Reaction-Diffusion Waves in a Simple Isothermal Chemical System

  • J. H. Merkin
  • D. J. Needham
Part of the Research Reports in Physics book series (RESREPORTS)

Abstract

Previous work on reaction-diffusion waves in the simple isothermal autocatalytic systems
$$\begin{array}{*{20}{c}} {quadratic:}&{A + B \Rightarrow 2B}&{rate}&{{k_1}ab} \end{array}$$
(1a)
$$\begin{array}{*{20}{c}} {cubic:}&{A + 2B \Rightarrow 3B}&{rate}&{{k_2}a{b^2}} \end{array}$$
(1b)
(and mixed schemes combining (la) and (lb)) has assumed that the catalyst B can exist indefinitely, see, for example [1–8]. Here we allow for the finite lifetime of catalyst B which we assume decays to the inert product C via
$$\begin{array}{*{20}{c}} {linear:}&{B \Rightarrow C}&{rate}&{{k_2}b} \end{array}$$
(2a)
or
$$\begin{array}{*{20}{c}} {quadratic:}&{B + B \Rightarrow C}&{rate}&{{k_q}{b^2}} \end{array}$$
(2b)
(here a and b are the concentrations of A and B and the ki are the rate constants).

Keywords

Rosen Acoustics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Fisher, R.A. Ann. of Eugenics 7 (1937) 355–369.CrossRefGoogle Scholar
  2. 2.
    Kolmogorov, A, Petrovsky, I and Piscounov, N Moscow Universitet. Bull. Math. 1 (1937) 1–25.Google Scholar
  3. 3.
    McKean, H.P. Comm. Pure Appl. Math. 28 (1975) 323–331.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bramson, M.D. Comm. Pure Appl. Math. 31 (1978) 531–581.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bramson, M.D. American Math. Soc. Memoir 285 (1983).Google Scholar
  6. 6.
    Rosen, G. Bull. Math. Biology 42 (1980) 95–106.MATHGoogle Scholar
  7. 7.
    Gray, P, Showalter, K. and Scott, S.K. J. de Chemie Physique 84 (1987) 1329–1333.Google Scholar
  8. 8.
    Merkin, J.H. and Needham, D.J. Submitted to J. Engng. Math. (1988).Google Scholar
  9. 9.
    Gray, P. and Scott, S.K. Chem. Engng. Sci. 38 (1983) 29–43.CrossRefGoogle Scholar
  10. 10.
    Gray, P. and Scott, S.K. Chem. Engng. Sci. 39 (1984) 1087–1097.CrossRefGoogle Scholar
  11. 11.
    D’Anna, A., Lignola, P.G. and Scott, S.K. Proc. Roy. Soc. Lond. A403 (1986) 341–363.MathSciNetADSGoogle Scholar
  12. 12.
    Gray, B.F. and Roberts, M.J. Proc. Roy. Soc. Lond. A416 (1988) 403–424.MathSciNetADSGoogle Scholar
  13. 13.
    Merkin, J.H., Needham, D.J. and Scott, S.K. J. Engng. Math. 21 (1987) 115–127.CrossRefGoogle Scholar
  14. 14.
    Merkin, J.H., Needham, D.J. and Scott, S.K. S.I. A.M. J. Appl. Math. 47 (1987) 1040–1060.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Merkin, J.H., Needham, D.J. and Scott, S.K. Proc. Roy. Soc. Lond. A406 (1986) 299–323.ADSGoogle Scholar
  16. 16.
    Gray, B.F., Roberts, M.J. and Merkin, J.H. J. Engng. Math. 22 (1988) 267–284.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gray, P. Proc. Roy. Soc. Lond. A415 (1988) 1–34.ADSGoogle Scholar
  18. 18.
    Slater, L. Confluent Hypergeometric Functions, Cambridge University Press (1960).Google Scholar
  19. 19.
    Protter, M.H. and Weinberger, H.F. Maximum Principles in Differential Equations. Prentice-Hall (1967).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • J. H. Merkin
    • 1
  • D. J. Needham
    • 2
  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.School of MathematicsUniversity of East AngliaNorwichUK

Personalised recommendations