Ordinary differential equation numerical stability

The approach outlined, considering the trajectories in the i-rj plane instead of f(/) and T)(t) individually, has numerous applications. For an introduction to the theory of trajectory classification see D. A. Sanchez, Ordinary Differential Equations and Stability Theory (San Francisco Freeman, 1968). 

A marching-ahead solution to a parabolic partial differential equation is conceptually straightforward and directly analogous to the marching-ahead method we have used for solving ordinary differential equations. The difficulties associated with the numerical solution are the familiar ones of accuracy and stability. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

In this subsection, some commonly used numerical schemes that involve difference equations to solve ordinary differential equations are presented along with their stability characteristics. Simple examples to illustrate the effects of step size on the convergence of numerical methods are shown. A simple discretization of the first-order linear differential equation... 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

In modds of tubular reactors, material and energy balance are expressed by partial differential equations in time and space variaUes. Althou detailed numerical studies have been made in order to duddate the transient behaviour of tubular reactors, analytical studies have largely been confined to the question of existence, multiplicity, and stability of the reactm steady-state profiles, since the dimination of transirait behaviour often reduces tbe balance equations to a system of ordinary differential equations. 

The stability of the system, in other words the solution of the ordinary differential equation (Eq. 3), can be calculated, based on the models of the machine tool s structure and the machining process. According to Altintas (2012), such problems are mainly solved with numerical methods,... 

The mechanism of the methanol oxidation defines the time evolution of species coverage on the electrode surface, hence, defines also the set of ordinary differential equations which describes the time evolution for each coverage species [136] which is known as chemical network stability analyses (SNA). Accordingly, the unique instability which allows for oscillatory behavior is the Hopf bifurcation (HB) which most of the time is proofed by numerical solution. However, the methodology to solve analytically the set of differential equations was very recently applied to chemistry [137]. 

Dahlquist GG (1963) Stability questions for some numerical methods for ordinary differential equations. Proc Symp Appl Math 15 147-158... 

A numerical tool, sensitivity analysis, which can be used to study the effects of parameter perturbations on systems of dynamical equations is briefly described. A straightforward application of the methods of sensitivity analysis to ordinary differential equation models for oscillating reactions is found to yield results which are difficult to physically interpret. In this work it is shown that the standard sensitivity analysis of equations with periodic solutions yields an expansion that contains secular terms. A Lindstedt-Poincare approach is taken instead, and it is found that physically meaningful sensitivity information can be extracted from the straightforward sensitivity analysis results, in some cases. In the other cases, it is found that structural stability/instability can be assessed with this modification of sensitivity analysis. Illustration is given for the Lotka-Volterra oscillator. 

Differential equations play a dominant role in mathematical modeling. In practical engineering applications, only a very limited number of them can be solved analytically. The purpose of this chapter is to give an introduction to the numerical methods needed to solve differential equations, and to explain how solution accuracy can be controlled and how stability can be ensured by selecting the appropriate methods. The mathematical ftamework needed to solve both ordinary and partial differential equations is presented. A guideline for selecting numerical methods is presented at the end of the chapter. 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

The numerical solutions of ordinary and partial differential equations are based on the finite difference formulation of these differential equations. Therefore, the stability and convergence considerations of finite difference solutions have important implications on the numerical solutions of differential equations. This topic will be discussed in more detail in Chaps. 5 and 6. 

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