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21: 9.10 Integrals
22: 4.45 Methods of Computation
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►Initial approximations are obtainable, for example, from the power series (4.13.6) (with ) when is close to , from the asymptotic expansion (4.13.10) when is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of .
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23: 23.3 Differential Equations
§23.3 Differential Equations
… ►The lattice roots satisfy the cubic equation … ►§23.3(ii) Differential Equations and Derivatives
…24: Daniel W. Lozier
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►Lozier received a degree in mathematics from Oregon State University in 1962 and his Ph.
…in applied mathematics from the University of Maryland, College Park, in 1979.
…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.
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25: 36.15 Methods of Computation
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►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7.
Close to the bifurcation set but far from
, the uniform asymptotic approximations of §36.12 can be used.
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►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints.
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§36.15(v) Differential Equations
►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).26: 31.12 Confluent Forms of Heun’s Equation
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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i).
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Confluent Heun Equation
… ►Biconfluent Heun Equation
… ►Triconfluent Heun Equation
…27: 21.9 Integrable Equations
§21.9 Integrable Equations
►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ► … ►Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemann surface; see Dubrovin (1981, §IV.4). …28: Peter A. Clarkson
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►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations.
… Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M.
…His well-known book Solitons, Nonlinear Evolution Equations and Inverse Scattering (with M.
…He is also coauthor of the book From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics (with J.
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29: 10.61 Definitions and Basic Properties
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►Most properties of , , , and follow straightforwardly from the above definitions and results given in preceding sections of this chapter.
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§10.61(ii) Differential Equations
…30: Bibliography C
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The Numerical Treatment of Differential Equations.
3rd edition, Die Grundlehren der Mathematischen Wissenschaften, Vol. 60, Springer, Berlin.
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