# asymptotic solutions of differential equations

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## 11—20 of 64 matching pages

##### 11: 2.11 Remainder Terms; Stokes Phenomenon

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###### §2.11(v) Exponentially-Improved Expansions (continued)

…##### 12: 2.6 Distributional Methods

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►For rigorous derivations of these results and also order estimates for ${\delta}_{n}(x)$, see Wong (1979) and Wong (1989, Chapter 6).

##### 13: 16.25 Methods of Computation

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►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations.
…There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions.
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##### 14: 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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##### 15: Bibliography W

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Asymptotic solutions of a fourth order differential equation.
Stud. Appl. Math. 118 (2), pp. 133–152.
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##### 16: 9.16 Physical Applications

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►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.
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►In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation).
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).
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►Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010).
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##### 17: 33.23 Methods of Computation

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###### §33.23(ii) Series Solutions

… ►###### §33.23(iii) Integration of Defining Differential Equations

… ►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►This implies decreasing $\mathrm{\ell}$ for the regular solutions and increasing $\mathrm{\ell}$ for the irregular solutions of §§33.2(iii) and 33.14(iii). … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …##### 18: Bibliography T

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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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##### 19: Bibliography B

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions.
SIAM J. Math. Anal. 17 (2), pp. 422–450.
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