Moments orthogonal polynomials

If the functions 1, t,. .. t are chosen, then the already mentioned moments of f(t) are found. However, convergence is not guaranteed in this case. Moreover, the calculation of the coefficients a requires the solution of N equations with N imknowns and the values a are dependent ofN. The introduction of a specialized set of orthogonal polynomials can be advantageous and circumvents some problems. Suppose that the following integral exists ... 

The expression (94) can easily be established in an inductive manner, as pointed out in Ref. [56]. Hereafter, we use the notation Q ,s(u) for the sth derivative of Qn(u) with respect to u instead of the more standard notation Q (u). The reason for this is that the notation Q (u) is customarily employed in the theory and applications of the so-called delayed orthogonal polynomials constructed from power moments, autocorrection functions, or signal points // +s = C +s = c +s, where the first s < N points /a, = C, = cr (0 < r < s - 1) are either skipped or simply missing from the full set of N elements. Delayed time signals c +s are thoroughly studied in Refs. [2, 25]. If the expression (84) is differentiated m times and Eq. (94) is used, the following recursion is obtained inductively for the derivatives lQn,m(U) [47] ... 

If we have some overall information on n( ), it is convenient to consider a weight function n E), different from zero in the same interval as n E), whose orthogonal polynomials (P (E) and parameters a , are known. It is then convenient to introduce the modified moments... 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

With the known 2n moments, the singularities of Ig e), in accordance with Eqn. (2.34), may be examined with a respective function q Z). Standard relations from the classical problem of moments and formulas from the Kristoffel-Darbu theory of orthogonal polynomials [14] allow us to establish the functions... 

Sack Donovan (1971) proposed an alternative approach for the calculation of the coefficient of the recursive formula reported in Eq. (3.5) (and appearing also in the Jacobi matrix) that resulted in higher stability. This approach is based on the idea of using a different set of basis functions naif) to represent the orthogonal polynomials, rather than the usual powers of f. The improved stability results from the ability of the new polynomial basis to better sample the integration interval. The coefficients are calculated from the modified moments defined as follows ... 

The integral terms in parentheses are known for the family of orthogonal polynomials. With finite N and known moments, this system of linear equations has the form M = AC and can be solved to find fhe expansion coefficients Ca(t, x). Thus, the presumed NDF n (t, X, Vp) is a unique function of a finite set of moments, and the latter are found by solving the moment-transport equations using n to close the unclosed terms. The fact that... 

For size-dependent growth [i.e., kc = kc(L)], the mathematical treatment of Eq. (10.11) becomes more complicated. One solution method, which included the use of moments and an orthogonal polynomial to simulate the population density function, was discussed by Wey (1985). Tavare et al. (1980) developed a... 

Introducing the expansion [25] into the Fokker-Plank equation [24] and using the orthogonal properties of the Hermite polynomials, the following moment equations are obtained ... 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

Va /VH-1. The shape of the distribution does not change with time, only N and Vo change. The distribution (11.27) allows us to look for the solution of equations (11.15) for moments in the form of expansion of n(V, t) over a system of orthogonal Laguerre associated polynomials [5] 1 ... 

To do this, the PCLD is expanded in an orthogonal series of special functions, in this case Laguerre polynomials. The coefficients of the series are given in terms of the moments of the original CLD, as illustrated below. 

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