Nonstandard finite difference schemes for Michaelis–Menten type reaction‐diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens' rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related reaction‐diffusion equations, advection‐reaction equations, and advection‐reaction‐diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Square-root dynamics of a giving up smoking model

The paper presents a new giving up smoking model for which interaction term is square root of potential and occasional smokers of model presented in Zaman (2011) [15]. First, we will show formulation of the model. Then we will discuss local and global stability of the model and its general solutions. The non-standard finite difference method (NSFD) is used to solve the new giving up smoking model. Both non-negativity and conservative law for differential equations system are discussed. Numerical results are presented graphically and compared well with those obtained by Runge–Kutta fourth-order method (RK4) and ODE45.     

A note on an NSFD scheme for a mathematical model of respiratory virus transmission

We construct a non-standard finite difference (NSFD) scheme for an SIRS mathematical model of respiratory virus transmission. This discretization is in full compliance with the NSFD methodology as formulated by Mickens. By use of an exact conservation law satisfied by the SIRS differential equations, we are able to determine the corresponding denominator function for the discrete first-order time derivatives. Our scheme is dynamically consistent with the SIRS differential equations, since the conservation laws are preserved. Furthermore, the scheme is shown to satisfy a positivity condition for its solutions for all values of the time step size.     

具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式

构造了一种解具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式．基本思想是离散Landau-Lifshitz方程成常微分方程组,应用李群方法和显式Runge-Kutta方法解常微分方程组．数值试验比较了两方法的保平方守恒特性和精度,得出李群方法(RK-Cayley方法)比相应的Runge-Kutta(RK)方法有更好的精度和保平方守恒特性．     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

Global Aspects of Age-Structured Cigarette Smoking Model

Smoking impacts health and as a result creates several problems related to age which means smoking has a strong correlation with age. Keeping this problem in view, we consider the global asymptotic properties of age-structured smoking model. First, we formulate the model and present the existence and uniqueness of solution. Then we discuss the equilibrium points and construct the Lyapunov function to examine global stability of the free smoking and positive smoking equilibrium points. Finally, we fixed the age factor and use the non-standard finite difference (NSFD) scheme for numerical solutions and compare our results obtained with RK4 and ODE45 graphically with the help of MATLAB.     

Exact finite-difference schemes for first order differential equations having three distinct fixed-points

We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixed-points.     

Optimal m-stage Runge-Kutta schemes for steady-state solutions of hyperbolic equations

A numerical scheme is developed to find optimal parameters and time step of m-stage Runge-Kutta (RK) schemes for accelerating the convergence to -steady-state solutions of hyperbolic equations. These optimal RK schemes can be applied to a spatial discretization over nonuniform grids such as Chebyshev spectral discretization. For each m given either a set of all eigenvalues or a geometric closure of all eigenvalues of the discretization matrix, a specially structured nonlinear minimax problem is formulated to find the optimal parameters and time step. It will be shown that each local solution of the minimax problem is also a global solution and therefore the obtained m-stage RK scheme is optimal. A numerical scheme based on a modified version of the projected Lagrangian method is designed to solve the nonlinear minimax problem. The scheme is generally applicable to any stage number m. Applications in solving nonsymmetric systems of linear equations are also discussed. © 1993 John Wiley & Sons, Inc.     

Analysis of numerical schemes of a mathematical model for sickle cell depolymerization

The purpose of this paper is to present and discuss numerical schemes for a mathematical model that describes carbon monoxide mediated sickle cell polymer melting. Two Runge-Kutta methods are analyzed and shown to be unstable by calculating the first failure value of step size and displaying the bifurcation diagram of RK4. Two nonstandard finite difference (NSFD) schemes are proposed and analyzed; one is shown to be stable subject to a predictable bound on step size, while the second one is unconditionally stable.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

