Abstract
Combustion processes are classified into three types depending upon the amount of fuel supply: two of them are the stationary states with either low or high temperatures and the other is the periodic state with relaxation oscillation type. We analyze the dependency of these processes on the amount of fuel supply by using the fast and slow dynamics approach.
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References
Bebernes, J., and Eberly, D. (1989). Mathematical problems from combustion theory.Appl. Math. Sci. Vol. 83, Springer-Verlag, New York.
Coppel, W. A., and Palmer, K. J. (1970). Averaging and integral manifolds.Bull. Austral. Math. Soc. 2, 197–222.
Dancer, E. N. (1980). On the structure of solutions of an equation in catalysis theory when a parameter is large.J. Diff. Eq. 37, 404–437.
Ei, S.-I. (1988). Two-timing methods with applications to heterogeneous reaction-diffusion systems.Hiroshima Math. J. 18, 127–160.
Frank-Kamenetzky, D. A. (1955).Diffusion and Heat Exchange in Chemical Kinetics (translated by N. Thon), Princeton University Press, Princeton, N.J.
Gavalas, G. R. (1968).Nonlinear Differential Equations of Chemically Reacting Systems, Springer-Verlag, New York.
Gelfand, I. M. (1963). Some problems in the theory of quasilinear equations.AMS Translat. Ser. 2(29), 295–381.
Giddas, B., Ni, W.-N., and Nirenberg, L. (1979). Symmetry and related properties via the maximum principle.Commun. Math. Phys. 68, 209–243.
Hale, J. K. (1988). Asymptotic behavior of dissipative systems.Math. Surv. Monogr. 25, AMS.
Hastings, S. P., and McLeod, J. B. (1985). The number of solutions to an equation from catalysis.Proc. Roy. Soc. Edinburgh 101A, 15–30.
Henry, D. (1981). Geometric theory of semilinear parabolic equations.Lecture Notes in Mathematics Vol. 840, Springer-Verlag, New York.
Hirsh, M., Pugh, C., and Shub, M. (1977). Invariant manifolds.Lectures Notes in Mathematics Vol. 583, Springer-Verlag, New York.
Matano, H., (1984). Existence of nontrivial unstable sets for equilibriums of strongly orderpreservinig systems.J. Fac. Sci. Univ. Tokyo Sec. IA 30(3), 645–673.
Nagasaki, K., and Suzuki, T. (1991). On the nonlinear eigenvalue problemΔu+γe u=0.Trans. Am. Math. Soc. (in press).
Nayfeh, A. H. (1973).Perturbation Methods, Wiley-Interscience, New York.
Parks, J. R. (1961). Criticality criteria for various configurations of a self-heating chemical as functions of activation energy and temperature of assembly.J. Chem. Phys. 34, 46–50.
Parter, S. (1974). Solutions of a differential equation arising in chemical reactor process.SIAM J. Appl. Math. 26, 686–715.
Parter, S. V., Stein, M. L., and Stein, P. R. (1975). On the multiplicity of solutions of a differential equation arising in chemical reactor theory.Stud. Appl. Math. 54, 293–314.
Sattinger, D. H. (1972). Stability of solutions of nonlinear equations.J. Math. Anal. Appl. 39, 1–12.
Sattinger, D. H. (1975). A nonlinear parabolic system in the theory of combustion.Q. Appl. Math. 33, 47–61.
Tam, K. K. (1979). Construction of upper and lower solutions for a problem in combustion theory.J. Math. Anal. Appl. 69, 131–145.
Wiebers, H. (1986). Critical behavior of nonlinear elliptic boundary value problems suggested by exothemic reactions.Proc. Roy. Soc. Edinburgh 102A, 19–36.
Williams, L. R., and Leggett, R. W. (1979). Multiple fixed point theorems for the problems in chemical reactor theory.J. Math. Anal. Appl. 69, 180–193.
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Ei, S.I., Mimura, M. Relaxation oscillations in combustion models of thermal self-ignition. J Dyn Diff Equat 4, 191–229 (1992). https://doi.org/10.1007/BF01048160
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DOI: https://doi.org/10.1007/BF01048160