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Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G^{′}/G)-Expansion Method

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## Abstract

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## 1. Introduction

## 2. Method and Materials

## 3. Results

**Problem 1:**Consider the fractional Klein-Gordon equation in Equation (1)

**Case 1:**$k=k,A=A,c=c,{d}_{0}=\frac{-3{a}^{2}{c}^{2}{A}^{2}+3{k}^{2}{A}^{2}-b}{2e},{d}_{1}=\frac{-6A(-{k}^{2}+{a}^{2}{c}^{2})}{e},$${d}_{2}=\frac{-6A(-{k}^{2}+{a}^{2}{c}^{2})}{e},$$B=\frac{-b+{a}^{2}{c}^{2}{A}^{2}-{k}^{2}{A}^{2}}{4(ac-k)(ac+k)}$

**Case 2:**$k=k,A=A,c=c,{d}_{0}=\frac{-3{a}^{2}{c}^{2}{A}^{2}+3{k}^{2}{A}^{2}+3b}{2e},{d}_{1}=\frac{-6A(-{k}^{2}+{a}^{2}{c}^{2})}{e},$${d}_{2}=\frac{-6A(-{k}^{2}+{a}^{2}{c}^{2})}{e},$$B=\frac{-b+{a}^{2}{c}^{2}{A}^{2}-{k}^{2}{A}^{2}}{4(ac-k)(ac+k)}$

**Family 1:**When ${A}^{2}-4B>0$ then Equation (14) with the help Equation (9) (if ${C}_{1}\ne 0$ and ${C}_{2}=0$; ${C}_{1}=0$ and ${C}_{2}\ne 0$) implies hyperbolic solitary wave solutions:

**Family 2:**When ${A}^{2}-4B<0$ then Equation (14) with the help of Equation (9) (if ${C}_{1}\ne 0$ and ${C}_{2}=0$; ${C}_{1}=0$ and ${C}_{2}\ne 0$) implies periodic solitary wave solutions:

**Family 3:**When ${A}^{2}-4B=0$ then Equation (14) with the help of Equation (9) (if ${C}_{1}\ne 0$ and ${C}_{2}=0$; ${C}_{1}=0$ and ${C}_{2}\ne 0$) implies rational solitary wave solutions:

**Family 4:**When ${A}^{2}-4B>0$ then Equation (14) with the help of Equation (9) (if ${C}_{1}\ne 0$ and ${C}_{2}=0$; ${C}_{1}=0$ and ${C}_{2}\ne 0$) implies hyperbolic solitary wave solutions:

**Family 5:**When ${A}^{2}-4B<0$ then Equation (14) with the help of Equation (9) (if ${C}_{1}\ne 0$ and ${C}_{2}=0$; ${C}_{1}=0$ and ${C}_{2}\ne 0$) implies periodic solitary wave solutions:

**Family 6:**When ${A}^{2}-4B=0$ then Equation (14) with the help of Equation (9) (if ${C}_{1}\ne 0$ and ${C}_{2}=0$; ${C}_{1}=0$ and ${C}_{2}\ne 0$) implies rational solitary wave solutions:

