Next Article in Journal
Intelligent Image Processing System for Detection and Segmentation of Regions of Interest in Retinal Images
Next Article in Special Issue
Analytic Solutions of Nonlinear Partial Differential Equations by the Power Index Method
Previous Article in Journal
Ensemble Genetic Fuzzy Neuro Model Applied for the Emergency Medical Service via Unbalanced Data Evaluation
Previous Article in Special Issue
Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients
Article Menu
Issue 3 (March) cover image

Export Article

Open AccessArticle
Symmetry 2018, 10(3), 72; https://doi.org/10.3390/sym10030072

Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion

1
Department of Mathematics and Statistics, La Trobe University, Bundoora, VIC 3086, Australia
2
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Fukuoka 819-0395, Japan
*
Author to whom correspondence should be addressed.
Received: 18 January 2018 / Revised: 6 March 2018 / Accepted: 12 March 2018 / Published: 17 March 2018
Full-Text   |   PDF [1258 KB, uploaded 17 March 2018]   |  

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 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. View Full-Text
Keywords: fourth-order diffusion; Allen–Cahn equation; Cahn–Hilliard equation; phase field; nonlinear reaction–diffusion fourth-order diffusion; Allen–Cahn equation; Cahn–Hilliard equation; phase field; nonlinear reaction–diffusion
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Broadbridge, P.; Triadis, D.; Gallage, D.; Cesana, P. Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion. Symmetry 2018, 10, 72.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top