boundary-value problems

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11: Bibliography S

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Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.

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Notes:

Available from the library, University of Surrey, UK

External Links:

LibraryRecord

Referenced by:

§31.9(i), §31.9(ii), Ch.31, Profile Brian D. Sleeman.

Encodings:

BibTeX

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Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.

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External Links:

MathReview (R. Shail), ZentralBlatt

Referenced by:

§10.22(iv).

Encodings:

BibTeX

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12: Bibliography

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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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Notes:

Corrected reprint of the 1988 original

External Links:

ISBN 0-89871-354-4, MathReview, ZentralBlatt

Referenced by:

§3.7(iii).

Encodings:

BibTeX

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13: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►For the Dirichlet boundary-value problem of the region ${\xi }_1\le \xi \le {\xi }_2$ between two ellipsoids, the eigenvalues are determined from …

14: 3.6 Linear Difference Equations

… ► … ►For a difference equation of order $k$ ($\ge 3$), …

15: 11.13 Methods of Computation

… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. … ►For ${𝐌}_{\nu }\left(x\right)$ both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►Sequences of values of ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$, with $z$ fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). There are similar problems to those described in §11.13(iv) concerning stability. In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

16: 28.34 Methods of Computation

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§28.34(ii) Eigenvalues

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Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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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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Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

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17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Ignoring the boundary value terms it follows that … ► A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …Boundary values and boundary conditions for the end point $b$ are defined in a similar way. If $n_1=1$ then there are no nonzero boundary values at $a$; if $n_1=2$ then the above boundary values at $a$ form a two-dimensional class. … ►See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.