# Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lie Symmetry Classification

**Proposition**

**1.**

**Theorem**

**1.**

**Proof.**

**Case 1: $k=0$.**In this case, the fifth equation of system Equation (9) imply that $\varphi =0$ and ${\tau}_{t}-2{\xi}_{x}=0$. With this condition, the system Equation (9) can be reduced to

**Case 2: $k=1$.**Here $K\in \{{e}^{\mu u},{u}^{\mu},\mu \ne 0\}$ mod ${\widehat{G}}^{\sim}$ and there exists $\mathbf{v}\in {A}^{max}$ with $\varphi \ne 0$, otherwise there is no additional extension of the maximal Lie invariance algebra in comparison with the case $k=0$.

**Case 3: $k=2$.**The assumption of two independent equations of form of the fifth equation of system Equation (9) for K yields $K=$ const, i.e. $K=1$ mod ${G}^{\sim}$. From the fifth equation of system Equation (9) we have ${\tau}_{t}=2{\xi}_{x}$. Thus, we can get ${\tau}_{tt}=0,{\xi}_{xx}=0,{\varphi}_{tu}={\varphi}_{xu}=0$, which implies system Equation (9) can be reduced to

## 3. Symmetry Reduction and Exact Solutions

## 4. Conclusions and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Ames, W.F. Nonlinear Partial Differential Equations in Engineering; Academic: New York, NY, USA, 1972; Volume II, pp. 50–52. [Google Scholar]
- Ames, W.F.; Adams, E.; Lohner, R.G. Group properties of u
_{tt}= [f(u)u_{x}]_{x}. Int. J. Non-Linear Mech.**1981**, 16, 439–447. [Google Scholar] [CrossRef] - Galaktionov, V.A. The formation of shocks and fundamental solution of a fourth-order quasilinear Boussinesq-type equation. Nonlinearity
**2009**, 22, 239–257. [Google Scholar] [CrossRef] - Favini, A.; Goldstein, G.R.; Goldstein, J.A.; Romanelli, S. Classification of general Wentzell boundary conditions for fourth order operators in one space dimension. J. Math. Anal. Appl.
**2007**, 335, 219–235. [Google Scholar] [CrossRef] - McKenna, P.J.; Walter, W. Travelling waves in a suspension bridges. SIAM J. Appl. Math.
**1990**, 50, 703–715. [Google Scholar] [CrossRef] - Ammann, O.H.; Karman, T.V.; Woodruff, G.B. The Failure of the Tacoma Narrow Bridge; Federal Works Agency: Washington, DC, USA, 1941. [Google Scholar]
- Champreys, A.R.; McKenna, P.J.; Zegeling, P.A. Solitary waves in nonlinear beam equations: Stability, fission and fusion. Nonlinear Dynam.
**2000**, 21, 31–53. [Google Scholar] [CrossRef] - McKenna, P.J.; Walter, W. Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal.
**1987**, 98, 167–177. [Google Scholar] [CrossRef] - Chen, Y.; McKenna, P.J. Traveling waves in a nonlinearly suspended beam: Theoretical results and numerical observations. J. Differ. Equ.
**1997**, 136, 325–355. [Google Scholar] [CrossRef] - Chen, Y.; McKenna, P.J. Traveling waves in a nonlinearly suspended beam: Some computational results and four open questions. Phil. Trans. R. Soc. Lond. A
**1997**, 355, 2175–2184. [Google Scholar] [CrossRef] - Choy, Y.S.; Jen, K.S.; McKenna, P.J. The structure of the solution set for periodic oscillations in a suspension bridge model. IMA J. Appl. Math.
**1991**, 47, 283–306. [Google Scholar] [CrossRef] - Humphreys, L.D. Numerical mountain pass solutions of a suspension bridge equation. Nonlinear Anal. TMA
**1997**, 35, 1811–1826. [Google Scholar] [CrossRef] - Lazer, A.C.; McKenna, P.J. Large Amplitude periodic oscillation in suspension bridges: Some new connections with nonlinear analysis. SIAM Rev.
**1990**, 32, 537–578. [Google Scholar] [CrossRef] - Humphreys, L.D.; McKenna, P.J. Multiple periodic solutions for a nonlinear suspension bridge equation. IMA J. Appl. Math.
**1999**, 63, 37–49. [Google Scholar] [CrossRef] - Doole, S.H.; Hogan, S.J. The nonlinear dynamics of suspension bridges under harmonic forcing. Appl. Nonlinear Math. Rep.
**1996**, 76, 127–128. [Google Scholar] - Doole, S.H.; Hogan, S.J. A piecewise linear suspension bridge model nonlinear dynamics and orbit continuation. Dynam. Stabil. Syst.
