…
►
Differential Equations: Spectral Methods
►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a
method originated by
Clenshaw (1957).
This process has been generalized to
spectral methods for solving partial differential equations.
…
►
Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
…
►These
methods have become known as
pseudo-spectral, and are overviewed in
Cerjan (1993), and
Shizgal (2015).
…
…
►
§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
►Shizgal (2015) gives a broad overview of techniques and applications of
spectral and
pseudo-
spectral methods to problems arising in theoretical chemistry, chemical kinetics, transport theory, and astrophysics.
…
►Shizgal (2015, Chapter 2), contains a broad-ranged discussion of
methods and applications for these, and other, non-classical weight functions.
…
►
§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods
…
►The technique to accomplish this follows the DVR idea, in which
methods are based on finding tridiagonal representations of the co-ordinate,
.
…
§34.9 Graphical Method
►The graphical
method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression.
…For an account of this
method see
Brink and Satchler (1993, Chapter VII).
For specific examples of the graphical
method of representing sums involving the
, and
symbols, see
Varshalovich et al. (1988, Chapters 11, 12) and
Lehman and O’Connell (1973, §3.3).
§17.18 Methods of Computation
…
►Method (2) is very powerful when applicable (
Andrews (1976, Chapter 5)); however, it is applicable only rarely.
Lehner (1941) uses
Method (2) in connection with the Rogers–Ramanujan identities.
►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §
3.9.
Shanks (1955) applies such
methods in several
-series problems; see
Andrews et al. (1986).
§12.18 Methods of Computation
►Because PCFs are special cases of confluent hypergeometric functions, the
methods of computation described in §
13.29 are applicable to PCFs.
…
§34.13 Methods of Computation
►Methods of computation for
and
symbols include recursion relations, see
Schulten and Gordon (1975a),
Luscombe and Luban (1998), and
Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see
Varshalovich et al. (1988, §§8.2.6, 9.2.1) and
Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see
Srinivasa Rao and Venkatesh (1978) and
Srinivasa Rao (1981).
►For
symbols,
methods include evaluation of the single-sum series (
34.6.2), see
Fang and Shriner (1992); evaluation of triple-sum series, see
Varshalovich et al. (1988, §10.2.1) and
Srinivasa Rao et al. (1989).
A review of
methods of computation is given in
Srinivasa Rao and Rajeswari (1993, Chapter VII, pp. 235–265).
…
§16.25 Methods of Computation
►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
…
§32.17 Methods of Computation
►The Painlevé equations can be integrated by Runge–Kutta
methods for ordinary differential equations; see §
3.7(v),
Hairer et al. (2000), and
Butcher (2003).
…
§29.20 Methods of Computation
…
►A second approach is to solve the continued-fraction equations typified by (
29.3.10) by Newton’s rule or other iterative
methods; see §
3.8.
…
►A third
method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the
methods of §§
3.2(vi) and
3.8(iv).
…The numerical computations described in
Jansen (1977) are based in part upon this
method.
►A fourth
method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree.
…
