# On Solutions of Fractional-Order Gas Dynamics Equation by Effective Techniques

Authors: Naveed Iqbal; Ali Akgül; Rasool Shah; Abdul Bariq; M. Mossa Al-Sawalha; Akbar Ali
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.


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## Body

## 1. Introduction

Fractional calculus (FC) has been there since classical calculus, but it has recently gained much attention due to its reaction to the requirements as mentioned above. The framework of Liouville and Riemann is used to analyze FC using differential and integral operators. Following that, it was widely used to investigate a variety of phenomena. Many academics, however, pointed out some limitations in engaging this operator, in particular, the physical meaning of the initial condition and the nonzero derivative of a constant. Caputo then presented a unique and new fractional operator that incorporated all of the abovementioned constraints. The Caputo operator is used to study most of the models studied and analyzed under the FC framework. For many ideas of FC, senior academics propose many pioneering directions, and they are the ones who provide the groundwork for the concept [1–5]. The theory and core ideas of FC have been applied to a variety of real-world problems, including biomathematics, financial models, chaos theory, optics, and other fields [6–12].Gas dynamic equations are mathematical representations defined as the physical laws of energy conservation, mass conservation, momentum conservation, etc. Nonlinear fractional-order gas dynamics equations are applied in shock fronts, unusual factions, and connection discontinuities. Gas dynamics is a study in the field of fluid dynamics that studies gas motion and its effect on physical constructions based on the concept of fluid mechanics and fluid dynamics. The science emerges from research of gas flows, mostly around or within human minds, several instances for these research involve, and not restricted to, choked flows in nozzles and pipes, gas fuel streams in a rocket engine, aerodynamic heating on atmospheric reentry cars, and shock waves around aircraft [13, 14].Consider the nonlinear fractional-order gas dynamics equation:(1)∂φυ∂ηφ+υ∂υ∂η−υ1−υ=0,η∈R,0<δ≤1.The initial condition isΦℑ,0=gℑ, where δ is a parameter that describes the fractional-order derivatives. When φ=1, (1) improves the equation of classical gas dynamics.Since certain physical processes, both in engineering and applied sciences, can be successfully explained by the creation of models with the aid of fractional calculus theory. The response of the fractional-order equations eventually converges to the equations of the integer order, attracting particular interest nowadays. Due to a broad variety of applications for mathematical modeling of real-world problems, fractional differentiations are very efficient, e.g., traffic flow models, earthquake modeling, regulation, diffusion model, and relaxation processes [15–17]. In the past decade, the gas dynamic equations are obtained by using different numerical analytical techniques [13, 14]. The homogeneous and nonhomogeneous nonlinear gas dynamics equations have been used the differential transformation technique [18]. Many techniques have been applied to the gas dynamics model such as fractional reduced differential transform technique [19], Elzaki transform homotopy perturbation technique [20], q-homotopy analysis technique [21], Adomian decomposition technique [22], variational iteration technique [23, 24], fractional homotopy analysis transform technique [25], homotopy perturbation algorithm using Laplace transform [26], and natural decomposition technique [27].The goal of this study is to show how, applying the novel iterative technique and the homotopy perturbation technique, the Elzaki transform can be used to obtain approximate solutions for linear and nonlinear fractional-order differential equations. The homotopy perturbation technique was developed by Chinese mathematician J.H. In 1998, he played an important role [28]. This approach is equitable, efficient, and effective, as it eliminates an unconditioned matrix, infinite series, and complicated integrals. This technique does not necessitate the use of any unique problem parameter. Tarig Elzaki, in 2010, develops a new transformation known as the Elzaki transform (E.T). E.T. is a new transform of Laplace and Sumudu transformations [29–32]. Many other researchers use HPTM to solve various equations, such as heat-like models [33], Navier–Stokes models [34], hyperbolic and Fisher’s equations [35], and gas dynamic problem [36].Jafari and Daftardar-Gejji presented a new iterative approach for solving nonlinear equations in 2006 [37]. Jafari et al. first apply the iterative technique and Laplace transformation and combine it. They developed an iterative Laplace transformation method, which is a modified straightforward method [38] to solve the FPDE system [39, 40].


## 2. Basic Definitions

### 2.1. Definition

The fractional-order Riemann operatorDφ of order φ is defined as [29](2)Dφνζ=dℓdζℓνζ,φ=ℓ,1Δℓ−φddζℓ∫0ζνζζ−ψφ−ℓ+1dψ,ℓ−1<φ<ℓ,where ℓ∈Z+, φ∈R+, and(3)D−φνζ=1Γφ∫0ζζ−ψφ−1νψdψ,0<φ≤1.


### 2.2. Definition

The Riemann fractional-order integral operatorJψ is presented by [29](4)Jφνζ=1Γφ∫0ζζ−ψφ−1νζdζ,ζ>0,φ>0.The basic properties of the operator are presented as(5)Jφζℓ=Γℓ+1Γℓ+φ+1ζℓ+ψ,Dφζℓ=Γℓ+1Γℓ−φ+1ζℓ−ψ.


### 2.3. Definition

The Caputo fractional operatorDφ of φ is defined as [29](6)CDφνζ=1Γℓ−φ∫0ζνℓψζ−ψφ−ℓ+1dψ,ℓ−1<φ<ℓ,dℓdζℓνζ,ℓ=φ.


### 2.4. Definition

The Elzaki transformation Caputo fractional-order operator is defined as(7)EDζφgζ=s−φDζφgζ−∑k=0ℓ−1s2−φ+kgk0,whereℓ−1<φ<ℓ.


## 2.1. Definition

The fractional-order Riemann operatorDφ of order φ is defined as [29](2)Dφνζ=dℓdζℓνζ,φ=ℓ,1Δℓ−φddζℓ∫0ζνζζ−ψφ−ℓ+1dψ,ℓ−1<φ<ℓ,where ℓ∈Z+, φ∈R+, and(3)D−φνζ=1Γφ∫0ζζ−ψφ−1νψdψ,0<φ≤1.


## 2.2. Definition

The Riemann fractional-order integral operatorJψ is presented by [29](4)Jφνζ=1Γφ∫0ζζ−ψφ−1νζdζ,ζ>0,φ>0.The basic properties of the operator are presented as(5)Jφζℓ=Γℓ+1Γℓ+φ+1ζℓ+ψ,Dφζℓ=Γℓ+1Γℓ−φ+1ζℓ−ψ.


## 2.3. Definition

The Caputo fractional operatorDφ of φ is defined as [29](6)CDφνζ=1Γℓ−φ∫0ζνℓψζ−ψφ−ℓ+1dψ,ℓ−1<φ<ℓ,dℓdζℓνζ,ℓ=φ.


## 2.4. Definition

The Elzaki transformation Caputo fractional-order operator is defined as(7)EDζφgζ=s−φDζφgζ−∑k=0ℓ−1s2−φ+kgk0,whereℓ−1<φ<ℓ.


## 3. New Iterative Transformation Technique

We consider(8)Dηφυℑ,η+Mυℑ,η+Nυℑ,η=hℑ,η,ℓ∈N,ℓ−1<φ≤ℓ,where M and N are linear and nonlinear terms. We consider the initial condition as(9)υℓℑ,0=gℓℑ.Using the Elzaki transformation of (8), we obtain(10)EDηφυℑ,η+EMυℑ,η+Nυℑ,η=Ehℑ,η.Implement the Elzaki differentiation property:(11)Eυℑ,η=∑ℓ=0ms2−φ+ℓυℓℑ,0+sφEhℑ,η−sφEMυℑ,η+Nυℑ,η.Using the inverse Elzaki transformation (11),(12)υℑ,η=E−1∑ℓ=0ms2−φ+ℓυℓℑ,0+sφEhℑ,η−E−1sφEMυℑ,η+Nυℑ,η.Then, we reach(13)υℑ,η=∑ℓ=0∞υℓℑ,η,(14)N∑ℓ=0∞υℓℑ,η=∑ℓ=0∞Nυℓℑ,η.We consider the nonlinear term N by(15)N∑ℓ=0∞υℓℑ,η=υ0ℑ,η+N∑ℓ=0mυℓℑ,η−M∑ℓ=0mυℓℑ,η.Replacing equations (12), (13), and (15) in (12) yields(16)∑ℓ=0∞υℓℑ,η=E−1sφ∑ℓ=0ms2−ℑ+ℓυℓℑ,η+Ehℑ,η−E−1sφEM∑ℓ=0mυℓℑ,η−N∑ℓ=0mυℓℑ,η.We describe the iterative method:(17)υ0ℑ,η=E−1sφ∑ℓ=0ℓs2−ℑ+ℓυℓℑ,0+sφEgℑ,η,υ1ℑ,η=−E−1sφEMυ0ℑ,η+Nυ0ℑ,η,υℓ+1ℑ,η=−E−1sφE−M∑ℓ=0ℓυℓℑ,η−N∑ℓ=0ℓυℓℑ,η,ℓ≥1.Finally, we can write as(18)υℑ,η≅υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+⋯+υℓℑ,η,m=1,2,….


## 4. Homotopy Perturbation Transform Method

In this section, we give the general solution of FPDEs via the homotopy perturbation method:(19)υDηφυℑ,η+Mυℑ,η+Nυℑ,η=hℑ,η,η>0,0<φ≤1,υℑ,0=gℑ,ν∈ℜ.Applying Elzaki transformation of (16),(20)EDηφυℑ,η+Mυℑ,η+Nυℑ,η=Ehℑ,η,η>0,0<φ≤1,υℑ,η=s2gℑ+sφEhℑ,η−sφEMυℑ,η+Nυℑ,η.Now, we use Elzaki inverse transformation, and we obtain(21)υℑ,η=Fx,η−E−1sφEMυℑ,η+Nυℑ,η,where(22)Fℑ,η=E−1s2gℑ+sφEhℑ,η=gν+E−1sφEhℑ,η.Now, the parameterp shows the producer of perturbation:(23)υℑ,η=∑ℓ=0∞pmυℓℑ,η.The nonlinear term can be defined as(24)Nυℑ,η=∑ℓ=0∞pℓHℓυℓ,where Hℓ are He’s polynomial in terms of υ0,υ1,υ2,…,υℓ and can be calculated as(25)Hnυ0,υ1,…,υn=1φn+1DpℓN∑ℓ=0∞pℓυℓp=0.Substituting (24) and (25) in (21), we achieve as(26)∑ℓ=0∞pℓυℓℑ,η=Fℑ,η−p×E−1sφEM∑ℓ=0∞pℓυℓℑ,η+∑ℓ=0∞pℓHℓυℓ.Comparison of coefficientsp on both sides, we obtain(27)p0:υ0ℑ,η=Fℑ,η,p1:υ1ℑ,η=E−1sφEMυ0ℑ,η+H0υ,p2:υ2ℑ,η=E−1sφEMυ1ℑ,η+H1υ,⋮pℓ:υℓℑ,η=E−1sφEMυℓ−1ℑ,η+Hℓ−1υ,ℓ>0,ℓ∈N.Theυℓℑ,η components can be calculated easily which is a fast convergence series. We can obtain p⟶1:(28)υℑ,η=limM⟶∞∑ℓ=1Mυℓℑ,η.


### 4.1. Example

Consider the fractional-order gas dynamics equation:(29)∂φυ∂ηφ+υ∂υ∂η−υ1−υ=0,0<φ≤1,with initial condition,(30)υℑ,0=e−ℑ.First on both sides apply Elzaki transformation in (29), we have(31)Eυℑ,η=s2e−ℑ−sφEυ∂υ∂η−υ1−υ.Using inverse Elzaki transform on the above equation,(32)υℑ,η=e−ℑ−E−1sφEυ∂υ∂η−υ1−υ.We use the NITM:(33)υ0ℑ,η=e−ℑ,υ1ℑ,η=−E−1sφEυ0∂υ0∂η−υ01−υ0=e−ℑηφΓφ+1,υ2ℑ,η=−E−1sφEυ1∂υ1∂η−υ11−υ1=e−ℑη2φΓ2φ+1,υ3ℑ,η=−E−1sφEυ2∂υ2∂η−υ21−υ2=e−ℑη3φΓ3φ+1,⋮υℓ+1ℑ,η=−E−1sφEυℓ∂υℓ∂η−υℓ1−υl=e−ℑηℓφΓlφ+1.The series solution form is given as(34)υℑ,η=υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+υ3ℑ,η+⋯υℓℑ,η.The approximate solution is achieved as(35)υℑ,η=e−ℑ+e−ℑηφΓφ+1+e−ℑη2φΓ2φ+1+e−ℑη3φΓ3φ+1+⋯+e−ℑηℓφΓlφ+1.Now, we apply the HPTM, and we obtain(36)∑ℓ=0∞pℓυℓℑ,η=e−ℑ+pE−1sφE∑ℓ=0∞pℓHℓυ.Then, we have(37)p0:υ0ℑ,η=e−ℑ,p1:υ1ℑ,η=E−1sφEH0υ=e−ℑηφΓφ+1,p2:υ2ℑ,η=E−1sφEH1υ=e−ℑη2φΓ2φ+1,p3:υ3ℑ,η=E−1sφEH2υ=e−ℑη3φΓ3φ+1,⋮pn:υnℑ,η=E−1sφEHn−1υ=e−ℑηnφΓηφ+1.Then, the series form solution of HPTM is presented:(38)υℑ,η=∑n=0∞pnυnℑ,η.The approximate solution of Example 1 is given by(39)υℑ,η=e−ℑ+e−ℑηφΓφ+1+e−ℑη2φΓ2φ+1+e−ℑη3φΓ3φ+1+⋯+e−ℑηmφΓmφ+1,υℑ,η=e−ℑ∑n=0∞ηφnΓnφ+1=e−ℑEφηφ.The exact result of (29):(40)υℑ,η=e−ℑ+η.In Figure1, the actual and analytical solutions are proved at φ=1 of Example 4.1. In Figure 2, the three-dimensional figure for numerous fractional orders are described which demonstrates that the modified decomposition technique and new iterative transform technique approximated obtained results are in close contact with the analytical and the exact results. In Figure 3, the analytical solution graph of fractional order φ=0.4 of problem 3.1. This comparative shows a strong connection among the homotopy perturbation transform method and actual solutions. Consequently, the homotopy perturbation transform method and new iterative transformation technique are accurate innovative techniques which need less calculation time and is very simple and more flexible as compared to other methods.Figure 1
Simulations of the solutions of problem 3.1.Figure 2
The fractional order ofφ=0.8 and 0.6 of problem 3.1.Figure 3
The fractional order ofφ=0.4 of problem 3.1.


