Michael Chapwanya

University of Pretoria | UP · Department of Mathematics and Applied Mathematics

A deterministic mathematical model for the dynamics of cannabis use in a South Africa metropolis of Durban is proposed and analysed. To the analysis the model, the important threshold parameter ℛ0 (the basic reproduction number), is determined. It is proved that the model exhibits multiple cannabis persistent equilibria. For ℛ0 < 1, the model exhib...

Crops are often subject to intense attacks by pests and diseases. Among them, Maize Lethal Necrosis (MLN) is a serious disease that impact maize crops in many Southern countries. It results from the synergistic interaction of two plant viruses, transmitted by two vectors. In this paper, we develop a general deterministic epidemic model of crop-vect...

We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms in the construction of nonstandard finite difference (NSFD) schemes. Firstly, we construct nonstandard one-stage and two-stage theta methods for a genera...

In this work, we consider numerical solutions of the FitzHugh-Nagumo system of equations describing the propagation of electrical signals in nerve axons. The system consists of two coupled equations: a nonlinear partial differential equation and a linear ordinary differential equation. We begin with a review of the qualitative properties of the non...

In this work we consider the Hodgkin Huxley model in the form of a coupled system of one singularly perturbed partial differential equation and three ordinary differential equations. The existence of a small parameter, the nonlinearity and the coupling makes the numerical approximations using explicit finite difference schemes very difficult. In pa...

Diseases, in particular, vector-borne diseases are very important issues in crop protection. However, despite their impact on food safety, very few mathematical models have been developed in order to improve control strategies. Motivated by existing literature, we begin by considering a temporal model of vector-borne diseases in (annual) crops. Usi...

Two new explicit finite difference schemes for the solution of the one-dimensional Korteweg-de-Vries equation are proposed. This equation describes the character of a wave generated by an incompressible fluid. We analyse the spectral properties of our schemes against two existing schemes proposed by Zabusky and Kruskal (1965) and Wang et al. (2008)...

A nonstandard finite difference method is proposed for the discretisation of the semilinear FitzHugh-Nagumo reaction diffusion equation. The equation has been useful in describing, for example, population models, biological models, heat and mass transfer models, and many other applications. The proposed approach involves splitting the equation into...

We design explicit nonstandard finite difference schemes for the nonlinear Allen–Cahn reaction diffusion equation in the limit of very small interaction length . In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimizing the restrictions normally associated with standard explicit finit...

The Schrödinger equation is a model for many physical processes in quantum physics. It is a singularly perturbed differential equation where the presence of the small reduced Planck's constant makes the classical numerical methods very costly and inefficient. We design two new schemes. The first scheme is the nonstandard finite volume method, where...

Mathematical modeling of transport phenomena in food processes is vital to understand the process dynamics. In this work, we study the process of double sided cooking of meat by developing a mathematical model for the simultaneous heat and mass transfer. The constitutive equations for the heat and mass transport are based on Fourier conduction, and...

Soft condensed matter (SCM) physics has recently gained importance for a large class of engineering materials. The treatment of food materials from a soft matter perspective, however, is only at the surface and is gaining importance for understanding the complex phenomena and structure of foods. In this work, we present a theoretical treatment of n...

We present a theory for the coupled flow of ice, subglacial water and subglacial sediment, which is designed to represent the processes which occur at the bed of an ice sheet. The ice is assumed to flow as a Newtonian viscous fluid, the water can flow between the till and the ice as a thin film, which may thicken to form streams or cavities, and th...

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonl...

We present a model of biofilm growth in a long channel where the biomass is assumed to have the rheology of a viscous polymer solution. We examine the competition between growth and erosion-like surface detachment due to the flow. A particular focus of our investigation is the effect of the biofilm growth on the fluid flow in the pores, and the iss...

We design nonstandard finite difference (NSFD) schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross-diffusion equations in biosciences. This settles a problem that was open for quite some time. The study is done in the setting of three concrete and highly relevant cross-diffusion sys...