**1996**, 11, 19–47. [Google Scholar] [CrossRef] - Bruzón, M.S.; Camacho, J.C.; Gandarias, M.L. Similarity reductions of a nonlinear model for vibrations of beams. PAMM Proc. Appl. Math. Mech.
**2007**, 7, 2040063–2040064. [Google Scholar] [CrossRef] - Camacho, J.C.; Bruzón, M.S.; Ramírez, J.; Gandarias, M.L. Exact travelling wave solutions of a beam equation. J. Nonlinear Math. Phys.
**2011**, 18, 33–49. [Google Scholar] [CrossRef] - Gao, Y.X. Quasi-periodic Solutions of the General Nonlinear Beam Equations. Commun. Math. Res.
**2012**, 28, 51–64. [Google Scholar] - Ovsiannikov, L.V. Group properties of the nonlinear heat-conduction equation. Dokl. Akad. Nauk SSSR
**1959**, V.125, 492–495. (In Russian) [Google Scholar] [PubMed] - Ovsiannikov, L.V. Group Analysis of Differential Equations; Academic Press: New York, NY, USA, 1982. [Google Scholar]
- Bluman, G.; Cheviakov, A.; Anco, S. Applications of Symmetry Methods to Partial Differential Equations; Springer: Berlin, Germany, 2010. [Google Scholar]
- Barone, A.; Esposito, F.; Magee, C.G.; Scott, A.C. Theory and applications of the sine-Gordon equation. Riv. Nuovo Cimento
**1971**, 1, 227–267. [Google Scholar] [CrossRef] - Arrigo, D.J. Group properties of u
_{xx}− ${u}_{y}^{m}$u_{yy}= f(u). Int. J. Non-Linear Mech.**1991**, 26, 619–629. [Google Scholar] [CrossRef] - Pucci, E.; Salvatori, M.C. Group properties of a class of semilinear hyperbolic equations. Int. J. Non-Linear Mech.
**1986**, 21, 147–155. [Google Scholar] [CrossRef] - Torrisi, M.; Valenti, A. Group properties and invariant solutions for infinitesimal transformations of a nonlinear wave equation. Int. J. Non-Linear Mech.
**1985**, 20, 135–144. [Google Scholar] [CrossRef] - Donato, A. Similarity analysis and nonlinear wave propagation. Int. J. Non-Linear Mech.
**1987**, 22, 307–314. [Google Scholar] [CrossRef] - Ibragimov, N.H.; Torrisi, M.; Valenti, A. Preliminary group classification of equations v
_{tt}= f(x, v_{x})v_{xx}+ g(x, v_{x}). J. Math. Phys.**1991**, 32, 2988–2995. [Google Scholar] [CrossRef] - Ibragimov, N.H. (Ed.) Lie Group Analysis of Differential Equations—Symmetries, Exact Solutions and Conservation Laws, V.1.; CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Oron, A.; Rosenau, P. Some symmetries of the nonlinear heat and wave equations. Phys. Lett. A
**1986**, 118, 172–176. [Google Scholar] [CrossRef] - Chikwendu, S.C. Non-linear wave propagation solutions by Fourier transform perturbation. Int. J. Non-Linear Mech.
**1981**, 16, 117–128. [Google Scholar] [CrossRef] - Gandarias, M.L.; Torrisi, M.; Valenti, A. Symmetry classification and optimal systems of a non-linear wave equation. Int. J. Non-Linear Mech.
**2004**, 39, 389–398. [Google Scholar] [CrossRef] - Pucci, E. Group analysis of the equation u
_{tt}+ λu_{xx}= g(u, u_{x}). Riv. Mat. Univ. Parma**1987**, 12, 71–87. [Google Scholar] - Bluman, G.W.; Cheviakov, A.F. Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation. J. Math. Anal. Appl.
**2007**, 333, 93–111. [Google Scholar] [CrossRef] - Bluman, G. W.; Kumei, S. Symmetries and Differential Equations; Springer: Berlin, Germany, 1989. [Google Scholar]
- Bluman, G.W.; Temuerchaolu; Sahadevan, R. Local and nonlocal symmetries for nonlinear telegraph equation. J. Math. Phys.
