# Lie Group Method for Solving the Generalized Burgers’, Burgers’–KdV and KdV Equations with Time-Dependent Variable Coefficients

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{xx}and Burgers’ Korteweg–de Vries Burgers’ (BKdV) equation involves both dispersion term u

_{xxx}and dissipation term u

_{xx}[6]. Typical examples describing the behavior of a long wave in shallow water and waves in plasma, also describing the behavior of flow of liquids containing gas bubbles and the propagation of waves on an elastic tube filed with a viscous fluid, [6] are considered.

_{t}+ uu

_{x}= g(t)u

_{xx}. In 2005, Wazwaz presented an analysis to generalized forms of Burgers’, Burgers’–KdV, and Burgers’–Huxley equations using the traveling wave method [12]. In 2011, Abd-el-Malek and Helal applied an analysis to the generalized forms of Burgers’ and Burgers’–KdV with variable coefficients and with initial and boundary conditions using the group theoretic approach [13].

## 2. The Generalized Burgers’ Equation

^{(2)}X on Equation (2) must vanish, where u is the solution of Equation (2), and then we find the following determining equations:

**Table 1.**Infinitesimals for ${u}_{t}+\text{\alpha}{({u}^{n})}_{x}=\text{\beta}g(t){({u}^{n})}_{xx}$.

No. | g | Infinitesimals |
---|---|---|

1 | $\forall $ | ${\xi}^{1}={c}_{2}$, ${\xi}^{2}=0$, ${\eta}^{1}=0$ |

2 | ${t}^{\lambda}$ | ${\xi}^{1}={c}_{1}\lambda x+{c}_{2}$, ${\xi}^{2}={c}_{1}t$, ${\eta}^{1}=\frac{\lambda -1}{n-1}{c}_{1}u$ |

3 | ${e}^{t}$ | ${\xi}^{1}={c}_{1}x+{c}_{2}$, ${\xi}^{2}={c}_{1}$, ${\eta}^{1}=\frac{\lambda -1}{n-1}{c}_{1}u$ |

4 | 1 | ${\xi}^{1}={c}_{2}$, ${\xi}^{2}={c}_{1}t+{c}_{3}$, ${\eta}^{1}=\frac{-1}{n-1}{c}_{1}u$ |

_{2}= 0, c

_{1}≠ 0 and $-t\frac{dr(t)}{dt}+\frac{\lambda -1}{n-1}r(t)=0$, from which we get:

^{*}.

^{λ}. While we succeeded to find other possible forms of g(t), the only case which keeps the boundary conditions invariant is a power function or constant.

**Figure 1.**(

**a**) Exact solution Equation (18) for α = 2, β = 2, γ = 0.6, and n = 2; (

**b**) Exact solution Equation (18) for α = 2, β = 2, γ = 0.6, and n = 3; (

**c**) Exact solution Equation (18) for α = 2, β = 2, γ = 0.6, and n = 5.

## 3. The Generalized Burgers’–KdV Equation “GBKdV”

^{(3)}X on Equation (19) must vanish, where u is the solution of Equation (19). Clearly, the determining Equations can be split with respect to different powers of u. Special cases of splitting cases arise if n ≠ 2 and if n = 2. Therefore, we investigate two cases, namely n ≠ 2 and n = 2

No. | g | Infinitesimals |
---|---|---|

1 | $\sqrt{t}$ | ${\xi}^{1}=\frac{{c}_{1}}{2}x+{c}_{2}$ ,${\xi}^{2}={c}_{1}t$, ${\eta}^{1}=\frac{-1}{2\left(n-1\right)}{c}_{1}u$ |

2 | 1 | ${\xi}^{1}={c}_{1}$ ,${\xi}^{2}={c}_{2}$, ${\eta}^{1}=0$ |

**Figure 2.**(

**a**) Bugle-shaped wave solution of BKdV Burgers’–KdV Equation (30) for α = 1, β = 10, and n = 5; (

**b**) Bugle-shaped wave solution of BKdV Burgers’–KdV Equation (30) for α = 1, β = 10, and n = 100.

No. | g | Infinitesimals |
---|---|---|

1 | $\sqrt{t}$ | ${\xi}^{1}=\frac{1}{2}{c}_{1}x+{c}_{2}t+{c}_{3}$, ${\xi}^{2}={c}_{1}t$, ${\eta}^{1}=\frac{{c}_{2}-{c}_{1}\text{\alpha}u}{2\text{\alpha}}$ |

