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Cubic autocatalysis in a reaction–diffusion annulus: semi-analytical solutions

Journal Article


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Abstract


  • Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction–diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.

Publication Date


  • 2016

Citation


  • Alharthi, M. R., Marchant, T. R. & Nelson, M. I. (2016). Cubic autocatalysis in a reaction–diffusion annulus: semi-analytical solutions. Zeitschrift fur Angewandte Mathematik und Physik, 67 (3), 65-1-65-13.

Scopus Eid


  • 2-s2.0-84969544813

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6744&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5716

Start Page


  • 65-1

End Page


  • 65-13

Volume


  • 67

Issue


  • 3

Abstract


  • Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction–diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.

Publication Date


  • 2016

Citation


  • Alharthi, M. R., Marchant, T. R. & Nelson, M. I. (2016). Cubic autocatalysis in a reaction–diffusion annulus: semi-analytical solutions. Zeitschrift fur Angewandte Mathematik und Physik, 67 (3), 65-1-65-13.

Scopus Eid


  • 2-s2.0-84969544813

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6744&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5716

Start Page


  • 65-1

End Page


  • 65-13

Volume


  • 67

Issue


  • 3