In this thesis, we study the mathematical modelling of some problems that
involve surface effects. These include an optical biosensor, which uses optical
principles qualitatively to convert chemical and biochemical concentrations
into electrical signals. A typical sensor of this type was constructed in
Badley et al., [6], and Jones et al., [18],but diffusion was considered in
only one direction in [18] to simulate the reaction between the antigen and
the antibody. For realistic applications, we propose the biosensor model in
R3. Our theoretical approach is explicitly presented since it is simple and
directly applicable to the numerical part of the thesis. In particular, we
present existence and uniqueness results based on Maximum Principle and
weak solution arguments. These ideas are later applied to systems and to
the numerical analysis of the approximate discretized problems.It should
be noted that without one dimensional symmetry, the equations can not be
decoupled in order to reduce the problem to a single equation. We also show
the long time monotonic convergence to the steady state. Next, a finite
volume method is applied to the equations, and we obtain existence and
uniqueness for the approximate solution as well as the convergence of the the
first order temporal norm and the L2 spatial norm. We illustrate the results
via some numerical simulations. Finally we consider a mathematically related
system motivated by lagoon ecology. We show that under suitable conditions
on the coecients, the system has a periodic solution under harvesting
conditions. The mathematical techniques now depend on estimates for
periodic parabolic problems. / Applied Mathematics
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:AEU.10048/668 |
| Date | 11 1900 |
| Creators | Zhang, Zhiyong |
| Contributors | Walter Allegretto, Mathematical Science, Yanping Lin, Mathematical Science, Yau Shu Wong, Mathematical Science, Xinwei Yu, Mathematical Science, Zihui Xia, Mechanical Engineering, Xiaoqiang Zhao, Mathematical Science, Memorial University of Newfoundland |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 6222481 bytes, application/pdf |
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