boundary conditions

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1: 32.5 Integral Equations
satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and the boundary condition
2: 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). …
3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Self-adjoint extensions of (1.18.28) and the Weyl alternative
A boundary value for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{*})$ of the form …Then, if the linear form $\mathcal{B}$ is nonzero, the condition $\mathcal{B}(f)=0$ is called a boundary condition at $a$. Boundary values and boundary conditions for the end point $b$ are defined in a similar way. … The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. …
4: 32.14 Combinatorics
and $w(x)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and boundary conditions
5: 10.73 Physical Applications
and on separation of variables we obtain solutions of the form $e^{\pm in\phi}e^{\pm\kappa z}J_{n}\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … With the spherical harmonic $Y_{\ell,m}\left(\theta,\phi\right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\ell}(k\rho)Y_{\ell,m}\left(\theta,\phi\right)$ with $g_{\ell}=\mathsf{j}_{\ell}$, $\mathsf{y}_{\ell}$, ${\mathsf{h}^{(1)}_{\ell}}$, or ${\mathsf{h}^{(2)}_{\ell}}$, depending on the boundary conditions. …
6: 1.13 Differential Equations
Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form …or periodic boundary conditionsFor a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in(0,c)$ as for $x\in(a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …
7: 28.32 Mathematical Applications
If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. …
8: 28.33 Physical Applications
The boundary conditions for $\xi=\xi_{0}$ (outer clamp) and $\xi=\xi_{1}$ (inner clamp) yield the following equation for $q$: …
9: 32.11 Asymptotic Approximations for Real Variables
with boundary conditionand with boundary condition
10: 30.14 Wave Equation in Oblate Spheroidal Coordinates
Equation (30.13.7) for $\xi\leq\xi_{0}$ together with the boundary condition $w=0$ on the ellipsoid given by $\xi=\xi_{0}$, poses an eigenvalue problem with $\kappa^{2}$ as spectral parameter. …