The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

Generally, the diﬀerential equations of integer order do not properly model various phenomena in diﬀerent areas of science and engineering as compared to diﬀerential equations of fractional order. The fractional-order diﬀerential equations provide the useful dynamics of the physical system and thus provide the innovative and eﬀective information about the given physical system. Keeping in view the above properties of fractional calculus, the present article is related to the analytical solution of the time-fractional system of equations which describe the unsteady ﬂow of polytropic gas dynamics. The present method provides the series form solution with easily computable components and a higher rate of convergence towards the targeted problem’s exact solution. The present techniques are straightforward and eﬀective for dealing with the solutions of fractional-order problems. The fractional derivatives are expressed in terms of the Caputo operator. The targeted problems’ solutions are calculated using the Adomian decomposition method and variational iteration methods along with Shehu transformation. In the current procedures, we ﬁrst applied the Shehu transform to reduce the problems into a more straightforward form and then implemented the decomposition and variational iteration methods to achieve the problems’ comprehensive results. The solution of the nonlinear equations of unsteady ﬂow of a polytropic gas at various fractional orders of the derivative is the core point of the present study. The solution of the proposed fractional model is plotted via two- and three-dimensional graphs. It is investigated that each problem’s solution-graphs are best ﬁtted with each other and with the exact solution. The convergence of fractional-order problems can be observed towards the solution of integer-order problems. Less computational time is the major attraction of the suggested methods. The present work will be considered a useful tool to handle the solution of fractional partial diﬀerential equations.


Introduction
In recent years, nonlinear fractional partial differential equations (FPDEs) have attracted researchers because of their useful applications in science and engineering [1][2][3]. e analysis of exact solutions to these nonlinear PDEs plays a very significant role in the Soliton theory since much of the information are provided on the description of the physical models, in the transmission of electrical signals, as a standard diffusion-wave equation, the transfer of neutrons by nuclear reactor, the theory of random walks, and so on [4][5][6][7][8][9][10][11][12][13][14].
In the present study, we consider the gas-dynamic equations fractional-order scheme describing the evolution of an ideal gas's two-dimensional unsteady flow. e polytropic gas in astrophysics is given as follows [36]: where ψ � (θ/ϕ) is the energy density, ϕ is the container volume, θ is the total energy of the gas, m is the polytropic index, and k is a constant. Degenerate adiabatic gas and electron gas are two instances of such gases. In astrophysics and cosmology, the analysis of polytropic gases plays a critical role, and these gases can perform like dark energy [37]. Now consider the gas-dynamic equations scheme, which describes the evolution of unstable flow of a perfect gas with fractional derivatives [36,38]: with initial conditions μ(ξ, ζ, 0) � α(ξ + ζ), where μ(ξ, ζ, η) and ](ξ, ζ, η) are the velocity components, ω(ξ, ζ, η) is the density, ψ(ξ, ζ, η) is the pressure, and τ is the ratio of the specific heat and it represents the adiabatic index. In past decade, the appropriate analytical results of several distinct types of gas-dynamic equations are achieved using many analytical and numerical methods. Various methods have been solved by a gas-dynamic model such as fractional reduced differential transform method [39], Elzaki transform homotopy perturbation method [40], q-homotopy analysis method [36], Adomian decomposition method [41], fractional homotopy analysis transform method [38], and natural decomposition method [42]. e variational iteration transform method (VITM) combines the variational iteration method and the Shehu transform. Many researchers commonly used this technique to solve linear and nonlinear models [43][44][45]. e method provides a reliable and effective procedure for a broad range of science. VIM does not need discretization, linearization, or perturbation. It provides quick convergence and successive approximations of the exact result [46][47][48]. Various equations solve the variational iteration method with the help of different transformations, such as Kuramoto-Sivashinsky equations [49] and fourth-order parabolic partial differential equation [45]. e ADM is an efficient and accurate technique that was suggested initially to solve analytically frontier physical models [50]. Since then, ADM has been implemented in nonlinear ODEs and PDEs without using perturbation or linearization procedure. e Shehu decomposition method (SDM) is a mixture of ADM and Shehu transform [51][52][53][54].
e motivation and novelty of the current research work are to modify the ADM and VIM along with Shehu transformation to investigate the solution of a nonlinear system of nonlinear FPDEs of unsteady flow of polytropic gas-dynamics equations. Besides the nonlinear system of four equations, the given problem's solution is calculated by an effortless and straightforward procedure. A higher degree of accuracy is achieved with a tiny number of calculations. e fractional-order solutions are achieved with some graphical justifications. e visual representation has confirmed the effectiveness and applicability of the suggested techniques. In the future, the proposed techniques are preferred to solve other nonlinear FPDEs that frequently arise in science and engineering.

