# variable boundaries

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## 1—10 of 24 matching pages

##### 1: 28.10 Integral Equations
###### §28.10(iii) Further Equations
For relations with variable boundaries see Volkmer (1983).
##### 2: 12.17 Physical Applications
By using instead coordinates of the parabolic cylinder $\xi,\eta,\zeta$, defined by …Setting $w=U(\xi)V(\eta)W(\zeta)$ and separating variables, we obtain … Buchholz (1969) collects many results on boundary-value problems involving PCFs. Miller (1974) treats separation of variables by group theoretic methods. … For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978). …
##### 3: 28.32 Mathematical Applications
If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. These are given by … This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta=\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). …
28.32.6 $w(z)=\int_{\mathcal{L}}K(z,\zeta)u(\zeta)\mathrm{d}\zeta$
##### 4: 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. …
##### 5: 12.15 Generalized Parabolic Cylinder Functions
12.15.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\nu+\lambda^{-1}-\lambda^{-2% }z^{\lambda}\right)w=0$
This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …
##### 6: 20.13 Physical Applications
20.13.1 $\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},$
For $\tau=it$, with $\alpha,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation …
20.13.3 $g(z,t)=\sqrt{\frac{\pi}{4\alpha t}}\exp\left(-\frac{z^{2}}{4\alpha t}\right)$
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). …
##### 7: 28.33 Physical Applications
• Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

• ###### §28.33(ii) Boundary-Value Problems
with $W(x,y,t)=e^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^{2}=\omega^{2}\rho/{\tau}$. …The boundary conditions for $\xi=\xi_{0}$ (outer clamp) and $\xi=\xi_{1}$ (inner clamp) yield the following equation for $q$: … For a visualization see Gutiérrez-Vega et al. (2003), and for references to other boundary-value problems see: …
##### 9: 2.11 Remainder Terms; Stokes Phenomenon
In order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable $x$ that is intended to be used. … In $\mathbb{C}$ both the modulus and phase of the asymptotic variable $z$ need to be taken into account. …Then numerical accuracy will disintegrate as the boundary rays $\operatorname{ph}z=\alpha$, $\operatorname{ph}z=\beta$ are approached. In consequence, practical application needs to be confined to a sector $\alpha^{\prime}\leq\operatorname{ph}z\leq\beta^{\prime}$ well within the sector of validity, and independent evaluations carried out on the boundaries for the smallest value of $|z|$ intended to be used. … Since the ray $\operatorname{ph}z=\frac{3}{2}\pi$ is well away from the new boundaries, the compound expansion (2.11.7) yields much more accurate results when $\operatorname{ph}z\to\frac{3}{2}\pi$. …
##### 10: 10.20 Uniform Asymptotic Expansions for Large Order
###### §10.20(i) Real Variables
10.20.1 $\left(\frac{\mathrm{d}\zeta}{\mathrm{d}z}\right)^{2}=\frac{1-z^{2}}{\zeta z^{2}}$
10.20.2 $\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\mathrm{d}t% =\ln\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\sqrt{1-z^{2}},$ $0,
###### §10.20(ii) Complex Variables
The equations of the curved boundaries $D_{1}E_{1}$ and $D_{2}E_{2}$ in the $\zeta$-plane are given parametrically by …