Ordinary differential equations ODEs

Thus equation 4.45 is an ordinary differential equation (ODE) which can easily be solved for filter area, A (in the design problem) or filtrate collected, V (for performance mode calculation). 

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 partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

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). 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

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

In Chapter 6, the Gauss-Newton method for systems described by ordinary differential equations (ODE) is developed and is illustrated with three examples formulated with data from the literature. Simpler methods for estimating parameters in systems described by ordinary differential equations known as shortcut methods are presented in Chapter 7. Such methods are particularly suitable for systems in the field of biochemical engineering. 

Particle Size Development. Now that a general total property balance equation has been developed (equation (II-9)), one can use it to obtain ordinary differential equations (ode s) which will describe particle size development. What is needed with equation (II-9) is an expression for dp(t,t)/dt, where p denotes a specific property of the system (e.g. particle size). Such an expression can be written for the rate of change of polymer volume in a particle of a certain class. The analysis, which is general and described in Appendix III, will finally result in a set of ode s for Np(t), Dp(t), Ap(t) and Vp(t). 

With this transformation, the solution for the momentum can be converted into that of an ordinary differential equation (ODE) of Q ... 

Order-of-magnitude analysis, 11 145 Order-of-magnitude estimates, 9 529 Order parameters, of liquid crystalline materials, 15 82-85 Ordinary differential equation (ODE), 25 281... 

The transport rates fj will be determined by the turbulent flow field inside the reactor. When setting up a zone model, various methods have been proposed to extract the transport rates from experimental data (Mann et al. 1981 Mann et al. 1997), or from CFD simulations. Once the transport rates are known, (1.15) represents a (large) system of coupled ordinary differential equations (ODEs) that can be solved numerically to find the species concentrations in each zone and at the reactor outlet. 

DNS of homogeneous turbulence thus involves the solution of a large system of ordinary differential equations (ODEs see (4.3)) that are coupled through the convective and pressure terms (i.e., the terms involving T). 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

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. 

The Peclet number dependence of the soot cake microstructure has important implications for the morphology of the soot (and ash) deposits in the square channels of DPFs, as illustrated in Fig. 29, obtained by a mixed Ordinary Differential Equations (ODE)-Monte Carlo simulation (Rodriguez-Perez et al.,... 

Teorell studied the system (6.1-1)—(6.1.5) graphically, by the isocline method, and also numerically. He recovered most of the features observed experimentally. This study was further elaborated by several investigators. Thus, Kobatake and Fujita [5], [6] criticized the original model for invoking the ad hoc equation (6.1.5). These authors assumed instead instantaneous relaxation of the resistance to its stationary value while preserving the overall order of the relevant ordinary differential equation (ODE) system by including consideration of the mechanical inertia of the liquid column in the manometer tube. 

Develop two method-of-lines simulations to solve this problem. In the first, formulate the problem as standard-form ordinary differential equations, y7 = ff(f, y). In the second, formulate the problem in differential-algebraic (DAE) form, 0 = g(t, y, y ). Standard-form stiff, ordinary-differential-equation (ODE) solvers are readily avalaible. DAE solvers are less readily available, but Dassl is a good choice. The Fortran source code for Dassl is available at http //wwwjietlib.org. 

Thus the equations that we must solve are 12.196 and 12.197, which comprise a set of two coupled first-order differential equations, subject to the boundary conditions, Xj = 0.01395, and X2 = 0.00712 at z = 0 and Xj = X2 = 0 at z = Z, with the unknown fluxes Ni, N2 that must be found. This equation set could easily be solved as a two-point boundary-value problem using the spreadsheet-based iteration scheme discussed in Appendix D. However, for illustration purposes we choose to solve the equation set with a shooting method, mentioned in Section 6.3.4. We can solve the problem as an ordinary differential equation (ODE) initial-value problem, and iteratively vary Ni,N2 until the computed mole fractions X, X2 are both zero at z = Z. 

Many chemical-kinetics problems, such as the homogeneous mass-action kinetics problems discussed in Section 16.1, are easily posed as a system of standard-form ordinary differential equations (ODE),... 

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

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). 

The mathematical method that will be used to characterize the system response is the stroboscopic map that was used by Kai Tomita (1979) and Kevrekidis et al. (1984,1986). If a point in the phase plane is used as an initial condition for the integration of the ordinary differential equations (odes) (4), a trajectory will be traced out. After integrating for one forcing period (from t = 0 to t = T), the trajectory will arrive at a new point in the phase palne. This new point is defined as the stroboscopic map of the original point and integration for... 

Mathematical modeling of systems for which characteristic variables are time-dependent only and not space-dependent is done by ordinary differential equations (ODEs). The situation is found in a nearly well-mixed batch reactor. There one may find differences in temperature or concentrations from one site to another due to imperfect mixing. When space changes are not important to the model, the process variables can be approximated by means of lumped parameter models (LPMs). When the... 

Most commonly used ordinary differential equation (ODE) solvers provide options of several different integration techniques. Most solvers also automatically vary the integration step size during the simulation to allow the best trade-off between accuracy and solution time, based on user-specified numerical tolerances. There is no single best integration technique—different methods work better for various problems. 

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