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