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.
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]:where is the energy density, is the container volume, is the total energy of the gas, is the polytropic index, and 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 conditionswhere and are the velocity components, is the density, is the pressure, and 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]:
2.2. Definition 2
The Caputo’s fractional-order derivative of is defined as follows [55, 56]:
2.3. Definition 3
The Shehu transformation is the new and modern transformation which is described for exponential-order functions. In set A, we take a function represented as follows [51, 52, 57, 58]:
The Shehu transform which is given by for a function is defined as
The Shehu transform of a function is , and then, is called the inverse of , which is given as
2.4. Definition 4
Shehu transform for nth derivatives is given as follows [51, 52, 57, 58]:
2.5. Definition 5
Shehu transform for fractional-order derivatives [51, 52, 57, 58]:
2.6. Definition 6
The Mittag–Leffler function denoted by for is defined as
3. The Procedure of VITM
This section describes the VITM solution, the system of FPDEs:with initial conditionswhere is the Caputo fractional derivative of order ; , and , are linear and nonlinear functions, respectively; and are source operators.
The Shehu transformation is applied to equation (1):
Applying the differentiation property of Shehu transform, we have
The iteration method for equation (15) may be utilized to indicate the major iterative scheme requiring the Lagrange multiplier as
A Lagrange multiplierusing inverse Shehu transformation , and equation (16) can be written as
The initial value can be found as
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:with initial conditionswhere is the Caputo fractional derivative of order ; , and , are linear and nonlinear functions, respectively; and are source functions.
Apply Shehu transform to equation (20):
Applying the differentiation property of Shehu transform, we have
SDM solution of infinite series and is as follows:
Adomian polynomials of nonlinear terms and are given as
The 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
We define the following terms:in general for , and we have
5. Implementation of the Methods
Example 1. Consider fractional-order system of nonlinear equations of unsteady flow of a polytropic gas [36, 38]:with initial conditionswhere is the real constant.
First, SDM is used to solve equation (31).
For this applying Shehu transformation to equation (31),The above algorithm is reduced to simplified form:Applying inverse Shehu transformation, we getEquation (35) can be written in an operator form asAssume that the unknown functions , , , and have infinite series solution as follows:All forms of nonlinear Adomian polynomials can be defined asThe initial sources areFor ,For ,For ,In general, we haveThe Approximate Solution by VITM
According to equation (16) and the iteration formulas for system (31), we getwhereFor ,The exact solution of equation (31) at is
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 Figures 1–4, the solutions , , , and 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 , , and 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 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 , , and , respectively. In Table 1 and Figure 14, the combined graph for variables , , and is displayed at . 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.
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:
Nomenclature
| ST: | Shehu transform |
| LT: | Laplace transform |
| FPDEs: | Fractional partial differential equations |
| VITM: | Variational iteration transform method |
| SDM: | Shehu decomposition method |
| ADM: | Adomian decomposition method. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The 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.
Acknowledgments
The authors acknowledge Dr. Muhammad Arif, Academic Editor, Mathematical Problems in Engineering, Hindawi.