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

Authors: Pongsakorn Sunthrayuth; Rasool Shah; A. M. Zidan; Shahbaz Khan; Jeevan Kafle
Journal: Journal of Function Spaces
 (2021)

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## Abstract

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.


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## Body

## 1. 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 [1–5]. 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 [9–17].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
(1)Dηϖu+u.∇u=−1ρ∇p+v∇2u,0<ϖ≤1,∇u=0,where the Caputo fractional derivative is Dηϖu, 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, one-dimensional motion. The fluid flow in the tube of fractional NSEs in cylindrical coordinates that govern is defined by
(2)Dηϖu+P+v∂2u∂ζ2+1ζ∂u∂ζ,0<ϖ≤1,with initial condition
(3)uζ,0=gζ,where P=−1/ρ∂p/∂z and gζ is a function that depends only on ζ.Consider the two-dimensional fractional NSE is given as(4)Dηϖu=ρ0∂2∂ζ2u+∂2∂φ2u−u∂∂ζu−v∂∂φu+g,Dηϖv=ρ0∂2∂ζ2v+∂2∂φ2v−u∂∂ζv−v∂∂φv−g,with initial conditions
(5)uζ,φ,η=fζ,φ,vζ,φ,η=gζ,φ,where u=uζ,φ,η, v=vζ,φ,η, where time, pressure and constant density define by η, p, and ρ, respectively. ζ and φ are the spatial components and fζ,φ and gζ,φ 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 closed-form. 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 [35–37], 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 two-dimensional, fractional-order model of Navier-stokes equations by using RPSTM.The fractional calculus fundamental definitions are defined in Section2, 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. Basic Definitions

### 2.1. Definition

The fractional-order Riemann-Liouville integral is defined as [39, 40].
(6)I0δhη=1Γδ∫0ηη−sδ−1hsds.


### 2.2. Definition

The fractional-order Caputo’s derivative offη is given as [39, 40].
(7)Dηδfη=Ij−δfj,j−1<δ<j,j∈ℕdjdηjhη,δ=j,j∈ℕ.


### 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 [41–43]
(8)A=νη:∃,ρ1,ρ2>0,νη<Meηρi,ifη∈0,∞.The Shehu transform which is described asS. for a function νη is defined as
(9)Sνη=Vs,u=∫0∞e−sηuνηdη,η>0,s>0.The Shehu transform of a functionνη is Vs,u; then, νη is called the inverse of Vs,u which is given as
(10)S−1Vs,u=νη,forη≥0,S−1isinverseShehutransform.


### 2.4. Definition

Shehu transform fornth derivatives is given as [41–43]
(11)Sνjη=sjujVs,u−∑k=0j−1suj−k−1νk0.


### 2.5. Definition

The fractional-order derivatives of Shehu transformation are defined as [41–43]
(12)Sνδη=sδuδvs,u−∑k=0j−1suδ−k−1νk0,0<β≤n.


### 2.6. Definition

A power series is given as(13)∑k=0∞Pkη−ζkϖ=Po+P1η−ζ+P2η−ζ2ϖ+⋯,where 0≤k−1<ϖ≤k and η≤ζ is called fractional-order power series about ζ, where Pk 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 thatfo has a fractional power series expression of the form at tau=psi(14)foη=∑m=0∞Pmη−ζmϖ.IfDmϖfoη, for m=0,1,2,3,⋯ are continuous on ζ,ζ+Ro, then the coefficient Pm can be defined as
(15)Pm=DmϖfoζΓ1+mϖ.


### 2.8. Definition

The power series expansion is given as(16)∑m=0∞Gmϕη−ζmϖ,is called to be multi fractional power series at η=ζ, where Gmϕ are the fractional power series coefficients.


### 2.9. Theorem

Let us suppose thatuoϕ,η has the fractional-order power define at η=ζ is given as
(17)uoϕ,η=∑m=0∞Gmxη−ζmϖ.IfDηmϖuoϕ,η,m=0,1,2,3,⋯ are the continuous on Io×ζ,ζ+Ro, then the coefficient Gm can be written as
(18)Gmx=Dηmϖuoϕ,ηΓ1+mϖ,where ζ∈Io,ζ≤η<ζ+Ro,Ro.So we can write the fractional power expansion ofuoϕ,η of the form
(19)μϕ,η=∑m=0∞Dηmϖμϕ,ηΓ1+mϖη−ζmϖ,which is the generalized form of the Taylor expansion.


### 2.10. Corollary

Let assume thatμϕ,φ,η has a power series multi fractional represent about η=ζ as
(20)uoϕ,φ,η=∑m=0∞Gmϕ,φη−ζmϖ.IfDηmϖμϕ,φ,η, m=0,1,2,3.. are continuous on I1×I2×ζ,ζ+Ro, then
(21)Gmζ,φ=Dηmϖμϕ,φ,ηΓ1+mϖ,where ϕ,φ∈I1×I2,ζ≤η<ζ+Ro.


