# The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

Authors: Hassan Khan; Saeed Islam; Muhammad Arif
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

In recent years, nonlinear fractional partial differential equations (FPDEs) have attracted researchers because of their useful applications in science and engineering [1–3]. The 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 technique [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 the Laplace decomposition technique, to handle fractional-order Zakharov–Kuznetsov equations [35].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. The polytropic gas in astrophysics is given as follows [36]:(1)ψ=kω1+1/m,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]:(2)Dηδμ+μ∂μ∂ξ+ν∂μ∂ζ+1ω∂ψ∂ξ=0,Dηδν+μ∂ν∂ξ+ν∂ν∂ζ+1ω∂ψ∂ζ=0,Dηδω+μ∂ω∂ξ+ν∂ω∂ζ+ω∂μ∂ξ+∂ν∂ζ=0,Dηδψ+μ∂ψ∂ξ+ν∂ψ∂ζ+τψ∂μ∂ξ+∂ν∂ζ=0,with initial conditions(3)μξ,ζ,0=αξ+ζ,νξ,ζ,0=βξ+ζ,ωξ,ζ,0=γξ+ζ,ψξ,ζ,0=Φξ+ζ,where μξ,ζ,η and νξ,ζ,η are the velocity components, ωξ,ζ,η is the density, ψξ,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].The 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 [43–45]. The 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 [46–48]. 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].The 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. The Shehu decomposition method (SDM) is a mixture of ADM and Shehu transform [51–54].The 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. The fractional-order solutions are achieved with some graphical justifications. The 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.


## 2. Preliminaries Concepts

### 2.1. Definition 1

The Riemann–Liouville fractional integral is given as follows [55, 56]:(4)I0δhη=1Γδ∫0ηη−sδ−1hsds,δ>0,η>0,hη,δ>0.


### 2.2. Definition 2

The Caputo’s fractional-order derivative ofhη is defined as follows [55, 56]:(5)Dηδhη=In−δfn,n−1<δ<n,n∈ℕdndηnhη,δ=n,n∈ℕ.


### 2.3. Definition 3

The Shehu transformation is the new and modern transformation which is described for exponential-order functions. In setA, we take a function represented as follows [51, 52, 57, 58]:(6)A=νη:∃,ρ1,ρ2>0,νη<Meη/ρi,ifη∈0,∞.The Shehu transform which is given byS. for a function νη is defined as(7)Sνη=Vs,μ=∫0∞e−sη/μνηdη,η>0,s>0.The Shehu transform of a functionνη is Vs,μ, and then, νη is called the inverse of Vs,μ, which is given as(8)S−1Vs,μ=νη,forη≥0,S−1is inverse Shehu transformation.


### 2.4. Definition 4

Shehu transform fornth derivatives is given as follows [51, 52, 57, 58]:(9)Sνnη=snunVs,u−∑k=0n−1sun−k−1νk0.


### 2.5. Definition 5

Shehu transform for fractional-order derivatives [51, 52, 57, 58]:(10)Sνδη=sδuδVs,u−∑k=0n−1suδ−k−1νk0,0<β≤n.


### 2.6. Definition 6

The Mittag–Leffler function denoted byEδz for δ>0 is defined as(11)Eδz=∑m=0∞zmΓδm+1δ>0,z∈ℂ.


## 2.1. Definition 1

The Riemann–Liouville fractional integral is given as follows [55, 56]:(4)I0δhη=1Γδ∫0ηη−sδ−1hsds,δ>0,η>0,hη,δ>0.


## 2.2. Definition 2

The Caputo’s fractional-order derivative ofhη is defined as follows [55, 56]:(5)Dηδhη=In−δfn,n−1<δ<n,n∈ℕdndηnhη,δ=n,n∈ℕ.


## 2.3. Definition 3

The Shehu transformation is the new and modern transformation which is described for exponential-order functions. In setA, we take a function represented as follows [51, 52, 57, 58]:(6)A=νη:∃,ρ1,ρ2>0,νη<Meη/ρi,ifη∈0,∞.The Shehu transform which is given byS. for a function νη is defined as(7)Sνη=Vs,μ=∫0∞e−sη/μνηdη,η>0,s>0.The Shehu transform of a functionνη is Vs,μ, and then, νη is called the inverse of Vs,μ, which is given as(8)S−1Vs,μ=νη,forη≥0,S−1is inverse Shehu transformation.


