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1: 3.6 Linear Difference Equations
In practice, however, problems of severe instability often arise and in §§3.6(ii)3.6(vii) we show how these difficulties may be overcome. … However, w n can be computed successfully in these circumstances by boundary-value methods, as follows. … For a difference equation of order k ( 3 ), …or for systems of k first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …
2: 28.34 Methods of Computation
  • (d)

    Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

  • (c)

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

  • (d)

    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 methods3.6) 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).

  • 3: 11.13 Methods of Computation
    For M ν ( x ) both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … There are similar problems to those described in §11.13(iv) concerning stability. In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …
    4: 12.17 Physical Applications
    By using instead coordinates of the parabolic cylinder ξ , η , ζ , defined by … Buchholz (1969) collects many results on boundary-value problems involving PCFs. … 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). …
    5: 3.7 Ordinary Differential Equations
    §3.7(ii) Taylor-Series Method: Initial-Value Problems
    §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(iv) Sturm–Liouville Eigenvalue Problems
    6: 9.17 Methods of Computation
    In these cases boundary-value methods need to be used instead; see §3.7(iii). …
    7: 16.25 Methods of Computation
    Instead a boundary-value problem needs to be formulated and solved. …
    8: Brian D. Sleeman
    9: 29.19 Physical Applications
    Simply-periodic Lamé functions ( ν noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
    §29.19(ii) Lamé Polynomials
    Shail (1978) treats applications to solutions of elliptic crack and punch problems. …
    10: 11.12 Physical Applications
    §11.12 Physical Applications
    Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …