# Higher-Order Accurate and Conservative Hybrid Numerical Scheme for Relativistic Time-Fractional Vlasov-Maxwell System

Authors: Tamour Zubair; Muhammad Usman; Ilyas Khan; Nawaf N. Hamadneh; Tiao Lu; Mulugeta Andualem
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The historical analysis demonstrates that plasma scientists produced a variety of numerical methods for solving “kinetic” models, i.e., the Vlasov-Maxwell (VM) system. Still, on the other hand, a significant fact or drawback of most algorithms is that they do not preserve conservation philosophies. This is a crucial fact that cannot be disregarded since the Vlasov Maxwell system is associated with conservation rules and is capable of assessing after the accomplishment of certain helpful mathematical actions. To examine the fractional-order routine of charged particles, we constructed a fractional-order plasma model and proposed a higher-order conservative numerical approach based on operational matrices theory and Shifted Gegenbauer estimations. Numerical convergence is investigated to confirm its competence and compatibility. This concept may be used in problems involving variable order and multidimensionality, such as those involving Vlasov and Boltzmann systems.


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## Body

## 1. Introduction

The VM system is an important instrument due to its vast variety of applications [1–3]. Modern numerical plasma research is separated into two independent disciplines based on kinetic theory, notably the “particle” in cell technique (PICT) and the Vlasov equation (VE). The primary portion couple’s plasma “particle” motion equations to Maxwell equations and numerically cracks them using the particle in approach [1, 4], and [5]. The second approach employs FEM [6] and FDM [3] to discretize the VE. These are grid-constructed procedures. The main downside of debated numerical techniques is that they violate conservation commitments. Noteworthy key concerns with PICT are that in some conditions, complete energies rise dramatically in the lack of any valid source of energy. In 2010, a significant numerical study on this subject was conducted based on conservation rules. The terms “Crank” Nicolson and temporal integration are used to characterize the “conservative” arrangements of PICT [7–10].G. Lapenta [11] published research in 2017, in which he analyzed motion equations (ME) and “Maxwell” equations (ME) using “Crank” Nicolson and leap-frog discretization methods. The technique maintains “energy” conservation, but on the other hand, it contradicts Gauss's rule. As a result, we can conclude that formulating the PIC method, which adheres to all conservation rules, is difficult because two distinct types of the scheme are utilized, namely PICT and FDM. The forms of the “particles” are strongly retained by the integration approach used in PIC techniques, and they are also dispersion in the numerical sense, although one cannot state that FDM is dispersion-free. As a result of these mathematical conflicts, we may accomplish that it is dreadful to eloquent the PIC approach, which perfectly follows the conservation rules.Spectral algorithms are a powerful and groundbreaking tool for tackling several kinds of mathematical models. Different modules of “polynomials” in the orthogonal form are available in the prevailing literature for spectral estimates [12–14]. Two types of structures, i.e., VM [15] and Vlasov-Poisson (VP) [15, 16], are handled (numerically) using spectral methods which further linked Crank-Nicolson. These approaches assist us in overcoming difficulties, mainly numerical dispersion [17], that the PIC method encountered.The TFVMS is built and explored in this study utilizing a specific geometry and an improved structure of numerical scheme based on FD and SGPs approximations. Caputo sense's conservative time-fractional order finite difference approximations [18–24] has been utilized. Assuming that the “function” is constructed on multiple variables, we appropriately estimate stated “polynomials” and continue with the construction of the operational “matrices”. Recent advancement in this subject is detailed in ref [19, 20]. The project tightly enforces conservation laws, which will be discussed in further detail in the next segment. The numerical framework exhibits virtuous convergence, which is also discussed in the paper. Finally, we conclude our research.


## 2. Mathematical Modelling and Conservative Numerical Scheme

### 2.1. Generalized Form

The most recent and generalized form of VMS with important laws [19–24] are explained as:(1)D0CtαF+∂∂r⋅pγmF+∂∂p⋅qE+p×BγmcF,γ=1+pmc21/2,1cD0CtαE−∇×B+4πcJ=0,∇⋅E=4πρ,−1cD0CtαB−∇×E=0,∇⋅B=0,D0Ctαρ+∇⋅J=0.Table1 contains an explanation of all the symbols used in the preceding system.Table 1
Important symbols used in the theory.
SymbolsName of symbolsFDistribution functiontTimex,y,zVelocity componentspx,py,pzMomentum componentsγRelativistic parameterGjμGegenbauer polynomialsGjμShifted gegenbauer℘fFVector norms ofF,℘ExExVector norms ofEx,EElectric fieldBMagnetic fieldD0CtαCaputo time fractionalJCurrent densityρCharge densityhiμNormalizing factorϑμWeight function℘BzBzVector norms ofBz,℘EyEyVector norms ofEy


### 2.2. Mathematical Assumptions and Conversion

The significant assumptions for the problems are:(2)r=xi^+yj^,p=pxi^+pyj^,F=Ft,x,y,px,py,E=Exx,yi^+Eyx,yj^,J=jxi^+jyj^,B=Bzx,yk^.Therefore, we get the VMS as:(3)D0CtαF+1γmpx∂F∂x+py∂F∂y+qEx∂F∂px+Ey∂F∂py+qmcpyBzγ∂F∂px−pxBzγ∂F∂py=0,1cD0CtαEx+4πcjx=∂Bz∂y,1cD0CtαEy+4πcjy=−∂Bz∂x,1cD0CtαBz=∂Ex∂y−∂Ey∂x,∂Ex∂x+∂Ey∂y=4πρ,ρ=q∫Ω∈−∞,∞FdΩ,D0Ctαρ+∂∂xjx+∂∂yjy=0,jx=qm∫Ω∈−∞,∞pxγFdΩ,jy=qm∫Ω∈−∞,∞pyγFdΩ.


### 2.3. Scheme Discretization

According to the defined process [19, 20], we presented our system into discretized such matrix form as follow:(4)¶tαFη+1/2G+pxγmC1η+1Dx1G+pyγmC1η+1Dy1G+qC2ηG′C1η+1Dpx1G+qC3ηG′C1η+1Dpy1G+qmcpyγC4ηG′C1η+1Dpx1G−qmcpxγC4ηG′C1η+1Dpy1G=0,1c¶tαExη+1/2G′+4πcjxη+1=C4η+1Dy1G′,1c¶tαBzη+1/2G′=C2η+1Dy1G′−C3η+1Dx1G′,1c¶tαEyη+1/2G′+4πcjyη+1=−C4η+1Dx1G′,C2η+1Dx1G′+C3η+1Dy1G′=4πρη+1,¶tα∫Ω∈−∞,∞Fη+12dΩ+qm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0,ρη+1=q∫Ω∈−∞,∞C1η+1GdΩ,jxη+1=qm∫Ω∈−∞,∞pxγC1η+1GdΩ,jyη+1=qm∫Ω∈−∞,∞pyγC1η+1GdΩ.The definition of operator is [18].(5)¶tαFη+1/2=¶t¶tαCη+1/2+∑s=1ηCη+1/2−s=Δt−αΓ2−αCη+1−Cη+Δt−αΓ2−α∑s=1ηCη+1−s−Cη−sbsα.We also havebsα=s+11−α−s1−α. We use GM1,M2,M3,M4=G, and GM1,M2=G′ for simplicity in discretization form and also ¶tα is the operator and is specified by(6)¶tFη+1/2=δtCη+1/2+∑s=1ηδtCη+1/2−s=Cη+1−Cη+∑s=1ηCη+1−s−Cη−s.


### 2.4. Conservativeness of the Scheme

#### 2.4.1. Charge Conservation

Takingq∫Ω∈−∞,∞dΩ of VE (5) and once obtained:(7)∫Ω∈−∞,∞q¶tαFη+1/2G+pxγmC1η+1Dx1G+pyγmC1η+1Dy1GdΩ=0,¶tαq∫Ω∈−∞,∞Fη+1/2GdΩ+qm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0.Hence proved that charge conservation accordingly.


#### 2.4.2. Gauss Law and Solenoidal Constrains

(1) Differential Arrangement. Using the concepts of divergence and equation (2), we obtained: (8)1cdivD0CtαE−div∇×B+4πcdivJ=0⇒1cD0CtαdivE−4πρ=0.Hence, we can say that the above relation will satisfy if we have(9)divE=4πρ.(2) Discretization Arrangement. Apply the divergence based on the assumptions, and we contract:(10)1c¶tαDx1Exη+1/2G′+4πcqm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ=C4η+1Dx1Dy1G′,1c¶tαDy1Eyη+1/2G′+4πcqm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=−C4η+1Dx1Dx1G′,1c¶tαDx1Exη+1/2G′+Dy1Eyη+1/2G′+4πcqm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0,1c¶tαDx1Exη+1/2G′+Dy1Eyη+1/2G′−4πρη+1=0.As a result, we may claim that the Gauss rule is true if and only if the following conditions are met:(11)Dx1Ex0G′+Dy1Ey0G′=−4πρ0.It is not necessary to establish solenoidal limitations.


#### 2.4.3. Momentum Conservation

(1) Differential Arrangement. (12)Dtα∫Ω∈−∞,∞pFdΩ+∂∂r⋅∫Ω∈−∞,∞ppγmFdΩ=−∫Ω∈−∞,∞p∂∂p⋅qE+p×BγmcFdΩ,Dtα∫Ω∈−∞,∞pFdΩ+∂∂r⋅∫Ω∈−∞,∞ppγmFdΩ=∫Ω∈−∞,∞qEF+qFp×BγmcdΩ.For the case ofx-component, we obtained(13)Dtα∫Ω∈−∞,∞FpxdΩ+Ex∫Ω∈−∞,∞qFdΩ+∂∂r⋅∫Ω∈−∞,∞pxpγmFdΩ=qmc∫Ω∈−∞,∞1γFBzpydΩ,Dtα∫Ω∈−∞,∞pxFdΩ+∂∂r⋅∫Ω∈−∞,∞pxpγmFdΩ=ρEx+1cBzjy.Additionally, we shall express the “Maxwell” equation in its momentum form as:(14)Bz4π1cDtαEy+∂Bz∂x−∂Bx∂z+Ey4π1cDtαBz+∂Ey∂x−∂Ex∂y=−Bz4π4πcjy,−By4π1cDtαEz+∂Bx∂y−∂By∂x−Ez4π1cDtαBy+∂Ex∂z−∂Ez∂x=By4π4πcjz,Simplified form is(15)Bz4π1cDtαEy+∂Bz∂x−∂Bx∂z+Ey4π1cDtαBz+∂Ey∂x−∂Ex∂y=−Bz4π4πcjy.Additionally, we get(16)14πcDtαBzEy+18π∂∂xE2+B2−14πdivExE+BxB+14πBxdivB+ExdivE=−jyBzc.From (2), we get(17)14πcDtαBzEy+18π∂∂xE2+B2−14πdivBxE+BxB=−Exρ−jyBzc.Further from equations (14) and (17), we get(18)Dtα∫Ω∈−∞,∞pxF+BzEy4πcdΩ+∂∂x∫Ω∈−∞,∞pxpxγmF−Ex2+Bx24π+Ex2+Ey2+Bx2+By28πdΩ+∂∂y∫Ω∈−∞,∞pypyγmF−EyEx+ExEy4πdΩ∂∂x∫Ω∈−∞,∞pxpxγmF−Ex2+Bx24π+Ex2+Ey2+Bx2+By28πdΩ+∂∂y∫Ω∈−∞,∞pypyγmF−EyEx+ExEy4πdΩ=0.(2) Discretization Arrangement:(19)∫Ω∈−∞,∞px¶tαFη+12GdΩ+∫Ω∈−∞,∞pxpxγmC1η+1Dx1GdΩ+∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=qC2ηG′C1η+1GdΩ+∫Ω∈−∞,∞qpyγmcC4η+1G′C1η+1GdΩ,∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=C2ηG′ρη+1+1cC4η+1G′jyη+1.px¶tαFη+12GdΩ+∫Ω∈−∞,∞pxpxγmC1η+1Dx1GdΩ+∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=C2ηG′ρη+1+1cC4η+1G′jyη+1.Moreover; we have(20)14πC4η+1G′¶tαEyη+1/2G′+C4η+1Dx1G′+14πC3ηG′δtαBzη+1/2G′−C2η+1Dy1G′+C3η+1Dx1G′=−14πC4η+1G′4πcjyη+1,14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−C3η+1G′C2η+1Dy1G′=−1cC4η+1G′jyη+1,14πC2η+1G′C2η+1Dx1G′+C3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−14πC2η+1G′C2η+1Dx1G′+C3η+1Dy1G′=−1cC4η+1G′jyη+1.From (15), we can write as:(21)14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC2η+1G′C2η+1Dx1G′+C3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−14π2C2η+1G′C2η+1Dx1G′+C2η+1G′C3η+1Dy1G′+C3η+1G′C2η+1Dy1G′=−1cC4η+1G′jyη+1−14πC2η+1G′4πρη+1.As a result of (20) and (21), we may conclude that the scheme follows the momentum conservation rule.


#### 2.4.4. Energy Conservation

(1) Differential Arrangement. We shall use ∫Ω∈−∞,∞γmc2dΩ and obtained further:(22)Dtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=∫Ω∈−∞,∞pγm⋅qE+p×BγmcFdΩ,Dtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=∫Ω∈−∞,∞1γmqp⋅EF+1mcp⋅p×BγFdΩDtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=J⋅E,In the next we have;(23)c4πEx1cDtαEx−∂Bz∂y+c4πEy1cDtαEy+∂Bz∂x=−4πcc4πExjx−4πcc4πEyjy,c4πBz1cDtαBz+∂Ey∂x−∂Ex∂y=0,18πDtαEx2+Ey2+Bz2+c4π−Ex∂Bz∂y+Ey∂Bz∂x+Bz∂Ey∂x−Bz∂Ex∂y=−j⋅E,Following a series of specific stages, we get to(24)18πDtαE2+B2+c4πdivE×B=−j⋅E.We have obtained from equations (23) and (24) as follow:(25)Dtα∫Ω∈−∞,∞γmc2F+E2+B28πdΩ+div∫Ω∈−∞,∞c2pf+cdivE×B4πdΩ=0.(2) Discretization Arrangement:(26)Dtα∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=qpxmγC2ηGC1η+1GdΩ+∫Ω∈−∞,∞qpymγC3ηGC1η+1GdΩDtα,∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=C2ηGjx+C3ηGjy,Dtα∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=C2ηGjx+C3ηGjy.The discretization procedure is elucidated as:(27)c4πC2ηG1c¶tαExη+1/2G−C4ηDy1G+c4πC3ηG1c¶tαEyη+1/2G+C4ηDx1G=−4πcc4πC2ηGjx,i,jη+4πcc4πC3ηGjy,i,jη,c4πC4ηG1c¶tαBzη+1/2+C3ηDx1G−C2ηDy1G=0,14πC2ηG¶tαExη+1/2G+C3ηG¶tαEyη+1/2G+C4ηG¶tαBzη+1/2+c4πC3ηGC4ηDx1G+c4πC4ηGC3ηDx1G−c4πC2ηGC4ηDy1G−c4πC4ηGC2ηDy1G=−C2ηGjx,i,jη+C3ηGjy,i,jη.Hence proved.


