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1: 32.4 Isomonodromy Problems
§32.4 Isomonodromy Problems
P I P VI  can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. Suppose …Isomonodromy problems for Painlevé equations are not unique. …
2: Bibliography I
  • Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J 0 ( z ) - i J 1 ( z ) and of Bessel functions J m ( z ) of any real order m . Linear Algebra Appl. 194, pp. 35–70.
  • A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.
  • 3: 29.19 Physical Applications
    Simply-periodic Lamé functions ( ν noninteger) can be used to solve boundary-value problems for Laplace’s equation in elliptical cones. …
    §29.19(ii) Lamé Polynomials
    Shail (1978) treats applications to solutions of elliptic crack and punch problems. …
    4: 11.12 Physical Applications
    §11.12 Physical Applications
    Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …
    5: 14.31 Other Applications
    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)). … The conical functions P - 1 2 + i τ m ( x ) 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)). …
    §14.31(iii) Miscellaneous
    6: 28.33 Physical Applications
    §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: …
    7: Bibliography Q
  • H. Qin and Y. Lu (2008) A note on an open problem about the first Painlevé equation. Acta Math. Appl. Sin. Engl. Ser. 24 (2), pp. 203–210.
  • S.-L. Qiu and J.-M. Shen (1997) On two problems concerning means. J. Hangzhou Inst. Elec. Engrg. 17, pp. 1–7 (Chinese).
  • 8: 26.20 Physical Applications
    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). … Other applications to problems in engineering, crystallography, biology, and computer science can be found in Beckenbach (1981) and Graham et al. (1995).
    9: 6.17 Physical Applications
    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. …
    10: 16.25 Methods of Computation
    Instead a boundary-value problem needs to be formulated and solved. …