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Analysis and Numerical Computations of the Multi-Dimensional, Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation

Yuming Chu1, Saima Rashid2, Khadija Tul Kubra2, Mustafa Inc3,4,*, Zakia Hammouch5, M. S. Osman6,*

1 Department of Mathematics, Huzhou University, Huzhou, China
2 Department of Mathematics, Government College University, Faisalabad, Pakistan
3 Department of Mathematics, Firat University, Elazig, Turkey
4 Medical Research, China Medical University, Taichung, Taiwan
5 Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Vietnam
6 Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

* Corresponding Authors: Mustafa Inc. Email: email; M. S. Osman. Email: email,email

(This article belongs to this Special Issue: Applications of Fractional Operators in Modeling Real-world Problems: Theory, Computation, and Applications)

Computer Modeling in Engineering & Sciences 2023, 136(3), 3025-3060. https://doi.org/10.32604/cmes.2023.025470

Abstract

The analytical solution of the multi-dimensional, time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transform decomposition method is presented in this article. The aforesaid model is analyzed by employing Caputo fractional derivative. We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods, respectively. The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems. The exact and estimated solutions are delineated via numerical simulation. The proposed analysis indicates that the projected configuration is extremely meticulous, highly efficient, and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.

Keywords


1  Introduction

Physical and technical workflows are recognized by fractional calculus (FC) and are succinctly explained by fractional differential equations (FDEs) [14]. Admittedly, classical simulation results of integer-order derivatives, which include empirical systems, do not underlie well in numerous situations [59]. FC has contributed substantially to quantum physics, nuclear magnetic resonance, image recognition, circuit theory, transport phenomena, chaos, and epidemiology. Diverse aspects of FC have been researched by Kilbas et al. [1014].

Fractional differential equations (FDEs) are viewed as the most vital and effective mechanism for describing and modeling complex anomalies in general, such as seismic nonlinear vibration. Because of the revelation of fractal frameworks in finance, many researchers have examined fractional configurations in recent years. FDEs are also employed to simulate computational anatomy, biochemical mechanisms, and a variety of other natural or physical structures [1519]. Nonlinear problems are interesting to architects, astronomers, and cosmologists since most physical processes in nature are nonlinear. On the other hand, nonlinear equations are complicated to fix and can yield fascinating results [2026]. In the analysis of high-order nonlinear equations, the actual solutions of transition models contribute significantly.

Even so, humanity is constantly seeking improvements in correctness or accuracy, computation complexity, trustworthiness and appropriateness. Recently, Tarig Elzaki envisioned the Elazki transform (E-transform) by merging well-noted transforms, the Laplace and Sumudu transformations. It can indubitably strengthen the quantitative expression of differential equations in the same way that Laplace and Sumudu transformations have. The E-transform is developed from the traditional Fourier integral, with emphasis on its computational adaptability and promising effects. The E-transform is aimed at resolving ordinary and PDEs in the time realm. The Fourier, Laplace, and Sumudu transforms, in particular, are impacting, pragmatic computational methodologies for addressing FDEs. It is worth noting that the E-transformation was formerly recommended over an ancient, more intricate methodology, including the Sumudu technique. Nonetheless, we need to show that the Elzaki transformation can resolve issues that Laplace cannot [27]. In this article, the E-transform is used to reconfigure the iterative algorithm, and the novel paradigm is known as the “Elzaki Adomian Decomposition Method (EADM).”

Mathematicians have progressively focused their consideration on approximate and analytical solutions to PDEs for developing advanced mathematical strategies address PDEs. Some well acquainted strategies concerning the solution of PDEs are the homotopy analysis method (HAM), new iteration method (NIM), Laplace transform algorithm (LTA), the Haar wavelet technique (HWT) and many more.

In 1982, the Navier-Stokes equation (NSe) was first devised by Navier [28]. The equation is a confluence of the momentum equation, the continuity equation, and the energy equation, and it can be considered Newton’s second law of motion for fluid substances. The NS-model is significant because it explains a variety of interesting physical phenomena that occur in various areas of applied sciences, such as ocean circulation, atmosphere, air circulation across awing, and hydrological cycle in tubes, see [29,30].

