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

School of Mathematical Sciences, Peking University, Beijing 100871, China Department of Mathematics, National University of Modern Languages (NUML), Islamabad 44000, Pakistan Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia Department of Basic Sciences, College of Science and 2eoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia HEDPS and CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China Department of Mathematics, Bonga University, Bonga, Ethiopia


Introduction
e VM system is an important instrument due to its vast variety of applications [1][2][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). e primary portion couple's plasma "particle" motion equations to Maxwell equations and numerically cracks them using the particle in approach [1,4], and [5]. e second approach employs FEM [6] and FDM [3] to discretize the VE. ese are grid-constructed procedures. e 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. e terms "Crank" Nicolson and temporal integration are used to characterize the "conservative" arrangements of PICT [7][8][9][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. e 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. e 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][13][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.
ese approaches assist us in overcoming difficulties, mainly numerical dispersion [17], that the PIC method encountered.
e 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][19][20][21][22][23][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]. e project tightly enforces conservation laws, which will be discussed in further detail in the next segment. e numerical framework exhibits virtuous convergence, which is also discussed in the paper. Finally, we conclude our research.

Mathematical Modelling and Conservative
Numerical Scheme e most recent and generalized form of VMS with important laws [19][20][21][22][23][24] are explained as: (1) Table 1 contains an explanation of all the symbols used in the preceding system.

Mathematical Assumptions and Conversion.
e significant assumptions for the problems are: (2) erefore, we get the VMS as: Shifted gegenbauer

Scheme Discretization.
According to the defined process [19,20], we presented our system into discretized such matrix form as follow: Journal of Function Spaces e definition of operator is [18].

Conservativeness of the Scheme
dΩ of VE (5) and once obtained: Hence proved that charge conservation accordingly.

Gauss Law and Solenoidal Constrains. (1) Differential
Arrangement. Using the concepts of divergence and equation (2), we obtained: Hence, we can say that the above relation will satisfy if we have (2) Discretization Arrangement. Apply the divergence based on the assumptions, and we contract: As a result, we may claim that the Gauss rule is true if and only if the following conditions are met: It is not necessary to establish solenoidal limitations.
For the case of x-component, we obtained Additionally, we shall express the "Maxwell" equation in its momentum form as:

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Simplified form is Additionally, we get From (2), we get Further from equations (14) and (17), we get

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(2) Discretization Arrangement: Moreover; we have From (15), we can write as:

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As a result of (20) and (21), we may conclude that the scheme follows the momentum conservation rule.

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(2) Discretization Arrangement: e discretization procedure is elucidated as: Hence proved.

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.

Numerical Convergence.
To demonstrate the numerical convergence, we use concepts of norm. erefore, we have the following relation as: 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 Figure 1(a)-1(d). Consequently, we can easily show that our technique is well-matched, competent, and appropriate for the models discussed before.

Behaviour of Charged
Particles at α ∈ (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. 10 Journal of Function Spaces Noticed that excited "plasma" particles are accessible in cluster form in the range 400 < 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)).
A thin layer is seen in Figure 2(b) around clusters of low "momentum" plasma particles. Additionally, the assortment of particles deviate position. ese 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 Figures 2(d)-2(f ). Some of the particles are clustered, while others are uncontrolled, covering the computing area. is procedure will continue indefinitely until the equilibrium state is reached.
When the fractional parameter is modified, we may see statistically significant variations. erefore, 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.

Conclusion
e 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. e 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. e 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.

Data Availability
No data was required to perform this research.

Conflicts of Interest
e authors declare that they have no conflicts of interest.