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1: 16.25 Methods of Computation

… ►Instead a boundary-value problem needs to be formulated and solved. …

2: 12.15 Generalized Parabolic Cylinder Functions

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

3: Brian D. Sleeman

… ► thesis was Some Boundary Value Problems Associated with the Heun Equation. …

4: 12.17 Physical Applications

… ►By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by … ►Buchholz (1969) collects many results on boundary-value problems involving PCFs. … ►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). …

5: 29.19 Physical Applications

… ►Simply-periodic Lamé functions ($\nu$ noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …

6: 32.5 Integral Equations

… ►satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$ and the boundary condition …

7: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. …

8: 11.13 Methods of Computation

… ►For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

9: 28.34 Methods of Computation

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Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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10: 28.33 Physical Applications

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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).

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§28.33(ii) Boundary-Value Problems

… ►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: …