Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension

1993 ◽  
Vol 44 (4) ◽  
pp. 707-721 ◽  
Author(s):  
J. H. Merkin ◽  
D. J. Needham
Keyword(s):  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Chemical System ◽  
Diffusion Waves

scholarly journals Coupled reaction-diffusion waves in a chemical system via fractional derivatives in Liouville-Caputo sense

2018 ◽  
Vol 64 (5) ◽  
pp. 539 ◽  
Author(s):  
Francisco Gomez ◽  
Khaled Saad
Keyword(s):  
Fractional Derivatives ◽  
Error Function ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Chemical System ◽  
Transform Method ◽  
Diffusion Waves ◽  
Homotopy Analysis ◽  
Optimal Values ◽  
Coupled Reaction

In this paper, we have generalized the fractional cubic isothermal auto-catalytic chemical system (FCIACS) with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional time derivatives, respectively. We apply the Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of FCIACS using these fractional derivatives. We study the convergence analysis of HATM by computing the residual error function. Also, we find the optimal values of h so we assure the convergence of the approximate solutions. Finally we show the behavior of the approximate solutions graphically. The results obtained are very effectiveness and accuracy.

The development of travelling waves in a simple isothermal chemical system. IV. Quadratic autocatalysis with quadratic decay

1991 ◽  
Vol 434 (1892) ◽  
pp. 531-554 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Travelling Wave ◽  
Chemical System ◽  
Governing Equations ◽  
Initial Value ◽  
Dimensional Parameter ◽  
Diffusion Waves ◽  
Permanent Form ◽  
Bounds On The Solution

The initiation of travelling reaction-diffusion waves in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate k 1 ab) coupled with the decay or termination step B + B → C (rate k 4 b 2 ) is discussed. The system is described by the non-dimensional parameter K - k 4 / k 1 and parameters representing the local initial input of B. It is shown that a travelling wave of permanent form will develop for all K (and no matter how small the initial input of B). Bounds on the solution of the initial-value problem are obtained as well as numerical integrations of the governing equations. The structure of the permanent form travelling waves that arise is discussed in some detail, as well as the asymptotic limits K → 0 and K → ∞. The behaviour of the solution for this problem is compared with solutions found previously for other related simple autocatalytic systems with autocatalyst decay.

The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves

1991 ◽  
Vol 334 (1633) ◽  
pp. 1-24 ◽  
Keyword(s):  
Travelling Waves ◽  
Chemical Species ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Spatial Dimension ◽  
Travelling Wave ◽  
Diffusion Waves ◽  
Long Time ◽  
Diffusion Rates ◽  
Cubic Autocatalysis

We study the isothermal autocatalytic system , A + n B → ( n + 1)B , where n = 1 or 2 for quadratic or cubic autocatalysis respectively. In addition, we allow the chemical species, A and B, to have unequal diffusion rates. The propagating reaction-diffusion waves that may develop from a local initial input of the autocatalyst, B, are considered in one spatial dimension. We find that travelling wave solutions exist for all propagation speeds v ≥ v * n ,where v * n is a function of the ratio of the diffusion rates of the species A and B and represents the minimum propagation speed. It is also shown that the concentration of the autocatalyst, B, decays exponentially ahead of the wavefront for quadratic autocatalysis. However, for cubic autocatalysis, although the concentration of the autocatalyst decays exponentially ahead of the wavefront for travelling waves which propagate at speed v = v * 2 , this rate of decay is only algebraic for faster travelling waves with v > v * 2 . This difference in decay rate has implications for the selection of the long time wave speed when such travelling waves are generated from an initial-value problem.

Analysis of reaction–diffusion waves in the ferroin-catalysed Belousov–Zhabotinsky reaction

1999 ◽  
Vol 1 (19) ◽  
pp. 4595-4599 ◽  
Author(s):  
Annette F. Taylor ◽  
Vilmos Gáspár ◽  
Barry R. Johnson ◽  
Stephen K. Scott
Keyword(s):  
Reaction Diffusion ◽  
Diffusion Waves ◽  
Belousov Zhabotinsky Reaction

The development of travelling waves in a simple isothermal chemical system III. Cubic and mixed autocatalysis

1990 ◽  
Vol 430 (1880) ◽  
pp. 509-524 ◽  
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Chemical System ◽  
Mixed System ◽  
Dimensional Parameter ◽  
Diffusion Wave ◽  
Symmetrical Form ◽  
Asymptotic Speed ◽  
Mixed Systems ◽  
Cubic Autocatalysis

Autocatalytic chemical reactions can support isothermal travelling waves of constant speed and form. This paper extends previous studies to cubic autocatalysis and to mixed systems where quadratic and cubic autocatalyses occur concurrently. A + B → 2B, rate = k q ab , (1) A + 2B → 3B, rate = k c ab 2 . (2) For pure cubic autocatalysis the wave has, at large times, a constant asymptotic speed v 0 (where v 0 = 1/√2 in the appropriate dimensionless units). This result is confirmed by numerical investigation of the initial-value problem. Perturbations to this stable wave-speed decay at long times as t -3/2 e -1/8 t . The mixed system is governed by a non-dimensional parameter μ = k q / k c a 0 which measures the relative rates of transformation by quadratic and cubic modes. In the mixed case ( μ ≠ 0) the reaction-diffusion wave has a form appropriate to a purely cubic autocatalysis so long as μ lies between ½ and 0. When μ exceeds ½, the reaction wave loses its symmetrical form, and all its properties steadily approach those of quadratic autocatalysis. The value μ = ½ is the value at which rates of conversion by the two paths are equal.

Reaction-diffusion in a simple pooled chemical system

1989 ◽  
Vol 4 (2) ◽  
pp. 141-167 ◽  
Author(s):  
J. H. Merkin ◽  
D. J. Needham
Keyword(s):  
Reaction Diffusion ◽  
Chemical System
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