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

Authors: M. A. Abdou; Saud Owyed; S. Saha Ray; Yu-Ming Chu; Mustafa Inc; Loubna Ouahid
Journal: Advances in Mathematical Physics
 (2020)

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## 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.


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## Body

## 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 Section2, 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 α in a brief conclusion at the end of the article.Definition 1.
Letψ:0,∞⟶R. Some definitions, useful properties, and a theorem about conformable derivatives are given as follows:
(1)Uαψt=limε⟶0ψt+εt1−α−ψtε,t>0,0<α<1,Uαbψ+ch=BUαψ+CUαh,B,C∈R,Uαtλ=λtλ−α,λ∈R,Uαψh=ψUαh+hUαψ,Uαψh=hUαψ−ψUαhh2.Ifψ is differentiable, then Uαψt=t1−αdψ/dh.Theorem 2.
Letψ:0,∞⟶R be a differentiable function. Then,
(2)Uαψ∗h=t1−αh′tψ′ht.


## 2. Solution Method

### 2.1. Extended Fractional Subequation Method

For a given nonlinear FPDE as(3)χu1uk,Dtαu1Dtαuk,Dx1αu1Dx1αuk=0,0<α<1.in which χ is a polynomial of u. Using wave transformation as
(4)uit,x1,x2,⋯xn=Uiξ,ξ=ct+⋯+knxn,Eq. (3) reads
(5)ϕU1⋯Uk,cαDξαU1⋯cαDξαUk,k1αDξαU1⋯k1αDξαUk⋯,knαDξαU1=0,0<α<1.Thus,(6)Uiξ=∑i=0MaiDξαwξwξi,where w=wξ satisfies
(7)AwξDξ2αwξ−BwξDξαwξ−CDξαwξ2−Ewξ2=0,where Dξαwξ is a RL fractional operator of order α. To solve Eq. (7), assume wξ=zη, with the fractional complex transformation, then
(8)Azηz′′η−Bzηz′η−Cz′η2−Ez2η=0.SinceDξαwξ=Dξαzη=z′η. The general solutions Eq. (7) is as follows:where ν1=B2+4EA−C, ν2=EA−C, L1, and L2 are arbitrary constants and η=ξα/Γ1+α. Inserting Eq. (6) into (5) knowing Eq.(7), collecting the same order terms Dξαwξ/wξ, then equating it to zero, k and c are obtained. As long as the solutions are obtained with the general expression Dξαwξ/wξ, admits several solutions of Eq. (3).Family 1.
As long asB≠0, ν1>0, admits to
(9)Dξαwξwξ=B2A−C+ν12A−CL1sinhν1η/2A−C+L2coshν1η/2A−CL1coshν1η/2A−C+L2sinhν1η/2A−C.Family 2.
Limiting caseB≠0, ν1<0 gains
(10)Dξαwξwξ=B2A−C+−ν12A−C−L1sin−ν1η/2A−C+L2cos−ν1η/2A−CL1cos−ν1η/2A−C+L2sin−ν1η/2A−C.Family 3.
ForB≠0, ν1=0,
(11)Dξαwξwξ=B2A−C+L2L1+L2η.Family 4.
WhenB=0, ν2>0,
(12)Dξαwξwξ=ν22A−CL1sinhν2η/2A−C+L2coshν2η/2A−CL1coshν2η/2A−C+L2sinhν2η/2A−C.Family 5.
WhenB=0, ν2<0, then
(13)Dξαwξwξ=−ν22A−C−L1sin−Δ2η/2A−C+L2cos−ν2η/2A−CL1cos−ν2η/2A−C+L2sin−ν2η/2A−C,