## 2.1. Definition

The fractional-order Riemann-Liouville integral is defined as [39, 40].
(6)I0δhη=1Γδ∫0ηη−sδ−1hsds.


## 2.2. Definition

The fractional-order Caputo’s derivative offη is given as [39, 40].
(7)Dηδfη=Ij−δfj,j−1<δ<j,j∈ℕdjdηjhη,δ=j,j∈ℕ.


## 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 [41–43]
(8)A=νη:∃,ρ1,ρ2>0,νη<Meηρi,ifη∈0,∞.The Shehu transform which is described asS. for a function νη is defined as
(9)Sνη=Vs,u=∫0∞e−sηuνηdη,η>0,s>0.The Shehu transform of a functionνη is Vs,u; then, νη is called the inverse of Vs,u which is given as
(10)S−1Vs,u=νη,forη≥0,S−1isinverseShehutransform.


## 2.4. Definition

Shehu transform fornth derivatives is given as [41–43]
(11)Sνjη=sjujVs,u−∑k=0j−1suj−k−1νk0.


## 2.5. Definition

The fractional-order derivatives of Shehu transformation are defined as [41–43]
(12)Sνδη=sδuδvs,u−∑k=0j−1suδ−k−1νk0,0<β≤n.


## 2.6. Definition

A power series is given as(13)∑k=0∞Pkη−ζkϖ=Po+P1η−ζ+P2η−ζ2ϖ+⋯,where 0≤k−1<ϖ≤k and η≤ζ is called fractional-order power series about ζ, where Pk 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 thatfo has a fractional power series expression of the form at tau=psi(14)foη=∑m=0∞Pmη−ζmϖ.IfDmϖfoη, for m=0,1,2,3,⋯ are continuous on ζ,ζ+Ro, then the coefficient Pm can be defined as
(15)Pm=DmϖfoζΓ1+mϖ.


## 2.8. Definition

The power series expansion is given as(16)∑m=0∞Gmϕη−ζmϖ,is called to be multi fractional power series at η=ζ, where Gmϕ are the fractional power series coefficients.


## 2.9. Theorem

Let us suppose thatuoϕ,η has the fractional-order power define at η=ζ is given as
(17)uoϕ,η=∑m=0∞Gmxη−ζmϖ.IfDηmϖuoϕ,η,m=0,1,2,3,⋯ are the continuous on Io×ζ,ζ+Ro, then the coefficient Gm can be written as
(18)Gmx=Dηmϖuoϕ,ηΓ1+mϖ,where ζ∈Io,ζ≤η<ζ+Ro,Ro.So we can write the fractional power expansion ofuoϕ,η of the form
(19)μϕ,η=∑m=0∞Dηmϖμϕ,ηΓ1+mϖη−ζmϖ,which is the generalized form of the Taylor expansion.


## 2.10. Corollary

Let assume thatμϕ,φ,η has a power series multi fractional represent about η=ζ as
(20)uoϕ,φ,η=∑m=0∞Gmϕ,φη−ζmϖ.IfDηmϖμϕ,φ,η, m=0,1,2,3.. are continuous on I1×I2×ζ,ζ+Ro, then
(21)Gmζ,φ=Dηmϖμϕ,φ,ηΓ1+mϖ,where ϕ,φ∈I1×I2,ζ≤η<ζ+Ro.


## 3. The General Methodologies of RPSTM

In this section, we will go through the steps that RPSTM takes to solve the FPDEs.(22)Dηϖuζ,η=aDζ2uζ,η+buζ,η−cuqζ,η,with initial condition
(23)uζ,0=f0ζ,we use the Shehu transformation to (22)
(24)SDηϖuζ,η=aSDζ2uζ,η+bSuζ,η−cSuqζ,η.By the fact thatSDηϖuζ,η=sϖ/νϖSuζ,η−ν1−ϖ/s1−ϖux,o and apply the initial condition (23), we can write (10) as
(25)Uζ,s=νsf0ζ+νϖsϖaDx2Uζ,s+bνϖsϖUζ,s−cνϖsϖS−1SUζ,sq,where Uζ,s=Suζ,η.Second, we expressed the transformation functionUζ,s as the following equation
(26)Uζ,s=∑n=0∞νsnϖ+1fnx.Thekth-truncated series of (26), we write as
(27)Uζ,s=∑n=0∞νsnϖ+1fnx=νsf0x+∑n=1∞νsnϖ+1fnx.As stated in [24], the definition of Shehu residual function to (26) is
(28)SResζ,s=Uζ,s−f0xνs−aνϖsϖDx2Uζ,s−bνϖsϖUζ,s+cνϖsϖS−1SUζ,sq,and the kth-Shehu residual function of (28) is
(29)SResζ,s=Ukζ,s−f0xνs−aνϖsϖDx2Ukζ,s−bνϖsϖUkζ,s+cνϖsϖSS−1Ukζ,sq,Third, we explain some properties of the residual power series method(i)
SResζ,s=0 and limk⟶∞SReskζ,s=SResζ,s for each s>0(30)limk⟶∞SResζ,s=0⟹limk⟶∞sSReskζ,s=0(ii)
limk⟶∞skϖ+1SResζ,s=limk⟶∞skϖ+1SReskζ,s=0, 0<ϖ≤1, k=1,2,3⋯Furthermore, to calculate the coefficient functionfnζ, the following system can be solved recursively.
(31)lims⟶∞skϖ+1SReskζ,s=0,0<ϖ≤1,k=1,2,3,⋯.Finally, we used the Shehu inverse toUkζ,s, to obtain the kth-approximated supportive result ukζ,η.


