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

Authors: Nehad Ali Shah; Mounirah Areshi; Jae Dong Chung; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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.


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## Body

## 1. 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 [1–3]. 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 fractional-order 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 fractional-order reduced differential transformation method (FRDTM) to achieve an approximated analytical result of fractional-order 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 fractional-order 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:
(1)Dταu+u·∇u=−1ρ∇p+v∇2u,0<α≤1,∇u=0,where Dταu 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(2)Dταu+P+v∂2u∂ψ2+1ψ∂u∂ψ,0<α≤1,with initial condition
(3)uψ,0=gψ,where P=−1/ρ∂p/∂z and gψ is a function depending only on ψ.Consider that the fractional-order two-dimensional Navier-Stokes equations is defined as(4)Dταu=ρ0∂2∂ψ2u+∂2∂φ2u−u∂∂ψu−v∂∂φu+g,Dταv=ρ0∂2∂ψ2v+∂2∂φ2v−u∂∂ψv−v∂∂φv−g,with initial conditions
(5)uψ,φ,τ=fψ,φ,vψ,φ,τ=gψ,φ,where u=uψ,φ,τ,v=vψ,φ,τ,ρ,τ,p denote as constant density, time, and pressure, respectively. ψ,φ are the spatial components, and fψ,φ and gψ,φ 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 [21–23], 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.


## 2. Preliminaries

Definition 1.
The Abel-Riemann of fractional operatorDα of order α is given as [25–27]
(6)Dανζ=djdζjνζ,α=j,1Γj−αddζj∫0ζνζζ−ψα−j+1dψ,j−1<α<j,where j∈ℤ+,α∈ℝ+ and
(7)D−ανζ=1Γα∫0ζζ−ψα−1νψdψ,0<α≤1.Definition 2.
The fractional-order Abel-Riemann integration operatorJα is defined as [25–27]
(8)Jανζ=1Γα∫0ζζ−ψα−1νζdζ,ζ>0,α>0.
The operator of basic properties(9)Jαζj=Γj+1Γj+α+1ζj+ψ,Dαζj=Γj+1Γj−α+1ζj−ψ.Definition 3.
The Caputo fractional operatorCDα of α is defined as [25–27]
(10)CDανζ=1Γj−α∫0ζνjψζ−ψα−j+1dψ,j−1<α<j,djdζjνζ,j=α.Definition 4.
The fractional-order Caputo operator of Elzaki transform is given as [25–27]
(11)EDζαgζ=s−αEgζ−∑k=0j−1s2−α+kgk0,wherej−1<α<j.Definition 5.
A power series definition of the form [14]
(12)∑m=0∞Pmτ−ψmα=P0+P1τ−ψ+P2τ−ψ2α+⋯,where 0≤m−1<α≤m and τ≤ψ is called fractional power series (FPS) about ψ, where Pm 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 thatfo has a fractional power series representation at τ=ψ of the form [14]
(13)foτ=∑m=0∞Pmτ−ψmα.
Form∈NU0, if Dmαfoτ are continuous on ψ,ψ+Ro, then the coefficients Pm can be written as
(14)Pm=DmαfoψΓ1+mα,where ψ≤τ<ψ+Ro, and Ro is the radius of convergence.Definition 7.
The expansion of power series of the form [14]
(15)∑m=0∞Gmϕτ−ψmαis said to be multifractional power series at τ=ψ, where Gmϕ are the coefficients of multifractional power series.Theorem 8.
Let us assume thatuoϕ,τ which has the multifractional power representation at τ=ψ can be written as [14]
(16)uoϕ,τ=∑m=0∞Gmxτ−ψmα.
Form∈NU0, if Dτmαuoϕ,τ are the continuous on Io×ψ,ψ+Ro, then the coefficients Gm are given by
(17)Gmx=Dτmαuoϕ,τΓ1+mα,where ψ∈Io and ψ≤τ<ψ+Ro.
So, we can write the fractional power expansion ofuoϕ,τ of the form
(18)μϕ,τ=∑m=0∞Dτmαμϕ,τΓ1+mατ−ψmα,which is the generalized Taylor expansion. If we consider α=1, then the generalized Taylor formula will be converted to classical Taylor series.Corollary 9.
Let us assume thatμϕ,φ,τ has a multifractional power series representation about τ=ψ as [14]
(19)uoϕ,φ,τ=∑m=0∞Gmϕ,φτ−ψmα.
Form∈NU0 if Dτmαμϕ,φ,τ are continuous on I1×I2×ψ,ψ+Ro, then
(20)Gmψ,φ=Dτmαμϕ,φ,τΓ1+mα,where ϕ,φ∈I1×I2,ψ≤τ<ψ+Ro.