组合RK-Rosenbrock方法及其稳定性分析

１．引言 在研究和设计宇航飞行器时,常常会遇到刚性大系统,他们具有特殊结构,系统的解分量有的变化很快,而有的变化很慢。我们可将其分解成两个耦合的子系统;其中（１）式为刚性子系统,（２）式为非刚性子系统。 由于子系统（１）是刚性的,因而整个系统也是刚性的,所以需要采用适合于求解刚性方程的隐式或半隐式方法来求解。但是,在很多情况中,刚性方程组（１）仅占整个方程组的很小一部分,而且右函数相当简单,因而整个右函数计算量主要集中在非刚性方程组（２）上。另一方面,这种对整个方程组采用同一个数值积分方法来处理的…     

Conjugate gradient-boundary element solution using multiple reciprocity method for distributed elliptic optimal control problems

This paper deals with the numerical solution of optimal control problems, where the state equations are given by the fourth order elliptic partial differential equations. An iterative algorithm for this class of problems is developed. This new proposal is obtained by combining the Conjugate Gradient Method (CGM) with the Boundary Element Method (BEM) and Multiple Reciprocity Method (MRM). The local error estimates based on the stability of this scheme in the H² norm, L² norm and L^∞ norm are obtained. Finally, the numerical results on a test case show that this method is correct and feasible.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback

In this article, by a nonstandard finite-difference (NSFD) scheme we study the dynamics of the delay differential equation with unimodal feedback. First, under three cases local stability of the equilibria is discussed according to Schur polynomial and Hopf bifurcation theory of discrete system. Then, the explicit algorithms for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the normal form method and center manifold theorem. In Section 4, numerical example using Nicholson’s blowflies equation is provided to illustrate the theoretical results. Finally, it demonstrates significant superiority of nonstandard finite-difference scheme than Euler method under the means of describing approximately the dynamics of the original system.     

Convergence of a class of runge-kutta methods for differential-algebraic systems of index 2

This paper deals with convergence results for a special class of Runge-Kutta (RK) methods as applied to differential-algebraic equations (DAE's) of index 2 in Hessenberg form. The considered methods are stiffly accurate, with a singular RK matrix whose first row vanishes, but which possesses a nonsingular submatrix. Under certain hypotheses, global superconvergence for the differential components is shown, so that a conjecture related to the Lobatto IIIA schemes is proved. Extensions of the presented results to projected RK methods are discussed. Some numerical examples in line with the theoretical results are included.     

Introducing variational iteration method to a biochemical reaction model

A basic enzyme kinetics is used to test the effectiveness of an analytical method, called the variational iteration method (VIM). This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations. We shall compare the classical VIM against a modified version called the multistage VIM (MVIM). Additional comparison will be made against the conventional numerical method, Runge–Kutta (RK4)(fourth-order). Numerical results were obtained for these three methods and we found that MVIM and RK4 are in excellent conformance.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Discretization of an eco-epidemiological model and its dynamic consistency

In this paper, we discretize a continuous-time eco-epidemiological model by non-standard finite difference (NSFD) scheme as well as standard Euler forward scheme. Dynamical properties of both the systems are explored and compared with their continuous-time model. We show that the solution of NSFD system remains positive for all positive initial values. Fixed points and their local stability properties are shown to be identical with that of the continuous model, indicating its dynamic consistency. Dynamics of the Euler model, however, depend on the step–size and therefore dynamically inconsistent. Solutions in this latter method may be negative and allows numerical instabilities, leading to chaos. Extensive numerical simulations have been performed to validate the theoretical results.     

DAE的Runge-Kutta方法在不可压NS方程求解中的应用

1．引言自然界中的流场通常是非定常复杂流场，要正确模拟和跟踪复杂流场的变化，计算格式的时间精度极为重要．对于常微分方程（＊＊q，一般采用＊K方法及线性多步法来提高格式的时间精度．前者是单步法，在计算过程中可以改变步长，可找到稳定性较好的高精度格式：近年来在发展到偏微分方程的数倩水解中也有很多应用．原始变量的INS方程（二维）为：其中u，v分别是x，y方向速度分量，r是压力，连续方程（1．幻可视为约束条件．从［1］，［2］可见，经空间差分化后（固定空间网格），它可看作带约束的微分方程组，即微分代数方程（DAE－…     

A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula

In this article, a novel numerical method is proposed for nonlinear partial differential equations with space- and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.