**Problem 2:**The fractional Gas Dynamics equation in Equation (2) is given by

**Case 1:**

**Case 2:**

**Family 1:**When $A>0$ then Equation (18) with the help of Equation (10) implies periodic solitary wave solutions:

**Family 2:**When $A<0$ then Equation (18) with the help of Equation (10) implies hyperbolic solitary wave solutions:

**Family 3:**When $A>0$ then Equation (18) with the help of Equation (10) implies periodic solitary wave solutions:

**Family 4:**When $A<0$ then Equation (18) with the help of Equation (10) implies hyperbolic solitary wave solutions:

## 4. Discussion

## 5. Description of Figures

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Almeida, R.; Bastos, N.R.O.; Teresa, M.; Monteiro, T. Modeling some real phenomena by fractional differential equationsh. Math. Methods Appl. Sci.
**2016**, 39, 4846–4855. [Google Scholar] [CrossRef] - Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 64, 213–231. [Google Scholar] [CrossRef] - Tarasov, V. Generalized memory: Fractional calculus approach. Fractal Fract.
**2018**, 2, 23. [Google Scholar] [CrossRef] - El-Misiery, A.E.M.; Ahmed, E. On a fractional model for earthquakes. Appl. Math. Comput.
**2006**, 178, 207–211. [Google Scholar] [CrossRef] - Wu, J. A wavelet operational method for solving fractional partial differential equations numerically. Appl. Math. Comput.
**2009**, 214, 31–40. [Google Scholar] [CrossRef] - Zhang, W.; Li, J.; Yang, Y. A fractional diffusion-wave equation with non-local regularization for image denoising. Signal Process.
**2014**, 103, 6–15. [Google Scholar] [CrossRef] - Hu, Y.; Øksendal, B. Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top.
**2003**, 6, 1–32. [Google Scholar] [CrossRef] - Nagy, A.M. Numerical solution of time fractional nonlinear Klein-Gordon equation using Sinc-Chebyshev collocation method. Appl. Math. Comput.
**2017**, 310, 139–148. [Google Scholar] [CrossRef] - Garrappa, R. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics
**2018**, 6, 16. [Google Scholar] [CrossRef] - Jiang, J.; Feng, Y.; Li, S. Exact Solutions to the Fractional Differential Equations with Mixed Partial Derivatives. Axioms
**2018**, 7, 10. [Google Scholar] [CrossRef] - Song, J.; Yin, F.; Cao, X.; Lu, F. Fractional variational iteration method versus Adomian’s decomposition method in some fractional partial differential equations. J. Appl. Math.
**2013**, 2013, 392567. [Google Scholar] [CrossRef] - Duan, J.; Rach, R.; Baleanu, D.; Wazwaz, A. A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fract. Calc.
**2012**, 3, 73–99. [Google Scholar] - Mahmood, S.; Shah, R.; Arif, M. Laplace Adomian Decomposition Method for Multi Dimensional Time Fractional Model of Navier-Stokes Equation. Symmetry
**2019**, 11, 149. [Google Scholar] [CrossRef] - Wang, Q. Homotopy perturbation method for fractional KdV-Burgers equation. Chaos Solitons Fractals
**2008**, 35, 843–850. [Google Scholar] [CrossRef] - Raslan, K.R.; Ali, K.K.; Shallal, M.A. The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations. Chaos Solitons Fractals
**2017**, 103, 404–409. [Google Scholar] [CrossRef] - Alzaidy, J.F. Fractional sub-equation method and its applications to the space-time fractional differential equations in mathematical physics. Br. J. Math. Comput. Sci.
**2013**, 3, 153. [Google Scholar] [CrossRef] - Tasbozan, O.; Çenesiz, Y.; Kurt, A. New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method. Eur. Phys. J. Plus
**2016**, 131, 244. [Google Scholar] [CrossRef] - Meerschaert, M.M.; Tadjeran, C. Finite difference approximations for fractional advection–dispersion flow equations. J. Comput. Appl. Math.
**2004**, 172, 65–77. [Google Scholar] [CrossRef] [Green Version] - Jiang, Y.; Ma, J. High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math.
**2011**, 235, 3285–3290. [Google Scholar] [CrossRef] [Green Version] - Zheng, B. Exp-function method for solving fractional partial differential equations. Sci. World J.
**2013**, 2013, 465723. [Google Scholar] [CrossRef] - Shang, N.; Zheng, B. Exact solutions for three fractional partial differential equations by the ($\frac{{G}^{\prime}}{G}$)-expansion method. Int. J. Appl. Math.
**2013**, 43, 114–119. [Google Scholar] - Zhang, Y. Solving STO and KD equations with modified Riemann–Liouville derivative using improved ($\frac{{G}^{\prime}}{G}$)-expansion function method. Int. J. Appl. Math.
**2015**, 45, 16–22. [Google Scholar] - Shakeel, M.; Mohyud-Din, S.T. New ($\frac{{G}^{\prime}}{G}$)-expansion method and its application to the zakharov-kuznetsov-benjamin-bona-mahony (ZK–BBM) equation. J. Assoc. Arab Univ. Basic Appl. Sci.
**2015**, 18, 66–81. [Google Scholar] [CrossRef] - Zheng, B. ($\frac{{G}^{\prime}}{G}$)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys.
**2012**, 58, 623. [Google Scholar] [CrossRef] - Zayed, E.M.E.; Amer, Y.A.; Shohib, R.M.A. Exact traveling wave solutions for nonlinear fractional partial differential equations using the improved ($\frac{{G}^{\prime}}{G}$)-expansion method. Int. J. Eng.
**2014**, 4, 8269. [Google Scholar] - Abuteen, E.; Freihat, A.; Al-Smadi, M.; Khalil, H.; Khan, R.A. Approximate series solution of nonlinear, fractional Klein-Gordon equations using fractional reduced differential transform method. arXiv
**2017**, arXiv:1704.06982. [Google Scholar] [CrossRef] - Acan, O.; Baleanu, D. A new numerical technique for solving fractional partial differential equations. arXiv
**2017**, arXiv:1704.02575. [Google Scholar] [CrossRef] - Chowdhury, M.S.H.; Hashim, I. Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations. Chaos Solitons Fractals
**2009**, 39, 1928–1935. [Google Scholar] [CrossRef] - Kheiri, H.; Shahi, S.; Mojaver, A. Analytical solutions for the fractional Klein-Gordon equation. Comput. Methods Differ. Equ.
**2014**, 2, 99–114. [Google Scholar] - Singh, J.; Kumar, D.; Kılıçman, A. Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform. Abst. Appl. Anal.
**2013**, 2013, 934060. [Google Scholar] [CrossRef] - Tamsir, M.; Srivastava, V.K. Revisiting the approximate analytical solution of fractional-order gas dynamics equation. Alex. Eng. J.
**2016**, 55, 867–874. [Google Scholar] [CrossRef] - Alam, M.; Rahman, M.; Islam, R.; Roshid, H. Application of the new extended (G’/G)-expansion method to find exact solutions for nonlinear partial differential equation. Comput. Methods Differ. Equ.
**2015**, 3, 59–69. [Google Scholar] - Ozis, T.; Aslan, I. Symbolic computation and construction of new exact traveling wave solutions to Fitzhugh-Nagumo and Klein-Gordon equations. Z. Naturforschung A
**2009**, 64, 15. [Google Scholar]

