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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2011 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2011
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147187 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction / S.C. Anco, S. Ali, T. Wolf // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862631312176709632 |
|---|---|
| author | Anco, S.C. Ali, S. Wolf, T. |
| author_facet | Anco, S.C. Ali, S. Wolf, T. |
| citation_txt | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction / S.C. Anco, S. Ali, T. Wolf // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-11-30T11:09:30Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147187 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T11:09:30Z |
| publishDate | 2011 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Anco, S.C. Ali, S. Wolf, T. 2019-02-13T18:41:09Z 2019-02-13T18:41:09Z 2011 Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction / S.C. Anco, S. Ali, T. Wolf // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 16 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35K58; 35C06; 35A25; 58J70; 34C14 DOI:10.3842/SIGMA.2011.066 https://nasplib.isofts.kiev.ua/handle/123456789/147187 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. This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
 S. Anco and T. Wolf are each supported by an NSERC research grant. S. Ali thanks the
 Mathematics Department of Brock University for support during the period of a research visit when this paper was written. Computations were partly performed on computers of the Sharcnet consortium (www.sharcnet.ca). The referees and the editor are thanked for valuable comments which have improved this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction Article published earlier |
| spellingShingle | Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction Anco, S.C. Ali, S. Wolf, T. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147187 |
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