The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform

This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three-dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models.


Introduction
Fractional calculus is an engineering and sciences area that deals with the integral and derivative of arbitrary orders. Fractional differential equations (FDEs) have gained popularity and significance, primarily due to their proven implementations in applied mathematics. For instance, these problems are more and more utilized to an equation in fluid flow, diffusion, rheology, oscillation, relaxation, reaction-diffusion, anomalous diffusion, diffusive transport akin to diffusion, turbulence, polymer physics, electric networks, electrochemistry of corrosion, chemical physics, propagation of seismic waves, relaxation processes in complex systems, porous structures, and dynamical processes in self-similar and various other physical procedure. The most significant benefit of FDEs in these and other uses is their nonlocal property. It is well-known that the differential operator of integer-order is a local operator, but the fractional-order differential operator is nonlocal [1][2][3][4][5]. For example, the nonlinear oscillation of earthquakes can be modeled with fractional derivatives [6], and the fluid dynamic traffic model with fractional derivatives [7] can calculate the insufficiency arising from the assumption of continuum traffic flow. Experimental data fractional partial differential equations for seepage flow in porous media are shown in [8]. Fractional differential equations have created attention among the researcher due to the exact description of nonlinear phenomena, especially in nanohydrodynamics where continuum assumption does not well, and fractional model can be considered the best candidate. These findings invoked the growing interest in studies of fractal calculus in many branches of engineering and science. This is more realistic and it is one reason why fractional calculus has become more and more popular [9][10][11][12][13][14][15][16][17].
A famous governing equation of motion of viscous fluid flow called the Navier-Stokes (NS) equation has been derived in 1822 [18]. The equation can be regarded as the Momentum equation and is a combination of Newton's second law of motion for fluid substances, the energy equation, and continuity equation. This equation describes many physical things such as ocean currents, liquid flow in pipes, blood flow, and airflow around the wings of an aircraft. The NSE is the computational fluid dynamics initial equation, relating external forces and pressure acting on a fluid to the fluid flow response. The Navier-Stokes and continuity models are described by where the Caputo fractional derivative is D ϖ η u, time is η, the velocity vector is u, kinematics viscosity is v, pressure is p, and density is ρ. In this article, we consider two special cases. First, a viscous fluid in a tube moves in an unsteady, onedimensional motion. The fluid flow in the tube of fractional NSEs in cylindrical coordinates that govern is defined by with initial condition where P = −1/ρð∂p/∂zÞ and gðζÞ is a function that depends only on ζ.
Consider the two-dimensional fractional NSE is given as with initial conditions where u = uðζ, φ, ηÞ, v = vðζ, φ, ηÞ, where time, pressure and constant density define by η, p, and ρ, respectively. ζ and φ are the spatial components and f ðζ, φÞ and gðζ, φÞ are two functions dependent only on ζ and φ. El-Shahed and Salem [19] generalized the classical NSEs by substituting a Caputo fractional derivative of order ϖ for the first time derivative, where 0 < ϖ ≤ 1. Momani and Odibat [20] use the Adomian decomposition approach to solve fractional-order NSEs. Ganji et al. [21] used the Homotopy perturbation procedure, an analytical technique, to solve fractional NSEs in polar coordinates, and the solutions were given in the closedform. Singh and Kumar [22] implemented the reduced differential transform method to obtain an analytical solution of fractional multidimensional NSEs. Ahmad and Jaber used the residual power series approach to solve the nonlinear fractional two-dimension NSEs [23]. For a class of NSEs with fractional-order derivatives, Wang and Zhang [24] proposed a numerical method. Some investigators combined two strong methods to produce a new method for solving fractional-order NSE equations. Below, we expressed some of these combinations: a mixture of the Adomian decomposition method and Laplace transform; Kumar et al. [25] introduce homotopy perturbation transformation technique, combine the Homotopy perturbation technique with Laplace transform, to solve fractional NSEs. Jena and Chakraverty [26] applied the homotopy perturbation transform technique and this method consists in the combination of homotopy perturbation technique and Elzaki transform; Prakash et al. [27] proposed q-homotopy analysis transform method to obtain a solution of fractional system NSEs. This method combined the homotopy analysis technique and Laplace transform.
The residual power series method (RPSM) is an efficient and straightforward method for constructed a Power Series solution for extremely linear and nonlinear equations without linearization, perturbation, and discretization. The RPS method, unlike the classical power series method, does not involve comparing the coefficients of the corresponding terms or a recursion relation. The power series coefficients are calculated using a series of algebraic equations in one or more variables. The methodology's key benefit is its dependence on derivation, which is much simpler and more accurate than integration, which is the basis of most other solutions techniques. This approach is an alternative technique for solving fractional-order partial differential equations theoretically [28].
The RPSM was used as an essential tool for determining power series method values for fuzzy differential equations [29]. It has been successes used in the analytical solution of the generalize Lane-Emden equations [30], which is significantly nonlinear singular differential equations, in the exact work of higher-order regular differential equations [31], in the result of composite and noncomposite time-fractional differential equations [32], in estimating and demonstrating the variety of solutions for fractional boundary value equation [33], in the approximate investigation of the nonlinear time-fractional Burgers and KdV equations [34], in further to few other implementations [35][36][37], and recently, it has been implemented to investigate the analytical solution of a fractional two-component evolutionary system [38]. This paper presents an analytical investigation of one-and twodimensional, fractional-order model of Navier-stokes equations by using RPSTM.
The fractional calculus fundamental definitions are defined in Section 2, write the general methodologies in Section 3, and many test models to show the effectiveness of suggested techniques are given in Section 4, and finally, the conclusion is given in Section 5.
2.3. Definition. Shehu transformation is new and identical to other integral transformations defined for exponential order functions. In Set A, the function is defined by [41][42][43] The Shehu transform which is described as Sð:Þ for a function νðηÞ is defined as The Shehu transform of a function νðηÞ is Vðs, uÞ; then, νðηÞ is called the inverse of Vðs, uÞ which is given as 2.5. Definition. The fractional-order derivatives of Shehu transformation are defined as [41][42][43] 2.6. Definition. A power series is given as where 0 ≤ k − 1 < ϖ ≤ k and η ≤ ζ is called fractional-order power series about ζ, where P k are the constant called the coefficient of the series. If ζ = 0, then the fractional-order power series will be reduced to the fractional-order Maclaurin series.