**Figure 1.**For problem 1, the graph of ${V}_{2}(x,t)$ for the values $\alpha =c=k=e=1$, $A=a=2$, $b=-1$ and $B=\frac{11}{12}$ shows the hyperbolic solitary wave solution.

**Figure 2.**For problem 1, the graph of ${V}_{4}(x,t)$ for the values $\alpha =c=k=e=A=b=1$, $a=2$ and $B=\frac{1}{3}$ shows the periodic solitary wave solution.

**Figure 3.**For problem 2, the graph of ${V}_{4}(x,t)$ for the values $\alpha =1$, $A=-1$, $c=0$ and $k=\frac{1}{2}$ shows the hyperbolic solitary wave solution.

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**MDPI and ACS Style**

Khan, H.; Barak, S.; Kumam, P.; Arif, M.
Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (*G*^{′}/*G*)-Expansion Method. *Symmetry* **2019**, *11*, 566.
https://doi.org/10.3390/sym11040566

**AMA Style**

Khan H, Barak S, Kumam P, Arif M.
Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (*G*^{′}/*G*)-Expansion Method. *Symmetry*. 2019; 11(4):566.
https://doi.org/10.3390/sym11040566

**Chicago/Turabian Style**

Khan, Hassan, Shoaib Barak, Poom Kumam, and Muhammad Arif.
2019. "Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (*G*^{′}/*G*)-Expansion Method" *Symmetry* 11, no. 4: 566.
https://doi.org/10.3390/sym11040566