asymptotic%20solutions%20of%20differential%20equations
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11—16 of 16 matching pages
11: Bibliography
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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation.
Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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Rational solutions of Painlevé equations.
Stud. Appl. Math. 61 (1), pp. 31–53.
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Perturbation solutions of the ellipsoidal wave equation.
Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.
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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.
Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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12: Bibliography M
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The 192 solutions of the Heun equation.
Math. Comp. 76 (258), pp. 811–843.
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The Airy Integral, Giving Tables of Solutions of the Differential Equation
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British Association for the Advancement of Science,
Mathematical Tables Part-Vol. B, Cambridge University Press, Cambridge.
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On the choice of standard solutions for a homogeneous linear differential equation of the second order.
Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.
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Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions.
2nd edition, Springer-Verlag, Berlin.
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Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank.
Methods Appl. Anal. 4 (3), pp. 250–260.
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13: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10 Uniform Asymptotic Expansions for Large Parameter
… ►Lastly, the function in (12.10.3) and (12.10.4) has the asymptotic expansion: … ►The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). … ►14: Bibliography G
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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.
Ann. of Math. (2) 48 (4), pp. 785–826.
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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments.
ACM Trans. Math. Software 30 (2), pp. 145–158.
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Special classes of solutions of Painlevé equations.
Differ. Uravn. 18 (3), pp. 419–429 (Russian).
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The solutions of Painlevé’s fifth equation.
Differ. Uravn. 12 (4), pp. 740–742 (Russian).
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One-parameter systems of solutions of Painlevé equations.
Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).
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15: 18.40 Methods of Computation
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►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree.
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►In what follows we consider only the simple, illustrative, case that is continuously differentiable so that , with real, positive, and continuous on a real interval The strategy will be to: 1) use the moments to determine the recursion coefficients of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas and weights (or Christoffel numbers) from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32).
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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►Equation (18.40.7) provides step-histogram approximations to , as shown in Figure 18.40.1 for and , shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein.
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16: Bibliography C
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Elementary Differential Equations.
Clarendon Press, Oxford.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Algorithm 352: Characteristic values and associated solutions of Mathieu’s differential equation.
Comm. ACM 12 (7), pp. 399–407.
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The numerical solution of linear differential equations in Chebyshev series.
Proc. Cambridge Philos. Soc. 53 (1), pp. 134–149.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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