## 2.4. Definition 4

Shehu transform fornth derivatives is given as follows [51, 52, 57, 58]:(9)Sνnη=snunVs,u−∑k=0n−1sun−k−1νk0.


## 2.5. Definition 5

Shehu transform for fractional-order derivatives [51, 52, 57, 58]:(10)Sνδη=sδuδVs,u−∑k=0n−1suδ−k−1νk0,0<β≤n.


## 2.6. Definition 6

The Mittag–Leffler function denoted byEδz for δ>0 is defined as(11)Eδz=∑m=0∞zmΓδm+1δ>0,z∈ℂ.


## 3. The Procedure of VITM

This section describes the VITM solution, the system of FPDEs:(12)Dηδμξ,ζ,η+G¯1μ,ν+N1μ,ν−G1ξ,ζ,η=0,Dηδνξ,ζ,η+G¯2μ,ν+N2μ,ν−G2ξ,ζ,η=0,m−1<δ≤m,with initial conditions(13)μξ,ζ,0=g1ξ,ζ,νξ,ζ,0=g2ξ,ζ,where Dηδ=∂δ/∂ηδ is the Caputo fractional derivative of order δ; G¯1, G¯2 and N1, N2 are linear and nonlinear functions, respectively; and G1,G2 are source operators.The Shehu transformation is applied to equation (1):(14)SDηδμξ,ζ,η+SG¯1μ,ν+N1μ,ν−G1ξ,ζ,η=0,SDηδνξ,ζ,η+SG¯2μ,ν+N2μ,ν−G2ξ,ζ,η=0.Applying the differentiation property of Shehu transform, we have(15)Sμξ,ζ,η−∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0=−SG¯1μ,ν+N1μ,ν−G1ξ,ζ,η,Sνξ,ζ,η−∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0=−SG¯2μ,ν+N2μ,ν−G2ξ,ζ,η.The iteration method for equation (15) may be utilized to indicate the major iterative scheme requiring the Lagrange multiplier as(16)Sμm+1ξ,ζ,η=Sμmξ,ζ,η+λssδuδμmξ,ζ,η−∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0−SG1ξ,ζ,η−SG¯1μ,ν+N1μ,ν,Sνm+1ξ,ζ,η=Sνmξ,ζ,η+λssδuδνmξ,ζ,η−∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0−SG2ξ,ζ,η−SG¯2μ,ν+N2μ,ν.A Lagrange multiplier(17)λs=−uδsδ,using inverse Shehu transformation S−1, and equation (16) can be written as(18)μm+1ξ,ζ,η=μmξ,ζ,η−S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0−SG1ξ,ζ,η−SG¯1μ,ν+N1μ,ν,νm+1ξ,ζ,η=νmξ,ζ,η−S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0−SG2ξ,ζ,η−SG¯2μ,ν+N2μ,ν.The initial value can be found as(19)μ0ξ,ζ,η=S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0,ν0ξ,ζ,η=S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0.The converge of this technique is as follows [59, 60].


## 4. The Procedure of SDM

In this section, we discuss the SDM solution for system of FPDEs:(20)Dηδμξ,ζ,η+G¯1μ,ν+N1μ,ν−G1ξ,ζ,η=0,Dηδνξ,ζ,η+G¯2μ,ν+N2μ,ν−G2ξ,ζ,η=0,m−1<δ≤m,with initial conditions(21)μξ,ζ,0=g1ξ,ζ,νξ,ζ,0=g2ξ,ζ,where Dηδ=∂δ/∂ηδ is the Caputo fractional derivative of order δ; G¯1, G¯2 and N1, N2 are linear and nonlinear functions, respectively; and G1,G2 are source functions.Apply Shehu transform to equation (20):(22)SDηδμξ,ζ,η+SG¯1μ,ν+N1μ,ν−G1ξ,ζ,η=0,SDηδνξ,ζ,η+SG¯2μ,ν+N2μ,ν−G2ξ,ζ,η=0.Applying the differentiation property of Shehu transform, we have(23)Sμξ,ζ,η=uδsδ∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0+uδsδSG1ξ,ζ,η−uδsδSG¯1μ,ν+N1μ,ν,Sνξ,ζ,η=uδsδ∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0+uδsδSG2ξ,ζ,η−uδsδSG¯2μ,ν+N2μ,ν.SDM solution of infinite seriesμξ,ζ,η and νξ,ζ,η is as follows:(24)μξ,ζ,η=∑m=0∞μmξ,ζ,η,νξ,ζ,η=∑m=0∞νmξ,ζ,η.Adomian polynomials of nonlinear termsN1 and N2 are given as(25)N1μ,ν=∑m=0∞Am,N2μ,ν=∑m=0∞ℬm.The nonlinear of Adomian polynomials can be defined as(26)Am=1m!∂m∂λmN1∑k=0∞λkμk,∑k=0∞λkνkλ=0,ℬm=1m!∂m∂λmN2∑k=0∞λkμk,∑k=0∞λkνkλ=0.Substituting equation (24) and equation (25) into (23) gives(27)S∑m=0∞μmξ,ζ,η=uδsδ∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0+uδsδSG1ξ,ζ,η−uδsδSG¯1∑m=0∞μm,∑m=0∞νm+∑m=0∞Am,S∑m=0∞νmξ,ζ,η=uδsδ∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0+uδsδSG2ξ,ζ,η−uδsδSG¯2∑m=0∞μm,∑m=0∞νm+∑m=0∞ℬm.Applying the inverse Shehu transformation to equation (20), we get(28)∑m=0∞μmξ,ζ,η=S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0+uδsδSG1ξ,ζ,η−uδsδSG¯1∑m=0∞μm,∑m=0∞νm+∑m=0∞Am,∑m=0∞νmξ,ζ,η=S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0+uδsδSG2ξ,ζ,η−uδsδSG¯2∑m=0∞μm,∑m=0∞νm+∑m=0∞ℬm.We define the following terms:(29)μ0ξ,ζ,η=S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kμξ,ζ,η∂kη|η=0+uδsδSG1ξ,ζ,η,ν0ξ,ζ,η=S−1uδsδ∑k=0m−1sδ−k−1uδ−k∂kνξ,ζ,η∂kη|η=0+uδsδSG2ξ,ζ,η,μ1ξ,ζ,η=−S−1uδsδSG¯1μ0,ν0+A0,ν1ξ,ζ,η=−S−1uδsδSG¯2μ0,ν0+ℬ0,in general for m≥1, and we have(30)μm+1ξ,ζ,η=−S−1uδsδSG¯1μm,νm+Am,νm+1ξ,ζ,η=−S−1uδsδSG¯2μm,νm+ℬm.