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

Let us first consider the fractional sine-Gordon equation as(14)Vxx−Dt2αV=m2sinV,m is constant.By using the transformationV=Vξ, ξ=ax−νtα/α. Then Eq. (14) yields
(15)d2V/2dξ2=m2a21−ν2sin2V/2+C,where C is an integration constant to be zero. Setting χξ=Vξ/2, b2=m2/a21−ν2. Then Eq. (15) reads
(16)dχwξdξ=bsinχ,Settingb=1, we have
(17)sinχξ=2deξd2e2ξ+1=sechξ,cosχξ=d2e2ξ−1d2e2ξ+1=tanhξ,(18)sinχξ=icschξ,cosχξ=cothξ,d=1.In view of this method, we assume the trail solutions by(19)Vξ=∑j=1Ntanhj−1ξBjsechξ+Ajsechξ+A0,(20)Vξ=∑j=1Ncothj−1ξjBjcschξ+Ajcothξ+A0.Making use of Eq. (18), then Eq. (19) can be rewritten as follows
(21)Vξ=∑j=1Ncosj−1ξBjsinξ+Ajcosξ+A0,where N can be obtained by balancing principle. Inserting Eq. (21) into (15) and the collecting the same power of cosiξsinjξ, admitting the system of algebraic equation, by solving them by Maple, the coefficient values Aj,Bj,ν can be determined. Inserting these values into Eq. (19), the exact solutions of Eq. (14) are determined.


## 2.1. Extended Fractional Subequation Method

For a given nonlinear FPDE as(3)χu1uk,Dtαu1Dtαuk,Dx1αu1Dx1αuk=0,0<α<1.in which χ is a polynomial of u. Using wave transformation as
(4)uit,x1,x2,⋯xn=Uiξ,ξ=ct+⋯+knxn,Eq. (3) reads
(5)ϕU1⋯Uk,cαDξαU1⋯cαDξαUk,k1αDξαU1⋯k1αDξαUk⋯,knαDξαU1=0,0<α<1.Thus,(6)Uiξ=∑i=0MaiDξαwξwξi,where w=wξ satisfies
(7)AwξDξ2αwξ−BwξDξαwξ−CDξαwξ2−Ewξ2=0,where Dξαwξ is a RL fractional operator of order α. To solve Eq. (7), assume wξ=zη, with the fractional complex transformation, then
(8)Azηz′′η−Bzηz′η−Cz′η2−Ez2η=0.SinceDξαwξ=Dξαzη=z′η. The general solutions Eq. (7) is as follows:where ν1=B2+4EA−C, ν2=EA−C, L1, and L2 are arbitrary constants and η=ξα/Γ1+α. Inserting Eq. (6) into (5) knowing Eq.(7), collecting the same order terms Dξαwξ/wξ, then equating it to zero, k and c are obtained. As long as the solutions are obtained with the general expression Dξαwξ/wξ, admits several solutions of Eq. (3).Family 1.
As long asB≠0, ν1>0, admits to
(9)Dξαwξwξ=B2A−C+ν12A−CL1sinhν1η/2A−C+L2coshν1η/2A−CL1coshν1η/2A−C+L2sinhν1η/2A−C.Family 2.
Limiting caseB≠0, ν1<0 gains
(10)Dξαwξwξ=B2A−C+−ν12A−C−L1sin−ν1η/2A−C+L2cos−ν1η/2A−CL1cos−ν1η/2A−C+L2sin−ν1η/2A−C.Family 3.
ForB≠0, ν1=0,
(11)Dξαwξwξ=B2A−C+L2L1+L2η.Family 4.
WhenB=0, ν2>0,
(12)Dξαwξwξ=ν22A−CL1sinhν2η/2A−C+L2coshν2η/2A−CL1coshν2η/2A−C+L2sinhν2η/2A−C.Family 5.
WhenB=0, ν2<0, then
(13)Dξαwξwξ=−ν22A−C−L1sin−Δ2η/2A−C+L2cos−ν2η/2A−CL1cos−ν2η/2A−C+L2sin−ν2η/2A−C,