## 3. The Procedure of RPSTM

In this section, we explain the steps of RPSTM for solving the fractional-order partial differential equation(21)Dταuψ,τ=aDψ2uψ,τ+buψ,τ−cuqψ,τ,with initial condition
(22)uψ,0=f0ψ.First, we use the Elzaki transform to (21); we get
(23)EDταuψ,τ=aEDψ2uψ,τ+bEuψ,τ−cEuqψ,τ.By the fact thatEDταuψ,τ=1/sαEuψ,τ−s1−αux,0 and using the initial condition (22), we can write (23) as
(24)Uψ,s=s2f0ψ+sαaDx2Uψ,s+bsαUψ,s−csαE−1EUψ,sq,where Uψ,s=Euψ,τ.Second, we define the transform functionUψ,s as the following formula:
(25)Uψ,s=∑n=0∞snα+1fnx.We write thekth truncated series of (25) as
(26)Uψ,s=∑n=0∞snα+1fnx=s2f0x+∑n=1∞snα+1fnx.As stated in [25], the definition of Elzaki residual function to (25) is
(27)EResψ,s=Uψ,s−f0xs2−asαDx2Uψ,s−bsαUψ,s+csαE−1EUψ,sq,and the kth Elzaki residual function of (27) is
(28)EResψ,s=Ukψ,s−f0xs2−asαDx2Ukψ,s−bsαUkψ,s+csαEE−1Ukψ,sq.Third, we expand a few of the properties arising in the basic RPSM to find out certain facts:(i)
EResψ,s=0 and limk→∞EReskψ,s=EResψ,s for each s>0(29)limk→∞EResψ,s=0⇒limk→∞sEReskψ,s=0(ii)
limk→∞skα+1EResψ,s=limk→∞skα+1EReskψ,s=0, 0<α≤1, k=1,2,3,⋯Furthermore, to evaluate the coefficient functionsfnψ, we can recursively solve the following scheme
(30)lims→∞skα+1EReskψ,s=0,0<α≤1,k=1,2,3,⋯.Finally, we implemented the Elzaki inverse toUkψ,s to achieve the kth approximate supportive solution ukψ,τ.


