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1: 16.8 Differential Equations

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§16.8(i) Classification of Singularities

►An ordinary point of the differential equation …If $z_0$ is not an ordinary point but ${(z-z_0)}^{n-j}{f}_j(z)$, $j=0,1,\mathrm{\dots },n-1$, 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

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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.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

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5: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

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Asymptotic Expansions for Ordinary Differential Equations. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney.

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Notes:

Reprinted by Krieger, New York, 1976 and Dover, New York, 1987.

External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

Ch.2, Ch.9.

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Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt

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Ch.9.

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An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities. Invent. Math. 108, pp. 575–633.

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ISSN 0020-9910, Document, MathReview (David R. Masson), ZentralBlatt

Referenced by:

§16.4(iii).

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6: Bibliography S

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

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On the double points of a Mathieu equation. J. Comput. Appl. Math. 107 (1), pp. 111–125.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§28.7.

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Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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ISBN 0-387-97825-9, MathReview (Andrew Bremner), ZentralBlatt

Referenced by:

§22.18(iv), §23.1, §23.20(ii), §23.20(ii), §23.20(v).

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Izbrannye Trudy. Analiticheskaya teoriya obyknovennykh differentsialnykh uravnenii. Izdatel’ stvo Sankt-Peterburgskogo Universiteta, St. Petersburg (Russian).

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Notes:

In Russian. Translated title: Selected Works. Analytic Theory of Ordinary Differential Equations

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ISBN 5-288-01337-3, MathReview (Alois Klíč)

Referenced by:

§31.16(i).

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Algorithm 693: A FORTRAN package for floating-point multiple-precision arithmetic. ACM Trans. Math. Software 17 (2), pp. 273–283.

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Notes:

Multiple-precision Fortran.

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ISSN 0098-3500, Document, GAMS (module 693; package TOMS), ZentralBlatt

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2nd item.

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7: 3.7 Ordinary Differential Equations

§3.7 Ordinary Differential Equations

… ►Consideration will be limited to ordinary linear second-order differential equations … ►For 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

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§1.18(v) Point Spectra and Eigenfunction Expansions

… ►Assume $T$ has no point spectrum, i. … ►Then, for $z\in ℂ\backslash ℝ$, $f\in N_z$ iff $f$ is an ordinary solution (i. … A boundary value for the end point $a$ is a linear form $ℬ$ on $𝒟({ℒ}^{\ast })$ of the form …Boundary values and boundary conditions for the end point $b$ are defined in a similar way. …

9: 22.19 Physical Applications

… ► $\theta$ being the angular displacement from the point of stable equilibrium, $\theta =0$. … ►for the initial conditions $\theta (0)=0$, the point of stable equilibrium for $E=0$, and $d\theta (t)/dt=\sqrt{2E}$. … ►

§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

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Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.

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External Links:

ISSN 0962-8444, FullText, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§13.4(i), §2.11(v), §2.7(ii).

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On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt

Referenced by:

§16.25, §2.7(ii), §3.7(iii).

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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.

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ISSN 0377-0427, Document, MathReview (Jorge Mozo Fernández), ZentralBlatt

Referenced by:

§2.11(iv).

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Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

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ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

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Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.

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ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.7(ii).

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