numerical solution of differential equations

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21: 11.13 Methods of Computation

… ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for

\mathbf{K}_{\nu}\left(z\right)

and

\mathbf{M}_{\nu}\left(z\right)

. … ►

§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. … ►

§11.13(v) Difference Equations

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22: 9.17 Methods of Computation

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§9.17(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods. As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist. …

23: 2.9 Difference Equations

… ►As in the case of differential equations (§§2.7(iii), 2.7(iv)) recessive solutions are unique and dominant solutions are not; furthermore, one member of a numerically satisfactory pair has to be recessive. …

24: 33.23 Methods of Computation

… ►

§33.23(ii) Series Solutions

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§33.23(iii) Integration of Defining Differential Equations

►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). … ►This implies decreasing

\ell

for the regular solutions and increasing

\ell

for the irregular solutions of §§33.2(iii) and 33.14(iii). … ►

§33.23(vi) Other Numerical Methods

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25: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►

§10.74(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) 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. … ►For further information, including parallel methods for solving the differential equations, see Lozier and Olver (1993). …

26: 13.29 Methods of Computation

… ►

§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 …

27: 28.34 Methods of Computation

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(a)

Direct numerical integration of the differential equation (28.2.1), with initial values given by (28.2.5) (§§3.7(ii), 3.7(v)).

… ►

(c)

Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

… ►

(e)

Solution of the continued-fraction equations (28.6.16)–(28.6.19) and (28.15.2) by successive approximation. See Blanch (1966), Shirts (1993a), and Meixner and Schäfke (1954, §2.87).

… ►

§28.34(iii) Floquet Solutions

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(b)

Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

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28: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

►

15.10.1

z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)
Permalink:: http://dlmf.nist.gov/15.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(i), §15.10 and Ch.15

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§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

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29: Frank W. J. Olver

… ►In 1945–1961 he was a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. … ►He is particularly known for his extensive work in the study of the asymptotic solution of differential equations, i. …, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►He also served on the Editorial Boards of several of the leading journals devoted to mathematical and/or numerical analysis, including founding Managing Editor of the SIAM Journal on Mathematical Analysis. …

30: Daniel W. Lozier

… ►Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. His research interests have centered on numerical analysis, special functions, computer arithmetic, and mathematical software construction and testing. ►Lozier has been a frequent speaker at conferences, and has published numerous papers in journals and conference proceedings. …