## 2.1. Generalized Form

The most recent and generalized form of VMS with important laws [19–24] are explained as:(1)D0CtαF+∂∂r⋅pγmF+∂∂p⋅qE+p×BγmcF,γ=1+pmc21/2,1cD0CtαE−∇×B+4πcJ=0,∇⋅E=4πρ,−1cD0CtαB−∇×E=0,∇⋅B=0,D0Ctαρ+∇⋅J=0.Table1 contains an explanation of all the symbols used in the preceding system.Table 1
Important symbols used in the theory.
SymbolsName of symbolsFDistribution functiontTimex,y,zVelocity componentspx,py,pzMomentum componentsγRelativistic parameterGjμGegenbauer polynomialsGjμShifted gegenbauer℘fFVector norms ofF,℘ExExVector norms ofEx,EElectric fieldBMagnetic fieldD0CtαCaputo time fractionalJCurrent densityρCharge densityhiμNormalizing factorϑμWeight function℘BzBzVector norms ofBz,℘EyEyVector norms ofEy


## 2.2. Mathematical Assumptions and Conversion

The significant assumptions for the problems are:(2)r=xi^+yj^,p=pxi^+pyj^,F=Ft,x,y,px,py,E=Exx,yi^+Eyx,yj^,J=jxi^+jyj^,B=Bzx,yk^.Therefore, we get the VMS as:(3)D0CtαF+1γmpx∂F∂x+py∂F∂y+qEx∂F∂px+Ey∂F∂py+qmcpyBzγ∂F∂px−pxBzγ∂F∂py=0,1cD0CtαEx+4πcjx=∂Bz∂y,1cD0CtαEy+4πcjy=−∂Bz∂x,1cD0CtαBz=∂Ex∂y−∂Ey∂x,∂Ex∂x+∂Ey∂y=4πρ,ρ=q∫Ω∈−∞,∞FdΩ,D0Ctαρ+∂∂xjx+∂∂yjy=0,jx=qm∫Ω∈−∞,∞pxγFdΩ,jy=qm∫Ω∈−∞,∞pyγFdΩ.


## 2.3. Scheme Discretization

According to the defined process [19, 20], we presented our system into discretized such matrix form as follow:(4)¶tαFη+1/2G+pxγmC1η+1Dx1G+pyγmC1η+1Dy1G+qC2ηG′C1η+1Dpx1G+qC3ηG′C1η+1Dpy1G+qmcpyγC4ηG′C1η+1Dpx1G−qmcpxγC4ηG′C1η+1Dpy1G=0,1c¶tαExη+1/2G′+4πcjxη+1=C4η+1Dy1G′,1c¶tαBzη+1/2G′=C2η+1Dy1G′−C3η+1Dx1G′,1c¶tαEyη+1/2G′+4πcjyη+1=−C4η+1Dx1G′,C2η+1Dx1G′+C3η+1Dy1G′=4πρη+1,¶tα∫Ω∈−∞,∞Fη+12dΩ+qm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0,ρη+1=q∫Ω∈−∞,∞C1η+1GdΩ,jxη+1=qm∫Ω∈−∞,∞pxγC1η+1GdΩ,jyη+1=qm∫Ω∈−∞,∞pyγC1η+1GdΩ.The definition of operator is [18].(5)¶tαFη+1/2=¶t¶tαCη+1/2+∑s=1ηCη+1/2−s=Δt−αΓ2−αCη+1−Cη+Δt−αΓ2−α∑s=1ηCη+1−s−Cη−sbsα.We also havebsα=s+11−α−s1−α. We use GM1,M2,M3,M4=G, and GM1,M2=G′ for simplicity in discretization form and also ¶tα is the operator and is specified by(6)¶tFη+1/2=δtCη+1/2+∑s=1ηδtCη+1/2−s=Cη+1−Cη+∑s=1ηCη+1−s−Cη−s.


## 2.4. Conservativeness of the Scheme

### 2.4.1. Charge Conservation

Takingq∫Ω∈−∞,∞dΩ of VE (5) and once obtained:(7)∫Ω∈−∞,∞q¶tαFη+1/2G+pxγmC1η+1Dx1G+pyγmC1η+1Dy1GdΩ=0,¶tαq∫Ω∈−∞,∞Fη+1/2GdΩ+qm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0.Hence proved that charge conservation accordingly.


### 2.4.2. Gauss Law and Solenoidal Constrains

(1) Differential Arrangement. Using the concepts of divergence and equation (2), we obtained: (8)1cdivD0CtαE−div∇×B+4πcdivJ=0⇒1cD0CtαdivE−4πρ=0.Hence, we can say that the above relation will satisfy if we have(9)divE=4πρ.(2) Discretization Arrangement. Apply the divergence based on the assumptions, and we contract:(10)1c¶tαDx1Exη+1/2G′+4πcqm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ=C4η+1Dx1Dy1G′,1c¶tαDy1Eyη+1/2G′+4πcqm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=−C4η+1Dx1Dx1G′,1c¶tαDx1Exη+1/2G′+Dy1Eyη+1/2G′+4πcqm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0,1c¶tαDx1Exη+1/2G′+Dy1Eyη+1/2G′−4πρη+1=0.As a result, we may claim that the Gauss rule is true if and only if the following conditions are met:(11)Dx1Ex0G′+Dy1Ey0G′=−4πρ0.It is not necessary to establish solenoidal limitations.


