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Numerical Analysis of the Geometric Model for Dendritic Growth of Crystals

  • J. M. Hammersley
  • G. Mazzarino
Part of the NATO ASI Series book series (NSSB, volume 284)

Abstract

This paper investigates the numerical properties of solutions θ = θ(t) of the third-order equation
$$\varepsilon {\theta _{3}} + {\theta _{1}} = \frac{{\cos \;\theta }}{{1 + \alpha \cos 4\theta }},\;\theta \left( {\pm \infty } \right) = \pm \pi /2$$
(1.1)
Suffices to θ in (1.1) denote differentiation with respect to t; that is to say θ n = dn θ/dtn; the boundary conditions θ(±∞) = ±π/2 are abbreviations for θ(t) → π±/2 as t → ±∞ respectively; and ε and α are prescribed parameters satisfying ε > 0 and 0 ≤ α < 1. It is convenient to write ε = 2k, and to tabulate results as functions of α and k = log2 ε. Kruskal and Segur [1] give references to the appearance of (1.1) as a model for the dendritic growth of crystals in a supercooled liquid. [Warning: there is some variation of notation in the literature; and, in particular, Kruskal and Segur write ε 2 for the coefficient of θ 3, thus entailing k = 21og2 ε for their use of ε.] A strictly monotonic solution of (1.1) is called a needle crystal solution: and interest centres upon the question of the existence or non-existence of needle solutions. Our earlier paper [2] proved that needle solutions could not exist for α = 0. Kruskal and Segur [1] concluded that, for sufficiently small 6, needle solutions would exist for certain discrete values of α = α(k) > 0. Thus we have an eigenvalue problem with a discrete spectrum. Our analysis of this problem is incomplete, and several interesting questions remain unresolved.

Keywords

Discrete Spectrum Dendritic Growth Primary Solution Manual Setting Monotonic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    M.D.KRUSKAL and H. SEGUR, “Asymptotics beyond all orders in a model of crystal growth,” Stud. Add. Math. (to appear).Google Scholar
  2. [2]
    J.M. HAMMERSLEY and G. MAZZARINO, “A differential equation connected with the dendritic growth of crystals,” IMA J. AddI. Math. (1989) 42, 43–75.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    J.M. HAMMERSLEY and G. MAZZARINO, “Computational aspects of some autonomous differential equations.” Proc. Rov. Soc. (1989) A242, 19–37.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • J. M. Hammersley
    • 1
  • G. Mazzarino
    • 2
  1. 1.Trinity CollegeOxfordUK
  2. 2.Institute of Economics and StatisticsUniversity of OxfordOxfordUK

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