boundary conditions

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1: 32.5 Integral Equations

… ►satisfies ${\text{P}}_{\text{II}}$ 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 $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …Then, if the linear form $ℬ$ is nonzero, the condition $ℬ(f)=0$ 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 ${\text{P}}_{\text{II}}$ with $\alpha =0$ and boundary conditions …

5: 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. …

6: 1.13 Differential Equations

… ►Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form …or periodic boundary conditions … ►For 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: 28.33 Physical Applications

… ►The boundary conditions for $\xi ={\xi }_0$ (outer clamp) and $\xi ={\xi }_1$ (inner clamp) yield the following equation for $q$: …

8: 28.32 Mathematical Applications

… ►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. …

9: 32.11 Asymptotic Approximations for Real Variables

… ►with boundary condition … ►and with boundary condition …

10: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►Equation (30.13.7) for $\xi \le {\xi }_0$ 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. …