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

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## Abstract

**:**

## 1. Introduction

## 2. Nonclassical Symmetry Reduction

#### 2.1. Role of a Fourth-Order Kirchhoff–Helmholtz Equation

#### 2.2. Amenable Diffusivity and Reaction Functions

## 3. Interior Solutions for Slabs, Cylinders, and Spheres

## 4. Interior Solutions for Rectangular Domains

## 5. Energy Formulation

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Iterated approximations to the diffusivity function. ${D}_{1}$ (solid), ${D}_{2}$ (dash-dot) with $\kappa =1$, $A=-5/2$.

**Figure 2.**Cubic reaction term R (solid). Exact matching partner ${R}_{1}$ (dash-dot) for ${D}_{1}$ with $A=-5/2$.

**Figure 5.**Critical combinations of ${x}_{0}$ and ${y}_{0}$ separating the feasible rectangular solution region from the region where ${x}_{0}$ and ${y}_{0}$ in combination do not admit physical solutions.

**Figure 6.**A non-negative solution $\varphi (x,y)$ with $\kappa =1$, ${x}_{0}=0.3$, and ${y}_{0}=1.2$ that exhibits a pronounced sub-structure.

**Figure 7.**A non-negative solution $\varphi (x,y)$, periodic in the y direction with $\kappa =1$, and ${x}_{0}=0.4$.

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**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.
https://doi.org/10.3390/sym10030072

**AMA Style**

Broadbridge P, Triadis D, Gallage D, Cesana P. Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion. *Symmetry*. 2018; 10(3):72.
https://doi.org/10.3390/sym10030072

**Chicago/Turabian Style**

Broadbridge, Philip, Dimetre Triadis, Dilruk Gallage, and Pierluigi Cesana. 2018. "Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion" *Symmetry* 10, no. 3: 72.
https://doi.org/10.3390/sym10030072