### 2.4.3. Momentum Conservation

(1) Differential Arrangement. (12)Dtα∫Ω∈−∞,∞pFdΩ+∂∂r⋅∫Ω∈−∞,∞ppγmFdΩ=−∫Ω∈−∞,∞p∂∂p⋅qE+p×BγmcFdΩ,Dtα∫Ω∈−∞,∞pFdΩ+∂∂r⋅∫Ω∈−∞,∞ppγmFdΩ=∫Ω∈−∞,∞qEF+qFp×BγmcdΩ.For the case ofx-component, we obtained(13)Dtα∫Ω∈−∞,∞FpxdΩ+Ex∫Ω∈−∞,∞qFdΩ+∂∂r⋅∫Ω∈−∞,∞pxpγmFdΩ=qmc∫Ω∈−∞,∞1γFBzpydΩ,Dtα∫Ω∈−∞,∞pxFdΩ+∂∂r⋅∫Ω∈−∞,∞pxpγmFdΩ=ρEx+1cBzjy.Additionally, we shall express the “Maxwell” equation in its momentum form as:(14)Bz4π1cDtαEy+∂Bz∂x−∂Bx∂z+Ey4π1cDtαBz+∂Ey∂x−∂Ex∂y=−Bz4π4πcjy,−By4π1cDtαEz+∂Bx∂y−∂By∂x−Ez4π1cDtαBy+∂Ex∂z−∂Ez∂x=By4π4πcjz,Simplified form is(15)Bz4π1cDtαEy+∂Bz∂x−∂Bx∂z+Ey4π1cDtαBz+∂Ey∂x−∂Ex∂y=−Bz4π4πcjy.Additionally, we get(16)14πcDtαBzEy+18π∂∂xE2+B2−14πdivExE+BxB+14πBxdivB+ExdivE=−jyBzc.From (2), we get(17)14πcDtαBzEy+18π∂∂xE2+B2−14πdivBxE+BxB=−Exρ−jyBzc.Further from equations (14) and (17), we get(18)Dtα∫Ω∈−∞,∞pxF+BzEy4πcdΩ+∂∂x∫Ω∈−∞,∞pxpxγmF−Ex2+Bx24π+Ex2+Ey2+Bx2+By28πdΩ+∂∂y∫Ω∈−∞,∞pypyγmF−EyEx+ExEy4πdΩ∂∂x∫Ω∈−∞,∞pxpxγmF−Ex2+Bx24π+Ex2+Ey2+Bx2+By28πdΩ+∂∂y∫Ω∈−∞,∞pypyγmF−EyEx+ExEy4πdΩ=0.(2) Discretization Arrangement:(19)∫Ω∈−∞,∞px¶tαFη+12GdΩ+∫Ω∈−∞,∞pxpxγmC1η+1Dx1GdΩ+∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=qC2ηG′C1η+1GdΩ+∫Ω∈−∞,∞qpyγmcC4η+1G′C1η+1GdΩ,∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=C2ηG′ρη+1+1cC4η+1G′jyη+1.px¶tαFη+12GdΩ+∫Ω∈−∞,∞pxpxγmC1η+1Dx1GdΩ+∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=C2ηG′ρη+1+1cC4η+1G′jyη+1.Moreover; we have(20)14πC4η+1G′¶tαEyη+1/2G′+C4η+1Dx1G′+14πC3ηG′δtαBzη+1/2G′−C2η+1Dy1G′+C3η+1Dx1G′=−14πC4η+1G′4πcjyη+1,14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−C3η+1G′C2η+1Dy1G′=−1cC4η+1G′jyη+1,14πC2η+1G′C2η+1Dx1G′+C3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−14πC2η+1G′C2η+1Dx1G′+C3η+1Dy1G′=−1cC4η+1G′jyη+1.From (15), we can write as:(21)14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC2η+1G′C2η+1Dx1G′+C3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−14π2C2η+1G′C2η+1Dx1G′+C2η+1G′C3η+1Dy1G′+C3η+1G′C2η+1Dy1G′=−1cC4η+1G′jyη+1−14πC2η+1G′4πρη+1.As a result of (20) and (21), we may conclude that the scheme follows the momentum conservation rule.


### 2.4.4. Energy Conservation

(1) Differential Arrangement. We shall use ∫Ω∈−∞,∞γmc2dΩ and obtained further:(22)Dtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=∫Ω∈−∞,∞pγm⋅qE+p×BγmcFdΩ,Dtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=∫Ω∈−∞,∞1γmqp⋅EF+1mcp⋅p×BγFdΩDtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=J⋅E,In the next we have;(23)c4πEx1cDtαEx−∂Bz∂y+c4πEy1cDtαEy+∂Bz∂x=−4πcc4πExjx−4πcc4πEyjy,c4πBz1cDtαBz+∂Ey∂x−∂Ex∂y=0,18πDtαEx2+Ey2+Bz2+c4π−Ex∂Bz∂y+Ey∂Bz∂x+Bz∂Ey∂x−Bz∂Ex∂y=−j⋅E,Following a series of specific stages, we get to(24)18πDtαE2+B2+c4πdivE×B=−j⋅E.We have obtained from equations (23) and (24) as follow:(25)Dtα∫Ω∈−∞,∞γmc2F+E2+B28πdΩ+div∫Ω∈−∞,∞c2pf+cdivE×B4πdΩ=0.(2) Discretization Arrangement:(26)Dtα∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=qpxmγC2ηGC1η+1GdΩ+∫Ω∈−∞,∞qpymγC3ηGC1η+1GdΩDtα,∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=C2ηGjx+C3ηGjy,Dtα∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=C2ηGjx+C3ηGjy.The discretization procedure is elucidated as:(27)c4πC2ηG1c¶tαExη+1/2G−C4ηDy1G+c4πC3ηG1c¶tαEyη+1/2G+C4ηDx1G=−4πcc4πC2ηGjx,i,jη+4πcc4πC3ηGjy,i,jη,c4πC4ηG1c¶tαBzη+1/2+C3ηDx1G−C2ηDy1G=0,14πC2ηG¶tαExη+1/2G+C3ηG¶tαEyη+1/2G+C4ηG¶tαBzη+1/2+c4πC3ηGC4ηDx1G+c4πC4ηGC3ηDx1G−c4πC2ηGC4ηDy1G−c4πC4ηGC2ηDy1G=−C2ηGjx,i,jη+C3ηGjy,i,jη.Hence proved.