## 4. Numerical Results

Example 1.
Consider the time-fractional-order one-dimensional NS equation of the form(31)Dταuψ,τ=P+∂2u∂ψ2+1ψ∂u∂ψ,0<α≤1.
Subject to the initial condition(32)uψ,0=1−ψ2.Applying Elzaki transform to (31) and using the initial condition given in (32), we get
(33)Uψ,s=s21−ψ2+sαEτP+sαEτEτ−1∂2∂ψ2Uψ,s+sαEτEτ−11ψ∂∂ψUψ,s.Thekth truncated term series of (33) is
(34)Ukψ,s=s21−ψ2+∑n=1ksnα+2fnψ,and the kth Elzaki residual function is
(35)EτResk=Ukψ,s−s21−ψ2−sα+2P−sαEτEτ−1∂2∂ψ2Ukψ,s−sαEτEτ−11ψ∂∂xUkψ,s.Now, to determinefkψ,k=1,2,3,⋯, we substitute the kth truncated series (34) into the kth Elzaki residual function (35), multiply the resulting equation by skα+2, and then solve recursively the relation lims→∞skα+2Reskψ,s=0,k=1,2,3,⋯, for fkψ. The following are the first several components of the series fkψ,φ:
(36)f1ψ=p−4,f2ψ=0,f3ψ=0,⋮.Putting the values offnψn≥1 in (34), we get
(37)Uψ,s=s21−ψ2+sα+2f1ψ+s2α+2f2ψ+s3α+2f3ψ+⋯,Uψ,s=s21−ψ2+sα+2P−4+s2α+20+s3α+20+⋯,(38)Uψ,s=s21−ψ2+sα+2P−4.Using inverse Elzaki transform to (38), we get
(39)uψ,τ=1−ψ2+P−4ταΓα+2.Puttingα=1, we have
(40)uψ,τ=1−ψ2+P−4τ.In Figure1, 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 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(41)Dταuψ,τ=∂2u∂ψ2+1ψ∂u∂ψ,0<α≤1.Figure 1
Graph of exact and analytical results of Problem 1.Figure 2
The fractional order ofα=0.8 and 0.6 of Problem 1.Subject to the initial condition,(42)uψ,0=ψ.Applying Elzaki transform to (41) and using the initial condition given in (42), we get
(43)Uψ,s=s2ψ+sαEτEτ−1∂2∂ψ2Uψ,s+sαEτEτ−11ψ∂∂xUψ,s.Thekth truncated term series of (43) is
(44)Ukψ,s=s2ψ+∑n=1ksnα+2fnψ,and the kth Elzaki residual function is
(45)EτResk=Ukψ,s−s2ψ−sαEτEτ−1∂2∂ψ2Ukψ,s−sαEτEτ−11ψ∂∂xUkψ,s.Now, to determinefkψ,k=1,2,3,⋯, we substitute the kth truncated series (44) into the kth Elzaki residual function (45), multiply the resulting equation by skα+2, and then solve recursively the relation lims→∞skα+2Reskψ,s=0,k=1,2,3,⋯, for fkψ. The following are the first several components of the series fkψ,φ:
(46)f1ψ=1ψ,f2ψ=1ψ3,f3ψ=331ψ5,f4ψ=521ψ7,⋮.Putting the values offnψn≥1 in (44), we get
(47)Uψ,s=s2ψ+sα+2f1ψ+s2α+2f2ψ+s3α+2f3ψ+⋯,(48)Uψ,s=s2ψ+sα+2ψ+s2α+2ψ3+32s3α+2ψ5+⋯.Using inverse Elzaki transform to (48), we get
(49)uψ,τ=ψ+1ψταΓα+2+1ψ3τ2αΓ2α+2+32ψ5τ3αΓ3α+2+⋯.Puttingα=1, we have
