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11: 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. …

12: 3.7 Ordinary Differential Equations

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

… ►The remaining two equations are supplied by boundary conditions of the form … ►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). …

13: 20.13 Physical Applications

… ►These two apparently different solutions differ only in their normalization and boundary conditions. …Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). …

14: 14.31 Other Applications

… ►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ 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)). …

15: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►The boundary of each region comprises the characteristic curves $a=a_n\left(q\right)$ and $a=b_n\left(q\right)$; compare Figure 28.2.1. …

16: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Ignoring the boundary value terms it follows that … ►Other choices of boundary conditions, identical for $f(x)$ and $g(x)$, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of $T$. … ►

Self-adjoint extensions of (1.18.28) and the Weyl alternative

… ►Boundary values and boundary conditions for the end point $b$ are defined in a similar way. If $n_1=1$ then there are no nonzero boundary values at $a$; if $n_1=2$ then the above boundary values at $a$ form a two-dimensional class. …

17: 9.17 Methods of Computation

… ►In these cases boundary-value methods need to be used instead; see §3.7(iii). …

18: 32.14 Combinatorics

… ►and $w(x)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$ and boundary conditions …

19: Publications

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Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

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20: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …