# Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation

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## Abstract

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## 1. Introduction

## 2. $(\mathbf{1}/{\mathit{G}}^{\prime})$-Expansion Method

## 3. $({\mathit{G}}^{\prime}/\mathit{G},\mathbf{1}/\mathit{G})$-Expansion Method

## 4. Solutions of the (Z-S) Equation Using $(\mathbf{1}/{\mathit{G}}^{\prime})$-Expansion Method

## 5. Solutions of the (Z-S) Equation Using $({\mathit{G}}^{\prime}/\mathit{G},\mathbf{1}/\mathit{G})$-Expansion Method

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**3D, contour and 2D graphs respectively for $p=-1,\phantom{\rule{0.277778em}{0ex}}\lambda =-0.8,\phantom{\rule{0.277778em}{0ex}}A=-3,\phantom{\rule{0.277778em}{0ex}}{a}_{0}=-1,\phantom{\rule{0.277778em}{0ex}}{a}_{1}=-1,\phantom{\rule{0.277778em}{0ex}}q=2$ values of Equation (24).

**Figure 2.**3D, contour and 2D graphs respectively for $p=-1,\phantom{\rule{0.277778em}{0ex}}\mu =-0.8,\phantom{\rule{0.277778em}{0ex}}A=-3,\phantom{\rule{0.277778em}{0ex}}{a}_{2}=-1,\phantom{\rule{0.277778em}{0ex}}{a}_{1}=-1,\phantom{\rule{0.277778em}{0ex}}q=2$ values of Equation (27).

**Figure 3.**3D, contour and 2D graphs respectively for $p=-0.5,\phantom{\rule{0.277778em}{0ex}}\lambda =-1,\phantom{\rule{0.277778em}{0ex}}{c}_{2}=-1,\phantom{\rule{0.277778em}{0ex}}{c}_{1}=5,\phantom{\rule{0.277778em}{0ex}}q=-1,\phantom{\rule{0.277778em}{0ex}}w=0.5$ values of Equation (34).

**Figure 4.**3D, contour and 2D graphs respectively for $\lambda =-1,\phantom{\rule{0.277778em}{0ex}}{c}_{2}=2,{c}_{1}=5,\phantom{\rule{0.277778em}{0ex}}q=-5,\phantom{\rule{0.277778em}{0ex}}r=3$ values of Equation (37).

**Figure 5.**3D, contour and 2D graphs respectively for ${c}_{2}=-0.5,\phantom{\rule{0.277778em}{0ex}}{c}_{1}=-1,\phantom{\rule{0.277778em}{0ex}}q=1,\phantom{\rule{0.277778em}{0ex}}{a}_{0}=0.5,\phantom{\rule{0.277778em}{0ex}}p=-0.2,\phantom{\rule{0.277778em}{0ex}}\lambda =2,\phantom{\rule{0.277778em}{0ex}}{a}_{2}=-2$ values of Equation (40).

**Figure 6.**3D, contour and 2D graphs respectively ${c}_{2}=-0.5,\phantom{\rule{0.277778em}{0ex}}{c}_{1}=-1,{a}_{0}=0.5,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p=-0.2,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\lambda =2,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{a}_{2}=-2$ values of Equation (43).

**Figure 7.**The real part of the 3D, contour and 2D graphics respectively for ${c}_{2}=5,\phantom{\rule{0.277778em}{0ex}}{c}_{1}=-1,\phantom{\rule{0.277778em}{0ex}}q=-1,\phantom{\rule{0.277778em}{0ex}}p=-0.5,\phantom{\rule{0.277778em}{0ex}}w=3$ values of Equation (46).

**Figure 8.**The imaginary part of 3D, contour and 2D graphs respectively for ${c}_{2}=5,\phantom{\rule{0.277778em}{0ex}}{c}_{1}=-1,\phantom{\rule{0.277778em}{0ex}}q=-1,\phantom{\rule{0.277778em}{0ex}}p=-0.5,\phantom{\rule{0.277778em}{0ex}}w=3$ values of Equation (46).

**Figure 9.**3D, contour and 2D graphs respectively for ${c}_{2}=-0.1,\phantom{\rule{0.277778em}{0ex}}{c}_{1}=2,\phantom{\rule{0.277778em}{0ex}}{a}_{2}=2,\phantom{\rule{0.277778em}{0ex}}p=-0.5,\phantom{\rule{0.277778em}{0ex}}{a}_{0}=5$ values of Equation (49).

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**MDPI and ACS Style**

Yokus, A.; Durur, H.; Ahmad, H.; Yao, S.-W.
Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation. *Mathematics* **2020**, *8*, 908.
https://doi.org/10.3390/math8060908

**AMA Style**

Yokus A, Durur H, Ahmad H, Yao S-W.
Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation. *Mathematics*. 2020; 8(6):908.
https://doi.org/10.3390/math8060908

**Chicago/Turabian Style**

Yokus, Asıf, Hülya Durur, Hijaz Ahmad, and Shao-Wen Yao.
2020. "Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation" *Mathematics* 8, no. 6: 908.
https://doi.org/10.3390/math8060908