35 research outputs found
New parameter-uniform discretisations of singularly perturbed Volterra integro-differential equations
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
Performance of Richardson extrapolation on some numerical methods for a singularly perturbed turning point problem whose solution has boundary layers
Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used
A new parameter-uniform discretization of semilinear singularly perturbed problems
In this paper, we present a numerical approach to solving singularly perturbed semilinear
convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization
technique. We then design and implement a fitted operator finite difference method to solve the
sequence of linear singularly perturbed problems that emerges from the quasilinearization process.
We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice
that the method is first-order uniformly convergent. Some numerical evaluations are implemented on
model examples to confirm the proposed theoretical results and to show the efficiency of the method
To use face masks or not after Covid-19 vaccination? An impact analysis using mathematical modeling
The question of whether to drop or to continue wearing face masks especially after being
vaccinated among the public is controversial. This is sourced from the efficacy levels
of COVID-19 vaccines developed, approved, and in use. We develop a deterministic
mathematical model that factors in a combination of the COVID-19 vaccination program
and the wearing of face masks as intervention strategies to curb the spread of the
COVID-19 epidemic. We use the model specifically to assess the potential impact
of wearing face masks, especially by the vaccinated individuals in combating further
contraction of COVID-19 infections. Validation of the model is achieved by performing its
goodness of fit to the Republic of South Africa’s reported COVID-19 positive cases data
using the Maximum Likelihood Estimation algorithm implemented in the fitR package.
We first consider a scenario where the uptake of the vaccines and wearing of the face
masks, especially by the vaccinated individuals is extremely low
Mathematical modeling and impact analysis of the use of COVID alert SA app
The human life-threatening novel Severe Acute Respiratory Syndrome Corona-virus-2
(SARS-CoV-2) has lasted for over a year escalating and posing simultaneous anxiety day-by-day globally since its first report in the late December 2019. The scientific arena has been kept animated
via continuous investigations in an effort to understand the spread dynamics and the impact of various mitigation measures to keep this pandemic diminished. Despite a lot of research works having
been accomplished this far, the pandemic is still deep-rooted in many regions worldwide signaling for
more scientific investigations. This study joins the field by developing a modified SEIR (SusceptibleExposed-Infectious-Removed) compartmental deterministic model whose key distinct feature is the
incorporation of the COVID Alert SA app use by the general public in prolific intention to control the
spread of the epidemic. Validation of the model is performed by fitting the model to the Republic of
South Africa’s COVID-19 cases reported data using the Maximum Likelihood Estimation algorithm
implemented in fitR package
A NSFD method for the singularly perturbed Burgers-Huxley equation
This article focuses on a numerical solution of the singularly perturbed Burgers-Huxley equation. The simultaneous presence of a singular perturbation parameter and the nonlinearity raise the challenge of finding a reliable and efficient numerical solution for this equation via the classical numerical methods. To overcome this challenge, a nonstandard finite difference (NSFD) scheme is developed in the following manner. The time variable is discretized using the backward Euler method. This gives rise to a system of nonlinear ordinary differential equations which are then dealt with using the concept of nonlocal approximation. Through a rigorous error analysis, the proposed scheme has been shown to be parameter-uniform convergent. Simulations conducted on two numerical examples confirm the theoretical result. A comparison with other methods in terms of accuracy and computational cost reveals the superiority of the proposed scheme
Mathematical study of transmission dynamics of SARS-CoV-2 with waning immunity
The aim of this work is to provide a new mathematical model that studies transmission
dynamics of Coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome
coronavirus 2 (SARS-CoV-2). The model captures the dynamics of the disease taking into
consideration some measures and is represented by a system of nonlinear ordinary differential
equations including seven classes, which are susceptible class (S), exposed class (E), asymptomatic
infected class (A), severely infected class (V), hospitalized class (H), hospitalized class but in ICU
(C) and recovered class (R). We prove positivity and boundedness of solutions, compute the basic
reproduction number, and investigate asymptotic stability properties of the proposed model
A NSFD discretization of two-dimensional singularly perturbed semilinear convection-diffusion problems
Despite the availability of an abundant literature on singularly perturbed problems,
interest toward non-linear problems has been limited. In particular, parameter-uniform
methods for singularly perturbed semilinear problems are quasi-non-existent. In this
article, we study a two-dimensional semilinear singularly perturbed convection-diffusion
problems. Our approach requires linearization of the continuous semilinear problem
using the quasilinearization technique. We then discretize the resulting linear problems
in the framework of non-standard finite difference methods. A rigorous convergence
analysis is conducted showing that the proposed method is first-order parameter-uniform
convergent. Finally, two test examples are used to validate the theoretical findings
Modeling the impact of combined use of Covid Alert SA app and vaccination to curb Covid-19 infections in South Africa
The unanticipated continued deep-rooted trend of the Severe Acute Respiratory Syndrome
Corona-virus-2 the originator pathogen of the COVID-19 persists posing concurrent anxiety
globally. More effort is affixed in the scientific arena via continuous investigations in a prolific
effort to understand the transmission dynamics and control measures in eradication of the
epidemic. Both pharmaceutical and non-pharmaceutical containment measure protocols
have been assimilated in this effort. In this study, we develop a modified SEIR deterministic
model that factors in alternative-amalgamation of use of COVID Alert SA app and vaccination against the COVID-19 to the Republic of South Africa’s general public in an endeavor to
discontinue the chain of spread for the pandemic. We analyze the key properties of the
model not limited to positivity, boundedness, and stability. We authenticate the model by fitting it to the Republic of South Africa’s cumulative COVID-19 cases reported data utilizing
the Maximum Likelihood Estimation algorithm implemented in fitR package
A nonstandard fitted operator finite difference method for two-parameter singularly perturbed time-delay parabolic problems
In this article, a class of singularly perturbed time-delay two-parameter second-order parabolic problems are considered. The presence of the two small parameters attached to the derivatives causes the solution of the given problem to exhibit boundary layer(s). We have developed a uniformly convergent nonstandard fitted operator finite difference method (NSFOFDM) to solve the considered problems. The Crank-Nicolson scheme with a uniform mesh is used for the discretization of the time derivative, while for the spatial discretization, we have applied a fitted operator finite difference method following the nonstandard methodology of Mickens. Moreover, the solution bounds of the governing equation are shown by asymptotic analysis. The convergence of the proposed numerical scheme is investigated using truncation error and the barrier function approach. The study shows that our proposed scheme is uniformly convergent independent of the perturbation parameters, quadratically in time, and linearly in space. Numerical experiments are carried out, and the results are presented in tables and graphically