Integral equations classification

Ideal reactors can be classified in various ways, but for our purposes the most convenient method uses the mathematical description of the reactor, as listed in Table 14.1. Each of the reactor types in Table 14.1 can be expressed in terms of integral equations, differential equations, or difference equations. Not all real reactors can fit neatly into the classification in Table 14.1, however. The accuracy and precision of the mathematical description rest not only on the character of the mixing and the heat and mass transfer coefficients in the reactor, but also on the validity and analysis of the experimental data used to model the chemical reactions involved. 

According to the lUPAC classification of pores, the size ranges are micropoies (<2 nm), tnesopores (2-50 nm), and macropores (>50 mn) (lUPAC, 1972). All useful sorbents have micropores. The quantitative estimation of pore size distribution (PSD), particularly for the micropores, is a crucial problem in the characterization of sorbents. Numerous methods exist, of which three main methods will be described Kelvin equation (and the BJH method), Horvath-Kawazoe approach, and the integral equation approach. 

There are both Irregular (I) and Regular (R) points at the intersections between the Smooth (S) and Critical (C) boundaries Bi and B2 in the WMBVP as illustrated in Fig. 2.6 where these two boundary intersection points are identified as Pi and P2. The classification of the boundary points Pi and P2 in Fig. 2.6 depends on (1) the boundary conditions Pi Pj) and (2) the continuity of the boundaries B and their derivatives where i, j, and m = 1 or 2. A conformal mapping of the semi-infinite wave channel strip in the physical plane will yield a Fredholm integral equation, where these critical points may be transformed to singular points that are integrable over a smooth continuous mapped boundary. 

A model formulated in terms of differential equations can often be rephrased in terms of integral equations (and vice versa) so that many additional models are essentially included in this classification scheme. Difference equations account for finite changes from one stage to another and have significance parallel to that given above for (continuous) differential equations. 

The set of the reaction-diffusion equations (78) can be solved by different methods, including bifurcation analysis [185,189-191], cellular automata simulations [192,193], or numerical integration [194—197], Recently, two-dimensional Turing structures were also successfully studied by Mecke [198,199] within the framework of integral geometry. In his works he demonstrated that using morphological measures of patterns facilitates their classification and makes possible to describe the pattern transitions quantitatively. 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

The orbital and vibrational components of the wave functions as expanded in equation (46) are functions of the Cartesian coordinates. They can generally be classified as being symmetric to inversion through the origin (g) or antisymmetric to this operation ( ). The integration implicit in equation (45), from -oo to +oo, yields two qualitatively different results on the basis of such a classification. As r has the u classification it gives a zero value if if/ and t/f have the same classification (both g or both u) but, possibly, a finite value if they differ in classification (one g, one u). We have then a further selection rule only transition between functions of opposite parity are allowed. 

Zsako [29] has suggested sub-classification of integral methods on the basis of the means of evaluation of the temperature integral in equation (5.4). The three main approaches are the use of (i) numerical values of/>(x) (ii) series approximations for p(x) and (iii) approximations to obtain an expression which can be integrated. 

The distinction of declarative and procedural representations is directly related to the classification of the models from the point of view of an external solver [303], Open-form model representations provide interfaces to access the full equation system of the model, for example, in form of a CAPE-OPEN equation set object [894], Alternatively, the closed-form model representation provides interfaces which only enable to set inputs and to retrieve outputs of the model. Typically, declarative representations require an external solver and use an open-form interface, whereas procedural representations come with an integrated solver and usually have an interface of the closed-form type. 

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