Loading...
3 results
Search Results
Now showing 1 - 3 of 3
- Modelling silicosis: dynamics of a model with piecewise constant rate coefficientsPublication . Antunes, Pedro R. S.; Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe study the dynamics about equilibria of an infinite dimensional system of ordinary differential equations of coagulation–fragmentation–death type that was introduced recently by da Costa et al. (Eur J Appl Math 31(6):950–967, 2020) as a model for the silicosis disease mechanism. For a class of piecewise constant rate coefficients an appropriate change of variables allows for the appearance of a closed finite dimensional subsystem of the infinite-dimensional system and the analysis of the eigenvalues of the linearizations of this finite dimensional subsystem about the equilibria is then used to obtain the results on the stability of the equilibria in the original infinite dimensional model.
- Steady state solutions in a model of a cholesteric liquid crystal samplePublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Grinfeld, Michael; Mottram, Nigel; Xayxanadasy, KedtysackMotivated by recent mathematical studies of Fréedericksz transitions in twist cells and helix unwinding in cholesteric liquid crystal cells [(da Costa et al. in Eur J Appl Math 20:269–287, 2009), (da Costa et al. in Eur J Appl Math 28:243–260, 2017), (McKay in J Eng Math 87:19–28, 2014), (Millar and McKay in Mol Cryst Liq Cryst 435:277/[937]–286/[946], 2005)], we consider a model for the director configuration obtained within the framework of the Frank-Oseen theory and consisting of a nonlinear ordinary differential equation in a bounded interval with non-homogeneous mixed boundary conditions (Dirichlet at one end of the interval, Neumann at the other). We study the structure of the solution set using the depth of the sample as a bifurcation parameter. Employing phase space analysis techniques, time maps, and asymptotic methods to estimate integrals, together with appropriate numerical evidence, we obtain the corresponding novel bifurcation diagram and discuss its implications for liquid crystal display technology. Numerical simulations of the corresponding dynamic problem also provide suggestive evidence about stability of some solution branches, pointing to a promising avenue of further analytical, numerical, and experimental studies.
- Modelling silicosis: existence, uniqueness and basic properties of solutionsPublication . Costa, Fernando Pestana da; Pinto, João Teixeira; Sasportes, RafaelWe present a model for the silicosis disease mechanism following the original proposal by Tran et al. (1995), as modified recently by da Costa et al. (2020). The model consists in an infinite ordinary differential equation system of coagulation–fragmentation–death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.