Ordinary Differential Equation Models

The formulation for the next three problems of the parameter estimation problem was given in Chapter 6. These examples were formulated with data from the literature and hence the reader is strongly recommended to read the original papers for a thorough understanding of the relevant physical and chemical phenomena. 

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ORDINARY DIFFERENTIAL EQUATION MODELS 17.3.1 Contact Inhibition in Microcarrier Cultures of MRC-5 Cells... 

It would be of considerable interest to construct and analyze a model that treats growth and consumption as in this chapter (following [Cu2], i.e., proportional to surface area) and that treats cell division as in [MD]. It seems unlikely that this marriage of the two approaches would yield a model that can easily be reduced to the ordinary differential equation models of Chapter 1. 

The similarity between these two equations is associated with the lack of bias. This sort of ordinary differential equation model relies on the calorimeter being adequately represented by a finite number of parts (here two) each of which has a uniform temperature. The heat transfer coefficients will be independent of temperature for a truly linear system (but the device can be regarded satisfactorily as linear as long as their values do not change significantly over the temperature range inside the calorimeter at any instant or from the minimum to the maximum of a modulation). Eliminating Tp, the model reduces to... 

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. 

Throughout this chapter I have taken the point of view that the meandering of spiral waves in excitable media can and should be examined from the perspective of bifurcation theory. With this approach, it has been possible to show that the organizing center for spiral dynamics is a particular codimension-two bifurcation resulting from the interaction of a Hopf bifurcation from rotating waves with symmetries of the plane. From this observation has followed a simple ordinary-differential-equation model for spiral meandering. 

For most of the oscillating reactions, the knowledge of the reaction mechanism and rate constants is generally very sketchy. Since the details of a particular kinetic model are not relevant close to bifurcation conditions, we will consider the simplest ordinary differential equation model which accounts for the characteristic features of the chlorite-iodide reaction and of its variants [69-71], namely bistability, excitability and relaxation oscillations. Our model of the reaction term is a two-variable Van der Pol-like system [82, 83] (C = u,v)) ... 

A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities the boundaiy or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial valueproblem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinaiy differential equations become two-point boundaiy value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. 

Several methods have been employed to study chemical reactions theoretically. Mean-field modeling using ordinary differential equations (ODE) is a widely used method [8]. Further extensions of the ODE framework to include diffusional terms are very useful and, e.g., have allowed one to describe spatio-temporal patterns in diffusion-reaction systems [9]. However, these methods are essentially limited because they always consider average environments of reactants and adsorption sites, ignoring stochastic fluctuations and correlations that naturally emerge in actual systems e.g., very recently by means of in situ STM measurements it has been demon-... 

The steady state TMB model equations are obtained from the transient TMB model equations by setting the time derivatives equal to zero in Equations (25) and (26). The steady state TMB model was solved numerically by using the COLNEW software [29]. This package solves a general class of mixed-order systems of boundary value ordinary differential equations and is a modification of the COLSYS package developed by Ascher et al. [30, 31]. 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

The resulting model of raulticonponent enulsion pjolymerization systems is consituted by the Pffil 17, an integro-differential equation, a set of ordinary differential equations (equation 18 and 25 and the equations for pjoiymer conposltlon) and the system of the remaining non linear algebraic equations. As expected the conputatlonal effor t is concentrated on the solution of the PBE therefore, let us examine this aspect with some detail. 

For fast reactions Da becomes large. Based on that assumption and standard correlations for mass transfer inside the micro channels, both the model for the micro-channel reactor and the model for the fixed bed can be reformulated in terms of pseudo-homogeneous reaction kinetics. Finally, the concentration profile along the axial direction can be obtained as the solution of an ordinary differential equation. 

The scope of this book deals primarily with the parameter estimation problem. Our focus will be on the estimation of adjustable parameters in nonlinear models described by algebraic or ordinary differential equations. The models describe processes and thus explain the behavior of the observed data. It is assumed that the structure of the model is known. The best parameters are estimated in order to be used in the model for predictive purposes at other conditions where the model is called to describe process behavior. 

The primary classification that is employed throughout this book is algebraic versus differential equation models. Namely, the mathematical model is comprised of a set of algebraic equations or by a set of ordinary (ODE) or partial differential equations (PDE). The majority of mathematical models for physical or engineered systems can be classified in one of these two categories. 

Let us first concentrate on dynamic systems described by a set of ordinary differential equations (ODEs). In certain occasions the governing ordinary differential equations can be solved analytically and as far as parameter estimation is concerned, the problem is described by a set of algebraic equations. If however, the ODEs cannot be solved analytically, the mathematical model is more complex. In general, the model equations can be written in the form... 

Gauss-Newton Method for Ordinary Differential Equation (ODE) Models... 

In this chapter we are concentrating on the Gauss-Newton method for the estimation of unknown parameters in models described by a set of ordinary differential equations (ODEs). 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

Let us now turn our attention to systems described by ordinary differential equations (ODEs). Namely, the mathematical model is of the form,... 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

Differential-Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are... 

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