Several researchers have attempted to solve the NSe problem using various approaches. To efficiently strengthen the algorithmic structure, Eltayeb et al. [31] proposed unifying the multi-Laplace transform decomposition method for NSe. Shah et al. [32] presented the analytical solution of NSe utilizing the natural transform decomposition method. Singh et al. [33] have expounded numerical simulation by considering fractional reduced differential transformation method for discovering an analytical solution of the time-fractional model of NSe. Khan et al. [34] introduced the Elzaki transform decomposition approach for finding the analytical solution of NSe.

Inspired by the work of [31,34], we aim to investigate the multi-dimensional time fractional model of NSe with the aid of triple Elzaki Adomian decomposition method (TEADM) and quadruple Elzaki Adomian decomposition method (QEADM). Elzaki transformation [35,36] and Adomian decomposition method (ADM) [37,38] are incorporated in the suggested framework. For the formulations of quantitative and qualitative analysis, the Elzaki transformation [3941] and ADM [37,38] have been utilized independently and supply the exact solutions in terms of convergent series. The exact and approximated solution of nonlinear NSe are addressed using advanced technique. With the assistance of graphs, the findings are illustrated and validated. The new approach is demonstrated to solve fractional-order equations in a quite clear and concise way. Other modulation problems in different fields of mathematical and physical sciences can be addressed using the current approach.

The structure of the article is as follows: Section 2 represents the basic definitions related to fractional calculus and the proposed Elzaki transform. Section 3 illustrates the road map for the analytical approximations of triple and quadruple Elzaki Adomian decomposition methods. Numerical experiments and their graphical illustrations concerning the TEADM and QEADM are conducted in Section 4. Theoretical findings are elaborated via comparison analysis with the previous findings demonstrated in Section 5. In a nutshell, we summarized the concluding remarks with open problems.

2  Prelude

In this unit, we have addressed several of the key aspects of FC and triple Elzaki transform. For more details, see [10,12].

The Elzaki transform [35] is a novel integral operator described for functions of exponential order. We shall look at mappings in the set G that are specified by

G={(ξ):,κ1,κ2>0;|(ξ)|<e|ξ|κ;ξ(1)ȷ×[0,+)}.

Definition 2.1. ([35]) Elzaki transform for function (ξ) is defined as

E[(ξ)]=(p)=p0+(ξ)eξpdξ,ξ>0,κ1<p<κ2.(1)

This transform has a stronger resemblance to the Laplace transform such as (s)=ηE(1η). The Elzaki transform is an efficient tool for providing the solution of integral equations, FDEs and PDEs.

Now we define the triple Elzaki transform as follows:

Definition 2.2. For ϖ1,ϖ2,ξ>0 and let be a mapping of three variables ϖ1,ϖ2 and ξ. Then the triple Elzaki transform of is stated as

Eϖ1Eϖ2Eξ[(ϖ1,ϖ2,ξ)]=(η,ζ,s)=ηζs0+0+0+(ϖ1,ϖ2,ξ)eϖ1ηϖ2ζξsdξdϖ2dϖ1,(2)

where η,ζ,s(κ1,κ2). Also, the triple Elzaki transforms of the first and second order partial derivatives are presented by

Eϖ1Eϖ2Eξ[ϖ1(ϖ1,ϖ2,ξ)]=1η(η,ζ,s)(0,ζ,s),Eϖ1Eϖ2Eξ[ϖ2(ϖ1,ϖ2,ξ)]=1ζ(η,ζ,s)(η,0,s),Eϖ1Eϖ2Eξ[ξ(ϖ1,ϖ2,ξ)]=1s(η,ζ,s)(η,ζ,0).(3)

Analogously, we have

Eϖ1Eϖ2Eξ[ϖ1ϖ1(ϖ1,ϖ2,ξ)]=1η2(η,ζ,s)(0,ζ,s)ηϖ1(0,ζ,s),Eϖ1Eϖ2Eξ[ϖ2ϖ2(ϖ1,ϖ2,ξ)]=1ζ2(η,ζ,s)(η,0,s)ζϖ1(η,0,s),Eϖ1Eϖ2Eξ[ξξ(ϖ1,ϖ2,ξ)]=1s2(η,ζ,s)(η,ζ,0)sϖ1(η,ζ,0).(4)

The inverse triple Elzaki transform Eη1Eζ1Es1[(η,ζ,s)]=(ϖ1,ϖ2,ξ) is stated as follows:

(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[(η,ζ,s)]=ηζs0+0+0+eηϖ1+ζϖ2+sξE(1η)E(1ζ)E(1s)dηdζds.(5)

Definition 2.3. ([13]) The Caputo fractional operator of a function of order ϕ>0 can be stated as follows:

c𝒟ϕ(ϖ1)={1Γ(ϕ)0ϖ1(ϖ1ξ)ϕ1(ξ)dξ,1<ϕ<,ddϖ1(ϖ1)=ϕ.