## 4. Numerical Result

### 4.1. Example

Consider the one dimensional fractional NS equation(32)Dηϖuζ,η=∂2u∂ζ2+1ζ∂u∂ζ,0<ϖ≤1.Subject to the initial condition(33)uζ,0=ζ.Applying Shehu transform to Eq. (32) and using the initial condition given in Eq. (33), we get
(34)Uζ,s=νsζ+νϖsϖSηSη−1∂2∂ζ2Uζ,s+νϖsϖSηSη−11ζ∂∂xUζ,s.Thekth truncated term series of Eq. (34) is
(35)Ukζ,s=νsζ+∑n=1kνsnϖ+1fnζ,and the kth Shehu residual function is
(36)SηResk=Ukζ,s−νsζ−νϖsϖSηSη−1∂2∂ζ2Ukζ,s−νϖsϖSηSη−11ζ∂∂xUkζ,s.Now, to determinefkζ, k=1,2,3,⋯, we put the kth-truncated series (35) into the kth-Shehu residual function (36), multiply the solution equation by skϖ+2 and, then, solve recursively the relation lims⟶∞ν/snϖ+1Reskζ,s=0, k=1,2,3,⋯ for fkζ. The first few component of the series fkζ,φ(37)f1ζ=1ζ,f2ζ=1ζ3,f3ζ=331ζ5,f4ζ=521ζ7,⋮Putting the values offnζn≥1 in Eq. (35), we get
(38)Uζ,s=νs1−ζ2+νϖsϖf1ζ+νϖ+1sϖ+1f2ζ+νϖ+2sϖ+2f3ζ+⋯,(39)Uζ,s=νsζ+νϖsϖ1ζ+νϖ+1sϖ+11ζ3+νϖ+2sϖ+232ζ5+⋯.Applying inverse Shehu transform to Eq. (39), we get
(40)uζ,η=ζ+1ζηϖΓϖ+2+1ζ3η2ϖΓ2ϖ+2+32ζ5η3ϖΓ3ϖ+2+⋯.The above equation putϖ=1, we get
(41)uζ,η=ζ+1ζη+1ζ3η22!+32ζ5η33!+⋯,uζ,η=ζ+∑n=1∞12×32×53×⋯×2n−32r2n−1ηnn!,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.Figure 1
Graphs of actual and RPSTM solutions of Example 1.Figure 2
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that ϖ=0.6 of Example 1.Figure 3
The first graph of fractional-order ofϖ=0.4 and second graph of the different fractional-order graph of Example 1.


### 4.2. Example

Consider the one-dimensional fractional NS equation(42)Dηϖuζ,η=P+∂2u∂ζ2+1ζ∂u∂ζ,0<ϖ≤1.Subject to the initial condition(43)uζ,0=1−ζ2.Applying Shehu transformation to Eq. (42) and applying the initial condition given in Eq. (43), we get
(44)Uζ,s=νs1−ζ2+νϖsϖSηP+νϖsϖSηSη−1∂2∂ζ2Uζ,s+νϖsϖSηSη−11ζ∂∂ζUζ,s.Thekth truncated term series of Eq. (44) is
(45)Ukζ,s=νs1−ζ2+∑n=1kνsnϖ+1fnζ,and the kth Shehu residual function is
(46)SηResk=Ukζ,s−νs1−ζ2−νϖsϖP−νϖsϖSηSη−1∂2∂ζ2Ukζ,s−νϖsϖSηSη−11ζ∂∂xUkζ,s.Now, to determinefkζ, k=1,2,3,⋯, we substitute the kth-truncated series (45) into the kth-Shehu residual function (46), multiply the resulting equation by skϖ+2 and, then, solve recursively the relation lims⟶∞ν/snϖ+1Reskζ,s=0, k=1,2,3,⋯ for fkζ. The first several component of the series fkζ,φ(47)f1ζ=p−4,f2ζ=0,f3ζ=0.⋮Substituting the value offnζn≥1 in Eq. (45), we obtain as
(48)Uζ,s=νs1−ζ2+νϖsϖf1ζ+νϖ+1sϖ+1f2ζ+νϖ+2sϖ+2f3ζ+⋯,Uζ,s=νs1−ζ2+νϖsϖP−4+νϖ+1sϖ+10+νϖ+2sϖ+20+⋯,(49)Uζ,s=νs1−ζ2+νϖsϖP−4.Using inverse Shehu transformation to Eq. (49), we get
(50)uζ,η=1−ζ2+P−4ηϖΓϖ+2.The above equation putϖ=1, we get
(51)uζ,η=1−ζ2+P−4η.Figure4 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.Figure 4
Graphs of actual and RPSTM solutions of Example 2.Figure 5
The different fractional-order ofϖ of Example 2.


### 4.3. Example

Consider the two-dimensional fractional NS equation(52)Dηϖu=ρ0∂2∂ζ2u+∂2∂φ2u−u∂∂ζu−v∂∂φu+g,Dηϖv=ρ0∂2∂ζ2v+∂2∂φ2v−u∂∂ζv−v∂∂φv−g,with initial condition
(53)uζ,φ,0=−sinζ+φ,vζ,φ,0=sinζ+φ.Applying Shehu transform to Eq. (52) and using the Eq. (53), we get
(54)Uζ,φ,s=νs−sinζ+φ+ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζUζ,φ,s−νϖsϖSηSη−1vζ,φ,s∂∂φUζ,φ,s+νϖsϖSηg,Vζ,φ,s=νssinζ+φ+ρ0νϖsϖSηSη−1∂2∂ζ2Vζ,φ,s+∂2∂φ2Vζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζVζ,φ,s−νϖsϖSηSη−1Vζ,φ,s∂∂φVζ,φ,s−νϖsϖSηg.Thekth truncated term series of Eq. (54) is
(55)Ukζ,φ,s=−sinζ+φνs+∑n=1kνsnϖ+1fnζ,φ,Vkζ,φ,s=sinζ+φνs+∑n=1kνsnϖ+1gnζ,φ,and the kth Shehu residual function is
(56)SηReskζ,φ,s=Ukζ,φ,s−−sinζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζUkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φUkζ,φ,s−νϖsϖg,SηReskζ,φ,s=Vkζ,φ,s−sinζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Vkζ,φ,s+∂2∂φ2Vkζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζVkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φVkζ,φ,s+νϖsϖg.Now, to determinefkζ,φ and gkζ,φ, k=1,2,3,⋯, we substitute the kth-truncated series (55) into the kth-Shehu residual function (56), multiply the resulting equation by skϖ+2, and then solve recursively the relation lims⟶∞skϖ+2Reskζ,φ,s=0, k=1,2,3,⋯ for fk and gk. The first few component of the series fkζ,φ and gkζ,φ(57)f1ζ,φ=2ρ0sinζ+φ+g,g1ζ,φ=−2ρ0sinζ+φ−g,f2ζ,φ=−2ρ02sinζ+φ,g2ζ,φ=2ρ02sinζ+φ,f3ζ,φ=2ρ03sinζ+φ,g3ζ,φ=−2ρ03sinζ+φ.⋮Putting the values offnζ,φ and gnζ,φ, n≥1 in Eq. (55), we get
(58)Uζ,φ,s=−sinζ+φνs+νϖsϖf1ζ,φ+νϖ+1sϖ+1f2ζ,φ+νϖ+2sϖ+2f3ζ,φ+⋯,Vζ,φ,s=sinζ+φνs+νϖsϖg1ζ,φ+νϖ+1sϖ+1g2ζ,φ+νϖ+2sϖ+2g3ζ,φ+⋯,Uζ,φ,s=−sinζ+φνs+2ρ0sinζ+φ+gνϖsϖ−2ρ02sinζ+φνϖ+1sϖ+1+2ρ03sinζ+φνϖ+2sϖ+2+⋯,Vζ,φ,s=sinζ+φνs−2ρ0sinζ+φ−gνϖsϖ+2ρ02sinζ+φνϖ+1sϖ+1−2ρ03sinζ+φνϖ+2sϖ+2+⋯,Uζ,φ,s=−sinζ+φνs−2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1−2ρ03νϖ+2sϖ+2+⋯+gνϖsϖ,Vζ,φ,s=sinζ+φνs−2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1−2ρ03νϖ+2sϖ+2+⋯−gνϖsϖ.Using inverse Shehu transformation, we get(59)uζ,φ,η=−sinζ+φ1−2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2−2ρ03η3ϖΓ3ϖ+2+⋯+gηϖΓϖ+2,vζ,φ,η=sinζ+φ1−2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2−2ρ03η3ϖΓ3ϖ+2+⋯−gηϖΓϖ+2.The exact solution is given as(60)uζ,φ,η=−sinζ+φe−2ρ0η+g,vζ,φ,η=sinζ+φe−2ρ0η−g.Figure6 shows that the actual and analytical solution graphs at ϖ=1 with respect to uζ,φ,η. 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, respectively. Similarly, Figure 8 shows that the actual and analytical solution graphs at ϖ=1 with respect to vζ,φ,η. 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.Figure 6
Graphs of actual and approximate solutions of Example 3.Figure 7
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 3.Figure 8
Graphs of actual and approximate solutions of Example 3.Figure 9
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 3.


### 4.4. Example

Consider the two-dimensional fractional NS equation(61)Dηϖu=ρ0∂2∂ζ2u+∂2∂φ2u−u∂∂ζu−v∂∂φu+g,Dηϖv=ρ0∂2∂ζ2v+∂2∂φ2v−u∂∂ζv−v∂∂φv−g,with initial condition
(62)uζ,φ,0=−eζ+φ,vζ,φ,0=eζ+φ.Applying Shehu transform to Eq. (61) and using the Eq. (62), we obtain as
(63)Uζ,φ,s=−eζ+φνs+ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζUζ,φ,s−νϖsϖSηSη−1vζ,φ,s∂∂φUζ,φ,s+νϖsϖSηg,Vζ,φ,s=eζ+φνs+ρ0νϖsϖSηSη−1∂2∂ζ2Vζ,φ,s+∂2∂φ2Vζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζVζ,φ,s−νϖsϖSηSη−1Vζ,φ,s∂∂φVζ,φ,s−νϖsϖSηg.Thekth truncated term series of Eq. (63) is
(64)Ukζ,φ,s=−eζ+φνs+∑n=1kνsnϖ+1fnζ,φ,Vkζ,φ,s=eζ+φνs+∑n=1kνsnϖ+1gnζ,φ,and the kth Shehu residual function is
(65)SηReskζ,φ,s=Ukζ,φ,s−−eζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζUkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φUkζ,φ,s−g1νϖ/sϖ,SηReskζ,φ,s=Vkζ,φ,s−eζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Vkζ,φ,s+∂2∂φ2Vkζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζVkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φVkζ,φ,s+g1νϖ/sϖ.Now, to determinefkζ,φ and gkζ,φ, k=1,2,3,⋯, we substitute the kth-truncated series (64) into the kth-Shehu residual function (65), multiply the resulting equation by skϖ+2, and then solve recursively the relation lims⟶∞ν/snϖ+1Reskζ,φ,s=0, k=1,2,3,⋯ for fk and gk. The first few component of the series fkζ,φ and gkζ,φ(66)f1ζ,φ=−2ρ0eζ+φ+g,g1ζ,φ=2ρ0eζ+φ−g,f2ζ,φ=−2ρ02eζ+φ,g2ζ,φ=2ρ02eζ+φ,f3ζ,φ=−2ρ03eζ+φ,g3ζ,φ=2ρ03eζ+φ.⋮Putting the values offnζ,φ and gnζ,φ, n≥1 in Eq. (64), we get
(67)Uζ,φ,s=−eζ+φνs+f1ζ,φνϖsϖ+f2ζ,φνϖ+1sϖ+1+f3ζ,φνϖ+2sϖ+2+⋯,Vζ,φ,s=eζ+φνs+g1ζ,φνϖsϖ+g2ζ,φνϖ+1sϖ+1+g3ζ,φνϖ+2sϖ+2+⋯,Uζ,φ,s=−eζ+φνs−2ρ0eζ+φ+gνϖsϖ−2ρ02eζ+φνϖ+1sϖ+1−2ρ03eζ+φνϖ+2sϖ+2+⋯,Vζ,φ,s=eζ+φνs+2ρ0eζ+φ−gνϖsϖ+2ρ02eζ+φνϖ+1sϖ+1+2ρ03eζ+φνϖ+2sϖ+2+⋯,Uζ,φ,s=−eζ+φνs+2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1+2ρ03νϖ+2sϖ+2+⋯+gνϖsϖ,Vζ,φ,s=eζ+φνs+2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1+2ρ03νϖ+2sϖ+2+⋯−gνϖsϖ.Using inverse Shehu transformation, we get(68)uζ,φ,η=−eζ+φ1+2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2+2ρ03η3ϖΓ3ϖ+2+⋯+gηϖΓϖ+2,vζ,φ,η=eζ+φ1+2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2+2ρ03η3ϖΓ3ϖ+2+⋯−gηϖΓϖ+2.The exact result is given as(69)uζ,φ,η=−eζ+φ+2ρ0η+g,vζ,φ,η=eζ+φ+2ρ0η−g.Figure10 shows that the actual and analytical solution graphs at ϖ=1 with respect to uζ,φ,η. 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 vζ,φ,η. The field of velocity behavior of the classical Navier-Stokes equations is show in Figure 13, and the fractional-order behavior of Navier-Stokes equations with ϖ=0.8 and 0.6 is shown in Figures 14 and 15, respectively.Figure 10
Graphs of actual and approximate solutions of Problem 4.Figure 11
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 4.Figure 12
The first graph of fractional-order ofϖ=0.4 and second graph of the different fractional-order graph of Example 4.Figure 13
Graphs of actual and approximate solutions of Example 4.Figure 14
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 4.Figure 15
The first graph of fractional-order ofϖ=0.4 and second graph of the different fractional-order graph of Example 4.


