asymptotic solutions of differential equations

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31: 18.40 Methods of Computation

… ►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. … … ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). … ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. …

32: Bibliography J

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

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Notes:

Second edition of Jones and Sleeman (2003)

External Links:

ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt

Referenced by:

Profile Brian D. Sleeman.

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

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On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§32.12(i).

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

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33: 32.5 Integral Equations

§32.5 Integral Equations

►Let $K(z,\zeta )$ be the solution of ►

32.5.1 $$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E1

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pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

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32.5.2 $$w(z)=K(z,z),$$

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Symbols:

$z$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E2

Encodings:

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See also:

Annotations for §32.5 and Ch.32

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32.5.3 $$w(z)\sim k\mathrm{Ai}\left(z\right),$$ $z\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $z$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.5.E3

Encodings:

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See also:

Annotations for §32.5 and Ch.32

34: 12.14 The Function $W(a,x)$

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§12.14(i) Introduction

►In this section solutions of equation (12.2.3) are considered. … ►

§12.14(viii) Asymptotic Expansions for Large Variable

… ►The differential equation … ►For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large $x$, see Miller (1955, pp. 87–88). …

35: Bibliography L

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

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Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.

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External Links:

ISSN 0022-0396, Document, MathReview (John Albert), ZentralBlatt

Referenced by:

§22.19(iii).

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On a set of solutions of the sixth Painlevé equation. Differ. Uravn. 3 (3), pp. 520–523 (Russian).

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Notes:

In Russian. English translation: Differential Equations 3(3), pp. 264–266

External Links:

MathReview (J. Rausen), ZentralBlatt

Referenced by:

§32.10(vi).

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Elementary solutions of certain Painlevé equations. Differ. Uravn. 1 (3), pp. 731–735 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(3), pp. 561–564

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.10(iii), §32.10(iv), §32.9(i).

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Solutions of the fifth Painlevé equation. Differ. Uravn. 4 (8), pp. 1413–1420 (Russian).

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Notes:

In Russian. English translation: Differential Equations 4(8), pp. 732–735

External Links:

MathReview (C. E. Langenhop), ZentralBlatt

Referenced by:

§32.10(v), §32.8(v).

Encodings:

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36: 16.23 Mathematical Applications

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§16.23(i) Differential Equations

►A variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations. These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … … ►In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_2{}^{}F_2^{}$ functions are applied to compute the expected value of the excess. …

37: Bibliography M

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

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The Airy Integral, Giving Tables of Solutions of the Differential Equation $y^{\prime \prime }=xy$ . British Association for the Advancement of Science, Mathematical Tables Part-Vol. B, Cambridge University Press, Cambridge.

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External Links:

MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

1st item, 1st item, §9.2(i), §9.2(ii), §9.2(v), §9.4, (9.6.2), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.8(i), §9.8(ii), §9.8(iv), §9.9(ii), Ch.9.

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On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.

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External Links:

Document, MathReview (W. R. Wasow), ZentralBlatt

Referenced by:

§12.2(vi), §2.7(iv), §2.7(iv).

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Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions. 2nd edition, Springer-Verlag, Berlin.

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Notes:

Reprinted in 1988.

External Links:

ISBN 3-540-18430-9, MathReview

Referenced by:

Table 28.1.1.

Encodings:

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Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. Methods Appl. Anal. 4 (3), pp. 250–260.

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External Links:

ISSN 1073-2772, MathReview (W. Balser), ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

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38: 13.29 Methods of Computation

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§13.29(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►with recessive solution … ►with recessive solution …

39: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►Solutions are called roots of the equation, or zeros of $f$. … ►This is useful when $f(z)$ satisfies a second-order linear differential equation because of the ease of computing $f^{\prime \prime }(z_n)$. … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). … ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …

40: 15.19 Methods of Computation

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§15.19(ii) Differential Equation

►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions. However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases. … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). …