2.7.
Theorem. Suppose that f o has a fractional power series expression of the form at tau = psi If D mϖ f o ðηÞ, for m = 0, 1, 2, 3, ⋯ are continuous on ðζ, ζ + R o Þ, then the coefficient P m can be defined as 2.8. Definition. The power series expansion is given as is called to be multi fractional power series at η = ζ, where G m ðϕÞ are the fractional power series coefficients.
2.9. Theorem. Let us suppose that u o ðϕ, ηÞ has the fractionalorder power define at η = ζ is given as where So we can write the fractional power expansion of u o ðϕ, ηÞ of the form which is the generalized form of the Taylor expansion.

The General Methodologies of RPSTM
In this section, we will go through the steps that RPSTM takes to solve the FPDEs.
Finally, we used the Shehu inverse to U k ðζ, sÞ, to obtain the kth-approximated supportive result u k ðζ, ηÞ.

Example. Consider the one dimensional fractional NS equation
Subject to the initial condition Applying Shehu transform to Eq. (32) and using the initial condition given in Eq. (33), we get The kth truncated term series of Eq. (34) is and the kth Shehu residual function is Now, to determine f k ðζÞ, k = 1, 2, 3, ⋯, we put the kthtruncated series (35) into the kth-Shehu residual function (36), multiply the solution equation by s kϖ+2 and, then, solve recursively the relation lim s⟶∞ ½ðν/sÞ nϖ+1 Res k ðζ, sÞ = 0, k = 1, 2, 3, ⋯ for f k ðζÞ. The first few component of the series 4 Journal of Function Spaces f k ðζ, φÞ Putting the values of f n ðζÞðn ≥ 1Þ in Eq. (35), we get Applying inverse Shehu transform to Eq. (39), we get The above equation put ϖ = 1, we get which is the actual result of classical Navier-Stokes equations for the velocity field. The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 1, and the fractional-order behavior of Navier-Stokes equations with ϖ = 0:8,0:6 and 0:4 is shown in Figures 2 and 3, respectively.

