# boundary conditions

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## 1—10 of 22 matching pages

##### 1: 32.5 Integral Equations

##### 2: 20.13 Physical Applications

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►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).
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##### 3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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###### 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 $\mathcal{D}({\mathcal{L}}^{\ast})$ 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

##### 5: 10.73 Physical Applications

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►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.
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►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 ${g}_{\mathrm{\ell}}={\U0001d5c3}_{\mathrm{\ell}}$, ${\U0001d5d2}_{\mathrm{\ell}}$, ${\U0001d5c1}_{\mathrm{\ell}}^{(1)}$, or ${\U0001d5c1}_{\mathrm{\ell}}^{(2)}$, depending on the boundary conditions.
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##### 6: 1.13 Differential Equations

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►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: 28.32 Mathematical Applications

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►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient.
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##### 8: 28.33 Physical Applications

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►The boundary conditions for $\xi ={\xi}_{0}$ (outer clamp) and $\xi ={\xi}_{1}$ (inner clamp) yield the following equation for $q$:
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