About the Project

uniform asymptotic solutions of differential equations

AdvancedHelp

(0.009 seconds)

11—20 of 31 matching pages

11: 2.9 Difference Equations
§2.9 Difference Equations
§2.9(ii) Coincident Characteristic Values
For error bounds see Zhang et al. (1996). … For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …
12: 11.9 Lommel Functions
For uniform asymptotic expansions, for large ν and fixed μ = 1 , 0 , 1 , 2 , , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). …
13: Bibliography O
  • 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 (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 (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.
  • 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.
  • F. W. J. Olver (1999) On the uniqueness of asymptotic solutions of linear differential equations. Methods Appl. Anal. 6 (2), pp. 165–174.
  • 14: 10.20 Uniform Asymptotic Expansions for Large Order
    §10.20 Uniform Asymptotic Expansions for Large Order
    Define ζ = ζ ( z ) to be the solution of the differential equation
    §10.20(iii) Double Asymptotic Properties
    For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of z see §10.41(v).
    15: 9.16 Physical Applications
    The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. …The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). … Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …
    16: Bibliography S
  • H. Segur and M. J. Ablowitz (1981) Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent. Phys. D 3 (1-2), pp. 165–184.
  • S. Yu. Slavyanov (1996) Asymptotic Solutions of the One-dimensional Schrödinger Equation. American Mathematical Society, Providence, RI.
  • 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.
  • F. Stenger (1966a) Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 167–186.
  • F. Stenger (1966b) Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.
  • 17: 12.14 The Function W ( a , x )
    §12.14(i) Introduction
    §12.14(ix) Uniform Asymptotic Expansions for Large Parameter
    The differential equation
    Airy-type Uniform Expansions
    18: Bibliography J
  • X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.
  • D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.
  • S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions p s ¯ n r ( η , h ) and q s ¯ n r ( η , h ) for large h . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.
  • N. Joshi and A. V. Kitaev (2001) On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.
  • N. Joshi and A. V. Kitaev (2005) The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.
  • 19: Bibliography L
  • Y. A. Li and P. J. Olver (2000) Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.
  • J. L. López (2001) Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.
  • N. A. Lukaševič and A. I. Yablonskiĭ (1967) On a set of solutions of the sixth Painlevé equation. Differ. Uravn. 3 (3), pp. 520–523 (Russian).
  • N. A. Lukaševič (1965) Elementary solutions of certain Painlevé equations. Differ. Uravn. 1 (3), pp. 731–735 (Russian).
  • N. A. Lukaševič (1968) Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).
  • 20: 36.15 Methods of Computation
    §36.15(ii) Asymptotics
    Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from 𝐱 = 𝟎 , the uniform asymptotic approximations of §36.12 can be used. …
    §36.15(v) Differential Equations
    For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).