• 대한수학회지
• /
• 제42권6호
• /
• pp.1121-1136
• /
• 2005

Nonstandard finite difference schemes are considered for a generalization of the Cahn-Hilliard equation with Neumann boundary conditions and the Kuramoto-Sivashinsky equation with periodic boundary conditions, which are of the type $$U_t\;+\;\frac{{\partial}^2}{{\partial}x^2} g(u,\;U_x,\;U_{xx})\;=\;\frac{{\partial}^{\alpha}}{{\partial}x^{\alpha}}f(u,\;u_x),\;{\alpha}\;=\;0,\;1,\;2$$. Stability and error estimate of approximate solutions for the corresponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem. Three examples are provided to apply the nonstandard finite difference schemes.

• 대한수학회보
• /
• 제54권6호
• /
• pp.1893-1912
• /
• 2017

In this paper, we aim to construct a nonstandard finite difference (NSFD) scheme to solve numerically a mathematical model for cholera epidemic dynamics. We first show that if the basic reproduction number is less than unity, the disease-free equilibrium (DFE) is locally asymptotically stable. Moreover, we mainly establish the global stability analysis of the DFE and endemic equilibrium by using suitable Lyapunov functionals regardless of the time step size. Finally, numerical simulations with different time step sizes and initial conditions are carried out and comparisons are made with other well-known methods to illustrate the main theoretical results.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제22권4호
• /
• pp.253-287
• /
• 2018

This paper studies the global stability of a class of discrete-time pathogen infection models with latently infected cells. The rate of pathogens infect the susceptible cells is taken as bilinear, saturation and general. The continuous-time models are discretized by using nonstandard finite difference scheme. The basic and global properties of the models are established. The global stability analysis of the equilibria is performed using Lyapunov method. The theoretical results are illustrated by numerical simulations.