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Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis
Thesis (PhD (Mathematical Science))--University of Pretoria, 2007.
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University of Pretoria
2013
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| _version_ | 1867613589234253824 |
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| access_status_str | Open Access |
| author2 | Anguelov, Roumen |
| author_browse | Anguelov, Roumen |
| author_facet | Anguelov, Roumen |
| collection | Thesis |
| dc_rights_str_mv | © University of Pretoria 20 |
| description | Thesis (PhD (Mathematical Science))--University of Pretoria, 2007. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/25363 |
| institution | University of Pretoria (South Africa) |
| last_indexed | 2026-06-10T12:38:33.011Z |
| 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:2263/25363 Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis Anguelov, Roumen minanif2006@yahoo.fr Minani, Froduald Viscosity solutions Hamilton-jacobi Hausdorff UCTD Thesis (PhD (Mathematical Science))--University of Pretoria, 2007. The theory of viscosity solutions was developed for certain types of nonlinear first-order and second-order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In its classical formulation, see [16], the theory deals with solutions which are continuous functions. The concept of continuous viscosity solutions was further generalized in various ways to include discontinuous solutions with the definition of Ishii given in [71] playing a pivotal role. In this thesis we propose a new approach for the treatment of discontinuous solutions of first-order Hamilton-Jacobi equations, namely, by involving Hausdorff continuous interval valued functions. The advantages of the proposed approach are justified by demonstrating that the main ideas within the classical theory of continuous viscosity solutions can be extended almost unchanged to the wider space of Hausdorff continuous functions and the existing theory of discontinuous viscosity solutions is a particular case of that developed in this thesis in terms of Hausdorff continuous interval valued functions. Two approaches to numerical solutions for Hamilton-Jacobi equations are presented. The first one is a monotone scheme for Hamilton-Jacobi equations while the second is based on preserving total variation diminishing property for conservation laws. In the first approach, we couple the finite element method with the nonstandard finite difference method which is based on the Mickens’ rule of nonlocal approximation [9]. The scheme obtained in this way is unconditionally monotone. In the second approach, computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size. Mathematics and Applied Mathematics unrestricted 2013-09-06T21:01:08Z 2008-07-03 2013-09-06T21:01:08Z 2008-04-11 2007 2008-06-09 Thesis a 2007 http://hdl.handle.net/2263/25363 http://upetd.up.ac.za/thesis/available/etd-06092008-113253/ © University of Pretoria 20 application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf University of Pretoria |
| spellingShingle | Viscosity solutions Hamilton-jacobi Hausdorff UCTD Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis |
| title | Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis |
| title_full | Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis |
| title_fullStr | Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis |
| title_full_unstemmed | Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis |
| title_short | Hausdorff continuous viscosity solutions of Hamilton-Jacobi equations and their numerical analysis |
| title_sort | hausdorff continuous viscosity solutions of hamilton jacobi equations and their numerical analysis |
| topic | Viscosity solutions Hamilton-jacobi Hausdorff UCTD |
| url | http://hdl.handle.net/2263/25363 http://upetd.up.ac.za/thesis/available/etd-06092008-113253/ |