Our next result is important for further investigation of this article.

Theorem 2.1. For ϕ1,ϕ2,ϕ3>0,1<ϕ1,n1<ϕ2n,q1<ϕ3q and ,n,qN, so that f1Ci(R+×R+×R+),i=max{,n,q},f1(i)L1[(0,a1)×(0,b1)×(0,c1)] for any a1,b1,c1>0,|f1(ϖ1,ϖ2,q)|eϖ1ξ1+ϖ2ξ2+ϖ3ξ3,0<a1<ϖ1,0<b1<ϖ2,0<c1<ξ, then the triple Elzaki transforms of Caputo’s fractional derivatives 𝒟ξϕ1(ϖ1,ϖ2,ξ),𝒟ξϕ2(ϖ1,ϖ2,ξ) and 𝒟ξϕ3(ϖ1,ϖ2,ξ) are described by

Eϖ1Eϖ2Eξ[𝒟ξϕ1(ϖ1,ϖ2,ξ)]=sϕ1(η,ζ,s)i=01s2ϕ1+iEϖ2Eϖ1[𝒟ξi(ϖ1,ϖ2,0)],1<ϕ1<,Eϖ1Eϖ2Eξ[𝒟ϖ2ϕ2(ϖ1,ϖ2,ξ)]=ζϕ2(η,ζ,s)j=0n1ζ2ϕ2+jEϖ1Eξ[𝒟ϖ2j(ϖ1,0,ξ)],n1<ϕ2<n

and

Eϖ1Eϖ2Eξ[𝒟ϖ1ϕ3(ϖ1,ϖ2,ξ)]=ηϕ3(η,ζ,s)κ=0q1η2ϕ3+κEϖ2Eξ[𝒟ϖ1κ(0,ϖ2,ξ)],q1<ϕ3<q.

3  Description of the Method for Triple Elzaki Transform Decomposition Method

The constructive approach of the TEADM for multi-dimensional time-fractional NSe is presented in this section. Assume the following framework of two-dimensional time-fractional NSe to demonstrate the underlying strategy of the TEADM:

ϕΦξϕ+ΦΦϖ1+ΨΦϖ2=ρ0(2Φϖ12+2Φϖ22)+1ρσϖ1,ϖ1,ϖ2,ξ>0,ϕΨξϕ+ΦΨϖ1+ΨΨϖ2=ρ0(2Ψϖ12+2Ψϖ22)1ρσϖ2,ϖ1,ϖ2,ξ>0,(6)

subject to

Φ(ϖ1,ϖ2,0)=f1(ϖ1,ϖ2),Ψ(ϖ1,ϖ2,0)=g1(ϖ1,ϖ2),

where ϕξϕ is the Caputo fractional derivative and σ is the pressure. Also, if σ is known, then 1=1ρσϖ1 and 2=1ρσϖ2. To employ the TEADM, multiply (6) by ϖ1, which leads us

1sϕEϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]κ=01Φ(κ)(η,ζ,0)s2ϕ+κ=Eϖ1Eϖ2Eξ(ΦΦϖ1+ΨΦϖ2)+Eϖ1Eϖ2Eξ(ρ0(2Φϖ12+2Φϖ22)+1),1sϕEϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]κ=01Ψ(κ)(η,ζ,0)s2ϕ+κ=Eϖ1Eϖ2Eξ(ΦΨϖ1+ΨΨϖ2)+Eϖ1Eϖ2Eξ(ρ0(2Ψϖ12+2Ψϖ22))Eϖ1Eϖ2Eξ(2).(7)

