# Conditionally Integrable Nonlinear Diffusion with Diffusivity 1/u

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Symmetry Group, after Rescaling

## 3. Infinite-Dimensional Class of Exact Solutions

## 4. Properties of Solutions

#### 4.1. Axisymmetric Solutions

#### 4.2. Image Source Solution

#### 4.3. A Solution That Is Bounded on ${\mathbb{R}}^{2}\times {\mathbb{R}}^{+}$

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zakharov, V.E. (Ed.) What Is Integrability? Springer: Berlin, Germany, 1991. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations; AMS No 107; Springer: New York, NY, USA, 1982. [Google Scholar]
- Bluman, G.W.; Kumei, S. Symmetries and Differential Equations; Springer: New York, NY, USA, 1989. [Google Scholar]
- Ibragimov, N.H. (Ed.) Lie Group Analysis of Differential Equations; CRC: London, UK, 1994; Volume 1. [Google Scholar]
- Galaktionov, V.A.; Dorodnitsyn, V.A.; Elenin, G.G.; Kurdyumov, S.P.; Samarskii, A.A. A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures. J. Sov. Math.
**1988**, 41, 1222–1292. [Google Scholar] [CrossRef] - Carslaw, H.S.; Jaeger, J.C. Conduction of Heat in Solids; Oxford University Press: London, UK, 1959. [Google Scholar]
- Adamyan, V.M.; Djuric, Z.; Ermolaev, A.M.; Mihajlov, A.A.; Tkachenko, I.M. Kinetic coefficients of fully ionized plasmas. J. Phys. D Appl. Phys.
**1994**, 27, 927–933. [Google Scholar] [CrossRef] - Goard, J.M.; Broadbridge, P. Nonclassical Symmetry Analysis of Nonlinear Reaction- Diffusion Equations in Two Spatial Dimensions. Nonlinear Anal. Theory Methods Appl.
**1996**, 26, 735–754. [Google Scholar] [CrossRef] - Ivanova, N.M.; Broadbridge, P. Solutions and reductions for radiative energy transport in laser-heated plasma. J. Math. Phys.
**2015**, 56, 011503. [Google Scholar] [Green Version] - Smith, R.E.; Smettem, K.R.J.; Broadbridge, P.; Woolhiser, D.A. Infiltration Theory for Hydrologic Applications; American Geophysical Union: Washington, DC, USA, 2002. [Google Scholar]
- Warrick, A.W. Soil Water Dynamics; Oxford University Press: New York, NY, USA, 2003. [Google Scholar]
- Broadbridge, P.; White, I. Constant Rate Rainfall Infiltration: A Versatile Non-linear Model: 1. Analytic Solution. Water Resour. Res.
**1988**, 24, 145–154. [Google Scholar] [CrossRef] - Knight, J.H.; Philip, J.R. Exact solutions in nonlinear diffusion. J. Eng. Math.
**1974**, 8, 219–227. [Google Scholar] [CrossRef] - Broadbridge, P.; Rogers, C. Exact Solutions for Vertical Drainage and Redistribution in Soils. J. Eng. Math.
**1990**, 24, 25–43. [Google Scholar] [CrossRef] - Munier, A.; Burgan, J.R.; Gutierrez, J.; Fijalkow, E.; Feix, M.R. Group Transformations and the Nonlinear Heat Diffusion Equation. SIAM J. Appl. Math.
**1981**, 40, 191–207. [Google Scholar] [CrossRef] - Kingston, J.G.; Rogers, C. Reciprocal Bäcklund transformations of conservation laws. Phys. Lett. A
**1982**, 92, 261–264. [Google Scholar] [CrossRef] - Bluman, G.W.; Kumei, S. Symmetry-based algorithms to relate partial differential equations: I. Local symmetries. Eur. J. Appl. Math.
**1990**, 1, 189–216. [Google Scholar] [CrossRef] [Green Version] - Broadbridge, P.; Bradshaw-Hajek, B.H.; Triadis, D. Exact non-classical symmetry solutions of Arrhenius reaction-diffusion. Proc. R. Soc. A
**2015**, 471, 20150580. [Google Scholar] [CrossRef] - Liouville, J. Sur l’équation aux differencées partielles ∂
^{2}logλ/∂u∂v ± 2λa^{2}= 0. J. Math.**1853**, 18, 71–72. [Google Scholar] - Crowdy, D.G. General solutions to the 2D Liouville equation. Int. J. Eng. Sci.
**1997**, 35, 141–149. [Google Scholar] [CrossRef] - Maple 18, Maplesoft; A Division of Waterloo Maple Inc.: Waterloo, ON, Canada, 2014.
- Rutherford, D.E. Fluid Dynamics; Oliver and Boyd: Edinburgh, UK, 1959. [Google Scholar]
- Popov, A.G. Exact formulas for constructing solutions of the Liouville equation Δ
_{2}u = e^{u}from solutions of the Laplace equation Δ_{2}v = 0. Dokl. Akad. Nauk**1993**, 333, 440–441. English translation: Acad. Sci. Dokl. Math.**1994**, 48, 570–572. (In Russian) [Google Scholar]

**Figure 1.**Radial flow from a point source to a container at ${r}_{0}$ = 0.33. p = 0.5. The analytic solution is compared to the numerical solution from the MAPLE routine pdsolve with flux boundary conditions prescribed at r = 0.1.

**Figure 2.**Contours of scaled density $u/({t}_{0}-t)$ from Equation (44) with p = 1.5. Contour levels vary from zero at $r=0$ and $(r,\varphi )=(1,\pi /3)$ to 29.24 at $(r,\varphi )=(0.585,0)$.

© 2019 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**

Broadbridge, P.; Goard, J.M.
Conditionally Integrable Nonlinear Diffusion with Diffusivity 1/*u*. *Symmetry* **2019**, *11*, 804.
https://doi.org/10.3390/sym11060804

**AMA Style**

Broadbridge P, Goard JM.
Conditionally Integrable Nonlinear Diffusion with Diffusivity 1/*u*. *Symmetry*. 2019; 11(6):804.
https://doi.org/10.3390/sym11060804

**Chicago/Turabian Style**

Broadbridge, Philip, and Joanna M. Goard.
2019. "Conditionally Integrable Nonlinear Diffusion with Diffusivity 1/*u*" *Symmetry* 11, no. 6: 804.
https://doi.org/10.3390/sym11060804