Essence of the Method

It is best to illustrate the essence of the method by way of a simple example. An iterative solution is sought which, in the limit of small grid size, should converge very closely to the exact solution. 

Let us start with the elementary second order differential equation 

This relation can be used to describe a chemical reaction occurring in a fixed bed. The equation, of course, can be handled by teehniques developed in Chapters 2 and 3, but we will solve it numerically for the purpose of demonstration. 

We first divide the domain into N intervals, with the grid size being Ax = 1/N. The /th point is the point having the coordinate i(Ax). Eq. 12.118 is valid at any point within the domain [0,1] thus, if we evaluate it at the point jc we have 

Using the discrete formula for the first and second derivatives, (Eqs. 12.116c and 12.117), which are both second order correct, we obtain 

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Brown et al. [494] developed a method for the production of hydrated niobium or tantalum pentoxide from fluoride-containing solutions. The essence of the method is that the fluorotantalic or oxyfluoroniobic acid solution is mixed in stages with aqueous ammonia at controlled pH, temperature, and precipitation time. The above conditions enable to produce tantalum or niobium hydroxides with a narrow particle size distribution. The precipitated hydroxides are calcinated at temperatures above 790°C, yielding tantalum oxide powder that is characterized by a pack density of approximately 3 g/cm3. Niobium oxide is obtained by thermal treatment of niobium hydroxide at temperatures above 650°C. The product obtained has a pack density of approximately 1.8 g/cm3. The specific surface area of tantalum oxide and niobium oxide is nominally about 3 or 2 m2/g, respectively. 

In this section we address formation of concentration shocks in reactive ion-exchange as an asymptotic phenomenon. The prototypical case of local reaction equilibrium of Langmuir type will be treated in detail, following [1], [51], For a treatment of the effects of deviation from local equilibrium the reader is referred to [51]. The methodological point of this section consists of presentation of a somewhat unconventional asymptotic procedure well suited for singular perturbation problems with a nonlinear degeneration at higher-order derivatives. The essence of the method proposed is the use of Newton iterates for the construction of an asymptotic sequence. 

The essence of the method of incomplete separation of variables consists in introducing the interelectronic distances (usually only in an open shell) in an atom. Then its wave function... 

Impedance spectroscopy is one of the most informative methods in electrochemistry research [1,2], The essence of the method consists in investigating the response of a target taking place in stationary conditions to weak influences of a variable voltage or to an electric current in a wide range of frequencies. It is possible... 

The essence of the method is very simple. We start from a base point a and by a series of exploratory moves build up a pattern from which it is possible to infer a direction in which it is likely that a successful move may be made. On the basis of our pattern we make such a pattern move, to establish a new base point for further exploration. We continue the cycle... 

In the factor label method, units may be canceled like variables x, y) in algebra. Placement of the units so that they cancel to give the desired units is the essence of the method. 

It is very important to understand non-isothermic kinetics to analyse the composition s technological properties, especially at very high sp s of cooling, which occur in thermoplasts and thermoplast-based compositions processing. The method for describing and forecasting non-isothermic crystallization kinetics is proposed in Ref [59]. Alternative methods for describing kinetics are in the studies cited in Ref [59] (see also Refs. [66, 67]). The essence of the method in Ref [59] is as follows. 

The indirect method was put forward by Stevenson in it the appearance potentials of the same ion produced from two different but related molecules are measured, and their difference is combined with thermochemical data to give the required dissociation energ) It is thus possible to calculate bond dissociation energies from appearance potentials and thermochemical data without a knowledge of ionization potentials. The essence of the method is to produce as the non-ionized partner in a dissociative ionization process of the type... 

Dynamic analysis of photochromic systems under continuous irradiation represents a powerful method of investigation of the reaction mechanisms. The characteristic kinetic and spectral parameters such as the quantum yields of the photochemical steps and the molar extinction coefficients of the transient species can be derived using this method. The essence of the method is the inverse treatment based on numerical simulation and fitting of the plots (Abs versus t) obtained under continuous irradiation. This also exploits the information contained in the irradiation kinetics. In order to extract one or more of the relevant parameters of a given process, specially designed experiments need to be carried out in which the effect of the process under consideration is conspicuous. 

Over 30 years ago Chuiko, Pavlik and Neimark [27] described briefly preparation by the sol-gel method of FPS which contained -(CH2)2COOH groups. The essence of the method consisted in an acidic hydrolysis (HCl) of 2-cyanoethyltriethoxysilane in the presence of TEOS. The resulting white product showed good sorbtion of NHEt2 and py. 

In response to this situation a much simplified technique for the handling of reactive materials has been developed. The essence of the method consists in protecting the sample from the atmosphere... 

The mapping of electron transfer pathways has been described in Refs. [44,46,47], The general idea of the approach is to examine dynamics of charge redistribution in the system clamped at the transition state of electron transfer reaction. Briefly, the essence of the method is as follows. The tunneling dynamics of a many-electron system is described by following wave function ... 

Another approximate method of solution of Eq. (6.118) that covers both longitudinal and radial diffusion of substance was proposed in [15]. The essence of the method is to reduce the diffusion equation to a system of equations for the moments, which have a simple physical meaning. Introduce the following dimensionless variables ... 

The term "inverse reflects the essence of the method more exactly than does stripping , since the analytical signal recorded is determined not by the solution composition, but by that of the electrode. 

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