asymptotic solutions of differential equations

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41: Bibliography V

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Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

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A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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External Links:

ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt

Referenced by:

§16.11(ii), §16.11(ii).

Encodings:

BibTeX

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

item (f), §28.34(ii), §29.20(i).

Encodings:

BibTeX

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On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

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42: 9.11 Products

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§9.11(i) Differential Equation

… ►where $w_1$ and $w_2$ are any solutions of (9.2.1). …Numerically satisfactory triads of solutions can be constructed where needed on $ℝ$ or $ℂ$ by inspection of the asymptotic expansions supplied in §9.7. … ►Let $w_1,w_2$ be any solutions of (9.2.1), not necessarily distinct. … ►For $\int z^n{w}_1{w}_{2}dz$, $\int z^n{w}_1{w}_2^{\prime }dz$, $\int z^n{w}_1^{\prime }{w}_2^{\prime }dz$, where $n$ is any positive integer, see Albright (1977). …

43: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

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§9.13(i) Generalizations from the Differential Equation

… ►are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). … ►As $z\to \mathrm{\infty }$ … ► …

44: 11.13 Methods of Computation

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§11.13(iv) Differential Equations

►A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. … ►The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. … ►

§11.13(v) Difference Equations

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45: 30.16 Methods of Computation

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§30.16(i) Eigenvalues

… ►If $|{\gamma }^2|$ is large we can use the asymptotic expansions in §30.9. Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93). … ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then we can compute ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime }(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime }(0)=1$ if $n-m$ is odd. … ►The coefficients $a_{n,r}^m({\gamma }^2)$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). …

46: 28.31 Equations of Whittaker–Hill and Ince

§28.31 Equations of Whittaker–Hill and Ince

… ►Hill’s equation with three terms … ►

§28.31(ii) Equation of Ince; Ince Polynomials

… ►They satisfy the differential equation … ►

Asymptotic Behavior

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47: 28.32 Mathematical Applications

… ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►Let $u(\zeta )$ be a solution of Mathieu’s equation (28.2.1) and $K(z,\zeta )$ be a solution of …defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. ►Kernels $K$ can be found, for example, by separating solutions of the wave equation in other systems of orthogonal coordinates. … ►The first is the $2\pi$-periodicity of the solutions; the second can be their asymptotic form. …

48: 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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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

Encodings:

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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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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

Encodings:

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The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

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Notes:

In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(v).

Encodings:

BibTeX

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One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).

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Notes:

In Russian. English translation: Differential Equations 14(12), pp. 1510–1513 (1979)

External Links:

ISSN 0374-0641, MathReview (M. Tvrdý), ZentralBlatt

Referenced by:

§32.10(i), §32.10(ii), §32.10(iii), §32.7(iv).

Encodings:

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49: 10.25 Definitions

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§10.25(i) Modified Bessel’s Equation

… ►Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. ►

§10.25(ii) Standard Solutions

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§10.25(iii) Numerically Satisfactory Pairs of Solutions

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50: 9.10 Integrals

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§9.10(ii) Asymptotic Approximations

… ►Let $w(z)$ be any solution of Airy’s equation (9.2.1). … ►

9.10.8 $$\int zw(z)dz=w^{\prime }(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.9 $$\int z^{2}w(z)dz=z{w}^{\prime }(z)-w(z),$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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9.10.10 $$\int z^{n+3}w(z)dz=z^{n+2}{w}^{\prime }(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)\int z^{n}w(z)dz,$$ $n=0,1,2,\mathrm{\dots }.$

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Defines:

$n$: index (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $w$: ODE solution

Proof sketch:

To verify, differentiate and refer to (9.2.1).

Permalink:

http://dlmf.nist.gov/9.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(iii), §9.10 and Ch.9

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