Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Nonmonotonic solutions of the Falkner-Skan boundary layer equation


Author: William C. Troy
Journal: Quart. Appl. Math. 37 (1979), 157-167
MSC: Primary 34B15; Secondary 76D10
DOI: https://doi.org/10.1090/qam/542988
MathSciNet review: 542988
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Abstract: We investigate the equation $ f''' + ff'' + \beta (1 - f'^2) = 0$ together with boundary conditions $ f\left( 0 \right) = f'\left( 0 \right) = 0$ and $ f'\left( \infty \right) = 1$. Here $ \beta $ is negative. Previous results are summarized which describe solutions which satisfy $ \left\vert f' \right\vert < 1$ for all $ \eta \ge 0$. It is shown that there is a sequence $ \left\{ \beta_j \right\}_{j \in \mathbb{N}}$ of decreasing, negative values of $ \beta $, and a corresponding sequence $ \left\{ f_j \right\}_{j \in \mathbb{N}}$ of solutions such that for each $ j \in \mathbb{N}$ the equation $ f_j' - 1 = 0$ has exactly $ j$ positive solutions and for some $ \mu_j > 0, f_j' = 1 + o\left( \exp \left( - \mu_j\eta \right) \right)$ as $ \eta \to \infty $.


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DOI: https://doi.org/10.1090/qam/542988
Article copyright: © Copyright 1979 American Mathematical Society


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