Numerical solution of Eq

To solve Eq. (4.86) we employ the Jacobi-Newton iteration technique, which proceeds iteratively in an alternating sequence of local and global minimization steps. Let p be the local density at lattice site i in the A th local and the /th global minimization step. A local estimate for the corresponding minimum value of fi is obtained via Newton s method (see Eq. (D.6)] that 

It is important to realize that throughout each local minimization cycle the densities at nearest-neighbor sites of site i represented by the set remain fixed at the initial values assigned to them at the beginning of the local cycle. The iterative solution of Eq. (D.14) is halted if [see Eq. (D.7)] 

To initiate the iterative scheme, suitable starting solutions for Eq. (D.14) are obviously required. These solutions are provided by the morphologies M at T = 0 for which /i can be calculated analytically from the expressions for Q compiled in Table 4.1. 

Here we work out expressions for the number Naa ( bb) of A-A (B-B) pairs. These pairs ar e directly connected sites, both of which are occupied by a molecule of species A (B). Likewise, expressions for Nab (s) and the total number of molecules of species A and B at the solid substrate, Naw (s) and yVew (s), may be derived easily. The resulting expressions presented in Eqs. (4.118)-(4.121) contain terms that can be cast as 

20) as well as (D.22) permit us to write down the mean-field expressions 

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Lane improved on these tables with accurate polynomial fits to numerical solutions of Eq. 11-17 [16]. Two equations result the first is applicable when rja 2... 

Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

It may happen that many steps are needed before this iteration process converges, and the repeated numerical solution of Eqs. III.21 and III.18 becomes then a very tedious affair. In such a case, it is usually better to try to plot the approximate eigenvalue E(rj) as a function of the scale factor rj, particularly since one can use the value of the derivative BE/Brj, too. The linear system (Eq. III. 19) may be written in matrix form HC = EC and from this and the normalization condition Ct C = 1 follows... 

Figure 16. The effective frequency 0(x) for two transversal excitations in HO2. Solid line shows the results of numerical solution of Eqs. (79) and (80). Dotted line represents the frequency in the adiabatic approximation, y = 1[3] corresponds to the mode (0,0,1) [(1,0,0)] (see Fig. 14). Taken from Ref. [32].

Alternately the numerical solution of Eqs (1) and (2) can be found directly by software ODE, without going through Eq (5). 

This could be solved by trial, using a numerical integration, to find C as a function of t. However, the graph is of a direct numerical solution of Eq (1). The degradation of the catalyst Is very rapid. In practice the catalyst will proceed from the transfer line to a reactivation zone and will be recycled. 

Numerical solution of Eq (1) by the shooting method automatically gives the derivative at the external surface required in Eq (2). 

In figure 3 we compare the analytical solution of Eq. (12) and Eq. (13) with the numerical solution of Eq. (14) for the SHG conversion efficiency for three different pulse profiles (hyperbolic secant, Gaussian and rectangular). It can be seen that the simplified model perfectly describes the process of SHG when L >. Further, the analytical model of Eq. (12)... 

For shapes whose boundaries are not simply described in a single coordinate system, numerical solution of Eq. (4-54) is required. However, it is possible to provide upper and lower bounds for the conductance (P8) in much the same way as for the drag. A lower bound for an arbitrary particle is the conductance of the sphere of the same volume, i.e.. 

Figure 1 Electron escape probability as a function of the applied electric field. The solid lines are obtained from Eq. (23) for different values of the initial electron cation distance ro- The broken lines are calculated for ro = 1 nm from the numerical solution of Eq. (16) with the sink term given by k r) = A exp[—a(r—

An example of numerical solutions of Eq. (29) for the attractive Coulomb potential, subject to the initial and the boundary conditions (30)-(32), is presented in Fig. 4 [26]. The figure shows how the initially uniform electron concentration gradually decreases in the vicinity of the cation until the steady state is established. The rate of reaction is determined by the diffusive flow of electrons from large distances toward the reaction sphere. 

Figure 5.6 shows the evolution of the concentration of active species predicted by the numerical solution of Eqs. (5.34) and (5.35), for runs carried out at constant heating rate. The lower the heating rate, the closer the... 

Figure 4.8. Eigenmode (p, (x) determined with the aid of the numerical solution of Eq. (4.130) for the dimensionless barrier height a 5 (dashed line), 10, 20, 25 (solid lines) the arrow shows the direction of a growth. Thick dashes show the step-wise function that is the limiting contour for tpj at...

Numerical solution of Eqs. (4.263) and (4.264) yields a representation for the distribution function (4.260) accurate up to the terms of the order J 0. Using it to evaluate the reduced magnetization (x) and taking into account expansion (4.250), one can present the magnetic response as a sum of the frequency-dependent contributions as... 

In Figure 4.20 we present the plot of Teff obtained from the numerical solution of Eq. (4.271). The coordinates used for this schematic representation... 

A numerical solution of Eqs. (6),(7) together with the corresponding boundary conditions yields temperature and concentration fields throughout the channel. 

The lower bound for the coagulation coefficient, ft, is calculated from Eq. [81] using the values of the constant A determined from the numerical solution of Eqs. [74], [77], and [78]. The upper bound for the coagulation coefficient, ft, is obtained from Eq. [97]. In the above calculations, the Philips slip correction factor (Eq. [6]), is used for the calculation of the diffusion coefficient of particles. The results are expressed in terms of the dimensionless coagulation coefficient y defined as the ratio between the coagulation coefficient and the Smoluchowski coagulation coefficient... 

If the bed of catalyst has a uniform cross section but is not a circular cylinder, another dimension enters into consideration. In cases where a fairly high symmetry is retained, the numerical solution of Eqs. (3-6) and (3-15) is certainly possible, but just as certainly not easy or fast. If the symmetry is low, or if the boundaries are complicated, as, for example, in a large bed through which cooling tubes are passed, the problem becomes very difficult. The best approach in such a case is probably to estimate the radius of an equivalent circular cylinder. A rough guide to such an estimate can be found in two-dimensional potential theory, but the fact that the rate is not a linear function of composition and temperature makes it impossible to form an accurate estimate in any general way. 

The decay rate constants of both Ni and N2 are independent of excitation power or Ni excitation density under these low-power excitation conditions, and the upconversion decay rate constant of N2 equals 2 ki. This analytical limiting decay curve is plotted in Fig. 9 c as the low-power dashed line, and compares very well to the numerical solution of Eq. (10) obtained for ki/lwsnjNi = 37 (low power in Fig. 9b). 

In this section we summarise the properties of the approximations to tc[M] discussed in Section 4 in applications to atoms. All results presented in the following [36] are based on the direct numerical solution of Eqs. (3.25-3.29) using a nuclear potential which corresponds to a homogeneously charged sphere [69]. Only spherical, i.e. closed subshell, atoms and ions are considered. Whenever suitable we use Hg as a prototype of all high-Z atoms. 

It is extremely important to note here that we Figure 2 Numerical solution of Eq. (1) with can easily shift from the one dimensional to three potential A (full curves) and theoretical results just by regarding (marked curves) from Eq. (26)... 

We numerically solve Eq. (4) by a method" similar to the cases of and B with a condition that the long-distance form of/ r) at large r is given by Eq. (3). Stokes theorem tells us that the friction exerted on the rotating sphere can be calculated from the long-distance behavior of/(r) With S in Eq (3) obtained by the numerical solution of Eq (4), is given by... 

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