Research Article The Fractional View Analysis of Polytropic Gas, Unsteady Flow System Hassan Khan , 1,2 Saeed Islam , 3,4 and Muhammad Arif 1 1 Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan 2 Department of Mathematics, Near East University TRNC, Mersin 10, Turkey 3 Faculty of Mathematics & Statistics, Ton Duc ang University, Ho Chi Minh City 70000, Vietnam 4 Informatics Research Group, Ton Duc ang University, Ho Chi Minh City 70000, Vietnam Correspondence should be addressed to Saeed Islam; saeed.islam@tdtu.edu.vn Received 20 August 2020; Revised 6 December 2020; Accepted 28 January 2021; Published 11 February 2021 Academic Editor: Lishan Liu Copyright © 2021 Hassan Khan et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. e 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. e present method provides the series form solution with easily computable components and a higher rate of convergence towards the targeted problem’s exact solution. e present techniques are straightforward and effective for dealing with the solutions of fractional-order problems. e fractional derivatives are expressed in terms of the Caputo operator. e 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. e 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. e 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. e 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. e present work will be considered a useful tool to handle the solution of fractional partial differential equations. 1. Introduction In recent years, nonlinear fractional partial differential equa- tions (FPDEs) have attracted researchers because of their useful applications in science and engineering [1–3]. 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 [4–14]. In recent decades, many researchers have used different approaches to analyze the solutions of nonlinear PDEs, such as Laplace transform [15], Akbari–Ganji’s method [16], homotopy analysis method [17], lattice Boltzmann method [18, 19], volume of fluid method [20, 21], Laplace homotopy analysis method [22, 23], Adomian decomposition tech- nique [24–27], the variational iteration technique [28], Adams–Bashforth–Moulton algorithm [29], homotopy perturbation Sumudu transform method [30], the tanh method [31], the sinh-cosh method [32], finite difference method [33], the homotopy perturbation method [34], and Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 5130136, 17 pages https://doi.org/10.1155/2021/5130136