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.
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:
The initial condition is , where is a parameter that describes the fractional-order derivatives. When , (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 operator of order is defined as [29]where , , and
2.2. Definition
The Riemann fractional-order integral operator is presented by [29]
The basic properties of the operator are presented as
2.3. Definition
The Caputo fractional operator of is defined as [29]
2.4. Definition
The Elzaki transformation Caputo fractional-order operator is defined as
3. New Iterative Transformation Technique
We considerwhere M and N are linear and nonlinear terms. We consider the initial condition as
Using the Elzaki transformation of (8), we obtain
Implement the Elzaki differentiation property:
Using the inverse Elzaki transformation (11),
Then, we reach
We consider the nonlinear term N by
Replacing equations (12), (13), and (15) in (12) yields
We describe the iterative method:
Finally, we can write as
4. Homotopy Perturbation Transform Method
In this section, we give the general solution of FPDEs via the homotopy perturbation method:
Applying Elzaki transformation of (16),
Now, we use Elzaki inverse transformation, and we obtainwhere
Now, the parameter shows the producer of perturbation:
The nonlinear term can be defined aswhere are He’s polynomial in terms of and can be calculated as
Substituting (24) and (25) in (21), we achieve as
Comparison of coefficients on both sides, we obtain
The components can be calculated easily which is a fast convergence series. We can obtain :
4.1. Example
Consider the fractional-order gas dynamics equation:with initial condition,
First on both sides apply Elzaki transformation in (29), we have
Using inverse Elzaki transform on the above equation,
We use the NITM:
The series solution form is given as
The approximate solution is achieved as
Now, we apply the HPTM, and we obtain
Then, we have
Then, the series form solution of HPTM is presented:
The approximate solution of Example 1 is given by
The exact result of (29):
In Figure 1, the actual and analytical solutions are proved at 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 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.
4.2. Example
We take into considerationwith initial condition,
Applying the Elzaki transformation in (40) gives
Using inverse Elzaki transform on the above equation,
We use the NITM:
The series solution form is presented by
The approximate solution is achieved as
Now, we apply the HPTM; we obtainwhere the polynomial signifying the nonlinear expressions is . For instance, the components of He’s polynomials are obtained through the recursive correlation , . Now, both sides as the equivalent power coefficient of are compared; the following calculation is obtain by
Thus, we obtain
The approximate solution of Example 2 is given as
The exact result of (40) is
In Figure 4, the actual and analytical solutions are proved at 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 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.
4.3. Example
We take into consideration the fractional-order nonlinear homogeneous gas dynamics equation:with initial condition,
Applying the Elzaki transformation in (52) yields
Using inverse Elzaki transform on the above equation,
We use the NITM:
The series solution form is given as
The approximate solution is achieved as
Now, we apply the HPTM, and we obtainwhere the polynomial signifying the nonlinear expressions is . For instance, the components of He’s polynomials are obtained through the recursive correlation , . Now, both sides of the equivalent power coefficient of is compared; the following calculation is obtain by
Then, the series-form solution of HPTM is given as
The approximate solution of example in this section is given as
The exact result of (52) is
In Figure 7, the actual and analytical solutions are proved at 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 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.
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.
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 study.
Acknowledgments
This research has been funded by Scientific Research Deanship at University of Ha’il ‐ Saudi Arabia through project number RG-21 005.