Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction
A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2011-07-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.3842/SIGMA.2011.066 |
| _version_ | 1818449270216327168 |
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| author | Stephen C. Anco Sajid Ali Thomas Wolf |
| author_facet | Stephen C. Anco Sajid Ali Thomas Wolf |
| author_sort | Stephen C. Anco |
| collection | DOAJ |
| description | A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation. |
| first_indexed | 2024-12-14T20:32:44Z |
| format | Article |
| id | doaj.art-dc9dd8a00dcf49369163998db4b43c12 |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-14T20:32:44Z |
| publishDate | 2011-07-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-dc9dd8a00dcf49369163998db4b43c122022-12-21T22:48:29ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592011-07-017066Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation ReductionStephen C. AncoSajid AliThomas WolfA novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.http://dx.doi.org/10.3842/SIGMA.2011.066semilinear heat equationsimilarity reductionexact solutionsgroup foliationsymmetry |
| spellingShingle | Stephen C. Anco Sajid Ali Thomas Wolf Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction Symmetry, Integrability and Geometry: Methods and Applications semilinear heat equation similarity reduction exact solutions group foliation symmetry |
| title | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction |
| title_full | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction |
| title_fullStr | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction |
| title_full_unstemmed | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction |
| title_short | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction |
| title_sort | exact solutions of nonlinear partial differential equations by the method of group foliation reduction |
| topic | semilinear heat equation similarity reduction exact solutions group foliation symmetry |
| url | http://dx.doi.org/10.3842/SIGMA.2011.066 |
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