### 4.2. Example

We take into consideration(41)∂φυ∂ηφ+υ∂υ∂η−υ1−υlogb=0,b>0,0<φ≤1,with initial condition,(42)υℑ,0=b−ℑ.Applying the Elzaki transformation in (40) gives(43)Eυℑ,η=s2b−ℑ−sφEυ∂υ∂η−υ1−υlogb.Using inverse Elzaki transform on the above equation,(44)υℑ,η=b−ℑ−E−1sφEυ∂υ∂η−υ1−υlogb.We use the NITM:(45)υ0ℑ,η=b−ℑ,υ1ℑ,η=−E−1sφEυ0∂υ0∂η−υ01−υ0logb=b−ℑlogbηφΓφ+1,υ2ℑ,η=−E−1sφEυ1∂υ1∂η−υ11−υ1logb=b−ℑlogb2η2φΓ2φ+1,υ3ℑ,η=−E−1sφEυ2∂υ2∂η−υ21−υ2logb=b−ℑlogb2η3φΓ3φ+1,⋮υn+1ℑ,η=−E−1sφEυℓ∂υn∂η−υn1−υnlogb=b−ℑlogb2ηnφΓnφ+1.The series solution form is presented by(46)υℑ,η=υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+υ3ℑ,η+⋯υnℑ,η.The approximate solution is achieved as(47)υℑ,η=b−ℑ+b−ℑlogbηφΓφ+1+b−ℑlogb2η2φΓ2φ+1+b−ℑlogb3η3φΓ3φ+1+⋯+b−ℑlogbηφmΓmφ+1,υℑ,η=b−ℑ∑m=0∞logbηφmΓmφ+1=b−ℑEφlogbηφ.Now, we apply the HPTM; we obtain(48)∑ℓ=0∞pℓυℓℑ,η=b−ℑ+pE−1sφE∑ℓ=0∞pℓHℓυ,where the polynomial signifying the nonlinear expressions is Hℓυ. For instance, the components of He’s polynomials are obtained through the recursive correlation Hℓυ=υℓ∂υℓ/∂η−υℓ1−υℓlogb, ∀ℓ∈N. Now, both sides as the equivalent power coefficient of p are compared; the following calculation is obtain by(49)p0:υ0ℑ,η=b−ℑ,p1:υ1ℑ,η=E−1sφEH0υ=b−ℑlogbηφΓφ+1,p2:υ2ℑ,η=E−1sφEH1υ=b−ℑlogb2η2φΓ2φ+1,p3:υ3ℑ,η=E−1sφEH2υ=b−ℑlogb3η3φΓ3φ+1,⋮pn:υnℑ,η=E−1sφEHn−1υ=b−ℑlogblηnφΓlφ+1.Thus, we obtain(50)υℑ,η=∑ℓ=0∞pℓυℓℑ,η.The approximate solution of Example 2 is given as(51)υℑ,η=b−ℑ+b−ℑlogbηφΓφ+1+b−ℑlogb2η2φΓ2φ+1+b−ℑlogb3η3φΓ3φ+1+⋯+b−ℑlogbηφmΓmφ+1,υℑ,η=b−ℑ∑m=0∞logbηφmΓmφ+1=b−ℑEφlogbηφ.The exact result of (40) is(52)υℑ,η=b−ℑ+η.In Figure4, the actual and analytical solutions are proved at φ=1 of Example 4.2. In Figure 5, the three-dimensional figure for numerous fractional order is described which demonstrates that the modified decomposition technique and new iterative transform technique approximated obtained results are in close contact with the analytical and the exact results. In Figure 6, the analytical solution graph of fractional order φ=0.4 of problem 3.2. This comparative result shows a strong connection between the homotopy perturbation transform method and actual solutions. Consequently, the homotopy perturbation transform method and new iterative transformation technique are accurate innovative techniques which needs less calculation time and is very simple and more flexible as compare to other methods.Figure 4
Simulations of the solutions of problem 3.2.Figure 5
The fractional order ofφ=0.8 and 0.6 of problem 3.2.Figure 6
The fractional order ofφ=0.4 of problem 3.2.