## 5. Implementation of the Methods

Example 1.
Consider fractional-order system of nonlinear equations of unsteady flow of a polytropic gas [36, 38]:(31)Dηδμ+μ∂μ∂ξ+ν∂μ∂ζ+1ω∂ψ∂ξ=0,Dηδν+μ∂ν∂ξ+ν∂ν∂ζ+1ω∂ψ∂ζ=0,Dηδω+μ∂ω∂ξ+ν∂ω∂ζ+ω∂μ∂ξ+∂ν∂ζ=0,Dηδψ+μ∂ψ∂ξ+ν∂ψ∂ζ+τψ∂μ∂ξ+∂ν∂ζ=0,with initial conditions(32)μξ,ζ,0=eξ+ζ,νξ,ζ,0=−1−eξ+ζ,ωξ,ζ,0=eξ+ζ,ψξ,ζ,0=c,where c is the real constant.
First, SDM is used to solve equation (31).
For this applying Shehu transformation to equation (31),(33)S∂δμ∂ηδ=S−μ∂μ∂ξ+ν∂μ∂ζ+1ω∂ψ∂ξ,S∂δν∂ηδ=S−μ∂ν∂ξ+ν∂ν∂ζ+1ω∂ψ∂ζ,S∂δω∂ηδ=S−μ∂ω∂ξ+ν∂ω∂ζ+ω∂μ∂ξ+∂ν∂ζ,S∂δψ∂ηδ=S−μ∂ψ∂ξ+ν∂ψ∂ζ+τψ∂μ∂ξ+∂ν∂ζ,sδuδSμξ,ζ,η−sδ−1uδμξ,ζ,0=S−μ∂μ∂ξ+ν∂μ∂ζ+1ω∂ψ∂ξ,sδuδSνξ,ζ,η−sδ−1uδνξ,ζ,0=S−μ∂ν∂ξ+ν∂ν∂ζ+1ω∂ψ∂ζ,sδuδSωξ,ζ,η−sδ−1uδωξ,ζ,0=S−μ∂ω∂ξ+ν∂ω∂ζ+ω∂μ∂ξ+∂ν∂ζ,sδuδSψξ,ζ,η−sδ−1uδψξ,ζ,0=S−μ∂ψ∂ξ+ν∂ψ∂ζ+τψ∂μ∂ξ+∂ν∂ζ.
The above algorithm is reduced to simplified form:(34)Sμξ,ζ,η=1sμξ,ζ,0−uδsδSμ∂μ∂ξ+ν∂μ∂ζ+1ω∂ψ∂ξ,Sνξ,ζ,η=1sνξ,ζ,0−uδsδSμ∂ν∂ξ+ν∂ν∂ζ+1ω∂ψ∂ζ,Sωξ,ζ,η=1sωξ,ζ,0−uδsδSμ∂ω∂ξ+ν∂ω∂ζ+ω∂μ∂ξ+∂ν∂ζ,Sψξ,ζ,η=1sψξ,ζ,0−uδsδSμ∂ψ∂ξ+ν∂ψ∂ζ+τψ∂μ∂ξ+∂ν∂ζ.
Applying inverse Shehu transformation, we get(35)μξ,ζ,η=μξ,ζ,0−S−1uδsδSμ∂μ∂ξ+ν∂μ∂ζ+1ω∂ψ∂ξ,νξ,ζ,η=νξ,ζ,0−S−1uδsδSμ∂ν∂ξ+ν∂ν∂ζ+1ω∂ψ∂ζ,ωξ,ζ,η=ωξ,ζ,0−S−1uδsδSμ∂ω∂ξ+ν∂ω∂ζ+ω∂μ∂ξ+∂ν∂ζ,ψξ,ζ,η=ψξ,ζ,0−S−1uδsδSμ∂ψ∂ξ+ν∂ψ∂ζ+τψ∂μ∂ξ+∂ν∂ζ.
Equation (35) can be written in an operator form as(36)μξ,ζ,η=μξ,ζ,0−S−1uδsδSA1μ,μξ+B1ν,μζ+C1ω,ψξ,νξ,ζ,η=νξ,ζ,0−S−1uδsδSA2μ,νξ+B2ν,νζ+C2ω,ψζ,ωξ,ζ,η=ωξ,ζ,0−S−1uδsδSA3μ,ωξ+B3ν,ωζ+C3ω,μξ+D3ω,νζ,ψξ,ζ,η=ψξ,ζ,0−S−1uδsδSA4μ,ψξ+B4ν,ψζ+τC4ψ,μξ+τD4ψ,νζ.
Assume that the unknown functionsμξ,ζ,η, νξ,ζ,η, ωξ,ζ,η, and ψξ,ζ,η have infinite series solution as follows:(37)μξ,ζ,η=∑m=0∞μmξ,ζ,η,νξ,ζ,η=∑m=0∞νmξ,ζ,η,ωξ,ζ,η=∑m=0∞ωmξ,ζ,η,ψξ,ζ,η=∑m=0∞ψmξ,ζ,η.
