Research Articles (Mathematics)http://hdl.handle.net/10566/1592019-10-24T03:58:40Z2019-10-24T03:58:40ZClosure, interior and neighbourhood in a categoryHolgate, DavidSlapal, Josefhttp://hdl.handle.net/10566/50592019-10-23T00:00:56Z2018-01-01T00:00:00ZClosure, interior and neighbourhood in a category
Holgate, David; Slapal, Josef
The natural correspondences in topology between closure, interior
and neighbourhood no longer hold in an abstract categorical setting
where subobject lattices are not necessarily Boolean algebras. We
analyse three canonical correspondences between closure, interior and
neighbourhood operators in a category endowed with a subobject
structure. While these correspondences coincide in general topology,
the analysis highlights subtle di erences which distinguish di erent
approaches taken in the literature.
2018-01-01T00:00:00ZNew parameter-uniform discretisations of singularly perturbed Volterra integro-differential equationsIragi, Bakulikira C.Munyakazi, Justin B.http://hdl.handle.net/10566/50462019-10-19T00:00:56Z2018-01-01T00:00:00ZNew parameter-uniform discretisations of singularly perturbed Volterra integro-differential equations
Iragi, Bakulikira C.; Munyakazi, Justin B.
We design and analyse two numerical methods namely a fitted mesh and a fitted operator finite difference methods for
solving singularly perturbed Volterra integro-differential equations. The fitted mesh method we propose is constructed using a finite
difference operator to approximate the derivative part and some suitably chosen quadrature rules for the integral part. To obtain a
parameter-uniform convergence, we use a piecewise-uniform mesh of Shishkin type. On the other hand, to construct the fitted operator
method, the Volterra integro-differential equation is discretised by introducing a fitting factor via the method of integral identity with
the use of exponential basis function along with interpolating quadrature rules [2]. The two methods are analysed for convergence and
stability. We show that the two methods are robust with respect to the perturbation parameter. Two numerical examples are solved to
show the applicability of the proposed schemes.
2018-01-01T00:00:00ZMathematical analysis and numerical simulation of a tumor-host model with chemotherapy applicationOwolab, Kolade M.Patidar, Kailash C.Shikongo, Alberthttp://hdl.handle.net/10566/47782019-08-16T00:00:30Z2018-01-01T00:00:00ZMathematical analysis and numerical simulation of a tumor-host model with chemotherapy application
Owolab, Kolade M.; Patidar, Kailash C.; Shikongo, Albert
In this paper, a system of non-linear quasi-parabolic partial differential system, modeling the chemotherapy application of spatial tumor-host interaction is considered. At some certain parameters, we derive the steady state of the anti-angiogenic therapy, baseline therapy and anti-cytotoxic therapy models as well as their local stability condition. We use the method of upper and lower solutions to show that the steady states are globally stable. Since the system of non-linear quasi-parabolic partial differential cannot be solved analytically, we formulate a robust numerical scheme based on the semi-fitted finite difference operator. Analysis of the basic properties of the method shows that it is consistent, stable and convergent. Our numerical results are in agreement with our theoretical findings.
2018-01-01T00:00:00ZA stochastic TB model for a crowded environmentVyambwera, Sibaliwe MakuWitbooi, Peterhttp://hdl.handle.net/10566/38192018-06-20T00:00:58Z2018-01-01T00:00:00ZA stochastic TB model for a crowded environment
Vyambwera, Sibaliwe Maku; Witbooi, Peter
We propose a stochastic compartmental model for the population dynamics of tuberculosis. The model is applicable to crowded
environments such as for people in high density camps or in prisons. We start off with a known ordinary differential equation
model, and we impose stochastic perturbation.We prove the existence and uniqueness of positive solutions of a stochastic model.
We introduce an invariant generalizing thebasic reproductionnumber andprove the stabilityof thedisease-free equilibriumwhen it
is below unity or slightly higher than unity and the perturbation is small. Ourmain theorem implies that the stochastic perturbation
enhances stability of the disease-free equilibrium of the underlying deterministic model. Finally, we perform some simulations to
illustrate the analytical findings and the utility of the model.
2018-01-01T00:00:00Z