### 4.3. Example

We take into consideration the fractional-order nonlinear homogeneous gas dynamics equation:(53)∂φυ∂ηφ+υ∂υ∂η−υ1−υ+e−ℑ+η=0,0<φ≤1,with initial condition,(54)υℑ,0=1−e−ℑ.Applying the Elzaki transformation in (52) yields(55)Eυℑ,η=s21−e−ℑ−sφEυ∂υ∂η−υ1−υ+e−ℑ+η.Using inverse Elzaki transform on the above equation,(56)υℑ,η=1−e−ℑ−E−1sφEυ∂υ∂η−υ1−υ+e−ℑ+η.We use the NITM:(57)υ0ℑ,η=1−e−ℑ,υ1ℑ,η=−E−1sφEυ0∂υ0∂η−υ01−υ0+e−ℑ+η=−e−ℑηφΓφ+1−e−ℑη2φΓ2φ+1,υ2ℑ,η=−E−1sφEυ1∂υ1∂η−υ11−υ1+e−ℑ+η=e−ℑη3φΓ3φ+1,υ3ℑ,η=−E−1sφEυ2∂υ2∂η−υ21−υ2+e−ℑ+η=b−ℑη4φΓ4φ+1,⋮υℓ+1ℑ,η=−E−1sφEυℓ∂υℓ∂η−υℓ1−υℓ+e−ℑ+η.The series solution form is given as(58)υℑ,η=υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+υ3ℑ,η+⋯υnℑ,η.The approximate solution is achieved as(59)υℑ,η=1−e−ℑ−e−ℑηφΓφ+1−e−ℑη2φΓ2φ+1−e−ℑη3φΓ3φ+1−⋯−e−ℑηmφΓmφ+1,υℑ,η=e−ℑ∑m=0∞ηφmΓmφ+1=e−ℑEφηφ.Now, we apply the HPTM, and we obtain(60)∑ℓ=0∞pℓυℓℑ,η=1−e−ℑ+pE−1sφE∑ℓ=0∞pℓHℓυ+e−ℑ+η,where the polynomial signifying the nonlinear expressions is Hℓυ. For instance, the components of He’s polynomials are obtained through the recursive correlation Hℓυ=υℓ∂υℓ/∂η−υℓ1−υℓlogb, ∀ℓ∈N. Now, both sides of the equivalent power coefficient of p is compared; the following calculation is obtain by(61)p0:υ0ℑ,η=1−e−ℑ,p1:υ1ℑ,η=E−1sφEH0υ+e−ℑ+η=−e−ℑηφΓφ+1,p2:υ2ℑ,η=E−1sφEH1υ+e−ℑ+η=−e−ℑη2φΓ2φ+1,p3:υ3ℑ,η=E−1sφEH2υ+e−ℑ+η=−e−ℑη3φΓ3φ+1,⋮pn:υnℑ,η=E−1sφEHn−1υ+e−ℑ+η=−e−ℑηnφΓηφ+1.Then, the series-form solution of HPTM is given as(62)υℑ,η=∑ℓ=0∞pℓυℓℑ,η.The approximate solution of example in this section is given as(63)υℑ,η=1−e−ℑ−e−ℑηφΓφ+1−e−ℑη2φΓ2φ+1−e−ℑη3φΓ3φ+1−⋯−e−ℑηmφΓmφ+1,υℑ,η=e−ℑ∑ℓ=0∞ηφℓΓℓφ+1=e−ℑEφηφ.The exact result of (52) is(64)uℑ,η=1−e−ℑ+η.In Figure7, the actual and analytical solutions are proved at φ=1 of Example 4.3. In Figure 8, the three-dimensional figure for numerous fractional order is described, which demonstrates that the modified decomposition technique and new iterative transform technique approximated obtained results are in close contact with the analytical and the exact results. In Figure 9, the analytical solution graph of fractional order φ=0.4 of problem 3.3. This comparative shows a strong connection among the homotopy perturbation transform method and actual solutions. Consequently, the homotopy perturbation transform method and new iterative transformation technique are accurate innovative techniques which need less calculation time and is very simple and more flexible as compared to other methods.Figure 7
Simulations of the solutions of Example 4.3.Figure 8
The fractional order ofφ=0.8 and 0.6 of problem 3.4.Figure 9
The fractional order ofφ=0.4 of problem 3.4.


