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)


This paper investigates the numerical properties of solutions θ = θ(t) of the third-order equation
$$\varepsilon {\theta _{3}} + {\theta _{1}} = \frac{{\cos \;\theta }}{{1 + \alpha \cos 4\theta }},\;\theta \left( {\pm \infty } \right) = \pm \pi /2$$
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.


Discrete Spectrum Dendritic Growth Primary Solution Manual Setting Monotonic Solution 
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  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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