boundary-value methods or problems
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11: 14.31 Other Applications
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►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)).
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►The conical functions appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)).
These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)).
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§14.31(iii) Miscellaneous
…12: 28.33 Physical Applications
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§28.33(ii) Boundary-Value Problems
►Physical problems involving Mathieu functions include vibrational problems in elliptical coordinates; see (28.32.1). …The general solution of the problem is a superposition of the separated solutions. … ►§28.33(iii) Stability and Initial-Value Problems
… ►References for other initial-value problems include: …13: Bibliography Q
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A note on an open problem about the first Painlevé equation.
Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.
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On two problems concerning means.
J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
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14: 26.20 Physical Applications
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►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993).
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►Other applications to problems in engineering, crystallography, biology, and computer science can be found in Beckenbach (1981) and Graham et al. (1995).
15: 6.17 Physical Applications
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►Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics.
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16: Bibliography X
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Prolate spheroidal wavefunctions, quadrature and interpolation.
Inverse Problems 17 (4), pp. 805–838.
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17: Alexander A. Its
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► Novokshënov), published by Springer in 1986, Algebro-geometric Approach to Nonlinear Integrable Problems (with E.
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18: Bibliography
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Unsteady lifting-line theory as a singular-perturbation problem.
J. Fluid Mech 153, pp. 59–81.
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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem.
Comm. Pure Appl. Math. 30 (1), pp. 95–148.
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The classical moment problem and some related questions in analysis.
Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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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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19: 9.16 Physical Applications
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►These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability.
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►Airy functions play a prominent role in problems defined by nonlinear wave equations.
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►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
…In the case of the rainbow, the scattering amplitude is expressed in terms of , the analysis being similar to that given originally by Airy (1838) for the corresponding problem in optics.
►An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).
20: 31.17 Physical Applications
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