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.
Highlights
Nonlinear fractional partial differential equations (FPDEs) have attracted researchers because of their useful applications in science and engineering [1,2,3]. 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 [4,5,6,7,8,9,10,11,12,13,14]
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 Shehu decomposition method (SDM) and variational iteration transform method (VITM)
It is observed that SDM and VITM solution-graphs are identical and within close
Summary
Nonlinear fractional partial differential equations (FPDEs) have attracted researchers because of their useful applications in science and engineering [1,2,3]. 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 [4,5,6,7,8,9,10,11,12,13,14].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 [24,25,26,27], 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 Mathematical Problems in Engineering the Laplace decomposition technique, to handle fractionalorder 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. e polytropic gas in astrophysics is given as follows [36]: ψ kω1+(1/m), (1)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. 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 [24,25,26,27], 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 Mathematical Problems in Engineering the Laplace decomposition technique, to handle fractionalorder Zakharov–Kuznetsov equations [35]. We consider the gas-dynamic equations fractional-order scheme describing the evolution of an ideal gas’s two-dimensional unsteady flow. Consider the gas-dynamic equations scheme, which describes the evolution of unstable flow of a perfect gas with fractional derivatives [36, 38]: Dδημ
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
