Research Articles (Applied Mathematics)

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Now showing 1 - 5 of 12

A note on exact solutions of the logistic map
(AIP Publishing, 2020-03) Maritz, Milton F
The logistic map, whose iterations lead to period doubling and chaos as the control parameter, is increased and has three cases of the control parameter where exact solutions are known. In this paper, we show that general solutions also exist for other values of the control parameter. These solutions employ a special function, not expressible in terms of known analytical functions. A method of calculating this function numerically is proposed, and some graphs of this function are given, and its properties are discussed. The logistic map is often studied as a model of the period doubling route to chaos as its control parameter is increased. For three particular values of the control parameter, exact solutions are known. In particular, for one of these values, when the solution is chaotic, an exact solution in terms of the square of the sine function is known. In this study, a general solution for the logistic map for all values of the control parameter in a continuous range is proposed. This solution employs a special function, not expressible in terms of known analytical functions. A method of calculating this function numerically, as well as calculating its inverse numerically is proposed and some of its properties are discussed. This study, therefore, contributes to the field of nonlinear dynamics by introducing a novel tool for visualization and for investigation. The technique may be extendable to other nonlinear maps.
Modelling the dispersion and deposition of solid wastes from fish farming with a continuum approach
(Stellenbosch : Stellenbosch University, 2022-04) Fourie, Samantha-Kerry; Goosen, Neill; Diedericks, Gerhardus Petrus Jacobus; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences (Applied Mathematics)
ENGLISH ABSTRACT: The sustainability of the environment in coastal ecosystems is of great concern due to aquaculture. There are many unknowns and constraints in the predictive modelling of fish farm waste due to the limitations of obtaining information in the farming area. By understanding the gaps in the information, the necessary research projects can be established, therefore improving the predictive modelling. This project presents an academic case for Delft3D-FLOW to predict the dispersion and deposition of organic mariculture waste using a continuum approach. The continuum approach allows us to model the waste interaction with the seabed as a cohesive sediment, which can be done using Delft3D-FLOW. More information is needed on how the wastes interact with the seabed as it has been observed that the erosion and deposition of the waste is sensitive to the cohesive properties. Delft3D-FLOW was used for the simulation as it includes modules of multi-dimensional hydrodynamic flows and transport phenomena, including the transport of cohesive and non-cohesive sediments. The simulation represents a growth cycle of six months for Atlantic salmon within a simple rectangular grid, using the background currents of a fjord in the Faroe Islands. A six month simulation period was chosen due to time constraints, allowing for a feasible simulation time and time to conduct a sensitivity analysis on the interaction of the waste with the seabed. The growth data of the simulation period was gathered from existing data in order to determine the amount of feed used, therefore representing the growth cycle of the Atlantic salmon. Using the feed cycle and information from literature, the corresponding waste output of the farm was calculated. Observing the output from the simulations, the waste has been completely dispersed from the grid at the end of the cycle, indication a highly dispersive site. These results correlates with the high currents that were calculated within the fjord. Further, an investigation was conducted into the critical deposition and erosion stresses in order to observe the affect the stresses have on the resultant deposition and dispersion of the waste at the seabed. The results of the sensitivity analysis of the shear stresses result in validation that low critical shear stresses were chosen
Geometric versus kinetic modelling approach for characterizing porous metal foams
(WIT Press, 2019) Mare, Esmari; Woudberg, Sonia
Knowledge of the geometric and kinematic parameters of porous foams are of great importance since it is used in a wide variety of industrial multiphase flow applications that require optimal functionality, e.g. gas filters, heat exchangers and catalyst supports. The large external surface area and high porosity of metal foams provide good chemical resistance, enhanced heat and mass transfer properties and low pressure drops. Four generic geometric models will be considered to characterize the metal foam geometry, namely the cubic unit cell, tetrakaidecahedron, dodecahedron and rectangular representative unit cell (RUC) models, as well as three kinetic approaches from the literature in order to predict the specific surface area (SSA). Two sets of experimental data from the literature will then be compared to the SSA model predictions of the geometric approach and to the SSA values obtained from the kinetic approach. A comparative analysis reveals that the most geometrically complex tetrakaidecahedron model indeed provides the best correspondence with the experimental data for the SSAs, followed by the geometrically simplest RUC model. The latter model, in addition, provides accurate results for the kinetic approach. The advantage of the RUC model is that it is the only geometric model that provides both a geometric and kinetic approach, and, as a result of its relatively simple geometry it is geometrically adaptable towards anisotropy. The Klinkenberg effect will also be considered to determine the influence on the predictions of the SSAs dependency on the permeability coefficients for different fluid phases.
Computational modelling and optimal control of Ebola virus disease with non-linear incidence rate
(IOP Publishing, 2017) Takaidza, I.; Makinde, O. D.; Okosun, O. K.
The 2014 Ebola outbreak in West Africa has exposed the need to connect modellers and those with relevant data as pivotal to better understanding of how the disease spreads and quantifying the effects of possible interventions. In this paper, we model and analyse the Ebola virus disease with non-linear incidence rate. The epidemic model created is used to describe how the Ebola virus could potentially evolve in a population. We perform an uncertainty analysis of the basic reproductive number R 0 to quantify its sensitivity to other disease-related parameters. We also analyse the sensitivity of the final epidemic size to the time control interventions (education, vaccination, quarantine and safe handling) and provide the cost effective combination of the interventions.
Minimum density hyperplanes
(Journal of Machine Learning Research, 2016) Pavlidis, Nicos G.; Hofmeyr, David P.; Tasoulis, Sotiris K.
Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classification which is motivated by this objective. The proposed minimum density hyperplane minimises the integral of the empirical probability density function along it, thereby avoiding intersection with high density clusters. We show that the minimum density and the maximum margin hyperplanes are asymptotically equivalent, thus linking this approach to maximum margin clustering and semi-supervised support vector classifiers. We propose a projection pursuit formulation of the associated optimisation problem which allows us to find minimum density hyperplanes efficiently in practice, and evaluate its performance on a range of benchmark data sets. The proposed approach is found to be very competitive with state of the art methods for clustering and semi-supervised classification.

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