Finite-difference methods for solving the reaction-diffusion equations of a simple isothermal chemical system

Numerical methods are proposed for the numerical solution of a system of reaction-diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well-stirred analogue of the chemical system. The first-order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well-known, first-order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling-wave problem. © 1994 John Wiley & Sons, Inc.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.     

Numerical Solutions for Ito Coupled System

In this paper, the classical fourth-order Runge-Kutta method and Heun’s method are applied to initial value problems for system of ordinary differential equations in nonlinear cases which we reach it by Painleve analysis, focusing our interest in Ito coupled nonlinear partial differential system. The equations are solved by scheme of one step methods. Numerical results for the velocity in three dimensions are obtained and reported graphically for various temperatures to show interesting aspects of the solution.     

Analytic approximate solution of three‐dimensional Navier‐Stokes equations of flow between two stretchable disks

The similarity transform for the steady three‐dimensional Navier‐Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations which is analytically solved by applying a newly developed method, namely, the homotopy analysis method. The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is analyzed. The validity of our solutions is verified by the numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

We study the class of nonlinear ordinary differential equations y″ y = F(z, y²), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.     

A new approach to fuzzy initial value problem by improved Euler method

This paper targets to investigate the solution of linear and nonlinear ordinary differential equations with fuzzy initial condition. Here, two improved Euler type methods have been proposed in order to obtain numerical solution of the problem. Along with this, an exact methodology is also discussed. The obtained results are depicted in term of plots to show the efficiency of the proposed methods. The solutions are compared with the known results and are found that those obtained by the proposed methods are tighter than the results from the existing method.     

Purely analytic approximate solutions for steady three-dimensional problem of condensation film on inclined rotating disk by homotopy analysis method

The similarity transform for the steady three-dimensional problem of a condensation film on an inclined rotating disk gives a system of nonlinear ordinary differential equations which are analytically solved by applying a newly developed method namely the homotopy analysis method (HAM). The analytic solutions of the system of nonlinear ordinary differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. The velocity and temperature profiles are shown and the influence of the Prandtl number on the heat transfer and the Nusselt number is discussed in detail. The validity of our results is verified by numerical results.     

Numerical methods for simulation of large systems

The numerical solution of large initial value problems, including those that are derived as approximations to systems of partial differential equations, may encounter difficulties using conventional numerical methods because of stiffness (large range of eigenvalues of the associated linear system). In a nonlinear system, the eigenvalues may change greatly during the solution and a system that is initially well behaved may become stiff, yielding increased computer cost or inaccuracies. This paper contains a discussion of various definitions of stiffness, and several methods for overcoming it, including a new method for identifying and partitioning a two-time-scale system into fast and slow sub-systems. Also included are some experiences using the DARE continuous system simulation language for systems as large as 200 coupled nonlinear ordinary differential equations.     

Mathematical study of two-variable systems with adaptive numerical methods

In this paper, we consider reaction–diffusion systems arising from two-component predator–prey models with Smith growth functional response. The mathematical approach used here is in two folds since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with the fourth-order difference operator and advance the resulting nonlinear system of ordinary differential equations with the two competing families of the exponential time differencing (ETD) schemes, and we analyze them for stability. Numerical comparison between these two methods for solving various predator–prey population models with functional responses are also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.     

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.     

求解非线性方程组的连续极小化方法

1.引言 求非线性方程组 F(x)=0 (1)(F:D?R~n→R~n)的各种方法中,牛顿法最为基本.但它只有局部收敛性和半局部收敛性,而且要求DF(x)~(-1)存在.为了扩大收敛范围及克服DF(x)奇异性带来的困难,用“连续化”的思想求方程(1)的解是一个有效的途径.这方面,已有许多工作,如[3—6].本文利用常微分方程几何理论,对连续化方法进行 些探讨,给出了沿积分曲线极小化求非线性方程组(1)的解的方法.考虑如下给定函数:     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Variational iteration method for solving three‐dimensional Navier–Stokes equations of flow between two stretchable disks

The similarity transform for the steady three‐dimensional Navier–Stokes equations of flow between two stretchable disks gives a system of nonlinear ordinary differential equations. In this article, the variational iteration method was used for solving these equations. The results have been compared with the numerical results. This article depicts that the VIM is an efficient and powerful method for solving nonlinear differential equations. This method is applicable to strongly and weakly nonlinear problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Numerical method of lines for parabolic functional differential equations

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given.     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 1: analysis

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations. The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations (ODEs). The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Meromorphic solutions of nonlinear ordinary differential equations

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

Natural convection on a vertical stretching cylinder

The flow and natural (or mixed) convection due to a vertical stretching cylinder is studied. Using similarity transforms, the Navier-Stokes and energy equations reduce to a set of nonlinear ordinary differential equations. Asymptotic analysis for large Reynolds numbers shows the relation between axisymmetric flow and two-dimensional flow. Due to the algebraic decay of the similarity functions, numerical integration is performed using a compressed coordinate. The axial velocity is composed of forced convection due to stretching and natural convection from the heated cylinder. The heat transfer increases with both the Reynolds number and the Prandtl number. The result is also a rare similarity solution of the free convection and Navier-Stokes equations.