**2005**, 46, 023505. [Google Scholar] [CrossRef] - Huang, D.J.; Ivanova, N.M. Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations. J. Math. Phys.
**2007**, 48, 073507. [Google Scholar] [CrossRef] - Huang, D.J.; Zhou, S.G. Group properties of generalized quasi-linear wave equations. J. Math. Anal. Appl.
**2010**, 366, 460–472. [Google Scholar] [CrossRef] - Huang, D.J.; Zhou, S.G. Group-theoretical analysis of variable coefficient nonlinear telegraph equations. Acta Appl. Math.
**2012**, 117, 135–183. [Google Scholar] [CrossRef] - Lahno, V.; Zhdanov, R.; Magda, O. Group classification and exact solutions of nonlinear wave equations. Acta Appl. Math.
**2006**, 91, 253–313. [Google Scholar] [CrossRef] - Sophocleous, C.; Kingston, J.G. Cyclic symmetries of one-dimensional non-linear wave equations. Int. J. Non-Linear Mech.
**1999**, 34, 531–543. [Google Scholar] [CrossRef] - Suhubi, E.S.; Bakkaloglu, A. Group properties and similarity solutions for a quasi-linear wave equation in the plane. Int. J. Non-Linear Mech.
**1991**, 26, 567–584. [Google Scholar] [CrossRef] - Vasilenko, O.F.; Yehorchenko, I.A. Group classification of multidimensional nonlinear wave equations. Proc. Inst. Math. NAS Ukr.
**2001**, 36, 63–66. [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] - Basarab-Horwath, P.; Lahno, V.I.; Zhdanov, R.Z. The structure of Lie algebras and the classification problem for partial differential equations. Acta Appl. Math.
**2001**, 69, 43–94. [Google Scholar] [CrossRef] - Zhdanov, R.Z.; Lahno, V.I. Group classification of heat conductivity equations with a nonlinear source. J. Phys. A
**1999**, 32, 7405–7418. [Google Scholar] [CrossRef] - Gazeau, J.P.; Winternitza, P. Symmetries of variable coefficient Korteweg-de Vries equations. J. Math. Phys.
**1992**, 33, 4087–4102. [Google Scholar] [CrossRef] - Popovych, R.O.; Kunzinger, M.; Eshraghi, H. Admissible point transformations and normalized classes of nonlinear Schrödinger equations. Acta Appl. Math.
**2010**, 109, 315–359. [Google Scholar] [CrossRef] - Nikitin, A.G.; Popovych, R.O. Group classification of nonlinear Schrödinger equations. Ukr. Math. J.
**2001**, 53, 1053–1060. [Google Scholar] [CrossRef] - Popovych, R.O.; Ivanova, N.M. New results on group classification of nonlinear diffusion-convection equations. J. Phys. A
**2004**, 37, 7547–7565. [Google Scholar] [CrossRef] - Ivanova, N.M.; Popovych, R.O.; Sophocleous, C. Group analysis of variable coefficient diffusion-convection equations. I. Enhanced group classification. Lobachevskii J. Math.
**2010**, 31, 100–122. [Google Scholar] [CrossRef] - Huang, D.J.; Yang, Q.M.; Zhou, S.G. Lie symmetry classification and equivalence transformation of variable coefficient nonlinear wave equations with power nonlinearities. Chin. J. Contemp. Math.
**2012**, 33, 205–214. [Google Scholar] - Huang, D.J.; Yang, Q.M.; Zhou, S.G. Conservation law classification of variable coefficient nonlinear wave equation with power Nonlinearity. Chin. Phys. B
**2011**, 20, 070202. [Google Scholar] [CrossRef] - Huang, D.J.; Ivanova, N.M. Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov-Kuznetsov equations. J. Differ. Equ.
**2016**, 260, 2354–2382. [Google Scholar] [CrossRef] - Cherniha, R.; King, J.R. Lie symmetries of nonlinear multidimensional reaction-diffusion systems: I. J. Phys. A
**2000**, 33, 267–282. [Google Scholar] [CrossRef] - 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. Lie symmetries and conservation laws of nonlinear multidimensional reaction-diffusion systems with variable diffusivities. IMA J. Appl. Math.
**2006**, 71, 391–408. [Google Scholar] [CrossRef] - Cherniha, R.; Myroniuk, L. Lie Symmetries and Exact Solutions of the Generalized Thin Film Equation. J. Phys. Math.
**2010**, 2, P100508:1–P100508:19. [Google Scholar] [CrossRef] - Olver, P.J. Application of Lie Groups to Differential Equations; Springer: New York, NY, USA, 1986. [Google Scholar]
- Patera, J.; Winternitz, P. Subalgebras of real three- and four-dimensional Lie algebras. J. Math. Phys.
**1977**, 18, 1449–1455. [Google Scholar] [CrossRef] - Malfliet, W.; Hereman, W. The tanh method. I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr.
**1996**, 54, 563–568. [Google Scholar] [CrossRef] - Fan, E.G. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A
**2000**, 277, 212–218. [Google Scholar] [CrossRef]

