| 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 coefficients, the system has a periodic solution under harvesting conditions. The mathematical techniques now depend on estimates for periodic parabolic problems. |
