Slowly varying function

Assuming that our LG is in a local equilibrium, it is reasonable to expect that the one-particle distribution functions should depend only on the macroscopic parameters u x,t) and p x,t) and their derivatives [wolf86c]. While there is no reason to believe that this dependence should be a particularly simple one, it is reasonable to expect that both u and p are slowly varying functions of x and t. Moreover, in the subsonic limit, we can assume that li << 1. 

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9. Moreover, as Eqs. (6-96) contain the small factor (i on the right-hand side, the quantities, a and 9 are small, that is, both a and 9 are slowly varying functions of time and one can assume that during one period T = 2nfca, the trigonometric functions vary but slightly. 

The dAc/dz term is usually zero since tubular reactors with constant diameter are by far the most important application of Equation (3.7). For the exceptional case, we suppose that Afz) is known, say from the design drawings of the reactor. It must be a smooth (meaning differentiable) and slowly varying function of z or else the assumption of piston flow will run into hydrodynamic as well as mathematical difficulties. Abrupt changes in A. will create secondary flows that invalidate the assumptions of piston flow. 

This modified density Is a more slowly varying function of x than the density. The domain of Interest, 0 < x < h, Is discretized uniformly and the trapezoidal rule Is used to evaluate the Integrals In Equations 8 and 9. This results In a system of nonlinear, coupled, algebraic equations for the nodal values of n and n. Newton s method Is used to solve for n and n simultaneously. The domain Is discretized finely enough so that the solution changes negligibly with further refinement. A mesh size of 0.05a was adopted In our calculations. 

In the approximation that the molecules can be replaced by equivalent noninterpenetrating spheres, it may be shown that F3 = (5/8)r2. This result is indicative of a strong dependence of the third coefficient on the second. A more detailed analysis shows that the numerical coefficient 5/8 should be replaced for polymer molecules by a slowly increasing function of F2—one that is less than 5/8 and that vanishes as F2 goes to zero. Calling this slowly varying function g... 

Since Inn is a slowly varying function of n, the error goes essentially as n. This is the typical behaviour of a discretization error for a numerical integration [23], but is atypical for the examples that we want to study. 

In general, the angular frequency (o(k) is a function of k, so that the angular frequencies in the composite wave (jc, t), as well as the wave numbers, vary from one plane wave to another. If (o k) is a slowly varying function of k and the values of k are confined to a small range AA , then (x) k) may be expanded in a Taylor series in k about some point ko within the interval AA ... 

In the presented form Eq. (8.13) is only valid, if Hj (s) is, indeed, constant over the whole angular range required for analysis. If this is not the case and numerical deconvolution is aimed at, the standard algorithm may be adapted by consideration of the fact that, in any case, the broadening is a slowly varying function of 29. 

In more refined calculations, 2 is replaced by its average over the Maxwellian velocity distribution and is a slowly varying function of t. 

This approach is based on the idea of SD mapping [37, 38] that exploits the linear relationship (Eqs. 1-3) between the measured relaxation parameters and the spectral densities. The spectral density components, 7(0), J(oN), and J(coH) could be determined directly from the relaxation data, J x, R2, and NOE, using the reduced spectral density approach [30, 31] based on the observation that the spectral densities are slowly varying functions of the frequency around wH ... 

Bronsted-Guggenheim approach to multicomponent equations are only exact if pR,x is constant, the basic equations are reasonable exact if 3r,x is a slowly varying function of concentration and Harned s Rule (7) for mixed electrolytes through an empirical correlation does show such of an effect. 

If it is assumed that AH° is a slowly varying function of T, one obtains... 

Here complex amplitude of the pulse envelope E x, z, t) is also a slowly varying function of z and t. Spatiotemporal distribution of the electric field is described by E x,z,t) = E x,z,t)ex-p[iujQt— i/Sz), f3 being the longitudinal wavenumber of the waveguide mode at the pulse peak. 

It is convenient to put the terms depending on the rate of change of with on the right hand side of (7) in order that, when is a slowly varying function of r, they may be dropped. The coupled equations therefore have the form... 

It is now seen that assumption (a) is, indeed, valid provided / Aay, and T are slowly varying functions of the energy. That is, by integration over the intermediate states in eq. (2-34) a function is obtained having the same energy dependence as the original (assumed) transition amplitude. Equation (2-39) may be solved for Aga. The result is ... 

To test the hypotheses (7.4.17) and (7.4.18), the kinetics of accumulation was simulated on a computer by the method described in [110]. For each of the values vp = 10,16,24, and 50, the process of accumulation was performed independently 200 times until the stage of steady-state values of no was reached. The relationships n(N), N = pt, and a(n) were constructed from the mean values obtained in this series. It was shown that within the limits of error of computer experiment ( 5%), the slowly varying function a(n) can be well approximated by the linear dependence of (7.4.18), which confirms the suitability of this approach for describing the accumulation of point defects in the discrete model. Analogous results are obtained for vp = 16 and 50 for which the values were found respectively, of 1.092 and 1.625 for n0 and 0.463 and 0.478 for f3(oo) = a(oo)vono. 

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