The New Semianalytical Technique for the Solution of Fractional- Order Navier-Stokes Equation

In this paper, we introduce a modified method which is constructed by mixing the residual power series method and the Elzaki transformation. Precisely, we provide the details of implementing the suggested technique to investigate the fractional-order nonlinear models. Second, we test the efficiency and the validity of the technique on the fractional-order Navier-Stokes models. Then, we apply this new method to analyze the fractional-order nonlinear system of Navier-Stokes models. Finally, we provide 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.


Introduction
The fractional-order Navier-Stokes equation (NSE) has been extensively analyzed. These equations model the fluid motion defined by several physical processes, such as the movement of blood, the ocean's current, the flow of liquid in vessels, and the airflow around an aircraft's arms [1][2][3]. The classical NSEs were generalized by El-Shahed and Salem [4] by replacing the first time derivative with a Caputo fractional derivative of order α, where 0 < α ≤ 1. Using Hankel transform, Fourier sine transform, and Laplace transform, the researchers achieved the exact solution for three different equations. In 2006, Momani and Odibat [5] solve fractionalorder NSEs using the Adomian decomposition method. Ganji et al. [6] applied an analytical technique, the homotopy perturbation method, for solving the fractional-order NSEs in polar coordinates, and the results achieved were expressed in a closed form. Singh and Kumar [7] solved the fractionalorder reduced differential transformation method (FRDTM) to achieve an approximated analytical result of fractionalorder multidimensional NSE. Oliveira and Oliveira [8] analyzed the residual power series method (RPSM) to find the result of the nonlinear fractional-order two-dimensional NSEs. Zhang and Wang [9] suggested numerical analysis for a class of NSEs with fractional-order derivatives; Ravindran, the exact boundary controllability of Galerkin approximations of a Navier-Stokes system for soret convection [10]; and Cibik and Yilmaz, the Brezzi-Pitkaranta stabilization and a priori error analysis for the Stokes control [11].
Some researchers mix two powerful techniques to achieve another result technique to solve equations and systems of fractional-order NSEs. Below, we define some of these combinations: a combination of the Laplace transformation and Adomian decomposition method; Kumar et al. [12] introduced the homotopy perturbation transform method (HPTM), combined Laplace transformation with the homotopy perturbation method, and solved fractionalorder NSEs in a tube. Jena and Chakraverty [13] implemented the homotopy perturbation transformation method (HPETM), and this technique consists in the mixture of Elzaki transformation technique and homotopy perturbation technique; Prakash et. al [1] suggested q-homotopy analysis transformation technique to achieve a result of coupled fractional-order NSEs. This technique mixture of the Laplace transformation and residual power series method is defined: where D α τ u is the Caputo derivative of order α, u is the velocity vector, τ is the time, v is kinematics viscosity, p is the pressure, and ρ is the density.
In this work, we consider two special cases. First, we consider unsteady, one-dimensional motion of a viscous fluid in a tube. The fractional-order Navier-Stokes equations in cylindrical coordinates that governs the flow field in the tube are given by with initial condition where P = −1/ρ∂p/∂z and gðψÞ is a function depending only on ψ.
Consider that the fractional-order two-dimensional Navier-Stokes equations is defined as with initial conditions where u = uðψ, φ, τÞ, v = vðψ, φ, τÞ, ρ, τ, p denote as constant density, time, and pressure, respectively. ψ, φ are the spatial components, and f ðψ, φÞ and gðψ, φÞ are two functions depending only on ψ and φ. The residual power series method (RPSM) is a simple and efficient technique for constructing a power series result for extremely linear and nonlinear equations without perturbation, linearization, and discretization. Unlike the classical power series technique, the RPS approach does not need to compare the coefficients of the corresponding terms and a recursion relation is not required. This approach calculates the power series coefficients by a series of algebraic equations of one or more variables, and its reliance on derivation, which is much simpler and more precise than integration, which is the basis of most other solution approaches, is the main advantage of this methodology. This method is, in effect, an alternative strategy for obtaining theoretical results for the fractional-order partial differential equations [14].
The RPSM was introduced as an essential tool for assessing the power series solution's values for the first and second-order fuzzy DEs [15]. It has been successfully implemented in the approximate result of the generalized Lane-Emden equation [16], which is a highly nonlinear singular DE, in the inaccurate work of higher-order regular DEs [17], in the solution of composite and noncomposite fractional-order DEs [18], in predicting and showing the diversity of results to the fractional-order boundary value equations [19], and in the numerical development of the nonlinear fractional-order KdV and Burgers equation [20], in addition to some other implementations [21][22][23], and recently, it has been applied to investigate the approximate result of a fractional-order two-component evolutionary scheme [24].
This paper introduces the modified analytical technique: the residual power series transform method (RPSTM) is implemented to investigate the fractional-order NS equations. The result of certain illustrative cases is discussed to explain the feasibility of the suggested method. The results of fractional-order models and integral-order models are defined by using the current techniques. The new approach has lower computing costs and higher rate convergence. The suggested method is also constructive for addressing other fractional orders of linear and nonlinear PDEs.

