A singular perturbation problem in needle crystals. (English) Zbl 0695.34060
The equation
\[
\epsilon \theta '''(x)+\theta '(x)=\cos \theta (x),\quad 0\leq x<\infty;\quad \epsilon >0;
\]
arises in the problem of dentritic solidification. The authors prove that it has a unique monotonic solution satisfying
\[
\theta (0)=0;\quad \lim_{x\to \infty}\theta (x)=\pi.
\]
The existence of needle-crystal solutions depends on whether \(\theta ''=0\). The authors show that this is not the case and give an asymptotic estimate of \(\theta ''(0)\) when \(\epsilon\) tends to zero.
Reviewer: J.Mika
MSC:
| 34E15 | Singular perturbations for ordinary differential equations |
| 34D15 | Singular perturbations of ordinary differential equations |
References:
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