inhomogeneous differential equations

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1: 1.13 Differential Equations

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§1.13(iii) Inhomogeneous Equations

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Variation of Parameters

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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

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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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External Links:

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

Referenced by:

§2.11(iv).

Encodings:

BibTeX

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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

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§11.2(ii) Differential Equations

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9: 9.12 Scorer Functions

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9.12.4 $$\mathrm{Gi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt+\mathrm{Ai}\left(z\right){\int }_0^z\mathrm{Bi}\left(t\right)dt,$$

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Defines:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Source:

Olver (1997b, (12.08), p. 430)

Referenced by:

(9.10.1)

Permalink:

http://dlmf.nist.gov/9.12.E4

Encodings:

pMML, png, TeX

See also:

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

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9.12.5 $$\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt-\mathrm{Ai}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt.$$

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Defines:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Source:

Olver (1997b, (12.09), p. 430)

Referenced by:

(9.10.3)

Permalink:

http://dlmf.nist.gov/9.12.E5

Encodings:

pMML, png, TeX

See also:

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

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9.12.19 $$\mathrm{Gi}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\sin \left(\frac{1}{3}{t}^3+xt\right)dt,$$ $x\in ℝ$.

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line, $\sin z$: sine function and $x$: real variable

Proof sketch:

Combine (9.5.3) and (9.12.11).

Permalink:

http://dlmf.nist.gov/9.12.E19

Encodings:

pMML, png, TeX

See also:

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

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

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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

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9.12.21 $$\mathrm{Gi}\left(z\right)=-\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3-\frac{1}{2}zt\right)\cos \left(\frac{1}{2}\sqrt{3}zt+\frac{2}{3}\pi \right)dt.$$

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Symbols:

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

Source:

Lee (1980, (2.20))

Permalink:

http://dlmf.nist.gov/9.12.E21

Encodings:

pMML, png, TeX

See also:

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

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10: 9.10 Integrals

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9.10.1 $${\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)\right),$$

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Symbols:

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

Proof sketch:

Combine (9.12.4) and its differentiated form with the first of (9.10.11)

Referenced by:

(9.10.2)

Permalink:

http://dlmf.nist.gov/9.10.E1

Encodings:

pMML, png, TeX

See also:

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

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9.10.2 $${\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right),$$

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Symbols:

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

Proof sketch:

Combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E2

Encodings:

pMML, png, TeX

See also:

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

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9.10.3 $${\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt={\int }_0^z\mathrm{Bi}\left(t\right)dt=\pi \left({\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)\right)=\pi \left(\mathrm{Bi}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right).$$

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Symbols:

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

Proof sketch:

Combine (9.12.5) and its differentiated form with (9.2.7).

Permalink:

http://dlmf.nist.gov/9.10.E3

Encodings:

pMML, png, TeX

See also:

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

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