# Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative

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## Abstract

**:**

## 1. Introduction

- (1)
- ${\mathcal{M}}_{i,z}^{\gamma ,\beta}(au+bv)=a{\mathcal{M}}_{i,z}^{\gamma ,\beta}\left(u\right)+b{\mathcal{M}}_{i,z}^{\gamma ,\beta}\left(v\right),\phantom{\rule{4pt}{0ex}}$
- (2)
- ${\mathcal{M}}_{i,z}^{\gamma ,\beta}(u\circ v)\left(z\right)={u}^{{}^{\prime}}\left(v\left(z\right)\right){\mathcal{M}}_{i,z}^{\gamma ,\beta}v\left(z\right),$
- (3)
- ${\mathcal{M}}_{i,z}^{\gamma ,\beta}\left(uv\right)=u{\mathcal{M}}_{i,z}^{\gamma ,\beta}v+v{\mathcal{M}}_{i,z}^{\gamma ,\beta}u,\phantom{\rule{4pt}{0ex}}$
- (4)
- $\phantom{\rule{4pt}{0ex}}{\mathcal{M}}_{i,z}^{\gamma ,\beta}u)\left(z\right)=\frac{{z}^{1-\gamma}}{\Gamma (\beta +1)}\frac{du}{dz},\phantom{\rule{4pt}{0ex}}$
- (5)
- ${\mathcal{M}}_{i,z}^{\gamma ,\beta}\left({z}^{\nu}\right)=\frac{\nu}{\Gamma (\beta +1)}{z}^{\nu -\gamma}.$

## 2. Exact Solutions of BLMPE-MTD

#### 2.1. He’s Semi-Inverse Method

#### 2.2. Extended Tanh Function Method

**First set:**

**Second set:**

**Third set:**

**First set:**The Equation (5) has the solution

**Case 1:**If $\vartheta >0,$ then we obtain by using (14)

**Case 2:**If $\vartheta <0,$ then we obtain by using (15)

**Case 3:**If $\vartheta =0,$ then we obtain by using (16)

**Second set:**When $\vartheta >0$ and $\vartheta <0,$ the solutions are identical to those in the first set. If $\vartheta =0$**,**the solution of BLMPE-MTD (1) is

**Third set:**The solution of Equation (5) is

**Case 1:**If $\vartheta >0,$ then by using (14) we obtain

**Case 2:**If $\vartheta <0,$ then by using (15) we have

## 3. Graphical Representation and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Al-Askar, F.M.; Cesarano, C.; Mohammed, W.W.
Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative. *Axioms* **2023**, *12*, 466.
https://doi.org/10.3390/axioms12050466

**AMA Style**

Al-Askar FM, Cesarano C, Mohammed WW.
Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative. *Axioms*. 2023; 12(5):466.
https://doi.org/10.3390/axioms12050466

**Chicago/Turabian Style**

Al-Askar, Farah M., Clemente Cesarano, and Wael W. Mohammed.
2023. "Abundant Solitary Wave Solutions for the Boiti–Leon–Manna–Pempinelli Equation with M-Truncated Derivative" *Axioms* 12, no. 5: 466.
https://doi.org/10.3390/axioms12050466