# Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonclassical Symmetries

#### 2.1. Nonlinear Diffusion–Convection Equation

#### 2.2. Nonlinear Wave Equation

#### Case (i) ${X}^{2}\ne F$

#### Case (ii) ${X}^{2}=F$

## 3. $\mathit{T}=\mathbf{0}$

#### 3.1. Nonlinear Diffusion Equation

#### 3.2. Nonlinear Wave Equation

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Lie, S. Klassifikation und Integration von gewohnlichen Differentialgleichen zwischen x, y die eine Gruppe von Transformationen gestatten. Math. Ann.
**1888**, 32, 213–281. [Google Scholar] [CrossRef] - Arrigo, D.J. Symmetries Analysis of Differential Equations—An Introduction; Wiley: New York, NY, USA, 2015. [Google Scholar]
- Bluman, G.; Kumei, S. Symmetries and Differential Equations; Springer: New York, NY, USA, 1989. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations, 2nd ed.; Springer: New York, NY, USA, 1993. [Google Scholar]
- Ovsiannikov, L.V. Group properties of nonlinear heat equation. Dokl. AN SSSR
**1959**, 125, 492–495. [Google Scholar] - Ovsiannikov, L.V. Group Analysis of Differential Equations; Academic Press: New York, NY, USA, 1982. [Google Scholar]
- Bluman, G.W.; Reid, G.J.; Kumei, S. New classes of symmetries for partial differential equations. J. Math. Phys.
**1988**, 29, 806–881. [Google Scholar] [CrossRef] - Bluman, G.W.; Cole, J.D. The general similarity solution of the heat equation. J. Math. Phys.
**1969**, 18, 1025–1042. [Google Scholar] - Bradshaw-Hajek, B.H.; Edwards, M.P.; Broadbridge, P.; Williams, G.H. Nonclassical symmetry solutions for reaction diffusion equations with explicit spatial dependence. Nonliner Anal.
**2007**, 67, 2541–2552. [Google Scholar] [CrossRef] - Cherniha, R. New Q-conditional symmetries and exact solutions of some reaction- diffusion-convection equations arising in mathematical biology. J. Math. Anal. Appl.
**2007**, 326, 783–799. [Google Scholar] [CrossRef] - Popovych, R.O.; Vaneeva, O.O.; Ivanova, N.M. Potential nonclassical symmetries and solutions of fast diffusion equation. Phys. Lett. A
**2007**, 362, 166–173. [Google Scholar] [CrossRef] - Bruzon, M.S.; Gandarias, M.L. Applying a new algorithm to derive nonclassical symmetries. Commun. Nonlinear Sci. Numer. Simul.
**2008**, 13, 517–523. [Google Scholar] [CrossRef] - Arrigo, D.J.; Ekrut, D.A.; Fliss, J.R.; Le, L. Nonclassical symmetries of a class of Burgers’ systems. J. Math. Anal. Appl.
**2010**, 371, 813–820. [Google Scholar] [CrossRef] - Bluman, G.W.; Tian, S.F.; Yang, Z. Nonclassical analysis of the nonlinear Kompaneets equation. J. Eng. Math.
**2014**, 84, 87–97. [Google Scholar] [CrossRef] - Cherniha, R.; Davydovych, V. Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with nonconstant diffusivities, Commun. Nonlinear Sci. Numer. Simulat.
**2012**, 17, 3177–3188. [Google Scholar] [CrossRef] - Hashemi, M.S.; Nucci, M.C. Nonclassical symmetries for a class of reaction-diffusion equations: The method of heir-equations. J. Non. Math Phys.
**2012**, 20, 44–60. [Google Scholar] [CrossRef] - Huang, D.J.; Zhou, S. Group-theoretical analysis of variable coefficient nonlinear telegraph equations. Acta Appl. Math.
**2012**, 117, 135–183. [Google Scholar] [CrossRef] - Vaneeva, O.O.; Popovych, R.O.; Sophocleous, C. Extended group analysis of variable coefficient reaction diffusion equations with exponential nonlinearities. J. Math. Anal. Appl.
**2012**, 396, 225–242. [Google Scholar] [CrossRef] - Broadbridge, P.; Bradshaw-Hajek, B.H.; Triadis, D. Exact non-classical symmetry solutions of Arrhenius reaction-diffusion. Proc. R. Soc. Lond.
**2015**, 471. [Google Scholar] [CrossRef] - Louw, K.; Moitsheki, R.J. Group-invariant solutions for the generalised fisher type equation. Nat. Sci.