**Table 1.**Group classification of class Equation (1).

N | $\mathit{K}\left(\mathit{u}\right)$ | $\mathit{D}\left(\mathit{u}\right)$ | $\mathit{F}\left(\mathit{u}\right)$ | Basis of A${}^{\mathbf{max}}$ |
---|---|---|---|---|

1 | ∀ | ∀ | ∀ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x}$ |

2 | ∀ | 0 | 0 | ${\partial}_{t},\partial x,x{\partial}_{x}+2t\partial t$ |

3 | ${e}^{\mu u}$ | 0 | 0 | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}-2t{\partial}_{t}-x{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}\frac{\mu}{4}x{\partial}_{x}+{\partial}_{u}$ |

4 | ${e}^{\mu u}$ | 0 | $f{e}^{\gamma u}(f\ne 0)$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}-\frac{\gamma}{2}t{\partial}_{t}+\frac{\mu -\gamma}{4}x{\partial}_{x}+{\partial}_{u}$ |

5 | ${e}^{\mu u}$ | $d{e}^{\nu u}(d\ne 0)$ | $f{e}^{(2\nu -\mu )u}$ | ${\partial}_{t},{\partial}_{x},\frac{\mu -2\nu}{2}t{\partial}_{t}+\frac{\mu -\nu}{2}x{\partial}_{x}+{\partial}_{u}$ |

6 | ${u}^{-4}$ | $d{u}^{-4}$ | $f{u}^{-3}$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}t{\partial}_{t}+\frac{1}{2}u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}{t}^{2}{\partial}_{t}+tu{\partial}_{u}$ |

7 | ${u}^{-4}$ | $d{u}^{-4}$ | $\frac{1}{4}\omega u+f{u}^{-3}(\omega >0)$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}{e}^{\sqrt{\omega}t}{\partial}_{t}+\frac{\sqrt{\omega}}{2}{e}^{\sqrt{\omega}t}u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}{e}^{-\sqrt{\omega}t}{\partial}_{t}-\frac{\sqrt{\omega}}{2}{e}^{-\sqrt{\omega}t}u{\partial}_{u}$ |

8 | ${u}^{-4}$ | $d{u}^{-4}$ | $\frac{1}{4}\omega u+f{u}^{-3}(\omega <0)$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}sin\left(\sqrt{-\omega}t\right){\partial}_{t}+\frac{\sqrt{-\omega}}{2}cos\left(\sqrt{-\omega}t\right)u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}$ |

$cos\left(\sqrt{-\omega}t\right){\partial}_{t}-\frac{\sqrt{-\omega}}{2}sin\left(\sqrt{-\omega}t\right)u{\partial}_{u}$ | ||||

9 | ${u}^{-4}$ | 0 | 0 | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}t{\partial}_{t}+\frac{1}{2}u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}{t}^{2}{\partial}_{t}+tu{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}-x{\partial}_{x}+u{\partial}_{u}$ |

10 | ${u}^{-4}$ | 0 | $fu(f>0)$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}{e}^{2\sqrt{f}t}{\partial}_{t}+\sqrt{f}{e}^{2\sqrt{f}t}u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}{e}^{-2\sqrt{f}t}{\partial}_{t}-\sqrt{f}{e}^{-2\sqrt{f}t}u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}$ |

$-x{\partial}_{x}+u{\partial}_{u}$ | ||||

11 | ${u}^{-4}$ | 0 | $fu(f<0)$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}sin\left(2\sqrt{-f}t\right){\partial}_{t}+\sqrt{-f}cos\left(2\sqrt{-f}t\right)u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}$ |

$cos\left(2\sqrt{-f}t\right){\partial}_{t}-\sqrt{-f}sin\left(2\sqrt{-f}t\right)u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}-x{\partial}_{x}+u{\partial}_{u}$ | ||||

