactclosedformsolutionsofnonlinearodes |