solutions of differential equations

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1: 16.25 Methods of Computation

… ►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …

2: T. Mark Dunster

… ►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …

3: 13.27 Mathematical Applications

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4: 3.7 Ordinary Differential Equations

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§3.7(ii) Taylor-Series Method: Initial-Value Problems

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§3.7(iii) Taylor-Series Method: Boundary-Value Problems

… ►The eigenvalues

\lambda_{k}

are simple, that is, there is only one corresponding eigenfunction (apart from a normalization factor), and when ordered increasingly the eigenvalues satisfy … ►

§3.7(v) Runge–Kutta Method

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5: 1.13 Differential Equations

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§1.13(i) Existence of Solutions

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Fundamental Pair

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Wronskian

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§1.13(v) Products of Solutions

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6: 10.72 Mathematical Applications

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§10.72(i) Differential Equations with Turning Points

►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►

7: 2.7 Differential Equations

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§2.7(ii) Irregular Singularities of Rank 1

… ►Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is,

a_{s,1}

a_{s,2}

with

s

large, are the “early” coefficients

a_{j,2}

a_{j,1}

with

j

small. …See §2.11(v) for other examples. … ►

§2.7(iii) Liouville–Green (WKBJ) Approximation

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§2.7(iv) Numerically Satisfactory Solutions

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8: 18.38 Mathematical Applications

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Differential Equations: Spectral Methods

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Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

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9: 15.11 Riemann’s Differential Equation

… ►The complete set of solutions of (15.11.1) is denoted by Riemann’s $P$ -symbol: ►

15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation
Symbols:: $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

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15.11.4

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w$ : solution
Permalink:: http://dlmf.nist.gov/15.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

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§15.11(ii) Transformation Formulas

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15.11.6

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $t$ : mapping
Permalink:: http://dlmf.nist.gov/15.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(ii), §15.11 and Ch.15

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10: Daniel W. Lozier

… ►Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. …