Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

“…Nowadays, NSFD schemes have become an efficient approach for numerically solving real-world problems (see, for example, [1,2,4,5,10,13,14,15,31,32,47,48,57,58]). Recently, we have developed the Mickens' methodology to construct NSFD schemes for mathematical models of phenomena and processes coming from sciences and technology like biology, ecology, or other natural sciences [16,17,18,19,20,21,25,26,27,28,29].…”

Framework for solving dynamics of Ca2+ ion concentrations in liver cells numerically: Analysis of a non‐negativity‐preserving non‐standard finite‐difference method

“…For these reasons, we slightly propose a modified time discretization of system () which gives explicit solutions and remains non‐negative for every time step. Our modified time‐stepping method bases on ideas from non‐standard finite‐difference methods by Mickens [16] and Hoang [17]. There are different possibilities of constructing non‐standard finite‐difference schemes for a dynamical system $$$ {\mathbf{Y}}^{\prime }(t)=\mathbf{F}\left(t,\mathbf{Y}(t)\right) $$$.…”

“…There are different possibilities of constructing non‐standard finite‐difference schemes for a dynamical system $$$ {\mathbf{Y}}^{\prime }(t)=\mathbf{F}\left(t,\mathbf{Y}(t)\right) $$$. Firstly, we can approximate the left‐hand side time‐continuous first derivative by an approximation $$$ {y}^{\prime }(t)\approx \frac{y_{n+1}-{y}_n}{\varphi (h)} $$$ with an appropriate function $$$ \varphi (h) $$$ under appropriate conditions with regard to the function $$$ \varphi $$$ [16, 17]. Secondly, we can use non‐local approximations for the right‐hand side function $$$ \mathbf{F}\left(t,\mathbf{Y}(t)\right) $$$, that is, this term does not solely depend on one time point in the time‐discrete case, but we use at least two time‐discrete function values, for example, $$$ {y}_n $$$ and $$$ {y}_{n+1} $$$.…”

“…with an appropriate function 𝜑(h) under appropriate conditions with regard to the function 𝜑 [16,17]. Secondly, we can use non-local approximations for the right-hand side function F (t, Y(t)), that is, this term does not solely depend on one time point in the time-discrete case, but we use at least two time-discrete function values, for example, 𝑦 n and 𝑦 n+1 .…”

A Novel Second-Order Nonstandard Finite Difference Method for Solving One-Dimensional Autonomous Dynamical Systems

“…During the past several decades, NSFD schemes have been strongly developed by mathematicians and engineers and have become one of the most powerful methods for solving differential equations. Nowadays, NSFD schemes have been widely used for ordinary differential equations, partial differential equations, delay differential equations, fractional differential equations and integro-differential equations (see, for instance, [1,2,6,7,8,9,12,16,19,20,28,29,30,33,35,36,41,50,59,60,61,62,63,64,65,66,67,68,69,70,71,71,72,73,74,75,78,79,80,81,82,83,86,87,88,89] and references therein). Here, we refer the readers to [62,…”