## Nonlinear Waves and Differential Equations in Applied Mathematics and Physics

View this Special IssueResearch Article | Open Access

M. A. Abdou, Saud Owyed, S. Saha Ray, Yu-Ming Chu, Mustafa Inc, Loubna Ouahid, "Fractal Ion Acoustic Waves of the Space-Time Fractional Three Dimensional KP Equation", *Advances in Mathematical Physics*, vol. 2020, Article ID 8323148, 7 pages, 2020. https://doi.org/10.1155/2020/8323148

# Fractal Ion Acoustic Waves of the Space-Time Fractional Three Dimensional KP Equation

**Academic Editor:**Xiao-Ling Gai

#### Abstract

Methods known as fractional subequation and sine-Gordon expansion (FSGE) are employed to acquire new exact solutions of some fractional partial differential equations emerging in plasma physics. Fractional operators are employed in the sense of conformable derivatives (CD). New exact solutions are constructed in terms of hyperbolic, rational, and trigonometric functions. Computational results indicate the power of the method.

#### 1. Introduction

Nonlinear propagation of electrostatic excitations in electron-positron ion plasmas and nonthermal distribution of electrons is an important research area in astrophysical and space plasmas [1–6].

Many important phenomena such as the effective behavior of the ionized matter, magnetic field near the surfaces of the sun and stars, emission mechanisms of pulsars, the origin of cosmic rays and radio sources, dynamics of magnetosphere, and propagation of electromagnetic radiation through the upper atmosphere required the study of plasma physics. Equations such as Korteweg de Vries (KdV), Burgers, KdV-Burgers, and Kadomtsev-Petviashvili (KP) were highly used models in the description of plasma systems.

We study the physical phenomena for space-time fractional KP equation with the aid of fractional calculus and examine the resulting solutions in detail. The factional calculus [7–13] has a wide range of applications and is deeply rotted in the field of probability, mathematical physics, differential equations, and so on. Very recently, fractional differential equations have got a lot of consideration as they define many complex phenomena in various fields. Several fractional-order models play very important roles in different areas including physics, engineering, mechanics and dynamical systems, signal and image processing, control theory, biology, and materials [14–18].

The paper is summarized as follows. Definitions and properties of conformable derivatives are discussed. In Section 2, a discussion about the two algorithms method, namely, fractional subequation method and sine-Gordon expansion method for solving FPDEs arising in plasma physics are given. In Section 3, two schemes are employed for some new exact solutions for the FKPE. We presented a graphical description of some of the solutions with a fixed value of fractal order in a brief conclusion at the end of the article.

*Definition 1. *Let . Some definitions, useful properties, and a theorem about conformable derivatives are given as follows:

If is differentiable, then .

Theorem 2. *Let be a differentiable function. Then,
*

#### 2. Solution Method

##### 2.1. Extended Fractional Subequation Method

For a given nonlinear FPDE as in which is a polynomial of . Using wave transformation as

Eq. (3) reads

Thus, where satisfies where is a RL fractional operator of order . To solve Eq. (7), assume , with the fractional complex transformation, then

Since . The general solutions Eq. (7) is as follows:where , , , and are arbitrary constants and . Inserting Eq. (6) into (5) knowing Eq.(7), collecting the same order terms , then equating it to zero, and are obtained. As long as the solutions are obtained with the general expression , admits several solutions of Eq. (3).

*Family 1. *As long as , , admits to

*Family 2. *Limiting case , gains

*Family 3. *For , ,

*Family 4. *When , ,

*Family 5. *When , , then

##### 2.2. Analysis of the Fractional Sine-Gordon Expansion (FSGE) Method

Let us first consider the fractional sine-Gordon equation as is constant.

By using the transformation , . Then Eq. (14) yields where is an integration constant to be zero. Setting , . Then Eq. (15) reads

Setting , we have

In view of this method, we assume the trail solutions by

Making use of Eq. (18), then Eq. (19) can be rewritten as follows where can be obtained by balancing principle. Inserting Eq. (21) into (15) and the collecting the same power of , admitting the system of algebraic equation, by solving them by Maple, the coefficient values can be determined. Inserting these values into Eq. (19), the exact solutions of Eq. (14) are determined.

#### 3. New Applications

In this part of our research, we apply a novel computational approach mentioned above to illustrate the advantages for finding analytical solutions of ()-dimension space-time FKPE which is as follows where is the field function, , , , and . Let , where , , , , , then

Then, Eq. (22) reduces to

Now, we assume the solution of Eq. (24) as where . Using the proposed algorithm for Eq. (24), we have .

Then,

Inserting (26) into (24) and collecting the terms with a similar degree of , equating it to zero, we have two values of , , , , and

From Eqs. (28) and (26), we gain

In view of Family 1–5 in (26), we obtain the following

*Family 6. *When ,

*Family 7. *When ,

*Family 8. *When ,

*Family 9. *For ,

*Family 10. *In case of , where , , , and . It is clearly seen that the solutions depend on , and when , we have the solutions that are obtained for normal derivative. The results introduce free parameters. Hence, five solutions are essential in handling initial and boundary problems. To solve the reduced Eq. (24) by the sine-Gordon expansion (FSGE) method, assume the solution of Eq. (24) as

Inserting Eq. (35) into (24) and the collecting the same power of , admitting the system of algebraic equation, by solving them by Maple, admits to

*Set 1. *

*Set 2. *

Inserting Set 1 into (35), we obtain the exact solution of Eq. (22) as where . Knowing Set 2 and Eq. (35), we gain the exact solution of Eq. (22) as follows: where . It is to be noted that, the graph represent the obtained solutions with fixed of Eqs. (38) and (40) are shown graphically (see Figures 1–7) for fixed parameter with a different choice of fractal order .