12 | ${u}^{\mu}$ | 0 | 0 | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}2t{\partial}_{t}+x{\partial}_{x},-\frac{\mu}{2}t{\partial}_{t}+u{\partial}_{u}$ |

13 | ${u}^{\mu}$ | 0 | $f{u}^{\gamma}(f\ne 0)$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}\frac{1-\gamma}{2}t{\partial}_{t}+\frac{1-\gamma +\mu}{4}x{\partial}_{x}+u{\partial}_{u}$ |

14 | ${u}^{\mu}$ | $d{u}^{\nu}(d\ne 0)$ | $f{u}^{1-\mu +2\nu}$ | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}\frac{1}{2}(\mu -2\nu )t{\partial}_{t}+\frac{1}{2}(\mu -\nu )x{\partial}_{x}+u{\partial}_{u}$ |

15 | 1 | 0 | 0 | ${\partial}_{t},\phantom{\rule{0.166667em}{0ex}}{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}2t{\partial}_{t}+x{\partial}_{x},\phantom{\rule{0.166667em}{0ex}}u{\partial}_{u},\phantom{\rule{0.166667em}{0ex}}{\mathbf{v}}^{\infty}=b(x,t){\partial}_{u}$ |

where ${b}_{tt}+{b}_{xxxx}=0$ | ||||

16 | 1 | d | $fu$ | ${\partial}_{t},{\partial}_{x},u{\partial}_{u},{\mathbf{v}}^{\infty}=b(x,t){\partial}_{u}$ |

where ${b}_{tt}-d{b}_{xx}+{b}_{xxxx}-fb(t,x)=0$ |

N | $\mathit{Snbalgebra}$ | Ansatz for $\mathit{u}$ | $\mathit{y}$ | Reduced ODE |
---|---|---|---|---|

1 | $<{Q}_{1}>$ | $h\left(y\right)$ | x | $-\mu {h}^{\mu -1}{h}^{\prime}{h}^{\u2034}-{h}^{\mu}{h}^{\u2033\u2033}+\nu {h}^{\nu -1}{{h}^{\prime}}^{2}+{h}^{\nu}{h}^{\u2033}+{h}^{1-\mu +2\nu}=0$ |

2 | $<{Q}_{2}>$ | $h\left(y\right)$ | t | ${h}^{\u2033}={h}^{1-\mu +2\nu}$ |

3 | $<{Q}_{3}>$ | $h\left(y\right){t}^{\frac{2}{\mu -2\nu}}$ | $x{t}^{-\frac{\mu -\nu}{\mu -2\nu}}$ | ${(\mu -\nu )}^{2}{y}^{2}{h}^{\u2033}+(\mu -\nu )(2\mu -3\nu -4)y{h}^{\prime}+2(2-\mu +2\nu )h=$ |

${(\mu -2\nu )}^{2}(-\mu {h}^{\mu -1}{h}^{\prime}{h}^{\u2034}-{h}^{\mu}{h}^{\u2033\u2033}+\nu {h}^{\nu -1}{{h}^{\prime}}^{2}+{h}^{\nu}{h}^{\u2033}+{h}^{1-\mu +2\nu})$ | ||||

4 | $<{Q}_{1}+\alpha {Q}_{2}>$ | $h\left(y\right)$ | $x-\alpha t$ | ${h}^{\u2033}=-\mu {h}^{\mu -1}{h}^{\prime}{h}^{\u2034}-{h}^{\mu}{h}^{\u2033\u2033}+\nu {h}^{\nu -1}{{h}^{\prime}}^{2}+{h}^{\nu}{h}^{\u2033}+{h}^{1-\mu +2\nu}$ |

© 2017 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**

Huang, D.; Li, X.; Yu, S.
Lie Symmetry Classification of the Generalized Nonlinear Beam Equation. *Symmetry* **2017**, *9*, 115.
https://doi.org/10.3390/sym9070115

**AMA Style**

Huang D, Li X, Yu S.
Lie Symmetry Classification of the Generalized Nonlinear Beam Equation. *Symmetry*. 2017; 9(7):115.
https://doi.org/10.3390/sym9070115

**Chicago/Turabian Style**

Huang, Dingjiang, Xiangxiang Li, and Shunchang Yu.
2017. "Lie Symmetry Classification of the Generalized Nonlinear Beam Equation" *Symmetry* 9, no. 7: 115.
https://doi.org/10.3390/sym9070115