By the virtue of Theorem 2.1, we obtain

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=s21(η,ζ)sϕEϖ1Eϖ2Eξ(ΦΦϖ1+ΨΦϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(2Φϖ12+2Φϖ22))+sϕEϖ1Eϖ2Eξ(1),Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2𝒢1(η,ζ)sϕEϖ1Eϖ2Eξ(ΦΨϖ1+ΨΨϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(2Ψϖ12+2Ψϖ22))sϕEϖ1Eϖ2Eξ(2).(8)

By implementing the triple inverse Elzaki transformation for (8), we obtain

Φ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[s21(η,ζ)sϕEϖ1Eϖ2Eξ(ΦΦϖ1+ΨΦϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(2Φϖ12+2Φϖ22))+sϕEϖ1Eϖ2Eξ(1)],Ψ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[s2𝒢1(η,ζ)sϕEϖ1Eϖ2Eξ(ΦΨϖ1+ΨΨϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(2Ψϖ12+2Ψϖ22))sϕEϖ1Eϖ2Eξ(2)].(9)

The infinite series solutions Φ(ϖ1,ϖ2,ξ) and Ψ(ϖ1,ϖ2,ξ) are presented as follows:

Φ(ϖ1,ϖ2,ξ)==0+Φ(ϖ1,ϖ2,ξ),Ψ(ϖ1,ϖ2,ξ)==0+Ψ(ϖ1,ϖ2,ξ).(10)

Meanwhile, the non-linear expressions 𝒩1=ΦΦϖ1,𝒩2=ΨΦϖ2,𝒩3=ΦΨϖ1 and 𝒩4=ΨΨϖ2 are denoted by

𝒩1(Φ,Ψ)==0+𝒜,𝒩2(Φ,Ψ)==0+,𝒩3(Φ,Ψ)==0+𝒞,𝒩4(Φ,Ψ)==0+𝒟(11)

and are presented by the Adomian polynomials as

𝒜=1![θ{𝒩1(κ=0+θκΦκ,κ=0+θκΨκ)}]θ=0,=1![θ{𝒩2(κ=0+θκΦκ,κ=0+θκΨκ)}]θ=0,𝒞=1![θ{𝒩3(κ=0+θκΦκ,κ=0+θκΨκ)}]θ=0,𝒟=1![θ{𝒩4(κ=0+θκΦκ,κ=0+θκΨκ)}]θ=0.(12)

By inserting (12) into (9), we attain

Eϖ1Eϖ2Eξ[=0+Φ(ϖ1,ϖ2,ξ)]=s21(η,ζ)sϕEϖ1Eϖ2Eξ(=0+(𝒜+))+sϕEϖ1Eϖ2Eξ(ρ0(=0+2Φϖ12+=0+2Φϖ22))+sϕEϖ1Eϖ2Eξ(1)(13)

and

Eϖ1Eϖ2Eξ[=0+Ψ(ϖ1,ϖ2,ξ)]=s2𝒢1(η,ζ)sϕEϖ1Eϖ2Eξ(=0+(𝒞+𝒟))+sϕEϖ1Eϖ2Eξ(ρ0(=0+2Ψϖ12+=0+2Ψϖ22))sϕEϖ1Eϖ2Eξ(2).(14)

By implementing the inverse Elzaki transformation to (13) and (14) we have

=0+Φ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[s21(η,ζ)]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(=0+(𝒜+))]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(ρ0(=0+2Φϖ12+=0+2Φϖ22))+sϕEϖ1Eϖ2Eξ(1)](15)

and

=0+Ψ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[s2𝒢1(η,ζ)]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(=0+(𝒞+𝒟))]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(ρ0(=0+2Ψϖ12+=0+2Ψϖ22))sϕEϖ1Eϖ2Eξ(2)].(16)

Taking into account the TEADM, we develop iterative connections as follows:

Φ0(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[s21(η,ζ)],Ψ0(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[s2𝒢1(η,ζ)],(17)

and the rest of the components Φ+1,0 are presented by

Φ+1(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(𝒜+)]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(ρ0(2Φϖ12+2Φϖ22))+sϕEϖ1Eϖ2Eξ(1)].(18)

and

Ψ+1(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(𝒞+𝒟)]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ(ρ0(2Ψϖ12+2Ψϖ22))sϕEϖ1Eϖ2Eξ(2)].(19)

where Eϖ1Eϖ2Eξ is the triple E-transform in respecting ϖ1,ϖ2,ξ1 and Eη1Eζ1Es1 is the triple inverse E-transform in respecting η,ζ,s. For (17)(19), we assumed that the triple inverse E-transform exists.

