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- Mathematical models of chiral symmetry-breaking: a review of general theories, and adiabatic approximations of the APED systemPublication . Diniz, Priscila Costa; Wattis, Jonathan AD; Costa, Fernando Pestana daWe review the literature surrounding chiral symmetry-breaking in chemical systems, with a focus on understanding the mathematical models underlying these chemical processes. We comment in particular on the toy model of Sandars, Viedma’s crystal grinding systems and the APED model. We include a few new results based on asymptotic analysis of the APED system.
- Differential equations and applicationsPublication . Costa, Fernando Pestana daTexto utilizado na lecionação da disciplina homónima lecionada pelo autor no German-Mongolian Institute for Resources and Technology, Nalaikh, Ulaanbaatar, Mongolia, em setembro de 2024.
- Alguns exemplos de equações diferenciais ordinárias em economiaPublication . Costa, Fernando Pestana daEste texto foi elaborado para apoio à unidade curricular "Equações Diferenciais em Macroeconomia" da licenciatura em Matemática Aplicada à Gestão da Universidade Aberta. O texto apresenta um conjunto de exemplos de modelos económicos que se traduzem matematicamente em equações diferenciais ordinárias de tipos que são estudados na referida unidade curricular. São indicados onde, noutros textos, podem ser aprofundados os fundamentos económicos dos modelos apresentados e estudadas as técnicas matemáticas que permitem analisar as equações apresentadas.
- 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.
- Theoretical analysis of a discrete population balance model with sum kernelPublication . Kaushik, Sonali; Kumar, Rajesh; Costa, Fernando Pestana daThe Oort–Hulst–Safronov equation is a relevant population balance model. Its discrete form, developed by Pavel Dubovski, is the main focus of our analysis. The existence and density conservation are established for nonnegative symmetric coagulation rates satisfying V_{i;j} \leq i + j , \forall i, j \in N. Differentiability of the solutions is investigated for kernels with V_{i;j} \leq i^\apha + j^\alpha˛ where 0 \leq \alpha \leq 1 with initial conditions with bounded (1+\alpha)-th moments. The article ends with a uniqueness result under an additional assumption on the coagulation kernel and the boundedness of the second moment.
- Modelling silicosis: the structure of equilibriaPublication . Costa, Fernando Pestana da; Grinfeld, Michael; Drmota, MichaelWe analyse the structure of equilibria of a coagulation–fragmentation–death model of silicosis. We present exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibria in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients.
- Mathematics and online educationPublication . Costa, Fernando Pestana da
- 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.
- Mathematical investigations of a kinetic model for glycerol hydrogenolysis via heterogeneous catalysisPublication . Costa, Fernando Pestana da; Ndlovu, Thandokuhle Quinton; Shozi, MzamoIn this paper we report on some mathematical investigations of the chemical process for the hydrogenolysis of glycerol over a heterogeneous metal catalyst. The main interest of this process is related to the fact that glycerol is produced as a by-product in the production of biodiesel in huge amounts that are expected to exceed the projected demands. This makes the sustainability of biodiesel production depend on the conversion of the glycerol into useful products hence it is a desirable goal to have effective conversion methods. A reaction model from literature is used to derive a system of ordinary differential equations (ODE) which is then analysed using methods from qualitative analysis of ODEs. Numerical solutions of the system are simulated to try and find out the solution’s behaviour in the chemistry point of view. It was found that all solutions of the model converge to some stable limit point in a 2-dimensional plane in the positive cone of the R5 phase space, and the limit point depends on the values of rate constants ki as well as on the hydrogen to glycerol initial ratios. Even though the results are based on a specific kinetic model, it is hoped that they may help in providing tools for better understanding and description of the reaction.
- 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.