# First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### Conformable Derivative

**Definition:**

## 3. Feng’s First Integral Method (FIM)

**Step 1**: Consider a nonlinear conformable PDE of the given form:

**Step 2**: The following transformation is used:

**Step 3**: Now new independent variables will be introduced as:

**Step 4**: In this step, the first integrals of the plane independent (autonomous) system (c.f. Equation (8)) are obtained, these first integrals ultimately provide general solutions. The first integral of such systems is extremely challenging to get, as there is no precise or sound method for finding them. FIM uses the division theorem on (c.f. Equation (8)) to find the first integral. Now, the division theorem converts a non-linear ODE into an integrable first order ODE. Ultimately, we can find exact solutions to the problem.

**Division Theorem:**

## 4. Exact Solutions of Conformable mRLW Equation, (1+2) Dimensional Conformable pKP Equation and (1+2) Dimensional Conformable DLW System

#### 4.1. Exact Solutions of Conformable Space-Time mRLW Equation

**Case 1**: we have,

**Case 2**: We get,

#### 4.2. Exact Solutions of Conformable Space-Time pKP Equation

**Case 1**: we have,

**Case 2**: We get,

#### 4.3. Exact Solutions of Conformable Space-Time DLW System

**Case 1**: we have,

**Case 2**: we get,

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Exact solutions of conformable mRLW equation Case 1 and Case 2 using $\gamma =0.8$, $\beta =1$, $p=0.01$, $m=1$, $\nu =1$, $\tau =1$, $\mu =1.$

**Figure 2.**Exact solutions of conformable mRLW equation (Case 1) using $\beta =1$, $p=0.01$, $m=1$, $\nu =1$, $\tau =1$, $\mu =1.$, $\gamma =1$, $\gamma =0.8$, $\gamma =0.4$ and $\gamma =0.1$.

**Figure 3.**Exact solutions of conformable pKP equation Case 1 and Case 2 using $\gamma =0.8,\beta =1,p=1,m=1,l=1.$

**Figure 4.**Exact solutions of conformable pKP equation Case 1 using $\beta =1,p=1,m=1,l=1.$, $\gamma =1$, $\gamma =0.6$, and $\gamma =0.3$.

**Figure 5.**Exact solutions of the conformable DLW equation Case 1 and Case 2 using $p=0,m=0.3,l=0.09,\beta =1,\gamma =0.8.$

**Figure 6.**Exact solutions of the conformable DLW equation (Case 1) using $p=0,m=0.3,l=0.09,\beta =1,\gamma =1$, $\gamma =0.8$, $\gamma =0.4$ and $\gamma =0.1$.

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**MDPI and ACS Style**

Javeed, S.; Riaz, S.; Saleem Alimgeer, K.; Atif, M.; Hanif, A.; Baleanu, D.
First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models. *Symmetry* **2019**, *11*, 783.
https://doi.org/10.3390/sym11060783

**AMA Style**

Javeed S, Riaz S, Saleem Alimgeer K, Atif M, Hanif A, Baleanu D.
First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models. *Symmetry*. 2019; 11(6):783.
https://doi.org/10.3390/sym11060783

**Chicago/Turabian Style**

Javeed, Shumaila, Sidra Riaz, Khurram Saleem Alimgeer, M. Atif, Atif Hanif, and Dumitru Baleanu.
2019. "First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models" *Symmetry* 11, no. 6: 783.
https://doi.org/10.3390/sym11060783