**2015**, 7, 613–624. [Google Scholar] [CrossRef] - Pliukhin, O. Q-conditional symmetries and exact solutions of nonlinear reaction-diffusion systems. Symmetry
**2015**, 7, 1841–1855. [Google Scholar] [CrossRef] - Yun, Y.; Temuer, C. Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation. Appl. Math. Mech. Eng. Ed.
**2015**, 36, 365–378. [Google Scholar] [CrossRef] - Broadbridge, P.; Bradshaw-Hajek, B.H. Exact solutions for logistic reaction-diffusion equations in biology. ZAMP
**2016**. [Google Scholar] [CrossRef] - Bluman, G.W.; Yan, Y.S. Nonclassical potential solutions of partial differential equations. Eur. J. Appl. Math.
**2005**, 16, 239–261. [Google Scholar] [CrossRef] - Gandarias, M.L.; Bruzon, M.S. Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation. Math. Meth. Appl. Sci.
**2008**, 31, 753–767. [Google Scholar] [CrossRef] - Broadbridge, P.; White, I. Constant rate rainfall infiltration: A versatile nonlinear model 1. Analytic solution. Water Res. Res.
**1988**, 24, 145–154. [Google Scholar] [CrossRef] - Rogers, C.; Stallybrass, M.P.; Clements, D.L. On two phase filtration under gravity and with boundary infiltration: Application of a Ba äcklund transformation. J. Nonliner Anal. Meth. Appl.
**1983**, 7, 785–799. [Google Scholar] [CrossRef] - Katayev, I.G. Electromagnetic Shock Waves; Iliffe: London, UK, 1966. [Google Scholar]
- Jeffery, A. Acceleration wave propagation in hyperelastic rods of variable cross-section. Wave Motion
**1982**, 4, 173–180. [Google Scholar] [CrossRef] - Broadbridge, P.; Arrigo, D.J. All solutions of standard symmetric linear partial differential equations have classical Lie symmetry. J. Math. Anal. Appl.
**1999**, 234, 109–122. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M. Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms. Eur. J. Appl. Math.
**1998**, 72, 21–39. [Google Scholar] [CrossRef] - Arrigo, D.J.; Hill, J.M. Nonclassical symmetries for nonlinear diffusion and absorption. Stud. Appl. Math.
**1995**, 72, 21–39. [Google Scholar] [CrossRef] - Arrigo, D.J.; Beckham, J.R. Nonclassical symmetries of evolutionary partial differential equations and compatibility. J. Math. Anal. Appl.
**2004**, 289, 55–65. [Google Scholar] [CrossRef] - Niu, X.H.; Pan, Z.L. Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility. J. Math. Anal. Appl.
**2005**, 311, 479–488. [Google Scholar] [CrossRef] - Wan, W.T.; Chen, Y. A note on nonclassical symmetries of a class of nonlinear partial differential equations and compatibility. Commun. Theor. Phys.
**2009**, 52, 398–402. [Google Scholar] - El-Sabbagh, M.F.; Ali, A.T. Nonclassical symmetries for nonlinear partial differential equations via compatibility. Commun. Theor. Phys.
**2011**, 56, 611–616. [Google Scholar] [CrossRef] - Bluman, G.W.; Shtelen, V. Developments in similarity methods related to pioneering work of Julian Cole. In Mathematics Is for Solving Problems; Cook, S.L.P., Roytburd, V., Tulin, M., Eds.; SIAM: Philadelphia, PA, USA, 1996; pp. 105–118. [Google Scholar]
- Näslund, R.N. On Conditional Q-Symmetries of Some Quasi-Linear Hyperbolic Wave Equations; Reprint Department of Mathematics, Lulea University of Technology: Lulea, Sweden, 2003. [Google Scholar]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Arrigo, D.J.; Ashley, B.P.; Bloomberg, S.J.; Deatherage, T.W.
Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems. *Symmetry* **2016**, *8*, 140.
https://doi.org/10.3390/sym8120140

**AMA Style**

Arrigo DJ, Ashley BP, Bloomberg SJ, Deatherage TW.
Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems. *Symmetry*. 2016; 8(12):140.
https://doi.org/10.3390/sym8120140

**Chicago/Turabian Style**

Arrigo, Daniel J., Brandon P. Ashley, Seth J. Bloomberg, and Thomas W. Deatherage.
2016. "Nonclassical Symmetries of a Nonlinear Diffusion–Convection/Wave Equation and Equivalents Systems" *Symmetry* 8, no. 12: 140.
https://doi.org/10.3390/sym8120140