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Sequential eigenfunction expansion for a problem in combustion theory

Published online by Cambridge University Press:  17 February 2009

M. Al-Refai  and
K. K. Tam
M. Al-Refai
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke St. West, Monterals, Quebec H3A 2K5, Canada; e-mail: tam@math.mcgill.ca.
K. K. Tam
Affiliation:
Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke St. West, Monterals, Quebec H3A 2K5, Canada; e-mail: tam@math.mcgill.ca.
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Abstract

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A method of sequential eigenfunction expansion is developed for a semi-linear parabolic equation. It allows the time-dependent coefficients of the eigenfunctions to be determined sequentially and iterated to reach convergence. At any stage, only a single ordinary differential equation needs to be considered, in contrast to the Galerkin method which requires the consideration of a system of equations. The method is applied to a central problemin combustion theory to provide a definitive answer to the question of the influence of the initial data in determining whether the solution is sub- or super-critical, in the case of multiple steady-state solutions. It is expected this method will prove useful in dealing with similar problems.

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 1 , July 2003 , pp. 35 - 48
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Boddington, P. Gray and Wake, G. C., “Criteria for thermal explosions with and without reactant consumption”, Proc. Roy. Soc. London Ser. A 385 (1983) 289–311.Google Scholar
[2]Doedel, E. J. and Kernàves, J. P., “AUTO: Software for continuation and bifurcation problems in ordinary differential equations”, Appl. Math. Report, California Institute of Technology.Google Scholar
[3]Frank-Kamenetskĭ, D. A., Diffusion and heat transfer in chemical kinetics (Plenum Press, New York, 1959).Google Scholar
[4]Gelfand, I. M., “Some problems in the theory of quasi-linear equations”, Amer. Math. Soc. Trans. Ser. 2 29 (1963) 295–381.Google Scholar
[5]Tam, K. K., “Criticality dependence on data and parameters for a problem in combustion theory”, J. Austral. Math. Soc. Series B 27 (1986) 416–441.CrossRefGoogle Scholar
[6]Tam, K. K., “Critical parameters for some non-class A configurations in combustion theory, with non-uniform boundary temperatures”, J. Engin. Math. 21 (1987) 129–137.Google Scholar
[7]Parks, J.R., “Criticality criteria for various configurations of a self-heating chemical as functions of activation energy and temperature of assembly”, J. Chem. Phys. 34 (1961) 46–50.Google Scholar
[8]Sattinger, D. H., “A nonlinear parabolic system in the theory of combustion”, Quart. Appl. Math. 33 (1975) 47–61.Google Scholar
[9]Tam, K. K., “On the influence of the initial data in a combustion problem”, J. Austral. Math. Soc. Series B 22 (1980) 193–209.CrossRefGoogle Scholar
[10]Tam, K. K., “The temporal evolution of a system in combustion theory. II”, Stud. Appl. Math. 79 (1988) 159–171.CrossRefGoogle Scholar
[11]Tam, K. K., Andonowati, and Kiang, M. T., “Nonlinear eigenfunction expansion for a problem in microwave heating”, Canad. Appl. Math. Quart. 4 (1996) 311–325.Google Scholar