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

Let us first consider the fractional sine-Gordon equation as(14)Vxx−Dt2αV=m2sinV,m is constant.By using the transformationV=Vξ, ξ=ax−νtα/α. Then Eq. (14) yields
(15)d2V/2dξ2=m2a21−ν2sin2V/2+C,where C is an integration constant to be zero. Setting χξ=Vξ/2, b2=m2/a21−ν2. Then Eq. (15) reads
(16)dχwξdξ=bsinχ,Settingb=1, we have
(17)sinχξ=2deξd2e2ξ+1=sechξ,cosχξ=d2e2ξ−1d2e2ξ+1=tanhξ,(18)sinχξ=icschξ,cosχξ=cothξ,d=1.In view of this method, we assume the trail solutions by(19)Vξ=∑j=1Ntanhj−1ξBjsechξ+Ajsechξ+A0,(20)Vξ=∑j=1Ncothj−1ξjBjcschξ+Ajcothξ+A0.Making use of Eq. (18), then Eq. (19) can be rewritten as follows
(21)Vξ=∑j=1Ncosj−1ξBjsinξ+Ajcosξ+A0,where N can be obtained by balancing principle. Inserting Eq. (21) into (15) and the collecting the same power of cosiξsinjξ, admitting the system of algebraic equation, by solving them by Maple, the coefficient values Aj,Bj,ν 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 (3+1)-dimension space-time FKPE which is as follows
(22)DxαDtαux,y,z,t+dDxαux,y,z,tDxαux,y,z,t+νDyααααux,y,z,t+βDzααux,y,z,t+νDxααux,y,z,t=0,where uξ=Ux,y,z,t is the field function, ν, δ, d, and β∈ℝ. Let ux,y,z,t=Uξ, where ξ=kx+ct+ly+mz+ξ0, k, l, c, m, ξ0 then
(23)DxαU=DxαUξ=DξαUξx′α=kαDξαU,DtαU=DtαUξ=DξαUξt′α=cαDξαU.Then, Eq. (22) reduces to
(24)cαkαDξααUξ+dk2αDξαUξDξαUξ+νk4αDξααααUξ+δl2αDξααUξ+m2αβDξααUξ=0.Now, we assume the solution of Eq. (24) as
(25)Uξ=∑i=0MaiDξαwξwξi,where w=wξ. Using the proposed algorithm for Eq. (24), we have M=2.Then,(26)Uξ=a0+a1Dξαwξwξ+a2Dξαwξwξ2.Inserting (26) into (24) and collecting the terms with a similar degree of Dξαwξ/wξ, equating it to zero, we have two values of ai,i=0,1,2.., k, c, l, and m(27)a0=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+k2ανB2+8k2ανECdA2,(28)a1=12−C+Aνk4αBdk2αA2,a2=−12C2−2AC+A2νk4αdk2αA2,cα=cα,kα=kα.From Eqs. (28) and (26), we gain
(29)Uξ=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+k2ανB2+8k2ανECdA2,+12−C+Aνk4αBdk2αA2Dξαwξwξ+−12C2−2AC+A2νk4αdk2αA2Dξαwξwξ2.In view of Family1–5 in (26), we obtain the followingFamily 6.
WhenB≠0, ν1>0(30)U1ξ=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+k2ανB2+8k2ανECdA2+12−C+Aνk4αBdk2αA2B2A−C+ν12A−CL1sinhν1η/2A−C+L2coshν1η/2A−CL1coshν1η/2A−C+L2sinhν1η/2A−C+−12C2−2AC+A2νk4αdk2αA2B2A−C+ν12A−CL1sinhν1η/2A−C+L2coshν1η/2A−CL1coshν1η/2A−C+L2sinhν1η/2A−C2.Family 7.
WhenB≠0, ν1<0(31)U2ξ=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+k2ανB2+8k2ανECdA2+12−C+Aνk4αBdk2αA2B2A−C+−ν12A−C−L1sin−ν1η/2A−C+L2cos−ν1η/2A−CL1cos−ν1η/2A−C+L2sin−ν1η/2A−C+−12C2−2AC+A2νk4αdk2αA2B2A−C+−ν12A−C−L1sin−ν1η/2A−C+L2cos−ν1η/2A−CL1cos−ν1η/2A−C+L2sin−ν1η/2A−C2.Family 8.
WhenB≠0, ν1=0(32)U3ξ=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+k2ανB2+8k2ανECdA2+12−C+Aνk4αBdk2αA2B2A−C+L2L1+L2η+−12C2−2AC+A2νk4αdk2αA2B2A−C+L2L1+L2η2.Family 9.
ForB=0, ν2>0(33)U4ξ=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+8k2ανECdA2+−12C2−2AC+A2νk4αdk2αA2ν22A−CL1sinhν2η/2A−C+L2coshν2η/2A−CL1coshΔ2η/2A−C+L2sinhν2η/2A−C2.Family 10.
