Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion
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 te...
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doaj-dab4a4e6d6724730aac5fcda61b7f0d92020-11-24T22:48:17ZengMDPI AGSymmetry2073-89942018-03-011037210.3390/sym10030072sym10030072Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–DiffusionPhilip Broadbridge0Dimetre Triadis1Dilruk Gallage2Pierluigi Cesana3Department of Mathematics and Statistics, La Trobe University, Bundoora, VIC 3086, AustraliaInstitute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka 819-0395, JapanDepartment of Mathematics and Statistics, La Trobe University, Bundoora, VIC 3086, AustraliaInstitute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka 819-0395, JapanUsing 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.http://www.mdpi.com/2073-8994/10/3/72fourth-order diffusionAllen–Cahn equationCahn–Hilliard equationphase fieldnonlinear reaction–diffusion |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Philip Broadbridge Dimetre Triadis Dilruk Gallage Pierluigi Cesana |
spellingShingle |
Philip Broadbridge Dimetre Triadis Dilruk Gallage Pierluigi Cesana Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion Symmetry fourth-order diffusion Allen–Cahn equation Cahn–Hilliard equation phase field nonlinear reaction–diffusion |
author_facet |
Philip Broadbridge Dimetre Triadis Dilruk Gallage Pierluigi Cesana |
author_sort |
Philip Broadbridge |
title |
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion |
title_short |
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion |
title_full |
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion |
title_fullStr |
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion |
title_full_unstemmed |
Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion |
title_sort |
nonclassical symmetry solutions for fourth-order phase field reaction–diffusion |
publisher |
MDPI AG |
series |
Symmetry |
issn |
2073-8994 |
publishDate |
2018-03-01 |
description |
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. |
topic |
fourth-order diffusion Allen–Cahn equation Cahn–Hilliard equation phase field nonlinear reaction–diffusion |
url |
http://www.mdpi.com/2073-8994/10/3/72 |
work_keys_str_mv |
AT philipbroadbridge nonclassicalsymmetrysolutionsforfourthorderphasefieldreactiondiffusion AT dimetretriadis nonclassicalsymmetrysolutionsforfourthorderphasefieldreactiondiffusion AT dilrukgallage nonclassicalsymmetrysolutionsforfourthorderphasefieldreactiondiffusion AT pierluigicesana nonclassicalsymmetrysolutionsforfourthorderphasefieldreactiondiffusion |
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1725678799005155328 |