All forms of nonlinear Adomian polynomials can be defined as(38)A1μ,μξ=μ0μ0ξ+μ1μ0ξ+μ0μ1ξ+⋯,B1ν,μζ=ν0μ0ξ+ν1μ0ξ+ν0μ1ξ+⋯,C1ω,ψξ=ψ0ξω0+ω0ψ1ξ−ω1ψ0ξω0+⋯,A2μ,νξ=μ0ν0ξ+μ1ν0ξ+μ0ν1ξ+⋯,B2ν,νζ=ν0ν0ξ+ν1ν0ξ+ν0ν1ξ+⋯,C2ω,ψζ=ψ0ζω0+ω0ψ1ζ−ω1ψ0ζω0+⋯,A3μ,ωξ=μ0ω0ξ+μ1ω0ξ+μ0ω1ξ+⋯,B3ν,ωζ=ν0ω0ξ+ν1ω0ξ+ν0ω1ξ+⋯,C3ω,μξ=ω0μ0ξ+ω1μ0ξ+ω0μ1ξ+⋯,D3ω,νζ=ω0ν0ζ+ω1ν0ζ+ω0ν1ζ+⋯,A4μ,ψξ=μ0ψ0ξ+μ1ψ0ξ+μ0ψ1ξ+⋯,C4ψ,μξ=ψ0μ0ξ+ψ1μ0ξ+ψ0μ1ξ+⋯,D4ψ,νζ=ψ0ν0ζ+ψ1ν0ζ+ψ0ν1ζ+⋯.
The initial sources are(39)μ0ξ,ζ,η=eξ+ζ,ν0ξ,ζ,η=−1−eξ+ζ,ω0ξ,ζ,η=eξ+ζ,ψ0ξ,ζ,0=c.μmξ,ζ,η=−S−1uδsδSA1μ,μξ+B1ν,μζ+C1ω,ψξ,νmξ,ζ,η=−S−1uδsδSA2μ,νξ+B2ν,νζ+C2ω,ψζ,ωmξ,ζ,η=−S−1uδsδSA3μ,ωξ+B3ν,ωζ+C3ω,μξ+D3ω,νζ,ψmξ,ζ,η=−S−1uδsδSA4μ,ψξ+B4ν,ψζ+τC4ψ,μξ+τD4ψ,νζ.
Form=1,(40)μ1ξ,ζ,η=eξ+ζηδΓδ+1,ν1ξ,ζ,η=−eξ+ζηδΓδ+1,ω1ξ,ζ,η=eξ+ζηδΓδ+1,ψ1ξ,ζ,0=0.
Form=2,(41)μ2ξ,ζ,η=eξ+ζη2δΓ2δ+1,ν2ξ,ζ,η=−eξ+ζη2δΓ2δ+1,ω2ξ,ζ,η=eξ+ζη2δΓ2δ+1,ψ2ξ,ζ,0=0.
Form=3,(42)μ3ξ,ζ,η=eξ+ζη3δΓ3δ+1,ν3ξ,ζ,η=−eξ+ζη3δΓ3δ+1,ω3ξ,ζ,η=eξ+ζη3δΓ3δ+1,ψ3ξ,ζ,0=0.μmξ,ζ,η=μ0ξ,ζ,η+μ1ξ,ζ,η+⋯,νmξ,ζ,η=ν0ξ,ζ,η+ν1ξ,ζ,η+⋯,νmξ,ζ,η=ν0ξ,ζ,η+ν1ξ,ζ,η+⋯,ωmξ,ζ,η=ω0ξ,ζ,η+ω1ξ,ζ,η+⋯,ψmξ,ζ,0=ψ0ξ,ζ,η+ψ1ξ,ζ,η+⋯,μξ,ζ,η=eξ+ζ+eξ+ζηδΓδ+1+eξ+ζη2δΓ2δ+1+eξ+ζη3δΓ3δ+1+⋯,νξ,ζ,η=−1−eξ+ζ−eξ+ζηδΓδ+1−eξ+ζη2δΓ2δ+1−eξ+ζη3δΓ3δ+1−⋯,ωξ,ζ,η=eξ+ζ+eξ+ζηδΓδ+1+eξ+ζη2δΓ2δ+1+eξ+ζη3δΓ3δ+1+⋯,ψξ,ζ,0=c+0+⋯.
In general, we have(43)μmξ,ζ,η=eξ+ζηmδΓmδ+1,νmξ,ζ,η=−eξ+ζηmδΓmδ+1,ωmξ,ζ,η=eξ+ζηmδΓmδ+1,ψmξ,ζ,0=0,m=1,2,….
The Approximate Solution by VITM
According to equation (16) and the iteration formulas for system (31), we get(44)μm+1ξ,ζ,η=μmξ,ζ,η−N−uδsδSsδuδ∂μm∂η+μm∂μm∂ξ+νm∂μm∂ζ+1ωm∂ψm∂ξ,νm+1ξ,ζ,η=νmξ,ζ,η−S−1uδsδSsδuδ∂νm∂η+μm∂νm∂ξ+νm∂νm∂ζ+1ωm∂ψm∂ζ,ωm+1ξ,ζ,η=ωmξ,ζ,η−N−uδsδSsδuδ∂μm∂η+μm∂ωm∂ξ+νm∂ωm∂ζ+ωm∂μm∂ξ+∂νm∂ζ,ψm+1ξ,ζ,η=ψmξ,ζ,η−S−1uδsδSsδuδ∂νm∂η+μm∂ψm∂ξ+νm∂ψm∂ζ+τψm∂μm∂ξ+∂νm∂ζ,where(45)μ0ξ,ζ,η=eξ+ζ,ν0ξ,ζ,η=−1−eξ+ζ,ω0ξ,ζ,η=eξ+ζ,ψ0ξ,ζ,0=c.
Form=0,1,2,…,(46)μ1ξ,ζ,η=μ0ξ,ζ,η−N−uδsδSsδuδ∂μ0∂η+μ0∂μ0∂ξ+ν0∂μ0∂ζ+1ω0∂ψ0∂ξ,ν1ξ,ζ,η=ν0ξ,ζ,η−S−1uδsδSsδuδ∂ν0∂η+μ0∂ν0∂ξ+ν0∂ν0∂ζ+1ω0∂ψ0∂ζ,ω1ξ,ζ,η=ω0ξ,ζ,η−N−uδsδSsδuδ∂μ0∂η+μ0∂ω0∂ξ0+ν0∂ω0∂ζ0+ω0∂μ0∂ξ+∂ν0∂ζ,ψ1ξ,ζ,η=ψ0ξ,ζ,η−S−1uδsδSsδuδ∂ν0∂η+μ0∂ψ0∂ξ+ν0∂ψ0∂ζ+τψ0∂μ0∂ξ+∂ν0∂ζ,μ1ξ,ζ,η=eξ+ζ1+ηδΓδ+1,ν1ξ,ζ,η=−1−eξ+ζ1+ηδΓδ+1,ω1ξ,ζ,η=eξ+ζ1+ηδΓδ+1,ψ1ξ,ζ,0=c+0.μ2ξ,ζ,η=μ1ξ,ζ,η−N−uδsδSsδuδ∂μ1∂η+μ1∂μ1∂ξ+ν1∂μ1∂ζ+1ω1∂ψ1∂ξ,ν2ξ,ζ,η=ν1ξ,ζ,η−S−1uδsδSsδuδ∂ν1∂η+μ1∂ν1∂ξ+ν1∂ν1∂ζ+1ω1∂ψ1∂ζ,ω2ξ,ζ,η=ω1ξ,ζ,η−N−uδsδSsδuδ∂μ1∂η+μ1∂ω1∂ξ0+ν1∂ω1∂ζ1+ω1∂μ1∂ξ+∂ν1∂ζ,ψ2ξ,ζ,η=ψ1ξ,ζ,η−S−1uδsδSsδuδ∂ν1∂η+μ1∂ψ1∂ξ+ν1∂ψ1∂ζ+τψ1∂μ1∂ξ+∂ν1∂ζ,μ2ξ,ζ,η=eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1,ν2ξ,ζ,η=−1−eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1,ω2ξ,ζ,η=eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1,ψ2ξ,ζ,0=c+0.μ3ξ,ζ,η=μ2ξ,ζ,η−N−uδsδSsδuδ∂μ2∂η+μ2∂μ2∂ξ+ν2∂μ2∂ζ+1ω2∂ψ2∂ξ,ν3ξ,ζ,η=ν2ξ,ζ,η−S−1uδsδSsδuδ∂ν2∂η+μ2∂ν2∂ξ+ν2∂ν2∂ζ+1ω2∂ψ2∂ζ,ω3ξ,ζ,η=ω2ξ,ζ,η−N−uδsδSsδuδ∂μ2∂η+μ2∂ω2∂ξ0+ν2∂ω2∂ζ2+ω2∂μ2∂ξ+∂ν2∂ζ,ψ3ξ,ζ,η=ψ2ξ,ζ,η−S−1uδsδSsδuδ∂ν2∂η+μ2∂ψ2∂ξ+ν2∂ψ2∂ζ+τψ2∂μ2∂ξ+∂ν2∂ζ,μ3ξ,ζ,η=eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1,ω3ξ,ζ,η=eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1,ψ3ξ,ζ,0=c+0.ν3ξ,ζ,η=−1−eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1,νξ,ζ,η=∑m=0∞νmξ,ζ=−1−eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηmδΓmδ+1,ωξ,ζ,η=∑m=0∞ωmξ,ζ=eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηmδΓmδ+1,ψξ,ζ,0=∑m=0∞ψmξ,ζ=c+0.μξ,ζ,η=∑m=0∞μmξ,ζ=eξ+ζ1+ηδΓδ+1+η2δΓ2δ+1+η3δΓ3δ+1+⋯+ηmδΓmδ+1.
The exact solution of equation (31) at δ=1 is(47)μξ,ζ,η=eξ+ζ+η,νξ,ζ,η=−1−eξ+ζ+η,ωξ,ζ,η=eξ+ζ+η,ψξ,ζ,0=c.


