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1: 16.25 Methods of Computation

§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. …Instead a boundary-value problem needs to be formulated and solved. …

2: Brian D. Sleeman

… ► thesis was Some Boundary Value Problems Associated with the Heun Equation. … ►Sleeman was elected a Fellow of the Royal Society of Edinburgh in 1976 and is the founding editor of the journal Computational and Mathematical Methods in Medicine. …

3: 12.17 Physical Applications

… ►By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by … ►Buchholz (1969) collects many results on boundary-value problems involving PCFs. Miller (1974) treats separation of variables by group theoretic methods. … ►Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978). …

4: 28.34 Methods of Computation

§28.34 Methods of Computation

… ►Methods available for computing the values of $w_{\text{I}}(\pi ;a,\pm q)$ needed in (28.2.16) include: … ►

Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

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Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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5: 11.13 Methods of Computation

§11.13 Methods of Computation

… ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. … ►For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

6: 3.6 Linear Difference Equations

… ► … ►However, $w_n$ can be computed successfully in these circumstances by boundary-value methods, as follows. … ►For a difference equation of order $k$ ($\ge 3$), …or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …

7: 3.7 Ordinary Differential Equations

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§3.7(ii) Taylor-Series Method: Initial-Value Problems

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§3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). … ►

§3.7(v) Runge–Kutta Method

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8: 9.17 Methods of Computation

§9.17 Methods of Computation

… ►The former reference includes a parallelized version of the method. … ►In these cases boundary-value methods need to be used instead; see §3.7(iii). … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). For the second method see also Gautschi (2002a). …

9: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

10: 29.19 Physical Applications

… ►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …