# Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Lie Symmetry

- (1)
- the Lie symmetry algebras are maximal algebras of invariance (MAIs) of the relevant PDEs from the list obtained;
- (2)
- all PDEs from the list are inequivalent with respect to a set of ETs;
- (3)
- any other PDE from the class that admits a nontrivial Lie symmetry algebra is reduced by an ET from the set to one of those from the list.

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Remark**

**1.**

**Definition**

**3.**

**Theorem**

**2.**

**Remark**

**2.**

**Definition**

**4.**

**Theorem**

**3.**

**Remark**

**3.**

**Proof.**

- (1)
- $\lambda ={\lambda}_{1}=0,$
- (2)
- $\lambda {\lambda}_{1}\ne 0,$
- (3)
- $\lambda \ne 0,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}=0$

- (a)
- $({\lambda}_{3},{\lambda}_{4})=({\lambda}_{3},0),\phantom{\rule{4pt}{0ex}}{\lambda}_{3}\ne 0,$
- (b)
- $({\lambda}_{3},{\lambda}_{4})=(0,{\lambda}_{4}),\phantom{\rule{4pt}{0ex}}{\lambda}_{4}\ne 0,$
- (c)
- $({\lambda}_{3},{\lambda}_{4})\ne (0,0).$

**Theorem**

**4.**

## 3. Lie’s Solutions of an RDC Equation with Exponential Nonlinearities

**Remark**

**4.**

## 4. Q-Conditional Symmetries of an RDC Equation with Exponential Nonlinearities

**a**) in what follows. The natural reason to avoid examination of case (

**b**) (so-called no-go case) follows from the well-known fact (firstly proved in [32]) that a complete description of Q-conditional symmetries of the form (44) for scalar evolution equations is equivalent to solving the equation in question.

**Definition**

**5.**

**Theorem**

**5.**

**Proof.**

**Remark**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

**Remark**

**6.**

**Remark**

**7.**

**Theorem**

**8.**

**Proof.**

## 5. Non-Lie Solutions

## 6. Application of an Exact Solution for Solving a Boundary-Value Problem Arising in Population Dynamics

**Theorem**

**9.**

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Boussinesq, J. Recherches théoriques sur l’écoulement des nappes d’eau infiltrées dans le sol et sur débit de sources. J. Math. Pures Appl.
**1904**, 10, 5–78. [Google Scholar] - Storm, M.L. Heat conduction in simple metals. J. Appl. Phys.
**1951**, 22, 940–951. [Google Scholar] [CrossRef] - Burgers, J.M. The Nonlinear Diffusion Equation: Asymptotic Solutions and Statistical Problems; D. Reidel Publishing Company: Dordrecht, Holland, 1974; p. x+173. [Google Scholar]
- Cole, J.D. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math.
**1951**, 9, 225–236. [Google Scholar] [CrossRef] - Hopf, E. The partial differential equation u
_{t}+ uu_{x}= μu_{xx}. Commun. Pure Appl. Math.**1950**, 3, 201–230. [Google Scholar] [CrossRef] - Rosen, G. Nonlinear heat conduction in solid H
_{2}. Phys. Rev. B**1979**, 19, 2398–2399. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M.; Pliukhin, O. Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications; Chapman & Hall/CRC Monographs and Research Notes in Mathematics; CRC Press: Boca Raton, FL, USA, 2018; p. xx+240. [Google Scholar]
- Frank-Kamenetskii, D. Diffusion and Heat Transfer in Chemical Kinetics; Plenum Press: New York, NY, USA, 1969. [Google Scholar]
- Fujita, H. On the nonlinear equations Δu + e
^{u}= 0 and ∂v/∂t = Δv + e^{v}. Bull. Amer. Math. Soc.**1969**, 75, 132–135. [Google Scholar] [CrossRef] - Dorodnitsyn, V.A. Invariant solutions of the nonlinear heat equation with a source. Zh. Vychisl. Mat. Mat. Fiz.
**1982**, 22, 1393–1400. [Google Scholar] - Broadbridge, P.; Bradshaw-Hajek, B.H.; Triadis, D. Exact non-classical symmetry solutions of Arrhenius reaction-diffusion. Proc. R. Soc. A
**2015**, 471. [Google Scholar] [CrossRef] - Ovsiannikov, L.V. Group relations of the equation of non-linear heat conductivity. Dokl. Akad. Nauk SSSR
**1959**, 125, 492–495. [Google Scholar] - Dai, H.; Zhang, H. Energy decay and nonexistence of solution for a reaction-diffusion equation with exponential nonlinearity. Bound. Value Probl.
**2014**, 2014, 70. [Google Scholar] [CrossRef] - Pulkkinen, A. Blow-up profiles of solutions for the exponential reaction-diffusion equation. Math. Methods Appl. Sci.
**2011**, 34, 2011–2030. [Google Scholar] [CrossRef] - Ioku, N. The Cauchy problem for heat equations with exponential nonlinearity. J. Differ. Equ.
**2011**, 251, 1172–1194. [Google Scholar] [CrossRef] - Ovsiannikov, L.V. Group Analysis of Differential Equations; Chapovsky, Y., Translator; Ames, W.F., Ed.; Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers): New York, NY, USA, 1982; p. xvi+416. [Google Scholar]
- Bluman, G.W.; Cheviakov, A.F.; Anco, S.C. Applications of Symmetry Methods to Partial Differential Equations; Springer: New York, NY, USA, 2010; Volume 168. [Google Scholar]
- Cherniha, R.; Serov, M.; Rassokha, I. Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations. J. Math. Anal. Appl.
**2008**, 342, 1363–1379. [Google Scholar] [CrossRef] - Kingston, J.G. On point transformations of evolution equations. J. Phys. A
**1991**, 24, L769–L774. [Google Scholar] [CrossRef] - Kingston, J.G.; Sophocleous, C. On form-preserving point transformations of partial differential equations. J. Phys. A
**1998**, 31, 1597–1619. [Google Scholar] [CrossRef] - Gazeau, J.P.; Winternitz, P. Symmetries of variable coefficient Korteweg–de Vries equations. J. Math. Phys.
**1992**, 33, 4087–4102. [Google Scholar] [CrossRef] - Niederer, U. Schrödinger invariant generalized heat equations. Helv. Phys. Acta
**1978**, 51, 220–239. [Google Scholar] - Cherniha, R.; King, J.R. Lie symmetries of nonlinear multidimensional reaction-diffusion systems. II. J. Phys. A
**2003**, 36, 405–425. [Google Scholar] [CrossRef] - Cherniha, R.; King, J.R. Non-linear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansatz and exact solutions. J. Math. Anal. Appl.
**2005**, 308, 11–35. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M. Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms. II. Eur. J. Appl. Math.
**2006**, 17, 597–605. [Google Scholar] [CrossRef] - Knyazeva, I.V.; Popov, M.D. A system of two diffusion equations. In CRC Handbook of Lie Group Analysis of Differential Equations; Ibragimov, N.H., Ed.; CRC Press: Boca Raton, FL, USA, 1994; pp. 171–176. [Google Scholar]
- Hill, J.M. Similarity solutions for nonlinear diffusion – a new integration procedure. J. Eng. Math.
**1989**, 23, 141–155. [Google Scholar] [CrossRef] - Polyanin, A.D.; Zaitsev, V.F. Handbook of Nonlinear Partial Differential Equations, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2012; p. xxxvi+876. [Google Scholar]
- Samarskii, A.A.; Galaktionov, V.A.; Kurdyumov, S.P.; Mikhailov, A.P. Blow-Up in Quasilinear Parabolic Equations; De Gruyter Expositions in Mathematics; Walter de Gruyter & Co.: Berlin, Germany, 1995; Volume 19, p. xxii+535. [Google Scholar]
- Arrigo, D.J.; Hill, J.M. Nonclassical symmetries for nonlinear diffusion and absorption. Stud. Appl. Math.
**1995**, 94, 21–39. [Google Scholar] [CrossRef] - Yung, C.; Verburg, K.; Baveye, P. Group classification and symmetry reductions of the non-linear diffusion-convection equation u
_{t}= (D(u)_{ux})_{x}− K^{′}(u)u_{x}. Int. J. Non-Linear Mech.**1994**, 29, 273–278. [Google Scholar] [CrossRef] - Zhdanov, R.Z.; Lahno, V.I. Conditional symmetry of a porous medium equation. Phys. D Nonlinear Phenom.
**1998**, 122, 178–186. [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**, 9, 527–542. [Google Scholar] [CrossRef] - Hashemi, M.S.; Nucci, M.C. Nonclassical symmetries for a class of reaction-diffusion equations: the method of heir-equations. J. Nonlinear Math. Phys.
**2013**, 20, 44–60. [Google Scholar] [CrossRef] - Serov, M.I. Conditional invariance and exact solutions of the nonlinear equation. Ukr. Math. J.
**1990**, 42, 1216–1222. [Google Scholar] [CrossRef] - Arrigo, D.J.; Hill, J.M.; Broadbridge, P. Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source. IMA J. Appl. Math.
**1994**, 52, 1–24. [Google Scholar] [CrossRef] - Clarkson, P.A.; Mansfield, E.L. Symmetry reductions and exact solutions of a class of nonlinear heat equations. Phys. D
**1994**, 70, 250–288. [Google Scholar] [CrossRef] - Cherniha, R.; Pliukhin, O. New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities. J. Math. Anal. Appl.
**2013**, 403, 23–37. [Google Scholar] [CrossRef] - Cherniha, R.; Pliukhin, O. Nonlinear evolution equations with exponential nonlinearities: conditional symmetries and exact solutions. In Algebra, Geometry and Mathematical Physics; Banach Center Publications: Warsaw, Poland, 2011; Volume 93, pp. 105–115. [Google Scholar]
- Cherniha, R. New non-Lie ansatz and exact solutions of nonlinear reaction-diffusion-convection equations. J. Phys. A
**1998**, 31, 8179–8198. [Google Scholar] [CrossRef] - Kamke, E. Differentialgleichungen: Lösungsmethoden und Lösungen, 10th ed.; I Gewöhnliche Differentialgleichungen; B. G. Teubner: Stuttgart, Germany, 1983; p. xxvi+668. (In German) [Google Scholar]
- Qu, C. Group classification and generalized conditional symmetry reduction of the nonlinear diffusion-convection equation with a nonlinear source. Stud. Appl. Math.
**1997**, 99, 107–136. [Google Scholar] [CrossRef] - Galaktionov, V.A. On new exact blow-up solutions for nonlinear heat conduction equations with source and applications. Differ. Integral Equ.
**1990**, 3, 863–874. [Google Scholar] - King, J.R. Exact multidimensional solutions to some nonlinear diffusion equations. Quart. J. Mech. Appl. Math.
**1993**, 46, 419–436. [Google Scholar] [CrossRef] - Kuang, Y.; Nagy, J.D.; Eikenberry, S.E. Introduction to Mathematical Oncology; Chapman & Hall/CRC Mathematical and Computational Biology Series; CRC Press: Boca Raton, FL, USA, 2016; p. xi+470. [Google Scholar]
- Murray, J.D. Mathematical Biology; Springer: Berlin/Heidelberg, Germany, 1989; Volume 19, p. xiv+767. [Google Scholar]
- Fisher, R.A. The wave of advance of advantageous genes. Ann. Eugen.
**1937**, 7, 353–369. [Google Scholar] [CrossRef] - Cherniha, R.; Dutka, V. Exact and numerical solutions of the generalized Fisher equation. Rep. Math. Phys.
**2001**, 47, 393–411. [Google Scholar] [CrossRef] - Murray, J.D. Nonlinear Differential Equation Models in Biology; Clarendon Press: Oxford, UK, 1977; p. xiii+370. [Google Scholar]
- Murray, J.D. Mathematical Biology. I, 3rd ed.; Springer: New York, NY, USA, 2002; Volume 17, p. xxiv+551. [Google Scholar]
- Okubo, A.; Levin, S.A. Diffusion and Ecological Problems: Modern Perspectives, 2nd ed.; Springer: New York, NY, USA, 2001; Volume 14, p. xx+467. [Google Scholar]
- Cherniha, R.; King, J.R. Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)—Dimensional Boundary Value Problems. Symmetry
**2015**, 7, 1410–1435. [Google Scholar] [CrossRef]

**Figure 1.**Exact solution (120) with $\lambda =0,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}=1,\phantom{\rule{4pt}{0ex}}{c}_{1}=\frac{1}{2},\phantom{\rule{4pt}{0ex}}{c}_{2}=0$.

**Figure 2.**Exact solution (120) with $\lambda =\frac{1}{8},\phantom{\rule{4pt}{0ex}}{\lambda}_{1}=1,\phantom{\rule{4pt}{0ex}}{c}_{1}=\frac{1}{2},\phantom{\rule{4pt}{0ex}}{c}_{2}=0$.

**Table 1.**The complete Lie symmetry classification (LSC) of equations of the form (6) using the group $\mathcal{E}$. MAI, maximal algebras of invariance.

RDC Equations | MAI | Constraints | |
---|---|---|---|

1 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}$ | $<{\partial}_{t},{\partial}_{x},{D}_{0},{D}_{2}>$ | |

2 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+p$ | $<{\partial}_{t},{\partial}_{x},T,{D}_{2}>$ | $p=\pm 1$ |

3 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+{e}^{u}{u}_{x}+\frac{2}{9}{e}^{u}$ | $<{\partial}_{t},{\partial}_{x},{D}_{1},X>$ | |

4 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+{e}^{u}{u}_{x}+\frac{2}{9}{e}^{u}+p$ | $<{\partial}_{t},{\partial}_{x},T,X>$ | $p=\pm 1$ |

5 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+{\lambda}_{3}{e}^{u}{u}_{x}+{\lambda}_{6}{e}^{u}+p$ | $<{\partial}_{t},{\partial}_{x},T>$ | $|{\lambda}_{3}|+|{\lambda}_{6}|\ne 0,{\lambda}_{6}\ne \frac{2}{9}{\lambda}_{3}^{2},p=\pm 1$ |

6 | ${u}_{t}={({e}^{nu}{u}_{x})}_{x}+{\lambda}_{2}{e}^{mu}{u}_{x}+{\lambda}_{3}{e}^{(2m-n)u}$ | $<{\partial}_{t},{\partial}_{x},(n-2m){D}_{0}+{D}_{2}>$ | $|{\lambda}_{2}|+|{\lambda}_{3}|\ne 0$ |

**Table 2.**Form-preserving transformations (FPTs) of the class of reaction-diffusion-convection (RDC) Equation (6).

RDC Equation | FPT | RDC Equation | |
---|---|---|---|

1 | ${u}_{t}={u}_{xx}+\lambda {e}^{u}{u}_{x}+C(u)$ | (12) | ${w}_{\tau}={w}_{yy}+{c}_{2}^{-1}\lambda {e}^{w}{w}_{y}+{c}_{0}^{-2}C(w-\mathrm{ln}\frac{{c}_{2}}{{c}_{0}})$ |

2 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{mu}{u}_{x}+C(u)$ | (13) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{c}_{0}^{2(m-1)}{c}_{1}^{1-2m}\lambda {e}^{mw}{w}_{y}+{c}_{0}^{-2}C(w-\mathrm{ln}\frac{{c}_{1}^{2}}{{c}_{0}^{2}})$ |

3 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{u}{u}_{x}+\frac{2}{9}{\lambda}^{2}{e}^{u}$ | (14) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}$ |

4 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{u}{u}_{x}+{\lambda}_{2}{e}^{u}+{\lambda}_{3}$ | (15) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{c}_{1}^{-1}\lambda {e}^{w}{w}_{y}+{c}_{1}^{-2}{\lambda}_{2}{e}^{w}+{c}_{0}^{-2}{\lambda}_{3}$ |

5 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{u}{u}_{x}+{\lambda}_{2}{e}^{u}+{\lambda}_{3}$ | (16) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{c}_{1}^{-1}\lambda {e}^{w}{w}_{y}+{c}_{1}^{-2}{\lambda}_{2}{e}^{w}$ |

6 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{u}{u}_{x}+\frac{2}{9}{\lambda}^{2}{e}^{u}+{\lambda}_{3}$ | (17) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{c}_{1}^{-1}\lambda {e}^{w}{w}_{y}+\frac{2}{9}{c}_{1}^{-2}{\lambda}^{2}{e}^{w}+{c}_{0}^{-2}{\lambda}_{3}$ |

7 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{u}{u}_{x}+\frac{2}{9}{\lambda}^{2}{e}^{u}+{\lambda}_{3}$ | (18) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{c}_{1}^{-1}\lambda {e}^{w}{w}_{y}+\frac{2}{9}{c}_{1}^{-2}{\lambda}^{2}{e}^{w}$ |

8 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+\lambda {e}^{u}{u}_{x}+\frac{2}{9}{\lambda}^{2}{e}^{u}+{\lambda}_{3}$ | (19) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{c}_{0}^{-2}{\lambda}_{3}$ |

9 | (20) | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}$ |

**Table 3.**Simplification of the RDC equations from Table 1 by means of FPTs.

RDC Equation | FPT | Canonical Form of RDC Equation | |
---|---|---|---|

1 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+p$ | $\tau =\frac{{e}^{pt}}{p},$$y=x,$ | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}$ |

$w=u-pt$ | |||

2 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+{e}^{u}{u}_{x}+\frac{2}{9}{e}^{u}$ | $\tau =t,$$y=3{e}^{\frac{1}{3}x},$ | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}$ |

$w=u+\frac{2}{3}x$ | |||

3 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+{e}^{u}{u}_{x}+\frac{2}{9}{e}^{u}+{\lambda}_{4},$ | $\tau =\frac{{e}^{{\lambda}_{4}t}}{{\lambda}_{4}},$$y=3{e}^{\frac{1}{3}x},$ | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}$ |

${\lambda}_{4}\ne 0$ | $w=u-{\lambda}_{4}t+\frac{2}{3}x$ | ||

4 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}+{\lambda}_{3}{e}^{u}{u}_{x}+{\lambda}_{6}{e}^{u}+p,$ | $\tau =\frac{{e}^{pt}}{p},$$y=x,$ | ${w}_{\tau}={({e}^{w}{w}_{y})}_{y}+{\lambda}_{3}{e}^{w}{w}_{y}+{\lambda}_{6}{e}^{w},$ |

$|{\lambda}_{3}|+|{\lambda}_{6}|\ne 0,\phantom{\rule{4pt}{0ex}}{\lambda}_{6}\ne \frac{2}{9}{\lambda}_{3}^{2}$ | $w=u-pt$ |

**Table 4.**The LSC of the class of RDC Equation (6) using FPTs.

The RDC Equation | MAI | |
---|---|---|

1 | ${u}_{t}={({e}^{u}{u}_{x})}_{x}$ | $\langle {\partial}_{t},{\partial}_{x},{D}_{0},{D}_{2}=nx{\partial}_{x}+2{\partial}_{u}\rangle ,$ |

2 | ${u}_{t}={({e}^{nu}{u}_{x})}_{x}+\lambda {e}^{mu}{u}_{x}+{\lambda}_{1}{e}^{(2m-n)u}$, | $\langle {\partial}_{t},{\partial}_{x},(n-2m){D}_{0}+{D}_{2}\rangle $ |

$|\lambda |+|2m-n|\ne 0$ |

**Table 5.**Lie’s ansatz and reduced equations for Equation (36).

Ansatz | Reduced Equation | |
---|---|---|

1 | $u=\phi (\omega ),$ | ${({e}^{n\phi}{\phi}_{\omega})}_{\omega}+\lambda {e}^{m\phi}{\phi}_{\omega}+{\lambda}_{1}{e}^{(2m-n)\phi}=-\theta {\phi}_{\omega}$ |

$\omega =x-\theta t,$ | ||

2 | $u=\phi (\omega )+\frac{1}{m}\mathrm{ln}x$, | ${\alpha}^{2}{\left({e}^{2m\phi}{\phi}_{\omega}\right)}_{\omega}+\left(3\alpha {e}^{2m\phi}+\lambda \alpha {e}^{m\phi}-1\right){\phi}_{\omega}+$ |

$\omega =\alpha \mathrm{ln}x+t,$ | $+\frac{1}{m}({e}^{2m\phi}+\lambda {e}^{m\phi})+{\lambda}_{1}=0$ | |

$n=2m$ | ||

3 | $u=\phi (\omega )-\frac{1}{m}\mathrm{ln}t,$ | ${({e}^{m\phi}{\phi}_{\omega})}_{\omega}+\lambda {e}^{m\phi}{\phi}_{\omega}+{\lambda}_{1}{e}^{m\phi}=\alpha {\phi}_{\omega}-\frac{1}{m}$ |

$\omega =\alpha \mathrm{ln}t+x,$ | ||

$n=m$ | ||

4 | $u=\phi (\omega )+\frac{1}{n-2m}\mathrm{ln}t,$ | ${({e}^{n\phi}{\phi}_{\omega})}_{\omega}+\lambda {e}^{m\phi}{\phi}_{\omega}+{\lambda}_{1}{e}^{(2m-n)\phi}=$ |

$\omega =x{t}^{\frac{m-n}{n-2m}},$ | $=\frac{1}{2m-n}[(n-m)\omega {\phi}_{\omega}-1]$ | |

$n\ne m;\phantom{\rule{4pt}{0ex}}2m$ |

The Explicit Forms of a(t,x) and f(t,x) | |
---|---|

1. | $a=\frac{-\lambda \pm \sqrt{{\lambda}^{2}-4{\lambda}_{1}}}{2}$, |

$f={\lambda}_{0}$ | |

2. | $a=0,$ |

$f=\left\{\begin{array}{c}\frac{{\lambda}_{0}\pm \sqrt{D}}{2},\phantom{\rule{4pt}{0ex}}D\equiv {\lambda}_{0}^{2}-4{\lambda}_{1}{\lambda}_{2},\hfill \\ -\frac{1}{t}+\frac{{\lambda}_{0}}{2},\phantom{\rule{4pt}{0ex}}D=0,\hfill \\ \frac{\sqrt{-D}}{2}\mathrm{tan}\left(\frac{\sqrt{-D}}{2}t\right)+\frac{{\lambda}_{0}}{2},\phantom{\rule{4pt}{0ex}}D<0,\hfill \\ -\frac{\sqrt{D}}{2}\mathrm{coth}(\frac{\sqrt{D}}{2}t)+\frac{{\lambda}_{0}}{2},\phantom{\rule{4pt}{0ex}}{(2f-{\lambda}_{0})}^{2}>D>0,\hfill \\ -\frac{\sqrt{D}}{2}\mathrm{tanh}(\frac{\sqrt{D}}{2}t)+\frac{{\lambda}_{0}}{2},\phantom{\rule{4pt}{0ex}}D>{(2f-{\lambda}_{0})}^{2}>0\hfill \end{array}\right.$ | |

3. | $a=-{\partial}_{x}\mathrm{ln}\left|{c}_{0}{e}^{\lambda x}-{c}_{1}x+\theta (t)\right|$, |

$f={\lambda}_{0}-{\partial}_{t}\mathrm{ln}\left|{c}_{0}{e}^{\lambda x}-{c}_{1}x+\theta (t)\right|$ | |

4. | $a=-{\partial}_{x}\mathrm{ln}\left|\theta (t)+{e}^{\frac{\lambda}{2}x}({c}_{0}{e}^{\frac{1}{2}\sqrt{P}x}+{c}_{1}{e}^{-\frac{1}{2}\sqrt{P}x})\right|,$ |

$f={\lambda}_{0}-{\partial}_{t}\mathrm{ln}\left|\theta (t)+{e}^{\frac{\lambda}{2}x}({c}_{0}{e}^{\frac{1}{2}\sqrt{P}x}+{c}_{1}{e}^{-\frac{1}{2}\sqrt{P}x})\right|$ | |

5. | $a=-{\partial}_{x}\mathrm{ln}\left|\theta (t)+{e}^{\frac{\lambda}{2}x}\left[{c}_{0}\mathrm{cos}(\frac{\sqrt{-P}}{2}x)-{c}_{1}\mathrm{sin}(\frac{\sqrt{-P}}{2}x)\right]\right|,$ |

$f={\lambda}_{0}-{\partial}_{t}\mathrm{ln}\left|\theta (t)+{e}^{\frac{\lambda}{2}x}\left[{c}_{0}\mathrm{cos}(\frac{\sqrt{-P}}{2}x)-{c}_{1}\mathrm{sin}(\frac{\sqrt{-P}}{2}x)\right]\right|$ | |

6. | $a=-{\partial}_{x}\mathrm{ln}\left|\theta (t)-({c}_{0}x-{c}_{1}){e}^{\frac{\lambda}{2}x}\right|,$ |

$f={\lambda}_{0}-{\partial}_{t}\mathrm{ln}\left|\theta (t)-({c}_{0}x-{c}_{1}){e}^{\frac{\lambda}{2}x}\right|$ |

**Table 7.**A complete list of the solutions $(a,\phantom{\rule{4pt}{0ex}}f)=(-{\partial}_{x}\mathrm{ln}\Gamma ,\phantom{\rule{4pt}{0ex}}-{\partial}_{t}\mathrm{ln}\Gamma )$ of System (61).

The Explicit Forms of Γ(t,x) | |
---|---|

1. | $\Gamma ={\displaystyle \sum _{i=1}^{4}}{c}_{i}\mathrm{exp}\left[{p}_{i}(x-\frac{{p}_{i}^{2}+{\lambda}_{1}}{\lambda}t)\right]$ |

2. | $\Gamma ={c}_{3}\mathrm{exp}\left[{p}_{3}(x-\frac{{p}_{3}^{2}+{\lambda}_{1}}{\lambda}t)\right]+{c}_{4}\mathrm{exp}\left[{p}_{4}(x-\frac{{p}_{4}^{2}+{\lambda}_{1}}{\lambda}t)\right]+$ |

$+\left[{c}_{2}(x-(3{p}_{1}^{2}+{\lambda}_{1})t)+{c}_{1}\right]\mathrm{exp}\left[{p}_{1}(x-\frac{{p}_{1}^{2}+{\lambda}_{1}}{\lambda}t)\right]$ | |

3. | $\Gamma =\left[{c}_{3}{\left(\lambda x+3{p}_{1}^{2}t\right)}^{2}+{c}_{2}\left(\lambda x+3{p}_{1}^{2}t\right)-2{c}_{3}{p}_{1}\lambda t+{c}_{1}\right]\mathrm{exp}\left[{p}_{1}\left(x+\frac{5}{\lambda}{p}_{1}^{2}t\right)\right]+$ |

$+{c}_{4}\mathrm{exp}\left[-3{p}_{1}\left(x-\frac{3}{\lambda}{p}_{1}^{2}t\right)\right]$ | |

4. | $\Gamma ={c}_{4}\lambda {x}^{3}+{c}_{3}{x}^{2}+{c}_{2}x+{c}_{1}-6{c}_{4}t$ |

5. | $\Gamma =\left[{c}_{2}\left(\lambda x-{p}_{1}^{2}t\right)+{c}_{1}\right]\mathrm{exp}\left[{p}_{1}\left(x+\frac{1}{\lambda}{p}_{1}^{2}t\right)\right]+$ |

$+\left[{c}_{4}\left(\lambda x-{p}_{1}^{2}t\right)+{c}_{3}\right]\mathrm{exp}\left[-{p}_{1}\left(x+\frac{1}{\lambda}{p}_{1}^{2}t\right)\right]$ | |

6. | $\Gamma ={c}_{1}\mathrm{exp}\left[{p}_{1}(x-\frac{{p}_{1}^{2}+{\lambda}_{1}}{\lambda}t)\right]+{c}_{2}\mathrm{exp}\left[{p}_{2}(x-\frac{{p}_{2}^{2}+{\lambda}_{1}}{\lambda}t)\right]+$ |

$+{c}_{3}\mathrm{sin}\left[\beta (x-\frac{3{\alpha}^{2}-{\beta}^{2}+{\lambda}_{1}}{\lambda}t)-{c}_{4}\right]\mathrm{exp}\left[\alpha (x-\frac{{\alpha}^{2}-3{\beta}^{2}+{\lambda}_{1}}{\lambda}t)\right]$ | |

7. | $\Gamma ={c}_{3}\mathrm{sin}\left[\beta \left(x-{p}_{1}^{2}t\right)+{c}_{4}\right]\mathrm{exp}\left[-{p}_{1}(x+\frac{2{\beta}^{2}+{p}_{1}^{2}}{\lambda}t)\right]+$ |

$+\left[{c}_{2}(\lambda x-({\beta}^{2}+{p}_{1}^{2})t)+{c}_{1}\right]\mathrm{exp}\left[{p}_{1}(x+\frac{{p}_{1}^{2}-{\beta}^{2}}{\lambda}t)\right]$ | |

8. | $\Gamma ={c}_{1}\mathrm{exp}\left[{\alpha}_{1}(x+\frac{{\alpha}_{1}^{2}+2{\beta}_{1}^{2}-{\beta}_{2}^{2}}{\lambda}t)\right]\mathrm{sin}\left[{\beta}_{1}(x-\frac{{\alpha}_{1}^{2}+{\beta}_{2}^{2}}{\lambda}t)+{c}_{2}\right]+$ |

+${c}_{3}\mathrm{exp}\left[-{\alpha}_{1}(x+\frac{{\alpha}_{1}^{2}-{\beta}_{1}^{2}+2{\beta}_{2}^{2}}{\lambda}t)\right]\mathrm{sin}\left[{\beta}_{2}(x-\frac{{\alpha}_{1}^{2}+{\beta}_{1}^{2}}{\lambda}t)+{c}_{4}\right]$ | |

9. | $\Gamma ={c}_{1}\mathrm{sin}\left[\beta (x-\frac{{\beta}^{2}}{\lambda}t)+{c}_{2}\right]+{c}_{3}\left(\lambda x+{\beta}^{2}t\right)\mathrm{sin}\left[\beta (x-\frac{{\beta}^{2}}{\lambda}t)+{c}_{4}\right]$ |

© 2018 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**

Cherniha, R.; Serov, M.; Pliukhin, O.
Lie and *Q*-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions. *Symmetry* **2018**, *10*, 123.
https://doi.org/10.3390/sym10040123

**AMA Style**

Cherniha R, Serov M, Pliukhin O.
Lie and *Q*-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions. *Symmetry*. 2018; 10(4):123.
https://doi.org/10.3390/sym10040123

**Chicago/Turabian Style**

Cherniha, Roman, Mykola Serov, and Oleksii Pliukhin.
2018. "Lie and *Q*-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions" *Symmetry* 10, no. 4: 123.
https://doi.org/10.3390/sym10040123