#### 4. Concluding Remarks

In this article, the extended fractional subequation method and sine-Gordon expansion (FSGE) method have been proposed for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of conformable derivative. This paper studies ()-dimensions space-time FKPE which appears in plasma physics in the sense of conformable derivatives via two algorithms, namely, the extended fractional subequation method and FSGE method to obtain sets of exact solutions. Using suitable wave transform, the equations are reduced to some ODEs. Then, the admissible solutions are substituted into the resultant ODE. Equating the coefficients of in extended fractional subequation method and cosine and sine functions and their multiplications in FSGE method to zero leads to some algebraic system of equations. Solving this system gives the relations among the parameters. Some 3-D solution graphs are presented in some finite domains to comprehend the effects of .

The presence of parameters makes our results useful for the IVBVP with fractional order. For , our solutions go back to that previously obtained solution. The performance of these approaches shows the ability for applying on various space-time fractional nonlinear equations in nonlinear science.

#### Data Availability

No any data availability

#### Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

#### Acknowledgments

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, and 11601485).

#### References

- R. Sabry, W. M. Moslem, P. K. Shukla, and H. Saleem, “Cylindrical and spherical ion-acoustic envelope solitons in multicomponent plasmas with positrons,”
*Physical Review E*, vol. 79, no. 5, article 056402, 2009. View at: Publisher Site | Google Scholar - H. Schamel, “Stationary solitary, snoidal and sinusoidal ion acoustic waves,”
*Journal of Plasma Physics*, vol. 14, no. 10, pp. 905–924, 1972. View at: Publisher Site | Google Scholar - M. A. Zahran, E. K. El-Shewy, and H. G. Abdelwahed, “Dust-acoustic solitary waves in a dusty plasma with dust of opposite polarity and vortex-like ion distribution,”
*Journal of Plasma Physics*, vol. 79, no. 5, pp. 859–865, 2013. View at: Publisher Site | Google Scholar - E. K. El-Shewy, “Linear and nonlinear properties of electron-acoustic solitary waves with non-thermal electrons,”
*Chaos, Solitons and Fractals*, vol. 31, no. 4, pp. 1020–1023, 2007. View at: Publisher Site | Google Scholar - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, World Scientific, Series on Complexity Nonliearity and Chaos. Boston, 2012. View at: Publisher Site - A. Jajarmi, A. Yusuf, D. Baleanu, and M. Inc, “Theory and application for the system of fractional Burger equations with Mittag leffler kernel,”
*Physica A*, vol. 547, p. 123860, 2020. View at: Google Scholar - R. Khalil, M. Al Forani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,”
*Journal of Computational Apllied Mathematics*, vol. 264, pp. 65–70, 2014. View at: Publisher Site | Google Scholar - R. Cimpoiasu and R. Constantinescu, “The inverse symmetry problem for a 2D generalized second order evolutionary equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 57, pp. 147–154, 2010. View at: Google Scholar - S. Saha Ray, “New exact solutions of nonlinear fractional acoustic wave equations in ultrasound,”
*Computers and Mathematics with Applications*, vol. 71, no. 3, pp. 859–868, 2016. View at: Publisher Site | Google Scholar - Z. Korpinar, M. Inc, and M. Bayram, “Theory and application for the system of fractional Burger equations with Mittag leffler kernel,”
*Applied Mathematics and Computation*, vol. 367, p. 124781, 2020. View at: Google Scholar - S. Sahoo and S. S. Ray, “Improved fractional sub-equation method for (3+1) -dimensional generalized fractional KdV–Zakharov–Kuznetsov equations,”
*Computers and Mathematics with Applications*, vol. 70, no. 2, pp. 158–166, 2015. View at: Publisher Site | Google Scholar - M. A. Abdou and A. A. Soliman, “New exact travelling wave solutions for space-time fractional nonlinear equations describing nonlinear transmission lines,”
*Results in Physics*, vol. 9, pp. 1497–1501, 2018. View at: Publisher Site | Google Scholar - M. A. Abdou, “An analytical method for space time fractional nonlinear differential equations arising in plasma physics,”
*Journal of Ocean Engineering and Science*, vol. 2, no. 4, pp. 288–292, 2017. View at: Publisher Site | Google Scholar - S. Sahoo and S. S. Ray, “New exact solutions of fractional Zakharov—Kuznetsov and modified Zakharov—Kuznetsov equations using fractional sub-equation method,”
*Communications in Theoretical Physics*, vol. 63, no. 1, pp. 25–30, 2015. View at: Google Scholar - Z. Odibat and D. Baleanu, “Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives,”
*Applied Numerical Mathematics*, vol. 156, pp. 94–105, 2020. View at: Publisher Site | Google Scholar - J. Singh, D. Kumar, Z. Hammouch, and A. Atangana, “A fractional epidemiological model for computer viruses pertaining to a new fractional derivative,”
*Applied Mathematics and Computation*, vol. 316, pp. 504–515, 2018. View at: Publisher Site | Google Scholar - S. Uçar, E. Uçar, N. Özdemir, and Z. Hammouch, “Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative,”
*Chaos, Solitons and Fractals*, vol. 118, pp. 300–306, 2019. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2020 M. A. Abdou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.