## 6. 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 Figures1–4, the solutions μ, ν, ω, and ψ obtained by using SDM and VITM are compared by keeping one variable and other constants. The 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, ν, and ψ are plotted for Example 1. The 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 solutions are plotted in two dimensions at fractional order δ=0.4,0.6,0.8,1 for Example 1. The convergence phenomenon of the fractional solutions towards integer solution is observed. The three-dimensional graphs of the fractional-order solutions for Example 1 are represented in Figures 11–13 for variables μ, ν, and ω, respectively. In Table 1 and Figure 14, the combined graph for variables μ, ν, and ψ is displayed at δ=1. The solution comparison of the suggested methods, SDM and VITM, is discussed. The suggested techniques have provided the solutions with the desire degree of accuracy with the consideration of very few terms in its series form solutions.Figure 1
SDM and VITM solution graph ofμ at ζ=1 and η=0.5 of Example 1.Figure 2
SDM and VITM solution graph ofν at ζ=1 and η=0.5 of Example 1.Figure 3
SDM and VITM solution graph ofω at ζ=1 and η=0.5 of Example 1.Figure 4
SDM and VITM solution graph ofψ of Example 1.Figure 5
SDM and VITM solution 3d graph ofμξ,ζ,η of Example 1.Figure 6
SDM and VITM solution 3d graph ofνξ,ζ,η of Example 1.Figure 7
SDM and VITM solution 3d graph ofωξ,ζ,η of Example 1.Figure 8
SDM and VITM graph of different value ofδ for μξ,ζ,η of Example 1.Figure 9
SDM and VITM graph of different value ofδ for νξ,ζ,η of Example 1.Figure 10
SDM and VITM graph of different value ofδ for ωξ,ζ,η of Example 1.Figure 11
SDM and VITM 3d graph of different value ofδ for μξ,ζ,η of Example 1.Figure 12
SDM and VITM 3d graph of different value ofδ for νξ,ζ,η of Example 1.Figure 13
SDM and VITM solution graph of different value ofδ for ωξ,ζ,η of Example 1.Table 1
SDM and VITM solution of Example1 at δ=1, η=0.5, and ζ=1.
ξAE ofμAE ofνAE ofω0.51.22000E − 071.22000E − 071.22000E − 0712.01000E − 072.01000E − 072.01000E − 071.53.32000E − 073.32000E − 073.32000E − 0725.50000E − 075.50000E − 075.50000E − 072.59.00000E − 079.00000E − 079.00000E − 0731.49000E − 061.49000E − 061.49000E − 063.52.45000E − 062.45000E − 062.45000E − 0644.06000E − 064.06000E − 064.06000E − 064.56.70000E − 066.70000E − 066.70000E − 0651.11000E − 051.11000E − 051.11000E − 05Figure 14
Combine graph ofμξ,ζ,η, νξ,ζ,η, and ωξ,ζ,η of Example 1.


## 7. Conclusion

In this article, the analytical solution of the system of time-fractional partial differential equations of unsteady flow of polytrophic dynamics is investigated by using two different techniques:(i)
The proposed techniques are the mixture of Shehu transformation with Adomian decomposition method and variational iteration method, respectively.(ii)
The 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)
The 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.The following abbreviations are used in this article:



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