## 4.1. Example

Consider the fractional-order gas dynamics equation:(29)∂φυ∂ηφ+υ∂υ∂η−υ1−υ=0,0<φ≤1,with initial condition,(30)υℑ,0=e−ℑ.First on both sides apply Elzaki transformation in (29), we have(31)Eυℑ,η=s2e−ℑ−sφEυ∂υ∂η−υ1−υ.Using inverse Elzaki transform on the above equation,(32)υℑ,η=e−ℑ−E−1sφEυ∂υ∂η−υ1−υ.We use the NITM:(33)υ0ℑ,η=e−ℑ,υ1ℑ,η=−E−1sφEυ0∂υ0∂η−υ01−υ0=e−ℑηφΓφ+1,υ2ℑ,η=−E−1sφEυ1∂υ1∂η−υ11−υ1=e−ℑη2φΓ2φ+1,υ3ℑ,η=−E−1sφEυ2∂υ2∂η−υ21−υ2=e−ℑη3φΓ3φ+1,⋮υℓ+1ℑ,η=−E−1sφEυℓ∂υℓ∂η−υℓ1−υl=e−ℑηℓφΓlφ+1.The series solution form is given as(34)υℑ,η=υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+υ3ℑ,η+⋯υℓℑ,η.The approximate solution is achieved as(35)υℑ,η=e−ℑ+e−ℑηφΓφ+1+e−ℑη2φΓ2φ+1+e−ℑη3φΓ3φ+1+⋯+e−ℑηℓφΓlφ+1.Now, we apply the HPTM, and we obtain(36)∑ℓ=0∞pℓυℓℑ,η=e−ℑ+pE−1sφE∑ℓ=0∞pℓHℓυ.Then, we have(37)p0:υ0ℑ,η=e−ℑ,p1:υ1ℑ,η=E−1sφEH0υ=e−ℑηφΓφ+1,p2:υ2ℑ,η=E−1sφEH1υ=e−ℑη2φΓ2φ+1,p3:υ3ℑ,η=E−1sφEH2υ=e−ℑη3φΓ3φ+1,⋮pn:υnℑ,η=E−1sφEHn−1υ=e−ℑηnφΓηφ+1.Then, the series form solution of HPTM is presented:(38)υℑ,η=∑n=0∞pnυnℑ,η.The approximate solution of Example 1 is given by(39)υℑ,η=e−ℑ+e−ℑηφΓφ+1+e−ℑη2φΓ2φ+1+e−ℑη3φΓ3φ+1+⋯+e−ℑηmφΓmφ+1,υℑ,η=e−ℑ∑n=0∞ηφnΓnφ+1=e−ℑEφηφ.The exact result of (29):(40)υℑ,η=e−ℑ+η.In Figure1, the actual and analytical solutions are proved at φ=1 of Example 4.1. In Figure 2, the three-dimensional figure for numerous fractional orders are described which demonstrates that the modified decomposition technique and new iterative transform technique approximated obtained results are in close contact with the analytical and the exact results. In Figure 3, the analytical solution graph of fractional order φ=0.4 of problem 3.1. This comparative shows a strong connection among the homotopy perturbation transform method and actual solutions. Consequently, the homotopy perturbation transform method and new iterative transformation technique are accurate innovative techniques which need less calculation time and is very simple and more flexible as compared to other methods.Figure 1
Simulations of the solutions of problem 3.1.Figure 2
The fractional order ofφ=0.8 and 0.6 of problem 3.1.Figure 3
The fractional order ofφ=0.4 of problem 3.1.


