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21: 28.10 Integral Equations

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§28.10(iii) Further Equations

… ►For relations with variable boundaries see Volkmer (1983).

22: 1.6 Vectors and Vector-Valued Functions

… ►Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. … ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …

23: 1.9 Calculus of a Complex Variable

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Point Sets in $ℂ$

… ►Any point whose neighborhoods always contain members and nonmembers of $D$ is a boundary point of $D$. When its boundary points are added the domain is said to be closed, but unless specified otherwise a domain is assumed to be open. … ►A region is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called interior points. …

24: 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. …

25: Bibliography H

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A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.

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External Links:

ISSN 0003-9527, Document, MathReview (Richard Brown), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

BibTeX

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On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.

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External Links:

ISSN 0033-5614, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(i), §32.17.

Encodings:

BibTeX

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26: Mathematical Introduction

… ►Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9. This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries. …

27: 20.2 Definitions and Periodic Properties

… ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. …

28: Bibliography J

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

BibTeX

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29: 28.32 Mathematical Applications

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

30: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►Equation (30.13.7) for $\xi \le {\xi }_0$, and subject to the boundary condition $w=0$ on the ellipsoid given by $\xi ={\xi }_0$, poses an eigenvalue problem with ${\kappa }^2$ as spectral parameter. …For the Dirichlet boundary-value problem of the region ${\xi }_1\le \xi \le {\xi }_2$ between two ellipsoids, the eigenvalues are determined from …