Approximations of Higher Order

The calculation of viscoelastic functions by means of spectra calculated by first-order approximations may lead to values of these functions that are in error with respect to the true values. These errors are lower if correction 

It should be noted that the equation assumed for the spectrum should be valid over at least two decades in the time scale. Substituting Eq. (9.69) into Eq. (9.6) and making t/x = u, gives the relationship 

Accordingly, Hi %) is in error with respect to the assumed function. Ax by the factor r(m + 1), and consequently, 

Equation (9.72) provides a more precise method for determining the relaxation spectrum. The strategy to follow in the calculation of the relaxation spectrum involves the determination of provisional values of H x), at t = x, at a series of points equally spaced on the logarithmic scale using m = 0 in Eq. (9.72). Then from a double logarithmic plot of H x) against x, the slope —m is determined at each point. The reciprocal of r(w -f 1) multiplied by the provisional value of H gives the value of the relaxation spectra. 

This method can be used to obtain the retardation spectrum from the compliance function J(t). Equation (9.15) can be written as 

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It is important to note that in all these methods, the first term in the series solution constitutes the so-called approximation of zero order. This is generally the solution of a simple linear problem e.g., the harmonic oscillator the second term appears as the first approximation, and so on. The amount of labor increases very rapidly with the order of approximation, but the additional information obtained from approximations of higher orders (beginning with the second) does not increase our knowledge from the qualitative point of view. It merely adds small quantitative corrections to the first approximation, and in most applied problems, these corrections are scarcely worth the considerable complication in calculations. For that reason the first approximation is generally sufficient in exploring a new problem, or in investigating the qualitative aspect of a phenomenon. 

We have entered into some details of the method of Poincar6 because it opened an entirely new approach to nonlinear problems encountered in applications. Moreover, the method is very general, since by taking more terms in the series solution (6-65), one can obtain approximations of higher order. However, the drawback of the method is its complexity, which resulted in efforts being directed toward a simplification of the calculating procedure. 

However, there is one other point to be mentioned. In the early (1937) theory of K.B., the theory of the first approximation follows directly from the assumption of the sinusoidal solution, as explained above. In order to obtain approximations of higher order, it became necessary to use an auxiliary perturbation procedure. 

Other approximations of higher order can be used in the evaluation of the relaxation and retardation spectra. Let us define the function... 

This approach was further extended in reference Cangialosi et al. (2007) to estimate the absolute size of a CRR. Therefore, the parameter a has to be obtained quantitatively. As a result for a variety of polymers, the size of a CRR was found to be between 1 nm and 3 nm at the glass transition. These numbers are in agreement with the fluctuation approach by Donth (Donth et al. 2001b Hempel et al. 2000 Beiner et al. 1998 Kahle et al. 1997) as well as with more recent theories using approximations of higher order correlation functions (Berthier et al. 2005 Dalle-Ferrier et al. 2007). 

Approximation of Higher-Order Moments Skewness and Kurtosis... 

Differentials of higher orders are of little significance unless dx is a constant, in which case the first, second, third, etc. differentials approximate the first, second, third, etc. differences and may be used in constructing difference tables (see Algebra ). 

Because much experimental work has been stimulated by the quasi-chemical theory, it is important to gain proper perspective by first describing the features of this theory.12 The term, quasichemical will be used to include the Bragg-Williams approximation as the zeroth-order theory, the Bethe or Guggenheim pair-distribution approximations as the first-order theory, and the subsequent elaborations by Yang,69 Li,28 or McGlashan31 as theories of higher order. 

To the same order of approximation of the equations, that is, with only terms linear in / (v) kept, better approximations to the viscosity may be found by considering the equations of higher order than Eqs. (1-86) and (1-87). These new equations will, to this order of approximation, have zero on the left sides (since the higher order coefficients are taken equal to zero) on the right sides appears the factor (p/fii) multiplied by a series of terms like those in Eq. (1-110). Using these equations, and the first order terms of Eq. (1-86) for arbitrary v,... 

Assuming that at the initial instant the angular velocity dp/dt — 0, we conclude that the mass m, placed at any point around the point of equilibrium, remains at rest. Of course, it is only an approximation, because we preserved in the power series, (Equation (3.146)), only the linear term and discarded terms of higher orders. Formally, this case is characterized by infinitely large period of free vibrations... 

These are the first two terms in a cumulant expansion [50]. We note here that the convergence of cumulant expansions is a subtle issue. Generally, if the statistics are nearly Gaussian, the cumulant expansion yields a good approximation. If the statistical distribution is not Gaussian, however, the cumulant expansion diverges with the inclusion of higher-order terms. See [29] and references therein for more discussion of this point. 

The failure of first-order moment closures for the treatment of mixing-sensitive reactions has led to the exploration of higher-order moment closures (Dutta and Tarbell 1989 Heeb and Brodkey 1990 Shenoy and Toor 1990). The simplest closures in this category attempt to relate the covariances of reactive scalars to the variance of the mixture fraction (I 2). The latter can be found by solving the inert-scalar-variance transport equation ((3.105), p. 85) along with the transport equation for (f). For example, for the one-step reaction in (5.54) the unknown scalar covariance can be approximated by... 

The full characterization of the stochastic properties of a surface requires consideration of higher order correlations of the height function. However, it can be difficult to construct surfaces in this manner without experimental input. As an approximation, it may be reasonable to neglect the higher order terms. 

As done previously, in The Newton-Raphson Algorithm (p.48), we neglect all but the first two terms in the expansion. This leaves us with an approximation that is not very accurate but, since it is a linear equation, is easy to deal with. Algorithms that include additional higher terms in the Taylor expansion, often result in fewer iterations but require longer computation times due to the calculation of higher order derivatives. 

The resorting to polynomials of higher orders leads to success only in those instances where the shape can reasonably be represented by polynomial approximation. Other strategies include piecewise fitting of linear functions or the use of appropriate transformations with the aim of retaining... 

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