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: We obtain existence and uniqueness results for the boundary-value problem \[ y” = {x^2} - {y^2}, \qquad y\left ( x \right ) \sim \mp x \qquad as \qquad x \to \pm \infty \]. Our main result shows that there are precisely two solutions ${y_+} \left ( x \right ) > - \left | x \right |$ and ${y_-}\left ( x \right ) < - \left | x \right |$. Only the latter is of physical interest in the problem in combustion theory from which the equation arises.
R. Alexander, private communication (1980)
S. P. Burke and T. E. W. Schumann, Diffusion flames, Indust. Engrg. Chem. 20, 998–1004 (1928)
- 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, DOI https://doi.org/10.1007/BF00283254
- 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
D. A. Spence and P. J. Holmes, A boundary value problem involving the first Painlevé transcendent (in preparation)
R. Alexander, private communication (1980)
S. P. Burke and T. E. W. Schumann, Diffusion flames, Indust. Engrg. Chem. 20, 998–1004 (1928)
S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteveg-de Vries equation, Arch. Rat. Mech. Anal. 73(1980), 31–51
G. S. S. Ludford and D. S. Stewart, Mathematical questions from combustion theory, in Trans. 26th Conf. of Army Mathematicians, Lebanon, NH, June 1980, A. R. O. Rept. 81–1 (to appear)
D. A. Spence and P. J. Holmes, A boundary value problem involving the first Painlevé transcendent (in preparation)
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Article copyright:
© Copyright 1982
American Mathematical Society