## 4.1. Example

Consider the one dimensional fractional NS equation(32)Dηϖuζ,η=∂2u∂ζ2+1ζ∂u∂ζ,0<ϖ≤1.Subject to the initial condition(33)uζ,0=ζ.Applying Shehu transform to Eq. (32) and using the initial condition given in Eq. (33), we get
(34)Uζ,s=νsζ+νϖsϖSηSη−1∂2∂ζ2Uζ,s+νϖsϖSηSη−11ζ∂∂xUζ,s.Thekth truncated term series of Eq. (34) is
(35)Ukζ,s=νsζ+∑n=1kνsnϖ+1fnζ,and the kth Shehu residual function is
(36)SηResk=Ukζ,s−νsζ−νϖsϖSηSη−1∂2∂ζ2Ukζ,s−νϖsϖSηSη−11ζ∂∂xUkζ,s.Now, to determinefkζ, k=1,2,3,⋯, we put the kth-truncated series (35) into the kth-Shehu residual function (36), multiply the solution equation by skϖ+2 and, then, solve recursively the relation lims⟶∞ν/snϖ+1Reskζ,s=0, k=1,2,3,⋯ for fkζ. The first few component of the series fkζ,φ(37)f1ζ=1ζ,f2ζ=1ζ3,f3ζ=331ζ5,f4ζ=521ζ7,⋮Putting the values offnζn≥1 in Eq. (35), we get
(38)Uζ,s=νs1−ζ2+νϖsϖf1ζ+νϖ+1sϖ+1f2ζ+νϖ+2sϖ+2f3ζ+⋯,(39)Uζ,s=νsζ+νϖsϖ1ζ+νϖ+1sϖ+11ζ3+νϖ+2sϖ+232ζ5+⋯.Applying inverse Shehu transform to Eq. (39), we get
(40)uζ,η=ζ+1ζηϖΓϖ+2+1ζ3η2ϖΓ2ϖ+2+32ζ5η3ϖΓ3ϖ+2+⋯.The above equation putϖ=1, we get
(41)uζ,η=ζ+1ζη+1ζ3η22!+32ζ5η33!+⋯,uζ,η=ζ+∑n=1∞12×32×53×⋯×2n−32r2n−1ηnn!,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.Figure 1
Graphs of actual and RPSTM solutions of Example 1.Figure 2
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that ϖ=0.6 of Example 1.Figure 3
The first graph of fractional-order ofϖ=0.4 and second graph of the different fractional-order graph of Example 1.


## 4.2. Example

Consider the one-dimensional fractional NS equation(42)Dηϖuζ,η=P+∂2u∂ζ2+1ζ∂u∂ζ,0<ϖ≤1.Subject to the initial condition(43)uζ,0=1−ζ2.Applying Shehu transformation to Eq. (42) and applying the initial condition given in Eq. (43), we get
(44)Uζ,s=νs1−ζ2+νϖsϖSηP+νϖsϖSηSη−1∂2∂ζ2Uζ,s+νϖsϖSηSη−11ζ∂∂ζUζ,s.Thekth truncated term series of Eq. (44) is
(45)Ukζ,s=νs1−ζ2+∑n=1kνsnϖ+1fnζ,and the kth Shehu residual function is
(46)SηResk=Ukζ,s−νs1−ζ2−νϖsϖP−νϖsϖSηSη−1∂2∂ζ2Ukζ,s−νϖsϖSηSη−11ζ∂∂xUkζ,s.Now, to determinefkζ, k=1,2,3,⋯, we substitute the kth-truncated series (45) into the kth-Shehu residual function (46), multiply the resulting equation by skϖ+2 and, then, solve recursively the relation lims⟶∞ν/snϖ+1Reskζ,s=0, k=1,2,3,⋯ for fkζ. The first several component of the series fkζ,φ(47)f1ζ=p−4,f2ζ=0,f3ζ=0.⋮Substituting the value offnζn≥1 in Eq. (45), we obtain as
(48)Uζ,s=νs1−ζ2+νϖsϖf1ζ+νϖ+1sϖ+1f2ζ+νϖ+2sϖ+2f3ζ+⋯,Uζ,s=νs1−ζ2+νϖsϖP−4+νϖ+1sϖ+10+νϖ+2sϖ+20+⋯,(49)Uζ,s=νs1−ζ2+νϖsϖP−4.Using inverse Shehu transformation to Eq. (49), we get
(50)uζ,η=1−ζ2+P−4ηϖΓϖ+2.The above equation putϖ=1, we get
(51)uζ,η=1−ζ2+P−4η.Figure4 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.Figure 4
Graphs of actual and RPSTM solutions of Example 2.Figure 5
The different fractional-order ofϖ of Example 2.


## 4.3. Example

Consider the two-dimensional fractional NS equation(52)Dηϖu=ρ0∂2∂ζ2u+∂2∂φ2u−u∂∂ζu−v∂∂φu+g,Dηϖv=ρ0∂2∂ζ2v+∂2∂φ2v−u∂∂ζv−v∂∂φv−g,with initial condition
(53)uζ,φ,0=−sinζ+φ,vζ,φ,0=sinζ+φ.Applying Shehu transform to Eq. (52) and using the Eq. (53), we get
(54)Uζ,φ,s=νs−sinζ+φ+ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζUζ,φ,s−νϖsϖSηSη−1vζ,φ,s∂∂φUζ,φ,s+νϖsϖSηg,Vζ,φ,s=νssinζ+φ+ρ0νϖsϖSηSη−1∂2∂ζ2Vζ,φ,s+∂2∂φ2Vζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζVζ,φ,s−νϖsϖSηSη−1Vζ,φ,s∂∂φVζ,φ,s−νϖsϖSηg.Thekth truncated term series of Eq. (54) is
(55)Ukζ,φ,s=−sinζ+φνs+∑n=1kνsnϖ+1fnζ,φ,Vkζ,φ,s=sinζ+φνs+∑n=1kνsnϖ+1gnζ,φ,and the kth Shehu residual function is
(56)SηReskζ,φ,s=Ukζ,φ,s−−sinζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζUkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φUkζ,φ,s−νϖsϖg,SηReskζ,φ,s=Vkζ,φ,s−sinζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Vkζ,φ,s+∂2∂φ2Vkζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζVkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φVkζ,φ,s+νϖsϖg.Now, to determinefkζ,φ and gkζ,φ, k=1,2,3,⋯, we substitute the kth-truncated series (55) into the kth-Shehu residual function (56), multiply the resulting equation by skϖ+2, and then solve recursively the relation lims⟶∞skϖ+2Reskζ,φ,s=0, k=1,2,3,⋯ for fk and gk. The first few component of the series fkζ,φ and gkζ,φ(57)f1ζ,φ=2ρ0sinζ+φ+g,g1ζ,φ=−2ρ0sinζ+φ−g,f2ζ,φ=−2ρ02sinζ+φ,g2ζ,φ=2ρ02sinζ+φ,f3ζ,φ=2ρ03sinζ+φ,g3ζ,φ=−2ρ03sinζ+φ.⋮Putting the values offnζ,φ and gnζ,φ, n≥1 in Eq. (55), we get
(58)Uζ,φ,s=−sinζ+φνs+νϖsϖf1ζ,φ+νϖ+1sϖ+1f2ζ,φ+νϖ+2sϖ+2f3ζ,φ+⋯,Vζ,φ,s=sinζ+φνs+νϖsϖg1ζ,φ+νϖ+1sϖ+1g2ζ,φ+νϖ+2sϖ+2g3ζ,φ+⋯,Uζ,φ,s=−sinζ+φνs+2ρ0sinζ+φ+gνϖsϖ−2ρ02sinζ+φνϖ+1sϖ+1+2ρ03sinζ+φνϖ+2sϖ+2+⋯,Vζ,φ,s=sinζ+φνs−2ρ0sinζ+φ−gνϖsϖ+2ρ02sinζ+φνϖ+1sϖ+1−2ρ03sinζ+φνϖ+2sϖ+2+⋯,Uζ,φ,s=−sinζ+φνs−2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1−2ρ03νϖ+2sϖ+2+⋯+gνϖsϖ,Vζ,φ,s=sinζ+φνs−2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1−2ρ03νϖ+2sϖ+2+⋯−gνϖsϖ.Using inverse Shehu transformation, we get(59)uζ,φ,η=−sinζ+φ1−2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2−2ρ03η3ϖΓ3ϖ+2+⋯+gηϖΓϖ+2,vζ,φ,η=sinζ+φ1−2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2−2ρ03η3ϖΓ3ϖ+2+⋯−gηϖΓϖ+2.The exact solution is given as(60)uζ,φ,η=−sinζ+φe−2ρ0η+g,vζ,φ,η=sinζ+φe−2ρ0η−g.Figure6 shows that the actual and analytical solution graphs at ϖ=1 with respect to uζ,φ,η. 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, respectively. Similarly, Figure 8 shows that the actual and analytical solution graphs at ϖ=1 with respect to vζ,φ,η. 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.Figure 6
Graphs of actual and approximate solutions of Example 3.Figure 7
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 3.Figure 8
Graphs of actual and approximate solutions of Example 3.Figure 9
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 3.


## 4.4. Example

Consider the two-dimensional fractional NS equation(61)Dηϖu=ρ0∂2∂ζ2u+∂2∂φ2u−u∂∂ζu−v∂∂φu+g,Dηϖv=ρ0∂2∂ζ2v+∂2∂φ2v−u∂∂ζv−v∂∂φv−g,with initial condition
(62)uζ,φ,0=−eζ+φ,vζ,φ,0=eζ+φ.Applying Shehu transform to Eq. (61) and using the Eq. (62), we obtain as
(63)Uζ,φ,s=−eζ+φνs+ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζUζ,φ,s−νϖsϖSηSη−1vζ,φ,s∂∂φUζ,φ,s+νϖsϖSηg,Vζ,φ,s=eζ+φνs+ρ0νϖsϖSηSη−1∂2∂ζ2Vζ,φ,s+∂2∂φ2Vζ,φ,s−νϖsϖSηSη−1Uζ,φ,s∂∂ζVζ,φ,s−νϖsϖSηSη−1Vζ,φ,s∂∂φVζ,φ,s−νϖsϖSηg.Thekth truncated term series of Eq. (63) is
(64)Ukζ,φ,s=−eζ+φνs+∑n=1kνsnϖ+1fnζ,φ,Vkζ,φ,s=eζ+φνs+∑n=1kνsnϖ+1gnζ,φ,and the kth Shehu residual function is
(65)SηReskζ,φ,s=Ukζ,φ,s−−eζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Ukζ,φ,s+∂2∂φ2Ukζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζUkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φUkζ,φ,s−g1νϖ/sϖ,SηReskζ,φ,s=Vkζ,φ,s−eζ+φνs−ρ0νϖsϖSηSη−1∂2∂ζ2Vkζ,φ,s+∂2∂φ2Vkζ,φ,s+νϖsϖSηSη−1Ukζ,φ,s∂∂ζVkζ,φ,s+νϖsϖSηSη−1Vkζ,φ,s∂∂φVkζ,φ,s+g1νϖ/sϖ.Now, to determinefkζ,φ and gkζ,φ, k=1,2,3,⋯, we substitute the kth-truncated series (64) into the kth-Shehu residual function (65), multiply the resulting equation by skϖ+2, and then solve recursively the relation lims⟶∞ν/snϖ+1Reskζ,φ,s=0, k=1,2,3,⋯ for fk and gk. The first few component of the series fkζ,φ and gkζ,φ(66)f1ζ,φ=−2ρ0eζ+φ+g,g1ζ,φ=2ρ0eζ+φ−g,f2ζ,φ=−2ρ02eζ+φ,g2ζ,φ=2ρ02eζ+φ,f3ζ,φ=−2ρ03eζ+φ,g3ζ,φ=2ρ03eζ+φ.⋮Putting the values offnζ,φ and gnζ,φ, n≥1 in Eq. (64), we get
(67)Uζ,φ,s=−eζ+φνs+f1ζ,φνϖsϖ+f2ζ,φνϖ+1sϖ+1+f3ζ,φνϖ+2sϖ+2+⋯,Vζ,φ,s=eζ+φνs+g1ζ,φνϖsϖ+g2ζ,φνϖ+1sϖ+1+g3ζ,φνϖ+2sϖ+2+⋯,Uζ,φ,s=−eζ+φνs−2ρ0eζ+φ+gνϖsϖ−2ρ02eζ+φνϖ+1sϖ+1−2ρ03eζ+φνϖ+2sϖ+2+⋯,Vζ,φ,s=eζ+φνs+2ρ0eζ+φ−gνϖsϖ+2ρ02eζ+φνϖ+1sϖ+1+2ρ03eζ+φνϖ+2sϖ+2+⋯,Uζ,φ,s=−eζ+φνs+2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1+2ρ03νϖ+2sϖ+2+⋯+gνϖsϖ,Vζ,φ,s=eζ+φνs+2ρ0νϖsϖ+2ρ02νϖ+1sϖ+1+2ρ03νϖ+2sϖ+2+⋯−gνϖsϖ.Using inverse Shehu transformation, we get(68)uζ,φ,η=−eζ+φ1+2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2+2ρ03η3ϖΓ3ϖ+2+⋯+gηϖΓϖ+2,vζ,φ,η=eζ+φ1+2ρ0ηϖΓϖ+2+2ρ02η2ϖΓ2ϖ+2+2ρ03η3ϖΓ3ϖ+2+⋯−gηϖΓϖ+2.The exact result is given as(69)uζ,φ,η=−eζ+φ+2ρ0η+g,vζ,φ,η=eζ+φ+2ρ0η−g.Figure10 shows that the actual and analytical solution graphs at ϖ=1 with respect to uζ,φ,η. 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 vζ,φ,η. The field of velocity behavior of the classical Navier-Stokes equations is show in Figure 13, and the fractional-order behavior of Navier-Stokes equations with ϖ=0.8 and 0.6 is shown in Figures 14 and 15, respectively.Figure 10
Graphs of actual and approximate solutions of Problem 4.Figure 11
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 4.Figure 12
The first graph of fractional-order ofϖ=0.4 and second graph of the different fractional-order graph of Example 4.Figure 13
Graphs of actual and approximate solutions of Example 4.Figure 14
The first graph shows that the fractional-order ofϖ=0.8 and the second graph shows that the fractional-order ϖ=0.6 of Example 4.Figure 15
The first graph of fractional-order ofϖ=0.4 and second graph of the different fractional-order graph of Example 4.


## 5. 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.



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