## 2.4.1. Charge Conservation

Takingq∫Ω∈−∞,∞dΩ of VE (5) and once obtained:(7)∫Ω∈−∞,∞q¶tαFη+1/2G+pxγmC1η+1Dx1G+pyγmC1η+1Dy1GdΩ=0,¶tαq∫Ω∈−∞,∞Fη+1/2GdΩ+qm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0.Hence proved that charge conservation accordingly.


## 2.4.2. Gauss Law and Solenoidal Constrains

(1) Differential Arrangement. Using the concepts of divergence and equation (2), we obtained: (8)1cdivD0CtαE−div∇×B+4πcdivJ=0⇒1cD0CtαdivE−4πρ=0.Hence, we can say that the above relation will satisfy if we have(9)divE=4πρ.(2) Discretization Arrangement. Apply the divergence based on the assumptions, and we contract:(10)1c¶tαDx1Exη+1/2G′+4πcqm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ=C4η+1Dx1Dy1G′,1c¶tαDy1Eyη+1/2G′+4πcqm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=−C4η+1Dx1Dx1G′,1c¶tαDx1Exη+1/2G′+Dy1Eyη+1/2G′+4πcqm∫Ω∈−∞,∞pxγC1η+1Dx1GdΩ+qm∫Ω∈−∞,∞pyγC1η+1Dy1GdΩ=0,1c¶tαDx1Exη+1/2G′+Dy1Eyη+1/2G′−4πρη+1=0.As a result, we may claim that the Gauss rule is true if and only if the following conditions are met:(11)Dx1Ex0G′+Dy1Ey0G′=−4πρ0.It is not necessary to establish solenoidal limitations.


## 2.4.3. Momentum Conservation

(1) Differential Arrangement. (12)Dtα∫Ω∈−∞,∞pFdΩ+∂∂r⋅∫Ω∈−∞,∞ppγmFdΩ=−∫Ω∈−∞,∞p∂∂p⋅qE+p×BγmcFdΩ,Dtα∫Ω∈−∞,∞pFdΩ+∂∂r⋅∫Ω∈−∞,∞ppγmFdΩ=∫Ω∈−∞,∞qEF+qFp×BγmcdΩ.For the case ofx-component, we obtained(13)Dtα∫Ω∈−∞,∞FpxdΩ+Ex∫Ω∈−∞,∞qFdΩ+∂∂r⋅∫Ω∈−∞,∞pxpγmFdΩ=qmc∫Ω∈−∞,∞1γFBzpydΩ,Dtα∫Ω∈−∞,∞pxFdΩ+∂∂r⋅∫Ω∈−∞,∞pxpγmFdΩ=ρEx+1cBzjy.Additionally, we shall express the “Maxwell” equation in its momentum form as:(14)Bz4π1cDtαEy+∂Bz∂x−∂Bx∂z+Ey4π1cDtαBz+∂Ey∂x−∂Ex∂y=−Bz4π4πcjy,−By4π1cDtαEz+∂Bx∂y−∂By∂x−Ez4π1cDtαBy+∂Ex∂z−∂Ez∂x=By4π4πcjz,Simplified form is(15)Bz4π1cDtαEy+∂Bz∂x−∂Bx∂z+Ey4π1cDtαBz+∂Ey∂x−∂Ex∂y=−Bz4π4πcjy.Additionally, we get(16)14πcDtαBzEy+18π∂∂xE2+B2−14πdivExE+BxB+14πBxdivB+ExdivE=−jyBzc.From (2), we get(17)14πcDtαBzEy+18π∂∂xE2+B2−14πdivBxE+BxB=−Exρ−jyBzc.Further from equations (14) and (17), we get(18)Dtα∫Ω∈−∞,∞pxF+BzEy4πcdΩ+∂∂x∫Ω∈−∞,∞pxpxγmF−Ex2+Bx24π+Ex2+Ey2+Bx2+By28πdΩ+∂∂y∫Ω∈−∞,∞pypyγmF−EyEx+ExEy4πdΩ∂∂x∫Ω∈−∞,∞pxpxγmF−Ex2+Bx24π+Ex2+Ey2+Bx2+By28πdΩ+∂∂y∫Ω∈−∞,∞pypyγmF−EyEx+ExEy4πdΩ=0.(2) Discretization Arrangement:(19)∫Ω∈−∞,∞px¶tαFη+12GdΩ+∫Ω∈−∞,∞pxpxγmC1η+1Dx1GdΩ+∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=qC2ηG′C1η+1GdΩ+∫Ω∈−∞,∞qpyγmcC4η+1G′C1η+1GdΩ,∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=C2ηG′ρη+1+1cC4η+1G′jyη+1.px¶tαFη+12GdΩ+∫Ω∈−∞,∞pxpxγmC1η+1Dx1GdΩ+∫Ω∈−∞,∞pxpyγmC1η+1Dy1GdΩ=C2ηG′ρη+1+1cC4η+1G′jyη+1.Moreover; we have(20)14πC4η+1G′¶tαEyη+1/2G′+C4η+1Dx1G′+14πC3ηG′δtαBzη+1/2G′−C2η+1Dy1G′+C3η+1Dx1G′=−14πC4η+1G′4πcjyη+1,14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−C3η+1G′C2η+1Dy1G′=−1cC4η+1G′jyη+1,14πC2η+1G′C2η+1Dx1G′+C3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−14πC2η+1G′C2η+1Dx1G′+C3η+1Dy1G′=−1cC4η+1G′jyη+1.From (15), we can write as:(21)14πC4η+1G′¶tαEyη+1/2G′+14πC3ηG′δtαBzη+1/2G′+14πC2η+1G′C2η+1Dx1G′+C3η+1G′C3η+1Dx1G′+C4η+1G′C4η+1Dx1G′−14π2C2η+1G′C2η+1Dx1G′+C2η+1G′C3η+1Dy1G′+C3η+1G′C2η+1Dy1G′=−1cC4η+1G′jyη+1−14πC2η+1G′4πρη+1.As a result of (20) and (21), we may conclude that the scheme follows the momentum conservation rule.


