Study of Fuzzy Fractional Third-Order Dispersive KdV Equation in a Plasma under Atangana-Baleanu Derivative

Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University, Saudi Arabia Department of Physics, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia Faculty of Mathematics, Kabul Polytechnic University, Afghanistan


Introduction
In recent years, researchers and scientists have been particularly interested in fractional calculus (FC), which encompasses fractional-order integrals and derivatives. FC offers a wide range of applications, with accurate and precise solutions in various modern biological and physical phenomena. The integral differential operators in fractional differential calculus have a higher freedom of degree. The solution studies have given a keen attraction in this area, and papers, monographs, and book, among other things, have been published in current history, containing numerous inquiries into various aspects such as existing theories and analytic conclusions [1][2][3][4][5][6][7]. FC can be used to a variety of fields in both practical and pure mathematics. Physical models of real-world phenomenon usually contain considerable uncertainty due to a variety of variables. Fuzzy sets also actually be a good method for modelling the unpredictability brought up by imprecision and vagueness. We implement it here to areas where needs to be explained ambiguity, such as medical, ecologic, social, financial, and sciences [8][9][10][11][12][13][14][15]. In 1965, Zadeh proposed the set theory of fuzzy to investigate these issues [16]. The fuzzy set approach has been applied in other areas, i.e., topology, control system, fuzzy automata, fixed-point theories, etc. Chang and Zadeh expanded on the fuzzy set notion by introducing fuzzy control and mapping [17]. Several academics are generalized the concept of fuzzy mapping and control to develop basic fuzzy calculus [18][19][20]. In the last many centuries, fractional fuzzy integral and differential equations have attracted much attention from researchers in applied sciences. The fundamental concept of integral fuzzy equations was first described by Dubois and Prade [21]. However, in situations where information is ambiguous and inaccurate, the parameters are expressed by fuzzy numbers rather than crisp numbers. Fractional fuzzy integral and differential equations can be used to simulate these types of problems. The mathematical modelling of particular real-world models maintaining uncertainty in data has given rise to fuzzy PDEs. PDEs have important implementations in several areas of engineering, plasma physics, and many science branches, as we will observe in future. Because numerous schemes in mechanical and machines and aerospace technology areas of study are applicable to heat, heat transfer is an important area of aerospace and mechanical engineering investigation [22]. As a result, numerous scholars investigated the solution of fuzzy FDEs using these models [23]. The researchers [24] used an effective mathematical technique to analyze analytical solutions to nonlinear Lane-Emden models and an efficient numerical method for the fractional advection dispersion model and vibration equations that arise in porous media. Many fractional PDEs applying in hydro-magnetic waves in a cold plasma, ion-acoustic waves in plasma, and magneto acoustic wave numerical simulation and analytical approaches are used [25,26]. Nonlinear effects occur in many applied science areas, such as plasma physics, mathematical biology, nonlinear optics, quantum mechanics, chemical kinetics, solid-state physics, and fluid dynamics [27][28][29][30][31][32][33][34]. These processes are modeled on nonlinear partial differential equations (PDEs) of a various higher order. PDEs are commonly used in the description of physical processes. The nonlinear nature of most of the essential physical systems is hidden. The exact result of such nonlinear phenomena may not be possible for some physical problems. For instance in a plasma physics, there are many nonintegrable PDEs that can not support exact analytic solutions such as the integer and fractional damped thirdorder KdV-type equations and the damped integer and fractional fifth-order KdV-type equations (the family of damped Kawahara equation) and many other equations related to plasma physics [35][36][37][38][39][40]. Moreover, in non-Maxwellian plasma models that have trapped particles follow nonisothermal or Schamel distribution in addition to the particle kinematic viscosity, in this case, the fluid equations of the plasma model can be reduced to a nonintegrable damped Schamel KdV-Burgers equation [41,42]. On the other side, there are other types of nonintegrable PDEs that can be used for describing modulated envelope structures in a plasma like a damped integer and fractional cubic nonlinear Schrödinger-type equations (CNLSE). This family also does not have exact analytic solutions but can support some approximate analytical and numerical solutions. Anyway, most nonlinear phenomena that can exist in plasma physics, nonlinear optics, quantum physics, etc. can only be investigated using useful techniques to solve their evolution equations [43][44][45][46][47][48][49]. In 1895, Korteweg and de Vries proposed a KdV model to design Russells soliton phenomenon, such as small and huge water waves. Solitons are steady waves of solitary; this implies that these lone waves are particles. The mathematical model for exploring dispersive wave phenomena in several research areas is the KdV equations, such as quantum mechanics, fluid dynamics, optics, and plasma physics [44,45]. Fifth-order KdV/Kawahara form equations utilized to analyze different nonlinear phenomena in particle physics and in plasma physics [38][39][40]. It plays a vital function in the distribution of waves [50]. In their analysis, the KdV form equation has dispersive terms of the third and fifth-order relevant to the magneto acoustic wave problem in nearcritical angle propagation that appears to be cool collision-free plasma and dispersive terms [51]. Numerous number of researchers used this family of differential equations to model many nonlinear phenomena that arise and propagate in various plasma model. For example, El-Tantawy research group applied the Poincaré-Lighthill-Kuo (PLK) method for reducing the fluid equation of an ultracold neutral plasma (UNP) to two-coupled planar and nonplanar KdV equations for studying the face-toface planar and nonplanar soliton collisions and corresponding phase shifts after collisions. Moreover, the head-on collisions between the two-counterpart KdV and modified KdV (mKdV) planar solitons were investigated in different plasma models [52][53][54]. In addition to, the planar and nonplanar Gardner equations were used for modeling the several types of acoustic waves (AWs) in different plasma models [55][56][57][58]. All the abovementioned equations succeeded in giving a good description to many nonlinear phenomena that arise and propagate within different plasma systems and many other branches of science. On the other hand, many researchers tended to model most of the mysterious phenomena in plasma physics by describing them using fractional differential equations such as fractional KdV-type equations and many related equations in higher order because they give a more accurate and comprehensive description better than the integer differential equations [59][60][61]. As a result, numerous scholars investigated the solution of many fuzzy FDEs using these models of plasma physics. The researchers used an effective mathematical technique to analyze analytical solutions to nonlinear Lane-Emden models and an efficient numerical method for the fractional advection dispersion model and vibration equations that arise in porous media. Many fractional PDEs applying in hydro-magnetic waves in a cold plasma, ion-acoustic waves in plasma, and magneto acoustic wave numerical simulation and analytical approaches are used. Which as innovation is concerned, we suggest an initial value solitary wave solutions for the KdV equation under the fuzzy Caputo fractional operator. Because of the fuzzy number, we use an unidentified quantity's fuzziness and initial condition to estimate its result in fuzzy type with two section. Many researchers have been working on integer and fractional order diffusion equations, as well as the fuzzy heat equation. For the study of the given equation with various external source components, we examine both fuzziness and fractional order [62-66].

