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21: Peter A. Clarkson
Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M. …
22: 13.28 Physical Applications
§13.28(i) Exact Solutions of the Wave Equation
The reduced wave equation 2 w = k 2 w in paraboloidal coordinates, x = 2 ξ η cos ϕ , y = 2 ξ η sin ϕ , z = ξ η , can be solved via separation of variables w = f 1 ( ξ ) f 2 ( η ) e i p ϕ , where …and V κ , μ ( j ) ( z ) , j = 1 , 2 , denotes any pair of solutions of Whittaker’s equation (13.14.1). …
23: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …
24: Gerhard Wolf
 Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. …
  • 25: 28 Mathieu Functions and Hill’s Equation
    Chapter 28 Mathieu Functions and Hill’s Equation
    26: Simon Ruijsenaars
    His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …
  • 27: 14.31 Other Applications
    §14.31(i) Toroidal Functions
    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)). …
    §14.31(ii) Conical Functions
    §14.31(iii) Miscellaneous
    Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …
    28: Bibliography O
  • K. Okamoto (1987a) Studies on the Painlevé equations. I. Sixth Painlevé equation P VI . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.
  • K. Okamoto (1987b) Studies on the Painlevé equations. II. Fifth Painlevé equation P V . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
  • K. Okamoto (1987c) Studies on the Painlevé equations. IV. Third Painlevé equation P III . Funkcial. Ekvac. 30 (2-3), pp. 305–332.
  • A. B. Olde Daalhuis (2005b) Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
  • A. M. Ostrowski (1973) Solution of Equations in Euclidean and Banach Spaces. Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London.
  • 29: 28.34 Methods of Computation
    §28.34(i) Characteristic Exponents
    §28.34(ii) Eigenvalues
  • (c)

    Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

  • §28.34(iii) Floquet Solutions
  • (b)

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

  • 30: 31.14 General Fuchsian Equation
    §31.14 General Fuchsian Equation
    §31.14(i) Definitions
    Heun’s equation (31.2.1) corresponds to N = 3 .
    Normal Form
    An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …