inhomogeneous differential equations

SearchAdvancedHelp

(0.003 seconds)

1—10 of 18 matching pages

1: 1.13 Differential Equations

… ►

§1.13(iii) Inhomogeneous Equations

… ►

Variation of Parameters

…

2: 11.13 Methods of Computation

… ►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. …

3: 3.7 Ordinary Differential Equations

… ►If $h=0$ the differential equation is homogeneous, otherwise it is inhomogeneous. … … ►The latter is especially useful if the endpoint $b$ of $𝒫$ is at $\mathrm{\infty }$, or if the differential equation is inhomogeneous. …

4: 11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). …

5: Bibliography O

… ►

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

ⓘ

External Links:

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

Referenced by:

§2.11(iv).

Encodings:

BibTeX

…

6: 2.8 Differential Equations with a Parameter

… ►For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). …

7: 11.10 Anger–Weber Functions

… ►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation …

8: 11.2 Definitions

… ►

§11.2(ii) Differential Equations

…

9: 9.12 Scorer Functions

… ►

9.12.20 $$\mathrm{Hi}\left(z\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3+zt\right)dt,$$

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral and $z$: complex variable

Proof sketch:

Verify by showing that the right-hand side satisfies the differential equation (9.12.1) and the initial conditions given in §9.12(iii).

Referenced by:

(9.12.17), (9.12.31)

Permalink:

http://dlmf.nist.gov/9.12.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(vii), §9.12 and Ch.9

… ►

9.12.30 $${\int }_0^z\mathrm{Gi}\left(t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }-\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Integrate (9.12.25) and obtain the constant term by combining (9.12.12) and (9.12.31).(this equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Permalink:

http://dlmf.nist.gov/9.12.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

►

9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ can be found by replacing $\frac{1}{3}{t}^3$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.)

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

…

10: Bibliography L

… ►

The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

ⓘ

Notes:

In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)

External Links:

ISSN 0564-6162, Document, MathReview, ZentralBlatt

Referenced by:

§31.18.

Encodings:

BibTeX

… ►

Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

BibTeX

… ►

Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

ⓘ

External Links:

ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

Referenced by:

2nd item, 2nd item.

Encodings:

BibTeX

… ►

Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.

ⓘ

External Links:

MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

… ►

The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

…