Example. Consider the one-dimensional fractional NS equation
Subject to the initial condition Applying Shehu transformation to Eq. (42) and applying the initial condition given in Eq. (43), we get The kth truncated term series of Eq. (44) is and the kth Shehu residual function is Now, to determine f k ðζÞ, k = 1, 2, 3, ⋯, we substitute the kth-truncated series (45) into the kth-Shehu residual function (46), multiply the resulting equation by s kϖ+2 and, then, solve recursively the relation lim s⟶∞ ½ðν/sÞ nϖ+1 Res k ðζ, sÞ = 0, k = 1, 2, 3, ⋯ for f k ðζÞ. The first several component of the series f k ðζ, φÞ Substituting the value of f n ðζÞðn ≥ 1Þ in Eq. (45), we obtain as Using inverse Shehu transformation to Eq. (49), we get The above equation put ϖ = 1, we get   Journal of Function Spaces Figure 4 shows that the actual and analytical solutions graphs at ϖ = 1. The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 4, and the fractional-order behavior of Navier-Stokes equations with ϖ = 0:8,0:6 and 0:4 is shown in Figure 5, respectively.

Example. Consider the two-dimensional fractional NS equation
with initial condition Applying Shehu transform to Eq. (52) and using the Eq. (53), we get The kth truncated term series of Eq. (54) is and the kth Shehu residual function is Now, to determine f k ðζ, φÞ and g k ðζ, φÞ, k = 1, 2, 3, ⋯, we substitute the kth-truncated series (55) into the kth-Shehu residual function (56), multiply the resulting equation by s kϖ+2 , and then solve recursively the relation lim s⟶∞ ½s kϖ+2 Res k ðζ, φ, sÞ = 0, k = 1, 2, 3, ⋯ for f k and g k . The first few component of the series f k ðζ, φÞ and g k ðζ, φÞ Putting the values of f n ðζ, φÞ and g n ðζ, φÞ, ðn ≥ 1Þ in Eq.
(55), we get Using inverse Shehu transformation, we get The exact solution is given as Figure 6 shows that the actual and analytical solution graphs at ϖ = 1 with respect to uðζ, φ, ηÞ. The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 6, and the fractional-order behavior of Navier-Stokes equations with ϖ = 0:8 and 0:6 is shown in Figure 7,  Journal of Function Spaces respectively. Similarly, Figure 8 shows that the actual and analytical solution graphs at ϖ = 1 with respect to vðζ, φ, ηÞ. The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 8, and the fractional-order behavior of Navier-Stokes equations with ϖ = 0:8 and 0:6 is shown in Figure 9, respectively.

Example. Consider the two-dimensional fractional NS equation
with initial condition Applying Shehu transform to Eq. (61) and using the Eq. (62), we obtain as The kth truncated term series of Eq. (63) is and the kth Shehu residual function is Now, to determine f k ðζ, φÞ and g k ðζ, φÞ, k = 1, 2, 3, ⋯,  The exact result is given as u ζ, φ,η ð Þ = −e ζ+φ+2ρ 0 η + g, v ζ, φ, η ð Þ = e ζ+φ+2ρ 0 η − g: ð69Þ Figure 10 shows that the actual and analytical solution graphs at ϖ = 1 with respect to uðζ, φ, ηÞ. The field of velocity behavior of the classical Navier-Stokes equations is shown in Figure 10, and the fractional-order behavior of Navier-Stokes equations with ϖ = 0:8,0:6 and 0:4 is shown in Figures 11 and  12, respectively. Similarly, Figure 13 shows that the actual and analytical solution graphs at ϖ = 1 with respect to vðζ, φ, ηÞ. The field of velocity behavior of the classical Navier-Stokes equations is show in Figure 13, and the fractionalorder behavior of Navier-Stokes equations with ϖ = 0:8 and 0:6 is shown in Figures 14 and 15, respectively.

Conclusion
In this paper, the residual power series transform method is implemented to approximate analytical solutions of the fractional-order Navier-Stokes equations, which are widely applied as problems for spatial effects in applied sciences. The method gives a series form of solutions that converge very quickly in physical problems. It is predicted that this paper obtains solutions to analyze the complicated nonlinear physical problems. The calculations of this method are very simple and straightforward. Thus, we assume that this method can be applied to solve many systems of nonlinear fractional-order partial differential equations.

Data Availability
The numerical data used to support the findings of this study are included within the article.