Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A free boundary value problem related to the combustion of a solid: flux boundary conditions


Authors: John R. Cannon and Alec L. Matheson
Journal: Quart. Appl. Math. 55 (1997), 687-705
MSC: Primary 35R35; Secondary 35K57, 80A25
DOI: https://doi.org/10.1090/qam/1486543
MathSciNet review: MR1486543
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Abstract | References | Similar Articles | Additional Information

Abstract: We demonstrate the existence, uniqueness, and continuous dependence upon the data for the solution $ \left( u, v, s \right)$ of the free boundary value problem:

$\displaystyle {u_t} = \alpha {u_{xx}}, \qquad {v_t} = \beta {v_{xx,}} \qquad 0 < x < s\left( t \right), \qquad 0 < t \le T$

,

$\displaystyle u\left( x, 0 \right) = \phi \left( x \right), \qquad v\left( x, 0 \right) = \psi \left( x \right), \qquad 0 \le x \le s\left( 0 \right) = b$

.

$\displaystyle - \alpha {u_x}\left( 0, t \right) = f\left( t \right), \qquad - \beta {v_x}\left( 0, t \right) = g\left( t \right), \qquad 0 < t \le T$

,

$\displaystyle \alpha {u_x}\left( s\left( t \right), t \right) = - \left( \gamma... ... s\left( t \right), t \right) \right)\dot s\left( t \right), \qquad 0 < t \le T$

,

$\displaystyle \dot s\left( t \right) = \nu \left( v\left( s\left( t \right), t ... ...\right)F\left( u\left( s\left( t \right), t \right) \right), \qquad 0 < t \le T$

, where $ \alpha , \beta , \gamma , \delta $, and $ \mu $ are positive constants related to the physical constants.

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

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  • [CCF] John R. Cannon, James C. Cavendish, and Antonio Fasano, A free boundary-value problem related to the combustion of a solid, SIAM J. Appl. Math. 45 798-809 (1985) MR 804007
  • [CL] John R. Cannon and Yanping Lin, A free boundary-value problem related to the combustion of a solid: first initial-boundary value case, Proc. of the Intl. Conf. on Theory and Appl. of Diff. Eq. (Reza Aftabizadeh, ed.), Ohio University Press, Athens, Ohio, 1989, pp. 109-121 MR 1026123
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  • [LC] Xhi Yuan Liang and Hong Cheng, A class of free boundary problems with two boundaries arising in combustion of solids, J. Harbin Inst. Tech. 24 1-6 (1992) MR 1178246

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Additional Information

DOI: https://doi.org/10.1090/qam/1486543
Article copyright: © Copyright 1997 American Mathematical Society

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