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Generalized solutions of systems of nonlinear partial differential equations
Thesis (PhD)--University of Pretoria, 2009.
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University of Pretoria
2013
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| _version_ | 1867613514411016192 |
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| access_status_str | Open Access |
| author2 | Rosinger, Elemer E. |
| author_browse | Rosinger, Elemer E. |
| author_facet | Rosinger, Elemer E. |
| collection | Thesis |
| dc_rights_str_mv | © University of Pretoria 2009 |
| description | Thesis (PhD)--University of Pretoria, 2009. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/24933 |
| institution | University of Pretoria (South Africa) |
| last_indexed | 2026-06-10T12:37:21.635Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2013 |
| publishDateRange | 2013 |
| publishDateSort | 2013 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/24933 Generalized solutions of systems of nonlinear partial differential equations Rosinger, Elemer E. Anguelov, Roumen janharm.vanderwalt@up.ac.za Van der Walt, Jan Harm Nonlinear partial differential equations Order completion method UCTD Thesis (PhD)--University of Pretoria, 2009. In this thesis, we establish a general and type independent theory for the existence and regularity of generalized solutions of large classes of systems of nonlinear partial differential equations (PDEs). In this regard, our point of departure is the Order Completion Method. The spaces of generalized functions to which the solutions of such systems of PDEs belong are constructed as the completions of suitable uniform convergence spaces of normal lower semi-continuous functions. It is shown that large classes of systems of nonlinear PDEs admit generalized solutions in the mentioned spaces of generalized functions. Furthermore, the generalized solutions that we construct satisfy a blanket regularity property, in the sense that such solutions may be assimilated with usual normal lower semi-continuous functions. These fundamental existence and regularity results are obtain as applications of basic topological processes, namely, the completion of uniform convergence spaces, and elementary properties of real valued continuous functions. In particular, those techniques from functional analysis which are customary in the study of nonlinear PDEs are not used at all. The mentioned sophisticated methods of functional analysis are used only to obtain additional regularity properties of the generalized solutions of systems of nonlinear PDEs, and are thus relegated to a secondary role. Over and above the mentioned blanket regularity of the solutions, it is shown that for a large class of equations, the generalized solutions are in fact usual classical solutions of the respective system of equations everywhere except on a closed, nowhere dense subset of the domain of definition of the system of equations. This result is obtained under minimal assumptions on the smoothness of the equations, and is an application of convenient compactness theorems for sets of sufficiently smooth functions with respect to suitable topologies on spaces of such functions. As an application of the existence and regularity results presented here, we obtain for the first time in the literature an extension of the celebrated Cauchy-Kovalevskaia Theorem, on its own general and type independent grounds, to equations that are not analytic. Mathematics and Applied Mathematics unrestricted 2013-09-06T18:52:44Z 2009-06-02 2013-09-06T18:52:44Z 2009-04-28 2009-06-02 2009-05-24 Thesis Van der Walt, JH 2009, Generalized solutions of systems of nonlinear partial differential equations, PhD thesis, University of Pretoria, Pretoria, viewed yymmdd < http://hdl.handle.net/2263/24933 > D624/ag http://hdl.handle.net/2263/24933 http://upetd.up.ac.za/thesis/available/etd-05242009-122628/ © University of Pretoria 2009 application/pdf application/pdf application/pdf application/pdf application/pdf University of Pretoria |
| spellingShingle | Nonlinear partial differential equations Order completion method UCTD Generalized solutions of systems of nonlinear partial differential equations |
| title | Generalized solutions of systems of nonlinear partial differential equations |
| title_full | Generalized solutions of systems of nonlinear partial differential equations |
| title_fullStr | Generalized solutions of systems of nonlinear partial differential equations |
| title_full_unstemmed | Generalized solutions of systems of nonlinear partial differential equations |
| title_short | Generalized solutions of systems of nonlinear partial differential equations |
| title_sort | generalized solutions of systems of nonlinear partial differential equations |
| topic | Nonlinear partial differential equations Order completion method UCTD |
| url | http://hdl.handle.net/2263/24933 http://upetd.up.ac.za/thesis/available/etd-05242009-122628/ |