(50)uψ,τ=ψ+1ψt+1ψ3τ22!+32ψ5τ33!+⋯,uψ,τ=ψ+∑n=1∞12×32×53×⋯×2n−32r2n−1τnn!.In Figure3, 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 integer-order. The same phenomenon of convergence of fractional-order solutions towards integral-order solutions is observed.Example 3.
Consider the fractional-order two-dimensional NS equation of the form(51)Dταu=ρ0∂2∂ψ2u+∂2∂φ2u−u∂∂ψu−v∂∂φu+g,Dταv=ρ0∂2∂ψ2v+∂2∂φ2v−u∂∂ψv−v∂∂φv−g,with initial condition
(52)uψ,φ,0=−sinψ+φ,vψ,φ,0=sinψ+φ.Figure 3
Graph of exact and analytical results of Problem 2.Figure 4
The fractional order ofα=0.8 and 0.6 of Problem 2.Applying Elzaki transform to (51) and using (52), we get
(53)Uψ,φ,s=s2−sinψ+φ+ρ0sαEτEτ−1∂2∂ψ2Ukψ,φ,s+∂2∂φ2Ukψ,φ,s−sαEτEτ−1Uψ,φ,s∂∂ψUψ,φ,s−sαEτEτ−1vψ,φ,s∂∂φUψ,φ,s+sαEτg,Vψ,φ,s=s2sinψ+φ+ρ0sαEτEτ−1∂2∂ψ2Vψ,φ,s+∂2∂φ2Vψ,φ,s−sαEτEτ−1Uψ,φ,s∂∂ψVψ,φ,s−sαEτEτ−1Vψ,φ,s∂∂φVψ,φ,s−sαEτg.Thekth truncated term series of (53) is
(54)Ukψ,φ,s=−sinψ+φs2+∑n=1ksnα+2fnψ,φ,Vkψ,φ,s=sinψ+φs2+∑n=1ksnα+2gnψ,φ,and the kth Elzaki residual function is
(55)EτReskψ,φ,s=Ukψ,φ,s−−sinψ+φs2−ρ0sαEτEτ−1∂2∂ψ2Ukψ,φ,s+∂2∂φ2Ukψ,φ,s+sαEτEτ−1Ukψ,φ,s∂∂ψUkψ,φ,s+sαEτEτ−1Vkψ,φ,s∂∂φUkψ,φ,s−gsα+2,EτReskψ,φ,s=Vkψ,φ,s−sinψ+φs2−ρ0sαEτEτ−1∂2∂ψ2Vkψ,φ,s+∂2∂φ2Vkψ,φ,s+sαEτEτ−1Ukψ,φ,s∂∂ψVkψ,φ,s+sαEτEτ−1Vkψ,φ,s∂∂φVkψ,φ,s+gsα+2.Now, to determinefkψ,φ and gkψ,φ,k=1,2,3,⋯, we substitute the kth truncated series (54) into the kth Elzaki residual function (55), multiply the resulting equation by skα+2, and then solve recursively the relation lims→∞skα+2Reskψ,φ,s=0,k=1,2,3,⋯, for fk and gk. The following are the first several components of the series fkψ,φ and gkψ,φ:
(56)f1ψ,φ=2ρ0sinψ+φ+g,g1ψ,φ=−2ρ0sinψ+φ−g,f2ψ,φ=−2ρ02sinψ+φ,g2ψ,φ=2ρ02sinψ+φ,f3ψ,φ=2ρ03sinψ+φ,g3ψ,φ=−2ρ03sinψ+φ,⋮.Putting the values offnψ,φ and gnψ,φn≥1 in (54), we have
(57)Uψ,φ,s=−sinψ+φs2+f1ψ,φsα+2+f2ψ,φs2α+2+f3ψ,φs3α+2+⋯,Vψ,φ,s=sinψ+φs2+g1ψ,φsα+2+g2ψ,φs2α+2+g3ψ,φs3α+2+⋯,Uψ,φ,s=−sinψ+φs2+2ρ0sinψ+φ+gsα+2−2ρ02sinψ+φs2α+2+2ρ03sinψ+φs3α+2+⋯,Vψ,φ,s=sinψ+φs2−2ρ0sinψ+φ−gsα+2+2ρ02sinψ+φs2α+2−2ρ03sinψ+φs3α+2+⋯,Uψ,φ,s=−sinψ+φs2−2ρ0sα+2+2ρ02s2α+2−2ρ03s3α+2+⋯+gsα+2,Vψ,φ,s=sinψ+φs2−2ρ0sα+2+2ρ02s2α+2−2ρ03s3α+2+⋯−gsα+2.Using inverse Elzaki transform, we get(58)uψ,φ,τ=−sinψ+φ1−2ρ0ταΓα+2+2ρ02τ2αΓ2α+2−2ρ03τ3αΓ3α+2+⋯+gταΓα+2,vψ,φ,τ=sinψ+φ1−2ρ0ταΓα+2+2ρ02τ2αΓ2α+2−2ρ03τ3αΓ3α+2+⋯−gταΓα+2.Puttingα=1, we get the solution in closed form
(59)uψ,φ,τ=−sinψ+φe−2ρ0τ+g,vψ,φ,τ=sinψ+φe−2ρ0τ−g.In Figures5 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(60)Dταu=ρ0∂2∂ψ2u+∂2∂φ2u−u∂∂ψu−v∂∂φu+g,Dταv=ρ0∂2∂ψ2v+∂2∂φ2v−u∂∂ψv−v∂∂φv−g,with initial condition
(61)uψ,φ,0=−eψ+φ,vψ,φ,0=eψ+φ.Figure 5
