Applications - Numerical Results

The accuracy and efficiency of HO FDTD schemes in relation to the improved PMLs and the generalized integration schemes of Section 5.5 are verified by means of several 2- and 3-D realistic waveguide problems. These include inclined-slot coupled T-junctions, thin apertures, power-bus printed board circuits (PCBs), and multiconductor microstrip transmission lines. The majority of the discretized models involve consistent grids that are compared to the respective second-order FDTD realizations. 

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In this analysis, weight coefficients for rows and for columns have been defined as constants. They could have been made proportional to the marginal sums of Table 32.10, but this would weight down the influence of the earlier years, which we wished to avoid in this application. As with CFA, this analysis yields three latent vectors which contribute respectively 89, 10 and 1% to the interaction in the data. The numerical results of this analysis are very similar to those in Table 32.11 and, therefore, are not reproduced here. The only notable discrepancies are in the precision of the representation of the early years up to 1972, which is less than in the previous application, and in the precision of the representation of the category of women chemists which is better than in the previous analysis by CFA (0.960 vs 0.770). 

Typical examples such as the ones mentioned above, are used throughout this book and they cover most of the applications chemical engineers are faced with. In addition to the problem definition, the mathematical development and the numerical results, the implementation of each algorithm is presented in detail and computer listings of selected problems are given in the attached CD. 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

We investigate theoretically how the adsorption of the polymer varies with the displacer concentration. A simple analytical expression for the critical displacer concentration is derived, which is found to agree very well with numerical results from recent polymer adsorption theory. One of the applications of this expression is the determination of segmental adsorption energies from experimental desorption conditions and the adsorption energy of the displacer. Illustrative experiments and other applications are briefly discussed. 

This chapter is intended to provide basic understanding and application of the effect of electric field on the reactivity descriptors. Section 25.2 will focus on the definitions of reactivity descriptors used to understand the chemical reactivity, along with the local hard-soft acid-base (HSAB) semiquantitative model for calculating interaction energy. In Section 25.3, we will discuss specifically the theory behind the effects of external electric field on reactivity descriptors. Some numerical results will be presented in Section 25.4. Along with that in Section 25.5, we would like to discuss the work describing the effect of other perturbation parameters. In Section 25.6, we would present our conclusions and prospects. 

The Hartree-Fock approach derives from the application of a series of well defined approaches to the time independent Schrodinger equation (equation 3), which derives from the postulates of quantum mechanics [27]. The result of these approaches is the iterative resolution of equation 2, presented in the previous subsection, which in this case is solved in an exact way, without the approximations of semiempirical methods. Although this involves a significant increase in computational cost, it has the advantage of not requiring any additional parametrization, and because of this the FIF method can be directly applied to transition metal systems. The lack of electron correlation associated to this method, and its importance in transition metal systems, limits however the validity of the numerical results. 

Following a brief introduction of the basic concepts of semiclassical dynamics, in particular of the semiclassical propagator and its initial value representation, we discuss in this section the application of the semiclassical mapping approach to nonadiabatic dynamics. Based on numerical results for the... 

Nagaya, K. (1984). On a magnetic damper consisting of a circular magnetic flux and a conductor of arbitrary shape. Part II Applications and numerical results. Transactions of the ASME J. Dynam. Syst. Meets. Control. 106, 52-55. 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

We have chosen to concentrate on a specific system throughout the chapter, the methanation reaction system. Thus, although our development is intended to be generally applicable to packed bed reactor modeling, all numerical results will be obtained for the methanation system. As a result, some approximations that we will find to apply in the methanation system may not in other reaction systems, and, where possible, we will point this out. The methanation system was chosen in part due to its industrial importance, to the existence of multiple reactions, and to its high exothermicity. 

In Eq. (4.39), the upper sign refers to solution soluble product and the lower one to amalgam formation. When species R is amalgamated inside the electrode, the applicability of this analytical equation is limited by Koutecky approximation, which considers semi-infinite diffusion inside the electrode, neglecting its finite size and simplifying the calculations. Due to the limitations of this approximation, the analytical and numerical results coincide only for < 1 with a relative difference < 1.7% [20]. For higher values of c,. a numerical solution obtained with the condition (3cR/3r)r=0 = 0 should be used. 

As discussed in Appendix G, for the resolution of the problem we have supposed that the mathematical concentration profiles of the first pulse for the total time t + r2 are not disturbed by the application of the second one. This assumption is fully valid for any electrode radius in DDPV technique, where the duration of the second pulse is much shorter than that of the first one (t >> t2). It has been confirmed the validity of the analytical Eq. (4.134) in DDPV conditions by comparison with numerical results [45], since nonsignificant deviations are obtained (less than 0.5 %) around the peak potential when t /r2 > 50. 

Whereas there has been a huge interest in multimode calibration in the theoretical chemometrics literature, there are important limitations to the applicability of such techniques. Good, very high order, data are rare in analytical chemistry. Even three-way calibration, such as in DAD-HPLC, has to be used cautiously as there are frequent experimental difficulties with exact alignments of chromatograms in addition to interpretation of the numerical results. However, there have been some significant successes in areas such as sensory research and psychometrics. 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

This relation can be obtained from the fundamental equation (Chap. I (4)) for the form of a liquid surface under gravity and surface tension. Unfortunately this equation cannot be solved in finite terms. Approximate solutions have been obtained in several ways which are outside the scope of this book. Sufficient account must, however, be given of the methods of Bashforth and Adams,1 to enable the reader to use their tables of numerical results, which are the most complete and accurate ever compiled. Some other important approximate formulae will also be given, for applications of the fundamental equation to special cases. 

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