Polynomial expansions

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

V in the Maxwellian and in the arguments of the expansion polynomials, due to the fact that V C t = j2Tfm in many fusion plasma applications usually. Hence fa can equally well be expanded in polynomials in either v or in the random velocity v. ... 

First passage Karhunen Loeve expansions Polynomial chaos Random fatigue Smolyak s quadrature Sparse grid... 

Since the KL expansion is obtained as a linear sum of Gaussian random variables, the series can be used to represent only Gaussian random processes. For spectral representation of non-Gaussian random processes, one uses the more general form known as the polynomial chaos expansion. Polynomial chaos expansion is a spectral representation of the random process in terms of orthonormal basis functions and deterministic coefficients. Based on the Cameron and Martin theorem (Cameron and Martin 1947), it can be shown that a zero mean, second order random process can be represented as... 

Within the phenomenological study, the apparent view of the function (jj) is imknown, therefore it is represented as expansion to the degrees aceording to 7. With this expansion polynomial to the 4 degree is enough ... 

The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

Mandelshtam V A and Taylor H S 1995 A simple recursion polynomial expansion of the Green s function with absorbing boundary conditions. Application to the reactive scattering J. Chem. Phys. 102... 

We shall expand the polynomial of z. But recalling that only terms of the even power of z do not vanish, we can write the expansion in the following form ... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

Chebyshev Approximation The well known expansion of exp(— into Chebyshev polynomials T, [23] is one of the most frequently used integration technique in numerical quantum dynamics ... 

Inherent in the development of approximations by the described interpolation models is to assign polynomial variations for function expansions over finite elements. Therefore the shape functions in a given finite element correspond to a... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

The polynomial expansion used in this equation does not include all of the temis of a complete quadratic expansion (i.e. six terms corresponding to p = 2 in the Pascal triangle) and, therefore, the four-node rectangular element shown in Figure 2.8 is not a quadratic element. The right-hand side of Equation (2.15) can, however, be written as the product of two first-order polynomials in temis of X and y variables as... 

Equation (3-111) is the secular equation for this problem. [Pauling and Wilson discuss the meaning of the term secular in this context.] Expansion of the determinant gives a polynomial in X that is called the characteristic equation ... 

So, for some match point / to infinity, the atomic pseudo-orbital is identical to the valence HF atomic orbital. For radial distances less than / the pseudoorbital is defined by a polynomial expansion that goes to zero. The values of the polynomial are found by matching the value and first three derivatives of the HF orbital at / . 

These statements are a consequence of the recursion relations obtained by identifying the coefficients of the power series expansion on the right- and left-hand side of the equation. For example, in (4.6), the coefficient of x" is (n > 1) on the left-hand side, and on the right-hand side a polynomial in R, . [cf. (2.56)], which implies the uniqueness. The coefficients of the polynomial mentioned are non-negative the term occurs, coming from x/, thus Rj > n-i statements that the coefficients are... 

Bather than using the Chapman-Enskog procedure directly, we shall employ the technique of Burnett,12 which involves an expansion of the distribution function in a set of orthogonal polynomials in particle-velocity space. 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Polarization wave function of photon, 557 Polynomials expansion, 25 Sonine, 25... 

It is shown in Appendix 6 that the generalized Laguerre polynomials are eigenfunctions of the integral operator (3.26) with kernel (3.52). Let us search for the solution of (3.26) in the form of expansion over these eigenfunctions... 

G is then a generating function for these integrals, which occur as coefficients in its expansion in powers of u and and it can he evaluated with the use of the generating function for the associated Laguerre polynomials, given in equation (19). Thus we have... 

The random phase error of a wavefront which has passed through turbulence may be expressed as a weighted sum of orthogonal polynomials. The usual set of polynomials for this expansion is the Zernike polynomials, which... 

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