Solutions of Smooth Nonlinear Partial Differential Equations
The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significa...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/658936 |
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| _version_ | 1832550315427102720 |
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| author | Jan Harm van der Walt |
| author_facet | Jan Harm van der Walt |
| author_sort | Jan Harm van der Walt |
| collection | DOAJ |
| description | The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial
differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant
improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the
parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension. |
| format | Article |
| id | doaj-art-ede9c5e335a549558b8dd52bd4aef5f3 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-ede9c5e335a549558b8dd52bd4aef5f32025-02-03T06:07:05ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/658936658936Solutions of Smooth Nonlinear Partial Differential EquationsJan Harm van der Walt0Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South AfricaThe method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension.http://dx.doi.org/10.1155/2011/658936 |
| spellingShingle | Jan Harm van der Walt Solutions of Smooth Nonlinear Partial Differential Equations Abstract and Applied Analysis |
| title | Solutions of Smooth Nonlinear Partial Differential Equations |
| title_full | Solutions of Smooth Nonlinear Partial Differential Equations |
| title_fullStr | Solutions of Smooth Nonlinear Partial Differential Equations |
| title_full_unstemmed | Solutions of Smooth Nonlinear Partial Differential Equations |
| title_short | Solutions of Smooth Nonlinear Partial Differential Equations |
| title_sort | solutions of smooth nonlinear partial differential equations |
| url | http://dx.doi.org/10.1155/2011/658936 |
| work_keys_str_mv | AT janharmvanderwalt solutionsofsmoothnonlinearpartialdifferentialequations |