EXACT SOLUTIONS OF HYPERBOLIC REACTION-DIFFUSION EQUATIONS IN TWO DIMENSIONS

Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz eq...

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Bibliographic Details
Published in:The ANZIAM journal 2022-10, Vol.64 (4), p.338-354
Main Authors: BROADBRIDGE, P., GOARD, J.
Format: Article
Language:English
Subjects:
Boundary conditions
Exact solutions
Heat
Helmholtz equations
Mathematical analysis
Partial differential equations
Propagation
Reaction-diffusion equations
Response time
Speed limits
Symmetry
Temperature
Time lag
Variables
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Summary:Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay, $\tau $ , between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on $\tau $ . A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive $\tau $ .
ISSN:1446-1811
1446-8735
DOI:10.1017/S1446181123000093