Basic Concept of Fuzzy and Fractional Calculus
This section demonstrates various key components related to fractional calculus and fuzzy sets, as well as some major research on the Shehu transformation.
Moreover, consider Shehu transformation of classical form [67][68][69][70], we get: The aforementioned expressions can then be expressed as Definition 13. Assume that it is an integral fuzzy mapping value c gH D ν IΦ ðIÞ, and ΦðIÞ is the primitive of c gH D ν IΦ ðIÞ on ½0 , +∞Þ; then, the Caputo fractional order ν is shown as [67][68][69][70] Bokhari et al. described the fractional derivative ABC operator in the sense Shehu transform. Moreover, we apply the concept of fuzzy fractional derivative of ABC in a Shehu fuzzy transformation sense as follows.

Main Result
Consider the general fuzzy fractional partial differential equation where ℘∈ð0, 1; therefore, the Shehu transform of (21) is applying the initial condition, we achieved as Decompose the result asΦðζ, ξ, IÞ = ∑ ∞ n=0Φn ðζ, ξ, IÞ; then, (23) applies Parts of the result can be taken as a comparison Taking the inverse Shehu transform, we obtain Thus, the solution becomes Equation (27) is the series type solution.

Numerical Problems
Problem 15. Consider the fractional fuzzy KdV equation is given as with the initial conditioñ Applying the proposed method of Equation (29), we get Using Equation (27) In lower and upper portion types can be written as The exact result is Φ ζ, I ð Þ=k sin ζ + I ð Þ: ð34Þ Figure 1 shows that the accuracy of this method by lower and upper branches of fuzzy solution of problem 15 link with the fuzzy Shehu transformation and Atangana-Baleanu operator is shown in this article. Figure 2 shows the twodimensional upper and lower branch plots. The behaviour defines the variance in the mappings g %Φðζ, IÞ on the space coordinate ξ with the consider of I and the unpredictability parameters rε½0, 1. The graph shows that, as the passage of time, the mappingsΦðζ, IÞ will become much intricate.

Problem 16. Consider the fractional fuzzy KdV equation is given as
with the initial conditioñ Applying the proposed method of Equation (35) Using Equation (27)  In lower and upper portion types can be written as The exact result is

Journal of Function Spaces
Atangana-Baleanu operator is represented in this article. Figure 5 the approximate solution graph of fuzzy upper and lower branches and the 2nd figure is the various fractional order of ℘ with respect to time. The behaviour defines the variance in the mappingsΦðζ, IÞ on the space coordinate ξ with the consider of I and the unpredictability parameters rε½0, 1. The graph shows that, as the passage of time, the mappingsΦðζ, IÞ will become much intricate.
Problem 17. Consider the fractional fuzzy KdV equation is given as with the initial conditioñ Applying the proposed method of Equation (42), we get Using Equation (27) In lower and upper portion types can be written as The exact result isΦ Figure 6 shows that the accuracy of this method by upper and lower branches of fuzzy solution of problem 17 link with the fuzzy Shehu transformation and Atangana-Baleanu operator is represented in this article. Figure 7 the approximate solution graph of fuzzy upper and lower branches and the 2nd figure is the various fractional order of ℘. The behaviour defines the variance in the mappings Φðζ, IÞ on the space coordinate ξ with the consider of I and the unpredictability parameters rε½0, 1. The graph shows that, as the passage of time, the mappingsΦðζ, IÞ will become much intricate.

Conclusion
In this paper, an analytical result of fuzzy fractional thirdorder KdV equations has been successfully developed. The series form solution of the consider equation in sense of fractional the Atangana-Baleanu fractional derivatives has been introduced by Shehu transformation along with iterative method. By multiplying the number of fuzzy, we achieved the lower and upper portions of the required result. Figures of the analytical solutions at various noninteger order of the given models are also provided. Furthermore, the method used to solve the fuzzy fractional KdV equation in this article is novel and can be used to all areas of physical and natural sciences where uncertainty exists in various phenomena such as quantum mechanics and various theories of orbits or orbital around the nucleus. As a result, our study will

Data Availability
The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.