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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: 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. …
4: 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.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
  • P. J. Olver (1993b) Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.
  • 5: 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.
  • 6: 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.
  • 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. P. Boyd and A. Natarov (1998) A Sturm-Liouville eigenproblem of the fourth kind: A critical latitude with equatorial trapping. Stud. Appl. Math. 101 (4), pp. 433–455.
  • J. C. Butcher (1987) The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.
  • 7: Errata
  • Equation (2.3.6)
    2.3.6 𝒱 a , b ( f ( t ) ) = a b | f ( t ) | d t

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Juan Luis Varona on 2021-02-08

  • Equation (1.4.34)
    1.4.34 𝒱 a , b ( f ) = a b | f ( x ) | d x

    The integrand has been corrected so that the absolute value does not include the differential.

    Reported by Tran Quoc Viet on 2020-08-11

  • Equations (10.15.1), (10.38.1)

    These equations have been generalized to include the additional cases of J - ν ( z ) / ν , I - ν ( z ) / ν , respectively.

  • Subsection 1.16(vii)

    Several changes have been made to

    1. (i)

      make consistent use of the Fourier transform notations ( f ) , ( ϕ ) and ( u ) where f is a function of one real variable, ϕ is a test function of n variables associated with tempered distributions, and u is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

    2. (ii)

      introduce the partial differential operator D in (1.16.30);

    3. (iii)

      clarify the definition (1.16.32) of the partial differential operator P ( D ) ; and

    4. (iv)

      clarify the use of P ( D ) and P ( x ) in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

  • Subsections 1.15(vi), 1.15(vii), 2.6(iii)

    A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order α was more precisely identified as the Riemann-Liouville fractional integral operator of order α , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

  • 8: 2.6 Distributional Methods
    §2.6(ii) Stieltjes Transform
    The Stieltjes transform of f ( t ) is defined by … Corresponding results for the generalized Stieltjes transformThe Riemann–Liouville fractional integral of order μ is defined by … For rigorous derivations of these results and also order estimates for δ n ( x ) , see Wong (1979) and Wong (1989, Chapter 6).
    9: Bibliography D
  • K. Dekker and J. G. Verwer (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.
  • T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
  • T. M. Dunster (1996c) Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.
  • T. M. Dunster (2001a) Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.
  • T. M. Dunster (2014) Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.
  • 10: Bibliography L
  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
  • L. Lorch, M. E. Muldoon, and P. Szegő (1970) Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.
  • L. Lorch, M. E. Muldoon, and P. Szegő (1972) Higher monotonicity properties of certain Sturm-Liouville functions. IV. Canad. J. Math. 24, pp. 349–368.
  • L. Lorch and P. Szegő (1963) Higher monotonicity properties of certain Sturm-Liouville functions.. Acta Math. 109, pp. 55–73.
  • N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).