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Authors = Pierluigi Cesana

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18 pages, 1258 KiB  
Article
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion
by Philip Broadbridge, Dimetre Triadis, Dilruk Gallage and Pierluigi Cesana
Symmetry 2018, 10(3), 72; https://doi.org/10.3390/sym10030072 - 17 Mar 2018
Cited by 10 | Viewed by 4031
Abstract
Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing [...] Read more.
Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing phase of cylindrical or spherical inclusion, allowing for a “mushy” zone with a mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions. Full article
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