Modeling system differential equations

The resulting set of model partial differential equations (PDEs) were solved numerically according to the method of lines, applying orthogonal collocation techniques to the discretization of the unknown variables along both the z and x coordinates and integrating the resulting ordinary differential equation (ODE) system in time. 

A rigorous nonlinear dynamic model of the column is used on-line to predict compositions. The measured flowrates of the manipulated variables (reflux and heat input) are fed into the model. The differential equations describing the system are integrated to predict all compositions and tray temperatures. The predicted tray temperatures are compared with the actual measured tray temperatures, and the differences... 

The analysis of outlet peaks is based on the model of processes in the column. Today the Kubi n - Kucera model [14,15], which accounts for all the above-mentioned processes, as long as they can be described by linear (differential) equations, is used nearly exclusively. Several possibilities exist for obtaining rate parameters of intracolumn processes (axial dispersion coefficient, external mass transfer coefficient, effective diffusion coefficient, adsorption/desorption rate or equilibrium constants) from the column response peaks. The moment approach in which moments of the outlet peaks are matched to theoretical expressions developed for the system of model (partial) differential equations is widespread because of its simplicity [16]. The today s availability of computers makes matching of column response peaks to model equations the preferred analysis method. Such matching can be performed in the Laplace- [17] or Fourier-domain [18], or, preferably in the time-domain [19,20]. 

In the course of developing models for the impedance response of physical systems, differential equations are commonly encountered that have complex variables. For equations with constant coefficients, solutions may be obtained using the methods described in the previous sections. For equations with variable coefficients, a numerical solution may be required. The method for numerical solution is to separate the equations into real and imaginary parts and to solve them simultaneously. 

A mathematical model may be constructed representing a chemical reaction. Solutions of the mathematical model must be compatible with the observed behavior of this chemical reaction. Furthermore if some other solutions would indicate possible behaviors so far unobserved, of the reaction, experiments maybe designed to experimentally observe them, thus to reinforce the validity of the mathematical model. Dynamical systems such as reactions are modelled by differential equations. The chemical equilibrium states are the stable singular solutions of the mathematical model consisting of a set of differential equations. Depending on the format of these equations solutions vary in a number of possible ways. In addition to these stable singular solutions periodic solutions also appear. Although there are various kinds of oscillatory behavior observed in reactions, these periodic solutions correspond to only some of these oscillations. 

Reverse engineering has been successfully applied to relatively simple biomolecular systems. Using a combination of cross-correlation analysis and multidimensional scaling, the glycolytic pathway was reconstructed from metabolite activity data from an in vitro enzymatic reactor system [21]. A complete spatio-temporal model of developmental gene expression in Drosophila was constructed for a small gene set based on models of differential equations and protein expression data [22],... 

Traditionally, dynamical systems are modeled by differential equations. In the case of biochemical networks, Ordinary Differential Equations (ODEs) model the concentration (or activity) of proteins, RNA species, or metabolites by time-dependent... 

This method is simple in theory and is the most widely used, although it becomes quite cumbersome with large models. First we note that if a linear model is identifiable with some input in an experiment, it is identifiable from impulsive inputs into the same compartments. That allows one to use impulsive inputs in checking identifiability even if the actual input in the experiment is not an impulse. Take Laplace transforms of the system differential equations and solve the resulting algebraic equations for the transforms of the state variables. Then write the Laplace transform for the observation function (response function). That will be of the form... 

As the example of the lac operon illustrates, abstract models involving differential equations that do not necessarily reflect the detailed mechanism are sometimes used by scientists when the goal is primarily to explore possible system dynamics originating from the structure of the network. Associated with the process of abstraction is the problem of reducing the network into a smaller set of modules and their interactions. Modules can range from individual molecules or genes, to a set of genes or proteins, or to functional... 

Thus, the thermokinetic models employ differential equations of material and heat balances when electric current passes through the system. We have just considered the model where the heat equilibrium is fast achieved and the rate of the transient process related to the mass transport is rather slow. This allowed us to simplify the mathematic problem and to extend the analysis as far as to obtain the bifurcation diagram and estimate the time dependence of the variables. 