## 2.4.4. Energy Conservation

(1) Differential Arrangement. We shall use ∫Ω∈−∞,∞γmc2dΩ and obtained further:(22)Dtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=∫Ω∈−∞,∞pγm⋅qE+p×BγmcFdΩ,Dtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=∫Ω∈−∞,∞1γmqp⋅EF+1mcp⋅p×BγFdΩDtα∫Ω∈−∞,∞γmc2FdΩ+∂∂r⋅∫Ω∈−∞,∞c2pFdΩ=J⋅E,In the next we have;(23)c4πEx1cDtαEx−∂Bz∂y+c4πEy1cDtαEy+∂Bz∂x=−4πcc4πExjx−4πcc4πEyjy,c4πBz1cDtαBz+∂Ey∂x−∂Ex∂y=0,18πDtαEx2+Ey2+Bz2+c4π−Ex∂Bz∂y+Ey∂Bz∂x+Bz∂Ey∂x−Bz∂Ex∂y=−j⋅E,Following a series of specific stages, we get to(24)18πDtαE2+B2+c4πdivE×B=−j⋅E.We have obtained from equations (23) and (24) as follow:(25)Dtα∫Ω∈−∞,∞γmc2F+E2+B28πdΩ+div∫Ω∈−∞,∞c2pf+cdivE×B4πdΩ=0.(2) Discretization Arrangement:(26)Dtα∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=qpxmγC2ηGC1η+1GdΩ+∫Ω∈−∞,∞qpymγC3ηGC1η+1GdΩDtα,∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=C2ηGjx+C3ηGjy,Dtα∫Ω∈−∞,∞γmc2¶tαFη+1/2GdΩ+∫Ω∈−∞,∞c2px,kC1η+1Dx1GdΩ+∫Ω∈−∞,∞c2py,kC1η+1Dy1GdΩ=C2ηGjx+C3ηGjy.The discretization procedure is elucidated as:(27)c4πC2ηG1c¶tαExη+1/2G−C4ηDy1G+c4πC3ηG1c¶tαEyη+1/2G+C4ηDx1G=−4πcc4πC2ηGjx,i,jη+4πcc4πC3ηGjy,i,jη,c4πC4ηG1c¶tαBzη+1/2+C3ηDx1G−C2ηDy1G=0,14πC2ηG¶tαExη+1/2G+C3ηG¶tαEyη+1/2G+C4ηG¶tαBzη+1/2+c4πC3ηGC4ηDx1G+c4πC4ηGC3ηDx1G−c4πC2ηGC4ηDy1G−c4πC4ηGC2ηDy1G=−C2ηGjx,i,jη+C3ηGjy,i,jη.Hence proved.


## 3. Results and Discussion

Using MAPLE and Python, we generated generic code for assessing the numerical solution of the “model” using the specified numerical technique. We will explore the attitude of plasma particles for fractional concepts using the following initial perturbation.(28)Ft,x,y,px,px=12πe−px2+py221+εcoskxxcoskyx,


### 3.1. Numerical Convergence

To demonstrate the numerical convergence, we use concepts of norm. Therefore, we have the following relation as:(29)FL∞=max℘fF,F=Fη+1−Fn,BzL∞=max℘BzBz,Bz=Bzη+1−Bzη,ExL∞=max℘ExEx,Ex=Exη+1−Exη,EyL∞=max℘EyEy,Ey=Eyη+1−Eyη.Even under the most difficult conditions imposed by the challenge, the implemented system demonstrates efficient convergence and precision. In both cases, i.e., integer and fractional values of the fractional parameter, convergence rises steadily as the computing domain extends as shown in Figure1(a)–1(d). Consequently, we can easily show that our technique is well-matched, competent, and appropriate for the models discussed before.Figure 1
Convergence plots atx=y=px=py=5,10,15,N=50, M1=M2=M3=M4=5.
(a)(b)(c)(d)