## 4.2. Example

We take into consideration(41)∂φυ∂ηφ+υ∂υ∂η−υ1−υlogb=0,b>0,0<φ≤1,with initial condition,(42)υℑ,0=b−ℑ.Applying the Elzaki transformation in (40) gives(43)Eυℑ,η=s2b−ℑ−sφEυ∂υ∂η−υ1−υlogb.Using inverse Elzaki transform on the above equation,(44)υℑ,η=b−ℑ−E−1sφEυ∂υ∂η−υ1−υlogb.We use the NITM:(45)υ0ℑ,η=b−ℑ,υ1ℑ,η=−E−1sφEυ0∂υ0∂η−υ01−υ0logb=b−ℑlogbηφΓφ+1,υ2ℑ,η=−E−1sφEυ1∂υ1∂η−υ11−υ1logb=b−ℑlogb2η2φΓ2φ+1,υ3ℑ,η=−E−1sφEυ2∂υ2∂η−υ21−υ2logb=b−ℑlogb2η3φΓ3φ+1,⋮υn+1ℑ,η=−E−1sφEυℓ∂υn∂η−υn1−υnlogb=b−ℑlogb2ηnφΓnφ+1.The series solution form is presented by(46)υℑ,η=υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+υ3ℑ,η+⋯υnℑ,η.The approximate solution is achieved as(47)υℑ,η=b−ℑ+b−ℑlogbηφΓφ+1+b−ℑlogb2η2φΓ2φ+1+b−ℑlogb3η3φΓ3φ+1+⋯+b−ℑlogbηφmΓmφ+1,υℑ,η=b−ℑ∑m=0∞logbηφmΓmφ+1=b−ℑEφlogbηφ.Now, we apply the HPTM; we obtain(48)∑ℓ=0∞pℓυℓℑ,η=b−ℑ+pE−1sφE∑ℓ=0∞pℓHℓυ,where the polynomial signifying the nonlinear expressions is Hℓυ. For instance, the components of He’s polynomials are obtained through the recursive correlation Hℓυ=υℓ∂υℓ/∂η−υℓ1−υℓlogb, ∀ℓ∈N. Now, both sides as the equivalent power coefficient of p are compared; the following calculation is obtain by(49)p0:υ0ℑ,η=b−ℑ,p1:υ1ℑ,η=E−1sφEH0υ=b−ℑlogbηφΓφ+1,p2:υ2ℑ,η=E−1sφEH1υ=b−ℑlogb2η2φΓ2φ+1,p3:υ3ℑ,η=E−1sφEH2υ=b−ℑlogb3η3φΓ3φ+1,⋮pn:υnℑ,η=E−1sφEHn−1υ=b−ℑlogblηnφΓlφ+1.Thus, we obtain(50)υℑ,η=∑ℓ=0∞pℓυℓℑ,η.The approximate solution of Example 2 is given as(51)υℑ,η=b−ℑ+b−ℑlogbηφΓφ+1+b−ℑlogb2η2φΓ2φ+1+b−ℑlogb3η3φΓ3φ+1+⋯+b−ℑlogbηφmΓmφ+1,υℑ,η=b−ℑ∑m=0∞logbηφmΓmφ+1=b−ℑEφlogbηφ.The exact result of (40) is(52)υℑ,η=b−ℑ+η.In Figure4, the actual and analytical solutions are proved at φ=1 of Example 4.2. In Figure 5, the three-dimensional figure for numerous fractional order is described which demonstrates that the modified decomposition technique and new iterative transform technique approximated obtained results are in close contact with the analytical and the exact results. In Figure 6, the analytical solution graph of fractional order φ=0.4 of problem 3.2. This comparative result shows a strong connection between the homotopy perturbation transform method and actual solutions. Consequently, the homotopy perturbation transform method and new iterative transformation technique are accurate innovative techniques which needs less calculation time and is very simple and more flexible as compare to other methods.Figure 4
Simulations of the solutions of problem 3.2.Figure 5
The fractional order ofφ=0.8 and 0.6 of problem 3.2.Figure 6
The fractional order ofφ=0.4 of problem 3.2.