Preliminaries
Definition 1. The Abel-Riemann of fractional operator D α of order α is given as [25][26][27] where j ∈ ℤ + , α ∈ ℝ + and Definition 2. The fractional-order Abel-Riemann integration operator J α is defined as [25][26][27] The operator of basic properties Definition 4. The fractional-order Caputo operator of Elzaki transform is given as [25][26][27] where 0 ≤ m − 1 < α ≤ m and τ ≤ ψ is called fractional power series (FPS) about ψ, where P m are the constants called the coefficients of the series. If ψ = 0, then the fractional power series will be reduced to the fractional Maclaurin series.
Theorem 6. Assume that f o has a fractional power series representation at τ = ψ of the form [14] For m ∈ NUf0g, if D mα f o ðτÞ are continuous on ðψ, ψ + R o Þ, then the coefficients P m can be written as where ψ ≤ τ < ψ + R o , and R o is the radius of convergence.
is said to be multifractional power series at τ = ψ, where G m ðϕÞ are the coefficients of multifractional power series.
Theorem 8. Let us assume that u o ðϕ, τÞ which has the multifractional power representation at τ = ψ can be written as [14] For m ∈ NUf0g, if D mα τ u o ðϕ, τÞ are the continuous on I o × ðψ, ψ + R o Þ, then the coefficients G m are given by where ψ ∈ I o and ψ ≤ τ < ψ + R o . So, we can write the fractional power expansion of u o ðϕ, τÞ of the form which is the generalized Taylor expansion. If we consider α = 1, then the generalized Taylor formula will be converted to classical Taylor series.

The Procedure of RPSTM
In this section, we explain the steps of RPSTM for solving the fractional-order partial differential equation with initial condition First, we use the Elzaki transform to (21); we get By the fact that E½D α τ uðψ, τÞ = 1/s α E½uðψ, τÞ − s 1−α uðx , 0Þ and using the initial condition (22), we can write (23) as where Uðψ, sÞ = E½uðψ, τÞ: Second, we define the transform function Uðψ, sÞ as the following formula:

Journal of Function Spaces
We write the kth truncated series of (25) as As stated in [25], the definition of Elzaki residual function to (25) is and the kth Elzaki residual function of (27) is Third, we expand a few of the properties arising in the basic RPSM to find out certain facts: (ii) lim k→∞ s kα+1 EResðψ, sÞ = lim k→∞ s kα+1 E Re s k ðψ, sÞ = 0, 0 < α ≤ 1, k = 1, 2, 3, ⋯ Furthermore, to evaluate the coefficient functions f n ðψÞ, we can recursively solve the following scheme Finally, we implemented the Elzaki inverse to U k ðψ, sÞ to achieve the kth approximate supportive solution u k ðψ, τÞ.

Numerical Results
Subject to the initial condition Applying Elzaki transform to (31) and using the initial condition given in (32), we get The kth truncated term series of (33) is and the kth Elzaki residual function is Now, to determine f k ðψÞ, k = 1, 2, 3, ⋯, we substitute the kth truncated series (34) into the kth Elzaki residual function (35), multiply the resulting equation by s kα+2 , and then solve recursively the relation lim s→∞ ½s kα+2 Re s k ðψ, sÞ = 0, k = 1, 2 , 3, ⋯, for f k ðψÞ. The following are the first several components of the series f k ðψ, φÞ: Putting the values of f n ðψÞðn ≥ 1Þ in (34), we get Using inverse Elzaki transform to (38), we get Putting α = 1, we have In Figure 1, the RPSTM and the exact results of Example 1 at α = 1 are shown by plots (a) and (b), respectively. From the given figures, it can be seen that both the exact and the EDM results are in close contact with each other. Also, in the Figure 2 subgraph, the RPSTM results of Example 1 are calculated at different fractional-order α = 0:8 and 0:6. It is investigated that fractional-order problem results are 4 Journal of Function Spaces convergent to an integer-order effect as fractional-order analysis to integer-order. The same phenomenon of convergence of fractional-order solutions towards integral-order solutions is observed.
Example 2. Consider the fractional-order one-dimensional NS equation of the form Subject to the initial condition, Applying Elzaki transform to (41) and using the initial condition given in (42), we get The kth truncated term series of (43) is and the kth Elzaki residual function is Now, to determine f k ðψÞ, k = 1, 2, 3, ⋯, we substitute the kth truncated series (44) into the kth Elzaki residual function (45), multiply the resulting equation by s kα+2 , and then solve recursively the relation lim s→∞ ½s kα+2 Re s k ðψ, sÞ = 0, k = 1, 2 , 3, ⋯, for f k ðψÞ. The following are the first several components of the series f k ðψ, φÞ:

Journal of Function Spaces
Putting the values of f n ðψÞðn ≥ 1Þ in (44), we get Using inverse Elzaki transform to (48), we get Putting α = 1, we have In Figure 3, the RPSTM and the exact results of Example 2 at α = 1 are shown by graphs, respectively. From the given figures, it can be seen that both the exact and the EDM results are in close contact with each other. Also, in the Figure 4 subgraph, the RPSTM results of Example 2 are calculated at different fractional-order α = 0:8 and 0:6. It is investigated that fractional-order problem results are convergent to an integer-order effect as fractional-order analysis to integerorder. The same phenomenon of convergence of fractionalorder solutions towards integral-order solutions is observed.

Example 3. Consider the fractional-order two-dimensional NS equation of the form
with initial condition Applying Elzaki transform to (51) and using (52), we get The kth truncated term series of (53) is and the kth Elzaki residual function is Journal of Function Spaces Now, to determine f k ðψ, φÞ and g k ðψ, φÞ, k = 1, 2, 3, ⋯, we substitute the kth truncated series (54) into the kth Elzaki residual function (55), multiply the resulting equation by s kα+2 , and then solve recursively the relation lim s→∞ ½s kα+2 Re s k ðψ , φ, sÞ = 0, k = 1, 2, 3, ⋯, for f k and g k . The following are the first several components of the series f k ðψ, φÞ and g k ðψ, φÞ: Putting the values of f n ðψ, φÞ and g n ðψ, φÞðn ≥ 1Þ in (54), we have Using inverse Elzaki transform, we get Putting α = 1, we get the solution in closed form In Figures 5 and 6, the RPSTM and the exact results of Example 3 at α = 1 are shown by graphs, respectively. From the given figures, it can be seen that both the exact and the RPSTM results are in close contact with each other. Also, in the Figure 7 and 8 subgraph, the RPSTM results of Example 3 are calculated at different fractional-order α = 0:8 and 0:6. It is investigated that fractional-order problem results are convergent to an integer-order effect as fractional-order analysis to integer-order. The same phenomenon of convergence of fractional-order solutions towards integral-order solutions is observed.
Example 4. Consider the fractional-order two-dimensional NS equation as with initial condition Applying Elzaki transform to (60) and using (61), we get  Journal of Function Spaces The kth truncated term series of (62) is and the kth Elzaki residual function is Now, to determine f k ðψ, φÞ and g k ðψ, φÞ, k = 1, 2, 3, ⋯, we substitute the kth truncated series (63) into the kth Elzaki residual function (64), multiply the resulting equation by s kα+2 , and then solve recursively the relation lim s→∞ ½s kα+2 Re s k ðψ, φ, sÞ = 0, k = 1, 2, 3, ⋯, for f k and g k . The following are the first several components of the series f k ðψ, φÞ and g k ðψ, φÞ:

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Journal of Function Spaces Putting the values of f n ðψ, φÞ and g n ðψ, φÞðn ≥ 1Þ in (63), we have Putting α = 1, we get the solution in closed form In Figures 9 and 10, the RPSTM and the exact results of Example 4 at α = 1 are shown by graphs, respectively. From the given figures, it can be seen that both the exact and the RPSTM results are in close contact with each other. Also, in the Figure 11 and 12 subgraph, the RPSTM results of Example 4 are calculated at different fractional-order α = 0:8 and 0:6. It is investigated that fractional-order problem results are convergent to an integer-order effect as fractional-order analysis to integer-order. The same phenomenon of convergence of fractional-order solutions towards integral-order solutions is observed.

Conclusions
In this article, a modified method constructed by a mixture of the residual power series and Elzaki transformation operator is presented to solve fractional-order Navier-Stokes models. The merit of the modified technique is to reduce the size of computational work needed to find the result in a power series form whose coefficient to be calculated is in successive algebraic steps. The technique gives a series form of results that converges very fast in physical models. It is predicted that this article achieved results which will be useful for further analysis of the complicated nonlinear physical problems. The calculations of this technique are very straightforward and simple. Thus, we deduce that this technique can be implemented to solve several schemes 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.