In case ofB=0, ν2<0(34)U5ξ=−k−2αA2m2αβ+k−2αA2δl2α+k−2αA2cαkα−8k2ανEA+8k2ανECdA2+−12C2−2AC+A2νk4αdk2αA2−ν22A−C−L1sin−Δ2η/2A−C+L2cos−ν2η/2A−CL1cos−ν2η/2A−C+L2sin−ν2η/2A−C2,where ν1=4EA−C, ν2=EA−C, η=ξα/Γα+1, and ξ=kx+ct+mz+ly+ξ0. It is clearly seen that the solutions depend on α, and when α=1, 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
(35)Uξ=A0+B1sinξ+A1cosξ+A2cos2ξ+B2cosξsinξ.Inserting Eq. (35) into (24) and the collecting the same power of cosiχxisinjχxi, admitting the system of algebraic equation, by solving them by Maple, admits toSet 1.
(36)kα=kα,cα=cα,A1=0,B1=0,B2=±6ik2ανd,A2=−6νk4αdk2α,A0=5k2αν−k−2αδL2α−k−2αcαkα−k−2αm2αβd,Set 2.
(37)kα=kα,cα=cα,A1=0,B1=0,B2=0,A2=−12νk4αdk2α,A0=8k2αν−k−2αcαkα−k−2αδL2α−k−2αm2αβd.Inserting Set1 into (35), we obtain the exact solution of Eq. (22) as
(38)u1ξ=5k2αν−k−2αδL2α−k−2αcαkα−k−2αm2αβd−6νk4αdk2αtanh2ξ±6ik2ανdtanhξsechξ,(39)u2ξ=5k2αν−k−2αδL2α−k−2αcαkα−k−2αm2αβd−6νk4αdk2αcoth2ξ±6ik2ανdicothξcschξ,where ξ=kx+ct+ly+mz+ξ0. Knowing Set 2 and Eq. (35), we gain the exact solution of Eq. (22) as follows:
(40)u3ξ=8k2αν−k−2αcαkα−k−2αδL2α−k−2αm2αβd−12νk4αdk2αtanh2χξ,(41)u4ξ=8k2αν−k−2αcαkα−k−2αδL2α−k−2αm2αβd−12νk4αdk2αcoth2χξ,where ξ=kx+ct+ly+mz+ξ0. It is to be noted that, the 3D graph represent the obtained solutions with fixed y=z=1 of Eqs. (38) and (40) are shown graphically (see Figures 1–7) for fixed parameter with a different choice of fractal order α.Figure 1
The 3D graph represents of Eq. (38) via sine-Gordon expansion method under the fixed values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=1.Figure 2
The 3D graph represents of Eq. (38) via sine-Gordon expansion method with a fixed values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=0.55Figure 3
The 3D graph represents of Eq. (38) via sine-Gordon expansion method with a fixed values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=0.85Figure 4
The 3D graph represents of Eq. (38) via sine-Gordon expansion method under the values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=0.45Figure 5
The 3D graph represents of the wave solutions of Eq. (40) via sine-Gordon expansion method with a fixed values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=1Figure 6
The 3D graph represents of Eq. (40) via sine-Gordon expansion method with a fixed values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=0.35Figure 7
The 3D graph represents of Eq. (40) via sine-Gordon expansion method with a fixed values m=0.05,l=1;k=0.5,v=1,y=z=1,c=0.05,d=1,β=1,δ=1,L=0.5,10<x<10,10<t<10 with fractal order α=0.55


## 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 (3+1)-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 Dξαwξ/wξ 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 α.The presence of parameters makes our results useful for the IVBVP with fractional order. Forα=1, 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.



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