In order to finite-difference the model partial differential equations, we need values of the state variables at discrete distances, yi,V2, --yN, and zi,Z2,...Zn, and at discrete times, 0, Ai, 2At,..., (n/ — l)At. Here N is the number of grid blocks along the quadrant boundar>% At is the time step, and n/ = t/fAt. The reservoir quadrant is therefore replaced by a system of grid blocks shown in Figure 8.37. The integer i is used as the index in the y direction, and the integer j as the index in the z direction. In addition, the index n is used to denote time. Hence pe use the following notation to identify a process variable... 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

The bead and spring model is clearly based on mechanical elements just as the Maxwell and Voigt models were. There is a difference, however. The latter merely describe a mechanical system which behaves the same as a polymer sample, while the former relates these elements to actual polymer chains. As a mechanical system, the differential equations represented by Eq. (3.89) have been thoroughly investigated. The results are somewhat complicated, so we shall not go into the method of solution, except for the following observations ... 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

Mathematical models that represent the dynamic behaviour of physical systems are constructed using differential equations. A more accurate representation of the motor vehicle would be... 

In order to eompute the time response of a dynamie system, it is neeessary to solve the differential equations (system mathematieal model) for given inputs. There are a number of analytieal and numerieal teehniques available to do this, but the one favoured by eontrol engineers is the use of the Laplaee transform. 

High-pressure fluid flows into the low-pressure shell (or tube chaimel if the low-pressure fluid is on the tubeside). The low-pressure volume is represented by differential equations that determine the accumulation of high-pressure fluid within the shell or tube channel. The model determines the pressure inside the shell (or tube channel) based on the accumulation of high-pressure fluid and remaining low pressure fluid. The surrounding low-pressure system model simulates the flow/pressure relationship in the same manner used in water hammer analysis. Low-pressure fluid accumulation, fluid compressibility and pipe expansion are represented by pipe segment symbols. If a relief valve is present, the model must include the spring force and the disk mass inertia. 

The axial dispersion plug flow model is used to determine the performanee of a reaetor with non-ideal flow. Consider a steady state reaeting speeies A, under isothermal operation for a system at eonstant density Equation 8-121 reduees to a seeond order differential equation ... 

One of tlie limitations of dimensional similitude is tliat it shows no dueet quantitative information on tlie detailed meehanisms of the various rate proeesses. Employing the basie laws of physieal and eheiTtieal rate proeesses to matliematieally deseiibe tlie operation of tlie system ean avert this shorteoiTung. The resulting matliematieal model eonsists of a set of differential equations tliat are too eomplex to solve by analytieal metliods. Instead, numerieal methods using a eomputerized simulation model ean readily be used to obtain a solution of tlie matliematieal model. 

A differential equation deseribing the material balanee around a seetion of the system was first derived, and the equation was made dimensionless by appropriate substitutions. Seale-up eriteria were then established by evaluating the dimensionless groups. A mathematieal model was further developed based on the kineties of the reaetion, deseribing the effeet of the proeess variables on the eonversion, yield, and eatalyst aetivity. Kinetie parameters were determined by means of both analogue and digital eomputers. 

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]. 

Traditional control systems are in general based on mathematical models that describe the control system using one or more differential equations that define the system response to its inputs. In many cases, the mathematical model of the control process may not exist or may be too expensive in terms of computer processing power and memory. In these cases a system based on empirical rules may be more effective. In many cases, fuzzy control can be used to improve existing controller systems by adding an extra layer of intelligence to the current control method. 

Coupled-map Lattices. Another obvious generalization is to lift the restriction that sites can take on only one of a few discrete values. Coupled-map lattices are CA models in which continuity is restored to the state space. That is to say, the cell values are no longer constrained to take on only the values 0 and 1 as in the examples discussed above, but can now take on arbitrary real values. First introduced by Kaneko [kaneko83]-[kaneko93], such systems are simpler than partial differential equations but more complex than generic CA. Coupled-map lattices are discussed in chapter 8. 

LGs can also serve as powerful alternatives to PDEs themselves in modeling physical systems. The distinction is an important one. It must be remembered, however, that not all PDEs (and perhaps not all physical systems see chapter 12) are amenable to a LG simulation. Moreover, even if a candidate PDE is selected for simulation by a LG. there is no currently known cookbook recipe allowing a researcher to go from the PDE to a LG description (or vice versa). Nonetheless, by their very nature, LGs lend themselves to modeling any partial differential equation (PDE) for which the underlying physical basis for its construction involves a large number of particles with local interactions [wolf86c]. 

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