Evans function stability of combustion waves

SIAM Journal on Applied Mathematics

Volumn 63, Issue 4, 2003, Pages 1259-1275

  Gubernov, V ^a   Mercer, G N ^a   Sidhu, H S ^a   Weber, R O ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Combustion waves; Compound matrix; Evans function; Nyquist plot

Indexed keywords

ACTIVATION ENERGY; NYQUIST DIAGRAMS; PARTIAL DIFFERENTIAL EQUATIONS;

COMBUSTION WAVES;

COMBUSTION;

EID: 0042977385     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139901400240     Document Type: Article

References (25)

1. G. I. SIVASHINSKY, Structure of Busen flames, J. Chem. Phys., 62 (1975), pp. 638-643.
2. G. I. SIVASHINSKY, Diffusional-thermal theory of cellular flames, Combust. Sci. Technol., 15 (1977), pp. 137-146.
3. B. J. MATKOWSKY AND G. I. SIVASHINSKY, Propagation of a pulsating reaction front in solid fuel combustion, SIAM J. Appl. Math., 35 (1978), pp. 465-478.
4. B. J. MATKOWSKY AND D. O. OLAGUNJU, Propagation of a pulsating flame front in a gaseous combustible mixture, SIAM J. Appl. Math., 39 (1980), pp. 290-300.
5. S. B. MARGOLIS AND B. J. MATKOWSKY, Nonlinear stability and bifurcation in the transition from laminar to turbulent flame propagation, Combust. Sci. Technol., 34 (1983), pp. 45-77.
6. W. B. BUSH AND F. E. FENDELL, Asymptotic analysis of laminar flame propagation for general Lewis numbers, Combust. Sc. Technol., 1 (1970), pp. 421-428.
7. R. O. WEBER, G. N. MERCER, H. S. SIDHU, AND B. F. GRAY, Combustion waves for gases (Le = 1) and solids (Le → ∞), Proc. Roy. Soc. London Ser. A, 453 (1997), pp. 1105-1118.
8. A. C. MCINTOSH, R. O. WEBER, AND G. N. MERCER, Non-adiabatic combustion waves for general Lewis numbers: Wave speed and extinction conditions, ANZIAM J., submitted.
9. D. A. SCHULT, Matched asymptotic expansions and the closure problem for combustion waves, SIAM J. Appl. Math., 60 (1999), pp. 136-155.
10. J. W. EVANS, Nerve axon equations, IV: The stable and unstable impulse, Indiana Univ. Math. J., 24 (1975), pp. 1169-1190.
11. T. J. BRIDGES AND G. DERKS, The symplectic Evans matrix, and the instability of solitary waves and fronts with symmetry, Arch. Ration. Mech. Anal., 156 (2001), pp. 1-87.
12. A. L. AFENDIKOV AND T. J. BRIDGES, Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), pp. 1-16.
13. L. A. ALLEN AND T. J. BRIDGES, Numerical exterior algebra and the compound matrix method, Numer. Math., 92 (2002), pp. 197-232.
14. N. J. BALMFORTH, R. V. CRASTER, AND S. J. A. MALHAM, Unsteady fronts in an autocatalytic system, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), pp. 1401-1433.
15. D. TERMAN, Stability of planar wave solutions to a combustion model, SIAM J. Math. Anal., 21 (1990), pp. 1139-1171.
16. R. L. PEGO AND M. I. WEINSTEIN, Evans' function, Melnikov's integral, and solitary wave instabilities, in Differential Equations with Applications to Mathematical Physics, W. F. Ames, E. M. Harrell II, and J. V. Herod, eds., Academic Press, San Diego, 1993, pp. 273-286.
17. D. E. PELINOVSKY, Y. S. KIVSHAR, AND V.V. AFANASJEV, Internal modes of envelope solitons, Phys. D, 116 (1998), pp. 121-142.
18. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
19. R. L. PEGO, P. SMEREKA, AND M. I. WEINSTEIN, Oscillatory instability of traveling waves for a KdV-Burgers equation, Phys. D, 67 (1993), pp. 45-65.
20. B. S. NG AND W. H. REID, An initial value method for eigenvalue problems using compound matrices, J. Comput. Phys., 30 (1979), pp. 125-136.
21. B. S. NG AND W. H. REID, The compound matrix method for ordinary differential equation, J. Comput. Phys., 58 (1985), pp. 209-228.
22. P. C. DRAZIN AND W. H. REID, Hydrodynamic Stability, Cambridge University Press, London, 1981.
23. V. GUBERNOV, G. N. MERCER, H. S. SIDHU, AND R. O. WEBER, Numerical methods for the analysis of travelling waves in reaction-diffusion equations, ANZIAM J(E), to appear.
24. A. I. VOLPERT, V. A. VOLPERT, AND V. A. VOLPERT, Travelling Wave Solutions of Parabolic Systems. Transl. Math. Monogr. 140, AMS, Providence, RI, 1994.
25. J. ALEXANDER, R. GARDNER, AND C. JONES, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., 410 (1990), pp. 167-212.