2 | $t$ | ${\xi}^{1}=({c}_{1}x+{c}_{2})t+{c}_{3}$, ${\xi}^{2}={c}_{1}{t}^{2}$, ${\eta}^{1}=\frac{{c}_{1}x+{c}_{2}-2{c}_{1}t\text{\alpha}u}{2\text{\alpha}}$ |

3 | 1 | ${\xi}^{1}={c}_{2}t+{c}_{3}$, ${\xi}^{2}={c}_{1}$, ${\eta}^{1}=\frac{1}{2\text{\alpha}}{c}_{2}$ |

4 | $\forall $ | ${\xi}^{1}={c}_{2}t+{c}_{3}$, ${\xi}^{2}=0$, ${\eta}^{1}=\frac{1}{2\text{\alpha}}{c}_{2}$ |

## 4. The Generalized KdV Equation “GKdV”

^{(3)}X on Equation (41) vanishes where ν, the solution of Equation (41), is, and then we find the following determining equations:

No. | $\tilde{g}(\tau )$ | Infinitesimals |
---|---|---|

1 | $\forall $ | ${\xi}^{1}={c}_{2}$, ${\xi}^{2}=0$, ${\eta}^{1}=0$ |

2 | ${\tau}^{\lambda}$ | ${\xi}^{1}=\frac{{c}_{1}}{3}\left(\lambda +1\right)x+{c}_{2}$ ,${\xi}^{2}={c}_{1}\tau $, ${\eta}^{1}=\frac{\lambda -2}{3n}{c}_{1}v$ |

3 | ${e}^{\tau}$ | ${\xi}^{1}=\frac{{c}_{1}}{3}x+{c}_{2}$ ,${\xi}^{2}={c}_{1}$, ${\eta}^{1}=\frac{1}{3n}{c}_{1}v$ |

4 | $1$ | ${\xi}^{1}=\frac{{c}_{1}}{3}x+{c}_{2}$ ,${\xi}^{2}={c}_{1}\tau +{c}_{3}$, ${\eta}^{1}=\frac{-2}{3n}{c}_{1}v$ |

**Figure 3.**(

**a**) Wave solution of boundary-value problem (56)–(58) for $w(t)={t}^{-1}$, $n=5$, $\gamma =0.5$; (

**b**) Wave solution of boundary-value problem (56)–(58) for $n=5$, $\gamma =0.5$, and $\eta \epsilon \left[0,50\right]$; (

**c**) Wave solution of boundary-value problem (56)–(58) for $w(t)={t}^{-1}$, n = 100, $\gamma =0.5$; (

**d**) Wave solution of boundary-value problem (56)–(58) for n =100, $\gamma =0.5$, and $\eta \epsilon \left[0,50\right]$.

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Abd-el-Malek, M.B.; Amin, A.M.
Lie Group Method for Solving the Generalized Burgers’, Burgers’–KdV and KdV Equations with Time-Dependent Variable Coefficients. *Symmetry* **2015**, *7*, 1816-1830.
https://doi.org/10.3390/sym7041816

**AMA Style**

Abd-el-Malek MB, Amin AM.
Lie Group Method for Solving the Generalized Burgers’, Burgers’–KdV and KdV Equations with Time-Dependent Variable Coefficients. *Symmetry*. 2015; 7(4):1816-1830.
https://doi.org/10.3390/sym7041816

**Chicago/Turabian Style**

Abd-el-Malek, Mina B., and Amr M. Amin.
2015. "Lie Group Method for Solving the Generalized Burgers’, Burgers’–KdV and KdV Equations with Time-Dependent Variable Coefficients" *Symmetry* 7, no. 4: 1816-1830.
https://doi.org/10.3390/sym7041816