Graph of exact and analytical results of Problem 3.Figure 6
Graph of exact and analytical results of Problem 3.Figure 7
The fractional order ofα=0.8 and 0.6 of Problem 3.Figure 8
The fractional order ofα=0.8 and 0.6 of Problem 3.Applying Elzaki transform to (60) and using (61), we get
(62)Uψ,φ,s=−eψ+φs2+ρ0sαEτEτ−1∂2∂ψ2Ukψ,φ,s+∂2∂φ2Ukψ,φ,s−sαEτEτ−1Uψ,φ,s∂∂ψUψ,φ,s−sαEτEτ−1vψ,φ,s∂∂φUψ,φ,s+sαEτg,Vψ,φ,s=eψ+φs2+ρ0sαEτEτ−1∂2∂ψ2Vψ,φ,s+∂2∂φ2Vψ,φ,s−sαEτEτ−1Uψ,φ,s∂∂ψVψ,φ,s−sαEτEτ−1Vψ,φ,s∂∂φVψ,φ,s−sαEτg.Thekth truncated term series of (62) is
(63)Ukψ,φ,s=−eψ+φs2+∑n=1ksnα+2fnψ,φ,Vkψ,φ,s=eψ+φs2+∑n=1ksnα+2gnψ,φ,and the kth Elzaki residual function is
(64)EτReskψ,φ,s=Ukψ,φ,s−−eψ+φs2−ρ0sαEτEτ−1∂2∂ψ2Ukψ,φ,s+∂2∂φ2Ukψ,φ,s+sαEτEτ−1Ukψ,φ,s∂∂ψUkψ,φ,s+sαEτEτ−1Vkψ,φ,s∂∂φUkψ,φ,s−g1sα+2p,EτReskψ,φ,s=Vkψ,φ,s−eψ+φs2−ρ0sαEτEτ−1∂2∂ψ2Vkψ,φ,s+∂2∂φ2Vkψ,φ,s+sαEτEτ−1Ukψ,φ,s∂∂ψVkψ,φ,s+sαEτEτ−1Vkψ,φ,s∂∂φVkψ,φ,s+g1sα+2.Now, to determinefkψ,φ and gkψ,φ, k=1,2,3,⋯, we substitute the kth truncated series (63) into the kth Elzaki residual function (64), multiply the resulting equation by skα+2, and then solve recursively the relation lims→∞skα+2Reskψ,φ,s=0,k=1,2,3,⋯, for fk and gk. The following are the first several components of the series fkψ,φ and gkψ,φ:
(65)f1ψ,φ=−2ρ0eψ+φ+g,g1ψ,φ=2ρ0eψ+φ−g,f2ψ,φ=−2ρ02eψ+φ,g2ψ,φ=2ρ02eψ+φ,f3ψ,φ=−2ρ03eψ+φ,g3ψ,φ=2ρ03eψ+φ,⋮.Putting the values offnψ,φ and gnψ,φn≥1 in (63), we have
(66)Uψ,φ,s=−eψ+φs2+f1ψ,φsα+2+f2ψ,φs2α+2+f3ψ,φs3α+2+⋯,Vψ,φ,s=eψ+φs2+g1ψ,φsα+2+g2ψ,φs2α+2+g3ψ,φs3α+2+⋯,Uψ,φ,s=−eψ+φs2−2ρ0eψ+φ+gsα+2−2ρ02eψ+φs2α+2−2ρ03eψ+φs3α+2+⋯,Vψ,φ,s=eψ+φs2+2ρ0eψ+φ−gsα+2+2ρ02eψ+φs2α+2+2ρ03eψ+φs3α+2+⋯,Uψ,φ,s=−eψ+φs2+2ρ0sα+2+2ρ02s2α+2+2ρ03s3α+2+⋯+gsα+2,Vψ,φ,s=eψ+φs2+2ρ0sα+2+2ρ02s2α+2+2ρ03s3α+2+⋯−gsα+2.Using inverse Elzaki transform, we get(67)uψ,φ,τ=−eψ+φ1+2ρ0ταΓα+2+2ρ02τ2αΓ2α+2+2ρ03τ3αΓ3α+2+⋯+gταΓα+2,vψ,φ,τ=eψ+φ1+2ρ0ταΓα+2+2ρ02τ2αΓ2α+2+2ρ03τ3αΓ3α+2+⋯−gταΓα+2.Puttingα=1, we get the solution in closed form
(68)uψ,φ,τ=−eψ+φ+2ρ0τ+g,vψ,φ,τ=eψ+φ+2ρ0τ−g.In Figures9 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.Figure 9
Graph of exact and analytical results of Problem 4.Figure 10
Graph of exact and analytical results of Problem 4.Figure 11
The fractional order ofα=0.8 and 0.6 of Problem 4.Figure 12
The fractional order ofα=0.8 and 0.6 of Problem 4.


## 5. 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.



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*Source: 5588601-2021-04-20.xml*