Asymptotics beyond All Orders pp 37-66 | Cite as

# Numerical Analysis of the Geometric Model for Dendritic Growth of Crystals

Chapter

## Abstract

This paper investigates the numerical properties of solutions Suffices to

*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)

*θ*in (1.1) denote differentiation with respect to t; that is to say*θ*_{n}= d^{n}*θ*/dt^{n}; 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*= 2^{k}, and to tabulate results as functions of*α*and k = log_{2}*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 = 21og_{2}*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## Preview

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## References

- [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]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]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