### 3.2. Behaviour of Charged Particles atα∈0,1

We have revealed one critical sort of graphical diagrams, namely “density”. We picked two distinct time periods,t=1.33,3.66, and varied 0<α≤1. While we were executing initial data, “plasma” particles were unexpectedly displaced from their initial positions. “Particles” also acquire energy as a result of this perturbation. As a result, plasma particles began moving abruptly in order to stabilize themselves.Noticed that excited “plasma” particles are accessible in cluster form in the range400<x,y≤800. Formulation of clusters is due to its high rate of plasma particle. As we vary t=1.33,α=0.2 to t=1.33,α=0.6, the particles undergo modification, and the bunch magnitude and outline are repaired appropriately (see Figure 2(b)).Figure 2
. Plots for Density of particles atN=50,M1=5,M2=M3=M4,. (a) t = 1.33, α = 0.2. (b) t = 1.33, α = 0.6. (c) t = 1.33, α = 1.0. (d) t = 3.66, α = 0.2. (e) t = 3.66, α = 0.6. (f) t = 3.66, α = 1.0.
(a)(b)(c)(d)(e)(f)A thin layer is seen in Figure2(b) around clusters of low “momentum” plasma particles. Additionally, the assortment of particles deviate position. These clusters are translated into the final location and “momentum” by selecting t=1.33,α=1.0.As a result, we can examine the particles' locus as the fractional parameter is varied. We clearly noticed significant fluctuations in the location and velocity of plasma particles during the second time period, as seen in Figures2(d)–2(f). Some of the particles are clustered, while others are uncontrolled, covering the computing area. This procedure will continue indefinitely until the equilibrium state is reached.When the fractional parameter is modified, we may see statistically significant variations. Therefore, fractional concepts are utilized to indicate the density of plasma particles and the path taken as a consequence of this. It dives into the complex picture of plasma particle behaviour that has remained concealed.


## 3.1. Numerical Convergence

To demonstrate the numerical convergence, we use concepts of norm. Therefore, we have the following relation as:(29)FL∞=max℘fF,F=Fη+1−Fn,BzL∞=max℘BzBz,Bz=Bzη+1−Bzη,ExL∞=max℘ExEx,Ex=Exη+1−Exη,EyL∞=max℘EyEy,Ey=Eyη+1−Eyη.Even under the most difficult conditions imposed by the challenge, the implemented system demonstrates efficient convergence and precision. In both cases, i.e., integer and fractional values of the fractional parameter, convergence rises steadily as the computing domain extends as shown in Figure1(a)–1(d). Consequently, we can easily show that our technique is well-matched, competent, and appropriate for the models discussed before.Figure 1
Convergence plots atx=y=px=py=5,10,15,N=50, M1=M2=M3=M4=5.
(a)(b)(c)(d)


## 3.2. Behaviour of Charged Particles atα∈0,1

We have revealed one critical sort of graphical diagrams, namely “density”. We picked two distinct time periods,t=1.33,3.66, and varied 0<α≤1. While we were executing initial data, “plasma” particles were unexpectedly displaced from their initial positions. “Particles” also acquire energy as a result of this perturbation. As a result, plasma particles began moving abruptly in order to stabilize themselves.Noticed that excited “plasma” particles are accessible in cluster form in the range400<x,y≤800. Formulation of clusters is due to its high rate of plasma particle. As we vary t=1.33,α=0.2 to t=1.33,α=0.6, the particles undergo modification, and the bunch magnitude and outline are repaired appropriately (see Figure 2(b)).Figure 2
. Plots for Density of particles atN=50,M1=5,M2=M3=M4,. (a) t = 1.33, α = 0.2. (b) t = 1.33, α = 0.6. (c) t = 1.33, α = 1.0. (d) t = 3.66, α = 0.2. (e) t = 3.66, α = 0.6. (f) t = 3.66, α = 1.0.
(a)(b)(c)(d)(e)(f)A thin layer is seen in Figure2(b) around clusters of low “momentum” plasma particles. Additionally, the assortment of particles deviate position. These clusters are translated into the final location and “momentum” by selecting t=1.33,α=1.0.As a result, we can examine the particles' locus as the fractional parameter is varied. We clearly noticed significant fluctuations in the location and velocity of plasma particles during the second time period, as seen in Figures2(d)–2(f). Some of the particles are clustered, while others are uncontrolled, covering the computing area. This procedure will continue indefinitely until the equilibrium state is reached.When the fractional parameter is modified, we may see statistically significant variations. Therefore, fractional concepts are utilized to indicate the density of plasma particles and the path taken as a consequence of this. It dives into the complex picture of plasma particle behaviour that has remained concealed.


## 4. Conclusion

The current work accomplished two critical objectives: it developed a multi-order (integer and fractional) “fully relativistic” model based on several concepts, and it proposed a conservative “hybrid” numerical technique for solving the anticipated “plasma” model. To deal with the time-“fractional” derivative, the Caputo sense definition is used. The reported findings unequivocally illustrate that plasma particles exhibit significant variances over a range of fractional parameter values. By assigning specific positions, momentum, and energy to plasma particles, we may now determine their eventual state. As a consequence, the proposed model has the potential to considerably advance our understanding of plasma particles in the field of computing. The technique is adequate, well-matched, and effectual for the suggested model based on numerical convergence. It has a high rate of “convergence” for both derivatives, which grows gradually as the computing domain is enlarged.



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*Source: 7046579-2022-03-22.xml*