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Liouville transformation for differential equations

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1: 1.13 Differential Equations
Liouville Transformation
2: 2.8 Differential Equations with a Parameter
In Case III f ( z ) has a simple pole at z 0 and ( z z 0 ) 2 g ( z ) is analytic at z 0 . …
3: Bibliography E
  • U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.
  • A. Erdélyi (1942a) Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.
  • W. N. Everitt (1982) On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.
  • W. N. Everitt (2005a) A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.
  • W. N. Everitt (2005b) Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950. In Sturm-Liouville theory, pp. 45–74.
  • 4: 2.7 Differential Equations
    See §2.11(v) for other examples. … The transformed differential equation either has a regular singularity at t = , or its characteristic equation has unequal roots. …
    §2.7(iii) Liouville–Green (WKBJ) Approximation
    For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: … The first of these references includes extensions to complex variables and reversions for zeros. …
    5: 18.38 Mathematical Applications
    Differential Equations: Spectral Methods
    This process has been generalized to spectral methods for solving partial differential equations. …
    Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions
    The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. …
    6: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.
  • A. B. Olde Daalhuis and F. W. J. Olver (1995b) On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.
  • A. B. Olde Daalhuis and F. W. J. Olver (1998) On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • P. J. Olver (1993b) Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.
  • 7: Errata
  • Chapter 1 Additions

    The following additions were made in Chapter 1:

  • 8: Bibliography S
  • D. R. Smith (1986) Liouville-Green approximations via the Riccati transformation. J. Math. Anal. Appl. 116 (1), pp. 147–165.
  • R. Spigler, M. Vianello, and F. Locatelli (1999) Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.
  • R. Spigler and M. Vianello (1992) Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.
  • R. Spigler and M. Vianello (1997) A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 567–577.
  • R. Spigler (1984) The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.
  • 9: Bibliography B
  • A. W. Babister (1967) Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.
  • A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.
  • G. Birkhoff and G. Rota (1989) Ordinary differential equations. Fourth edition, John Wiley & Sons, Inc., New York.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • J. C. Butcher (1987) The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.
  • 10: 18.39 Applications in the Physical Sciences
    The finite system of functions ψ n is orthonormal in L 2 ( , d x ) , see (18.34.7_3). …The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … The Schrödinger equation with potential … An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with L 2 eigenfunctions vanishing at the end points, in this case ± see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. … Substitution of (18.39.24) into (18.39.23) then gives the ordinary differential equation for the radial wave function R n , l ( r ) , …