NSFD Scheme Research Articles

Nonstandard finite difference method for solving the multi-strain TB model

Abstract In this paper, numerical studies for the mathematical model of tuberculosis (TB), that incorporates three strains, i.e., drug - sensitive, emerging multi - drug resistant(MDR) and extensively drug - resistant (XDR), are presented. Special class of numerical methods, known as nonstandard finite difference method (NSFDM) is introduced to solve this model. Numerical stability analysis of fixed points are studied. The obtained results by NSFDM are compared with other known numerical methods such as implicit Euler method and fourth-order Runge–Kutta method (RK4). It is concluded that NSFD scheme preserves the positivity of the solution and numerical stability in larger region than the other methods.

Contributions to Constructive Treatment of SIS-Type Models II: Nonstandard Volterra Difference Equation

This work is divided in two parts. In the first part, we construct and investigate the reliability of various NSFD schemes for diffusion and space-independent models. We consider two new NSFD schemes for the SIS-ODE model, which faithfully replicate the property of the continuous model of having the value R_{0}=1 of the basic reproduction parameter as a forward bifurcation: the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when R_{0} 1, and there appears a unique locally asymptotically stable (LAS) endemic equilibrium (EE) in this case.The schemes are further used to derive NSFD schemes that are dynamically consistent with the positivity and boundedness properties of the SIS-diffusion model. In the second part, the contact rate is a function of the infective population and we incorporate a distributed infective period. The resulting SIS-model is a Volterra integral equation of the second kind. The qualitative analysis is now based on two threshold parameters Rc_{0} Rm_{0} with the DFE being unstable when R_{0} > 1; for Rc_{0} < R_{0} < 1; the DFE is LAS and co-exists with at least one LAS endemic equilibrium. We design a NSFD scheme that preserves positivity and boundedness of the solution as well as the above-stated stability properties of equilibria. Numerical simulations that support the theory are provided.

Contributions to Constructive Treatment of SIS-Type Models I: Nonstandard Schemes for Diffusion and Non-Diffusion Models

This work is divided in two parts. In the first part, we construct and investigate the reliability of various NSFD schemes for diffusion and space-independent models. We consider two new NSFD schemes for the SIS-ODE model, which faithfully replicate the property of the continuous model of having the value R_{0}=1 of the basic reproduction parameter as a forward bifurcation: the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when R_{0} 1, and there appears a unique locally asymptotically stable (LAS) endemic equilibrium (EE) in this case.The schemes are further used to derive NSFD schemes that are dynamically consistent with the positivity and boundedness properties of the SIS-diffusion model. In the second part, the contact rate is a function of the infective population and we incorporate a distributed infective period. The resulting SIS-model is a Volterra integral equation of the second kind. The qualitative analysis is now based on two threshold parameters Rc_{0} Rm_{0} with the DFE being unstable when R_{0} > 1; for Rc_{0} < R_{0} < 1; the DFE is LAS and co-exists with at least one LAS endemic equilibrium. We design a NSFD scheme that preserves positivity and boundedness of the solution as well as the above-stated stability properties of equilibria. Numerical simulations that support the theory are provided.

Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.

An NSFD scheme for a class of SIR epidemic models with vaccination and treatment

An non-standard finite difference scheme is employed to discuss a class of SIR epidemic model with vaccination and treatment. The dynamical properties of the discretized model are then analysed. The results demonstrate that the discretized epidemic model is dynamically consistent with the continuous model since it maintains essential properties of the corresponding continuous model, such as positivity property and boundness of solutions, equilibrium points and their local stability properties.

Numerical Solution of the 1D Advection-Diffusion Equation Using Standard and Nonstandard Finite Difference Schemes

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted byhandk, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to approximate the 1D advection-diffusion equation at some values ofkandh. Two optimisation techniques are then implemented to find the optimal values ofkwhenh=0.02for the Lax-Wendroff and NSFD schemes, and this is validated by numerical experiments.

A SIR-model with square-root dynamics: An NSFD scheme

We present a SIR-model for which the interaction term is expressed as the square-root of the S and I variables. General results on the behaviour of the solutions are discussed and a discrete-time, finite difference scheme is constructed using the NSFD methodology of Mickens. An important feature of the scheme is that both the positivity and a conservation law for the system of differential equations are maintained by the discretization.