The Procedure of VITM
is section describes the VITM solution, the system of FPDEs: with initial conditions where D δ η � (z δ /zη δ ) is the Caputo fractional derivative of order δ; G 1 , G 2 and N 1 , N 2 are linear and nonlinear functions, respectively; and G 1 , G 2 are source operators.
e Shehu transformation is applied to equation (1): Applying the differentiation property of Shehu transform, we have e iteration method for equation (15) may be utilized to indicate the major iterative scheme requiring the Lagrange multiplier as A Lagrange multiplier using inverse Shehu transformation S − 1 , and equation (16) can be written as e initial value can be found as Mathematical Problems in Engineering e converge of this technique is as follows [59,60].

The Procedure of SDM
In this section, we discuss the SDM solution for system of FPDEs: with initial conditions where D δ η � (z δ /zη δ ) is the Caputo fractional derivative of order δ; G 1 , G 2 and N 1 , N 2 are linear and nonlinear functions, respectively; and G 1 , G 2 are source functions.
Apply Shehu transform to equation (20): Applying the differentiation property of Shehu transform, we have SDM solution of infinite series μ(ξ, ζ, η) and ](ξ, ζ, η) is as follows: Adomian polynomials of nonlinear terms N 1 and N 2 are given as e nonlinear of Adomian polynomials can be defined as Substituting equation (24) and equation (25) into (23) gives Applying the inverse Shehu transformation to equation (20), we get ∞ m�0 μ m (ξ, ζ, η) We define the following terms: in general for m ≥ 1, and we have

Results Discussion
In this section, we discuss the solution-graphs of fractional-order system of nonlinear equations of unsteady flow of a polytropic gas which has been solved by using SDM and VITM. In Figures 1-4, the solutions μ, ], ω, and ψ obtained by using SDM and VITM are compared by keeping one variable and other constants. e dotted and line subgraphs are, respectively, denoted the SDM and VITM solutions. It is observed that SDM and VITM solution-graphs are identical and within close contact. In similar way, in Figures 5-7, the three-dimensional graphs for variables μ, ], and ψ are plotted for Example 1. e identical solution-graphs of the suggested methods are attained and confirmed that the results obtained by two different procedures are identical and verified the applicability of the proposed techniques. In Figures 8-10, the SDM and VITM  and ω, respectively. In Table 1 and Figure 14, the combined graph for variables μ, ], and ψ is displayed at δ � 1. e solution comparison of the suggested methods, SDM and VITM, is discussed. e suggested techniques have provided the solutions with the desire degree of accuracy with the consideration of very few terms in its series form solutions.

Conclusion
In this article, the analytical solution of the system of timefractional partial differential equations of unsteady flow of polytrophic dynamics is investigated by using two different techniques: (i) e proposed techniques are the mixture of Shehu transformation with Adomian decomposition method and variational iteration method, respectively. (ii) e obtained solutions of the suggested techniques for both fractional and integer orders are calculated and plotted via two-and three-dimensional graphs. (iii) A close contact between the actual and the derived results is observed. (iv) e fractional-order solutions provide various dynamics for a different fractional order of the derivative. (v) Using analytical solutions, the task can be done rather simple and effective as compared to numerical investigations that need larger calculations. (vi) After all, the researchers are now able to select the fractional-order problem whose solution is comparatively very close to the experimental results of any physical problem. (vii) Due to simple and straightforward implementation, the suggested techniques are considered to be preferable to solve other system of FPDEs.
e following abbreviations are used in this article:

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

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

Authors' Contributions
H. K. conceptualized the study, prepared the methodology, wrote the original draft, and did formal analysis. M. A. analyzed using the software and supervised the study. S. I. validated, investigated, and visualized the study and administrated the project. H. K. and M. A. managed resources and reviewed and edited the document.