# second-order

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## 1—10 of 41 matching pages

##### 1: 9.15 Mathematical Applications

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►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.
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##### 2: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I.
Methods Appl. Anal. 2 (2), pp. 173–197.
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On the calculation of Stokes multipliers for linear differential equations of the second order.
Methods Appl. Anal. 2 (3), pp. 348–367.
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Hyperasymptotic solutions of second-order linear differential equations. II.
Methods Appl. Anal. 2 (2), pp. 198–211.
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Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
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Second-order differential equations with fractional transition points.
Trans. Amer. Math. Soc. 226, pp. 227–241.
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##### 3: Mark J. Ablowitz

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►ODEs with the Painlevé property contain the well-known Painlevé equations which are special second order scalar equations; their solutions are often called Painlevé transcendents.
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##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

###### §1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►###### §1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators

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…##### 5: 31.14 General Fuchsian Equation

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►The general second-order

*Fuchsian equation*with $N+1$ regular singularities at $z={a}_{j}$, $j=1,2,\mathrm{\dots},N$, and at $\mathrm{\infty}$, is given by … ►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). …##### 6: Bibliography T

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Eigenfunction Expansions Associated with Second-Order Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations.
Clarendon Press, Oxford.
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Eigenfunction expansions associated with second-order differential equations. Part I.
Second edition, Clarendon Press, Oxford.
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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades.
J. Appl. Math. Mech. 23, pp. 1549–1565.
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##### 7: Bibliography D

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A simplified algorithm for the second-order sound fields.
J. Acoust. Soc. Amer. 108 (6), pp. 2759–2764.
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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.
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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point.
SIAM J. Math. Anal. 21 (6), pp. 1594–1618.
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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane.
SIAM J. Math. Anal. 25 (2), pp. 322–353.
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A survey on orthogonal matrix polynomials satisfying second order differential equations.
J. Comput. Appl. Math. 178 (1-2), pp. 169–190.
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##### 8: 2.9 Difference Equations

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►or equivalently the second-order homogeneous linear difference equation
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►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)).
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►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999).
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►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).
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##### 9: 22.13 Derivatives and Differential Equations

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