Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On a second-order boundary value problem arising in combustion theory


Author: Philip Holmes
Journal: Quart. Appl. Math. 40 (1982), 53-62
MSC: Primary 34B15; Secondary 80A25
DOI: https://doi.org/10.1090/qam/652049
MathSciNet review: 652049
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain existence and uniqueness results for the boundary-value problem

$\displaystyle y'' = {x^2} - {y^2}, \qquad y\left( x \right) \sim \mp x \qquad as \qquad x \to \pm \infty $

. Our main result shows that there are precisely two solutions $ {y_+} \left( x \right) > - \left\vert x \right\vert$ and $ {y_-}\left( x \right) < - \left\vert x \right\vert$. Only the latter is of physical interest in the problem in combustion theory from which the equation arises.

References [Enhancements On Off] (What's this?)

  • [1] R. Alexander, private communication (1980)
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  • [3] S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), no. 1, 31–51. MR 555581, https://doi.org/10.1007/BF00283254
  • [4] G. S. S. Ludford and D. S. Stewart, Mathematical questions from combustion theory, Transactions of the Twenty-Sixth Conference of Army Mathematicians (Hanover, N.H., 1980) ARO Rep. 81, vol. 1, U. S. Army Res. Office, Research Triangle Park, N.C., 1981, pp. 53–66. MR 605318
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DOI: https://doi.org/10.1090/qam/652049
Article copyright: © Copyright 1982 American Mathematical Society


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