numerical integration

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21: 13.29 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. …

22: 32.17 Methods of Computation

… ►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003). For numerical studies of ${\text{P}}_{\text{I}}$ see Holmes and Spence (1984), Noonburg (1995), and Fornberg and Weideman (2011). For numerical studies of ${\text{P}}_{\text{II}}$ see Rosales (1978), Miles (1978, 1980), Kashevarov (1998, 2004), and S. …For numerical studies of ${\text{P}}_{\text{IV}}$ see Bassom et al. (1993).

23: 3.7 Ordinary Differential Equations

… ►For an introduction to numerical methods for ordinary differential equations, see Ascher and Petzold (1998), Hairer et al. (1993), and Iserles (1996). ►

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

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§3.7(v) Runge–Kutta Method

… ►An extensive literature exists on the numerical solution of ordinary differential equations by Runge–Kutta, multistep, or other methods. …

24: Bibliography L

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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25: 18.39 Applications in the Physical Sciences

… ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►with an infinite set of orthonormal $L^2$ eigenfunctions … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …

26: 16.25 Methods of Computation

… ►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …

27: Bernard Deconinck

… ►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations. He is the coauthor of several Maple commands to work with Riemann surfaces and the command to compute multidimensional theta functions numerically. …

28: David M. Bressoud

… ►Bressoud has published numerous papers in number theory, combinatorics, and special functions. … Wagon), published by Key College Press in 2000, and A Radical Approach to Lebesgue’s Theory of Integration, published by the Mathematical Association of America and Cambridge University Press in 2007. …

29: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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§1.18(ii) $L^2$ spaces on intervals in $ℝ$

… ►For a Lebesgue–Stieltjes measure $d\alpha$ on $X$ let $L^2(X,d\alpha )$ be the space of all Lebesgue–Stieltjes measurable complex-valued functions on $X$ which are square integrable with respect to $d\alpha$, … ►We integrate by parts twice giving: … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►The well must be deep and broad enough to allow existence of such $L^2$ discrete states. …

30: 18.38 Mathematical Applications

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§18.38(i) Classical OP’s: Numerical Analysis

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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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Integrable Systems

►The Toda equation provides an important model of a completely integrable system. …