The Schrödinger equation is a singularly perturbed differential equation which gives strongly oscillatory solutions. In the present article, we design two numerical schemes which are able to reproduce the solution property of the equation even at very small values of the reduced Planck's constant with relatively small number of computation cells. W...

In this work we consider an operator splitting numerical scheme for advection dominated material flow equation as given by the advection-diffusion-reaction equation. A model for multiphase flow is used to motivate the model under investigation. Dynamically consistent and positivity preserving nonstandard finite difference schemes are used to handle...

Limerick Wave Ltd. has developed an innovative wave energy converter (WEC) technology. They use a recently-patented flywheel technology to use the power from the movement of the waves (via the movement of a cylindrical floatation device) to generate electricity. The use of flywheel technology in this area is novel in that its rotation is unidirecti...

We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens' rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related...

This work considers the numerical solution of the Kuramoto-Sivashinsky equation using the fractional time splitting method. We will investigate the numerical behavior of two categories of the traveling wave solutions documented in the literature (Hooper & Grimshaw (1998)), namely: the regular shocks and the oscillatory shocks. We will also illustra...

We consider the basic SIR epidemiological model with the Michaelis–Menten formulation of the contact rate. From the study of the Michaelis–Menten basic enzymatic reaction, we design two types of Nonstandard Finite Difference (NSFD) schemes for the SIR model: Exact-related schemes based on the Lambert WW function and schemes obtained by using Micken...

We develop numerical solutions of a theoretical model which has been proposed to explain the formation of subglacial bedforms. The model has been shown to have the capability of producing bedforms in two dimensions, when they may be interpreted as ribbed moraine. However, these investigations have left unanswered the question of whether the theory...

We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substanc as a polymer solution, whose viscoelastic rheology is described by the classical Flory–Huggins theory. We show that one-dimensiona solutions exist, which take the form at large times of travelling waves, and we characteriz...

We investigate a flow problem of relevance in bioremediation and develop a mathematical model for transport of contamination by groundwater and the spreading, confinement, and remediation of chemical waste. The model is based on the fluid mass and momentum balance equations and simultaneous transport and consumption of the pollutant (hydrocarbon) a...

This is a computational study of gravity-driven fingering instabilities in unsaturated porous media. The governing equations and corresponding numerical scheme are based on the work of Nieber et al. (2003) in which nonmonotonic saturation profiles are obtained by supplementing the Richards equation with a nonequilibrium capillary pressure-saturatio...

Wastewater treatment requires the elimination of pathogens and reduction of organic matter in the treated sludge to acceptable levels. One process used to achieve this is Autothermal Thermophylic Aerobic Digestion (ATAD), which relies on promoting non-pathogenic thermophilic bacteria to digest organic matter and kill pathogens through metabolic hea...

A mathematical model is developed that captures the transport of liquid water in hardened concrete, as well as the chemical reactions that occur between the imbibed water and the residual calcium silicate compounds residing in the porous concrete matrix. The main hypothesis in this model is that the reaction product -- calcium silicate hydrate gel...

In this paper we develop a mathematical model for the transport of liquid water in hardened concrete and the chemical reactions that occur between the water and the calcium silicate compounds. The model is based on the hypothesis that it is the ionic species in solution that advect, diffuse and precipitate to form calcium-silicate-hydrate or C-S-H...

Wastewater treatment requires the elimination of pathogens and reduction of organic matter in the treated sludge to acceptable levels. One process used to achieve this is Autothermal Ther-mophylic Aerobic Digestion (ATAD), which relies on promoting non-pathogenic thermophilic bacteria to digest organic matter and kill pathogens through metabolic he...

We consider the penetration of liquid into a porous medium which is, in part, impervious to the liquid. Corners arise at points where the medium becomes impervious and numerical difficulties can arise due to a conflict in the boundary conditions. In addition there are start-up problems associated with numerically modelling the flow problem because...