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1: 16.8 Differential Equations
§16.8(i) Classification of Singularities
An ordinary point of the differential equation …If z 0 is not an ordinary point but ( z z 0 ) n j f j ( z ) , j = 0 , 1 , , n 1 , are analytic at z = z 0 , then z 0 is a regular singularity. …
2: 2.7 Differential Equations
An ordinary point of the differential equation …All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. …
3: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
4: Bibliography D
  • 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.
  • 5: Bibliography W
  • Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.
  • Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.
  • W. Wasow (1965) Asymptotic Expansions for Ordinary Differential Equations. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney.
  • W. Wasow (1985) Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.
  • H. S. Wilf and D. Zeilberger (1992a) An algorithmic proof theory for hypergeometric (ordinary and “ q ”) multisum/integral identities. Invent. Math. 108, pp. 575–633.
  • 6: Bibliography S
  • J. Segura (2002) The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.
  • P. N. Shivakumar and J. Xue (1999) On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.
  • J. H. Silverman and J. Tate (1992) Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.
  • V. I. Smirnov (1996) Izbrannye Trudy. Analiticheskaya teoriya obyknovennykh differentsialnykh uravnenii. Izdatel’ stvo Sankt-Peterburgskogo Universiteta, St. Petersburg (Russian).
  • D. M. Smith (1991) Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.
  • 7: 3.7 Ordinary Differential Equations
    §3.7 Ordinary Differential Equations
    Consideration will be limited to ordinary linear second-order differential equationsFor an introduction to numerical methods for ordinary differential equations, see Ascher and Petzold (1998), Hairer et al. (1993), and Iserles (1996). …
    §3.7(v) Runge–Kutta Method
    An extensive literature exists on the numerical solution of ordinary differential equations by Runge–Kutta, multistep, or other methods. …
    8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    §1.18(v) Point Spectra and Eigenfunction Expansions
    Assume T has no point spectrum, i. … Then, for z \ , f N z iff f is an ordinary solution (i. … A boundary value for the end point a is a linear form on 𝒟 ( ) of the form …Boundary values and boundary conditions for the end point b are defined in a similar way. …
    9: 22.19 Physical Applications
    θ being the angular displacement from the point of stable equilibrium, θ = 0 . … for the initial conditions θ ( 0 ) = 0 , the point of stable equilibrium for E = 0 , and d θ ( t ) / d t = 2 E . …
    §22.19(iii) Nonlinear ODEs and PDEs
    Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. … The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …
    10: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
  • 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.
  • A. B. Olde Daalhuis (2004b) On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.
  • F. W. J. Olver (1993a) Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.
  • F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.