Polynomial approximation

One method is to approximate real and imaginary impedances as functions of the frequency by polynomials in ca [560, 561, 568, 569], 

Another problem is related to the discontinuity of the integrated function at X = CO. Several authors have proposed a distribution of l/(x — co ) in series around the point x = co and an integration of the sum of the elements [560,561,563,570], Although in principle Kramers-Kronig relations demand that the impedance have a finite value at = 0 and u — oo, it has been shown that the CPE 

System with negative resistance cannot be represented by a passive circuit with positive R, L, and C elements. The stability criterion demands that there be no negative impedances in the system (mle 3 of Kramers-Kronig conditions). For example, the system shown in Fig. 13.1 contains negative resistance, and its impedance is not transformable. However, Kramer-Kronig transforms can also be applied to admittances. Admittance calculated for this case is also shown in Fig. 13.1, and it is perfectly Kramers-Kronig transformable. Moreover, infinite impedance corresponds to zero admittance which is transformable. This topic will be further discussed in Sect. 13.3.4. 

In this chapter, we will present several alternatives, including polynomial approximations, singular perturbation methods, finite difference solutions and orthogonal collocation techniques. To successfully apply the polynomial approximation, it is useful to know something about the behavior of the exact solution. Next, we illustrate how perturbation methods, similar in scope to Chapter 6, can be applied to partial differential equations. Finally, finite difference and orthogonal collocation techniques are discussed since these are becoming standardized for many classic chemical engineering problems. 

Polynomial approximation involves two key steps. One is the selection of the form of the solution, normally presented as a polynomial expression in the spatial variable with time dependent coefficients. The second step is to convert 

Find an approximate solution, valid for long times, to the linear parabolic partial differential equation, describing mass or heat transport ffom/to a sphere with constant physical properties, such as thermal conductivity and diffusion coefficient 

This linear partial differential equation was solved analytically in Chapters 10 and 11 using Laplace transform, separation of variables, and finite integral transform techniques, respectively. A solution was given in the form of infinite series 

The solution for the average concentration, which is needed later for comparison with the approximate solutions, is 

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Vihadsen, J. V, and M. L. Michelsen. Solution of Differ ential Equation Models by Polynomial Approximation. Prentice Hall, Englewood Cliffs, NJ(1978). 

Polynomial Approximation, Prentice Hall, Englewood Cliffs, New Jersey, 1978],... 

When the underlying distribution is not known, tools such as histograms, probability curves, piecewise polynomial approximations, and general techniques are available to fit distributions to data. It may be necessary to assume an appropriate distribution in order to obtain the relevant parameters. Any assumptions made should be supported by manufacturer s data or data from the literature on similar items working in similar environments. Experience indicates that some probability distributions are more appropriate in certain situations than others. What follows is a brief overview on their applications in different environments. A more rigorous discussion of the statistics involved is provided in the CPQRA Guidelines. ... 

The simplest form of approximation to a continuous function is some polynomial. Continuous functions may be approximated in order to provide a simpler form than the original function. Truncated power series representations (such as the Taylor series) are one class of polynomial approximations. 

This section describes a number of finite difference approximations useful for solving second-order partial differential equations that is, equations containing terms such as d f jd d. The basic idea is to approximate f 2 z. polynomial in x and then to differentiate the polynomial to obtain estimates for derivatives such as df jdx and d f jdx -. The polynomial approximation is a local one that applies to some region of space centered about point x. When the point changes, the polynomial approximation will change as well. We begin by fitting a quadratic to the three points shown below. 

Assumptions Empirical polynomial approximation to r-tables. A good overall fit was attempted relative errors of less than 1% are irrelevant as far as practical consequences are concerned. The number of coefficients is a direct consequence of this approach. Polynomials were chosen in lieu of other functions in order to maximize programnung flexibility and speed of execution. 

Assumption, x -distribution the curvature of the x -functions versus df is not ideal for polynomial approximations various transformations on both axes, in different combinations, were tried, the best one by far being a logio(x ) vs. logio(df) plot. The 34 x -values used for the optimization of the coefficients (two decimal places) covered degrees of freedom 1-20, 22, 24, 26, 28, 30, 35, 40, 50, 60, 80, 100, 120, 150, and 200. 

Response surfaces are mostly described mathematically by polynomial approximations of 1st and 2nd degree. Grid search corresponds to a com-... 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

Another class of methods of unidimensional minimization locates a point x near x, the value of the independent variable corresponding to the minimum of /(x), by extrapolation and interpolation using polynomial approximations as models of/(x). Both quadratic and cubic approximation have been proposed using function values only and using both function and derivative values. In functions where/ (x) is continuous, these methods are much more efficient than other methods and are now widely used to do line searches within multivariable optimizers. 

Wu and Cinar (1996) use a polynomial approximation (ARMENSI) of the error density function / based on a generalized exponential family, such as... 

Over each interval to the true time function is now approximated by some polynomial approximating functions A number of types can be used, but the simplest is a first-order approximation. This corresponds to using a straight line between the data points. 

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... 

Hwang, J.-T., Sensitivity analysis in chemical kinetics by the method of polynomial approximations, Int. J. Chem. Kinetics 15, 959 (1983). 

X + i(f) = piecewise polynomial approximation of Z(t) over element i, = piecewise polynomial approximation of U(t) over element i, Zii = coefficient of polynomial approximation, + i(f),... 

A detailed analysis of polynomial approximation using collocation at Legendre roots by de Boor (1978) shows that the global error e(t) = Z(t) — z +i(t) satisfies the relation... 

The possibility of simulating the actual BWG ordering energy, rather than Cp, using a polynomial approximation was also examined by Inden (1976) using the disordered solid solution as a reference state. The following expression was suggested for a continuous second-order transformation such as A2/B2 ... 

In the point force approximation technique (see Section Ic), Burgers (BIO) suggested a polynomial approximation for the distributed line force along the axis of a body of large aspect ratio ... 

Furthermore, optimal design theory assumes that the model is true within the region defined by the candidate design points, since the designs are optimal in terms of minimizing variance as opposed to bias due to lack-of-fit of the model. In reality, the response surface model is only assumed to be a locally adequate polynomial approximation to the truth it is not assumed to be the truth. Consequently, the experimental design chosen should reflect doubt in the validity of the model by allowing for model lack-of-fit to be tested. 

Fig. 6. Chebyshev approjdmation used as a filter on part of a chromatogram (0.1 pg r anthracene, 0.5 pg r benzanthracene, reversed-phase HPLQ 40 terms Chebyshev polynomial approximation.

Valstar, J. M., Van Den Berg, P. J., and Oyserman, J., Chem. Eng. Sci. 30,723-728 (1975). Vatcha, S. R., Analysis and Design of Methanation Processes in the Production of Substitute Natural Gas from Coal, Ph.D. thesis, California Institute of Technology, Pasadena (1976). Villadsen, J. V., and Michelsen, M. L., Solution of Differential Equation Models by Polynomial Approximation." Prentice-Hall, New York, 1978. 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

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