# boundary-value methods or problems

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## 1—10 of 100 matching pages

##### 1: 3.6 Linear Difference Equations

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►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.
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►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. …##### 2: 28.34 Methods of Computation

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(d)
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(d)
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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).

##### 3: 11.13 Methods of Computation

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►For ${\mathbf{M}}_{\nu}\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)).
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►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.
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##### 4: 12.17 Physical Applications

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►By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta $, defined by
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►Buchholz (1969) collects many results on boundary-value problems involving PCFs.
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►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).
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##### 5: 3.7 Ordinary Differential Equations

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###### §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

##### 7: 16.25 Methods of Computation

##### 8: 29.19 Physical Applications

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

… ►Shail (1978) treats applications to solutions of elliptic crack and punch problems. …##### 9: 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)). …##### 10: 14.31 Other Applications

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►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)).
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►The conical functions ${\mathsf{P}}_{-\frac{1}{2}+\mathrm{i}\tau}^{m}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)).
These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)).
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