## 4.3. Example

We take into consideration the fractional-order nonlinear homogeneous gas dynamics equation:(53)∂φυ∂ηφ+υ∂υ∂η−υ1−υ+e−ℑ+η=0,0<φ≤1,with initial condition,(54)υℑ,0=1−e−ℑ.Applying the Elzaki transformation in (52) yields(55)Eυℑ,η=s21−e−ℑ−sφEυ∂υ∂η−υ1−υ+e−ℑ+η.Using inverse Elzaki transform on the above equation,(56)υℑ,η=1−e−ℑ−E−1sφEυ∂υ∂η−υ1−υ+e−ℑ+η.We use the NITM:(57)υ0ℑ,η=1−e−ℑ,υ1ℑ,η=−E−1sφEυ0∂υ0∂η−υ01−υ0+e−ℑ+η=−e−ℑηφΓφ+1−e−ℑη2φΓ2φ+1,υ2ℑ,η=−E−1sφEυ1∂υ1∂η−υ11−υ1+e−ℑ+η=e−ℑη3φΓ3φ+1,υ3ℑ,η=−E−1sφEυ2∂υ2∂η−υ21−υ2+e−ℑ+η=b−ℑη4φΓ4φ+1,⋮υℓ+1ℑ,η=−E−1sφEυℓ∂υℓ∂η−υℓ1−υℓ+e−ℑ+η.The series solution form is given as(58)υℑ,η=υ0ℑ,η+υ1ℑ,η+υ2ℑ,η+υ3ℑ,η+⋯υnℑ,η.The approximate solution is achieved as(59)υℑ,η=1−e−ℑ−e−ℑηφΓφ+1−e−ℑη2φΓ2φ+1−e−ℑη3φΓ3φ+1−⋯−e−ℑηmφΓmφ+1,υℑ,η=e−ℑ∑m=0∞ηφmΓmφ+1=e−ℑEφηφ.Now, we apply the HPTM, and we obtain(60)∑ℓ=0∞pℓυℓℑ,η=1−e−ℑ+pE−1sφE∑ℓ=0∞pℓHℓυ+e−ℑ+η,where the polynomial signifying the nonlinear expressions is Hℓυ. For instance, the components of He’s polynomials are obtained through the recursive correlation Hℓυ=υℓ∂υℓ/∂η−υℓ1−υℓlogb, ∀ℓ∈N. Now, both sides of the equivalent power coefficient of p is compared; the following calculation is obtain by(61)p0:υ0ℑ,η=1−e−ℑ,p1:υ1ℑ,η=E−1sφEH0υ+e−ℑ+η=−e−ℑηφΓφ+1,p2:υ2ℑ,η=E−1sφEH1υ+e−ℑ+η=−e−ℑη2φΓ2φ+1,p3:υ3ℑ,η=E−1sφEH2υ+e−ℑ+η=−e−ℑη3φΓ3φ+1,⋮pn:υnℑ,η=E−1sφEHn−1υ+e−ℑ+η=−e−ℑηnφΓηφ+1.Then, the series-form solution of HPTM is given as(62)υℑ,η=∑ℓ=0∞pℓυℓℑ,η.The approximate solution of example in this section is given as(63)υℑ,η=1−e−ℑ−e−ℑηφΓφ+1−e−ℑη2φΓ2φ+1−e−ℑη3φΓ3φ+1−⋯−e−ℑηmφΓmφ+1,υℑ,η=e−ℑ∑ℓ=0∞ηφℓΓℓφ+1=e−ℑEφηφ.The exact result of (52) is(64)uℑ,η=1−e−ℑ+η.In Figure7, the actual and analytical solutions are proved at φ=1 of Example 4.3. In Figure 8, the three-dimensional figure for numerous fractional order is described, which demonstrates that the modified decomposition technique and new iterative transform technique approximated obtained results are in close contact with the analytical and the exact results. In Figure 9, the analytical solution graph of fractional order φ=0.4 of problem 3.3. This comparative shows a strong connection among the homotopy perturbation transform method and actual solutions. Consequently, the homotopy perturbation transform method and new iterative transformation technique are accurate innovative techniques which need less calculation time and is very simple and more flexible as compared to other methods.Figure 7
Simulations of the solutions of Example 4.3.Figure 8
The fractional order ofφ=0.8 and 0.6 of problem 3.4.Figure 9
The fractional order ofφ=0.4 of problem 3.4.


## 5. Conclusion

In this paper, we analyzed the time factional of gas dynamics equation by applying two analytical techniques. It is also used that the suggested methods’ rate of convergence is sufficient for the solution of fractional-order partial differential equations. The computations of these methods are very straightforward and simple. Therefore, these methods can be applied to fractional partial differential equations.



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