4  Application of the Proposed Method

In this section, we evaluate the efficacy of our current techniques by introducing the decomposition method in combination with the triple and quadruple E-transform.

Problem 4.1. We assume the two dimensional time-fractional NSe:

{ϕΦξϕ+ΦΦϖ1+ΨΦϖ2=ρ(2Φϖ12+2Φϖ22)+,ϖ1,ϖ2,ξ>0,ϕΨξϕ+ΦΨϖ1+ΨΨϖ2=ρ(2Ψϖ12+2Ψϖ22),ϖ1,ϖ2,ξ>0,1<ϕ<;(20)

subject to

{Φ(ϖ1,ϖ2,0)=sin(ϖ1+ϖ2),Ψ(ϖ1,ϖ2,0)=sin(ϖ1+ϖ2).(21)

Proof. By applying the triple Elzaki transform on (20), we have

Eϖ1Eϖ2Eξ[ϕΦξϕ+ΦΦϖ1+ΨΦϖ2]=Eϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)+],Eϖ1Eϖ2Eξ[ϕΨξϕ+ΦΨϖ1+ΨΨϖ2]=Eϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)].

On making the use of differentiation property of the Elzaki transform, we obtain

1sϕEϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]κ=01Φ(κ)(η,ζ,0)s2ϕ+κ=Eϖ1Eϖ2Eξ[ΦΦϖ1+ΨΦϖ2]+Eϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)+],1sϕEϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]κ=01Ψ(κ)(η,ζ,0)s2ϕ+κ=Eϖ1Eϖ2Eξ[ΦΨϖ1+ΨΨϖ2]+Eϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)].(22)

According to initial conditions and simple computations yields

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=s2Eϖ1Eϖ2[sin(ϖ1+ϖ2)]sϕEϖ1Eϖ2Eξ[ΦΦϖ1+ΨΦϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)+],Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2Eϖ1Eϖ2[sin(ϖ1+ϖ2)]sϕEϖ1Eϖ2Eξ[ΦΨϖ1+ΨΨϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)].

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=s2η2ζ2(η+ζ)(η2+1)(ζ2+1)sϕEϖ1Eϖ2Eξ[ΦΦϖ1+ΨΦϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)+],Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2η2ζ2(η+ζ)(η2+1)(ζ2+1)sϕEϖ1Eϖ2Eξ[ΦΨϖ1+ΨΨϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)].(23)

Employing the inverse triple Elzaki transform for (23)

Φ(ϖ1,ϖ2,ξ)=sin(ϖ1+ϖ2)+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ()]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΦϖ1+ΨΦϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],Ψ(ϖ1,ϖ2,ξ)=sin(ϖ1+ϖ2)Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ()]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΨϖ1+ΨΨϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].(24)

It follows that

Φ(ϖ1,ϖ2,ξ)=sin(ϖ1+ϖ2)+Eη1Eζ1Es1[η2ς2sϕ+2]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΦϖ1+ΨΦϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],Ψ(ϖ1,ϖ2,ξ)=sin(ϖ1+ϖ2)Eη1Eζ1Es1[η2ς2sϕ+2]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΨϖ1+ΨΨϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].(25)

Consequently, we have

Φ(ϖ1,ϖ2,ξ)=sin(ϖ1+ϖ2)+ξϕΓ(ϕ+1)Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΦϖ1+ΨΦϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],Ψ(ϖ1,ϖ2,ξ)=sin(ϖ1+ϖ2)ξϕΓ(ϕ+1)Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΨϖ1+ΨΨϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].(26)

The infinite series solution for unknown function Φ(ϖ1,ϖ2,ξ) and Ψ(ϖ1,ϖ2,ξ) have the following form:

Φ(ϖ1,ϖ2,ξ)==0+Φ(ϖ1,ϖ2,ξ),Ψ(ϖ1,ϖ2,ξ)==0+Ψ(ϖ1,ϖ2,ξ).(27)

The Adomian approach is required to determine the zeroth elements Φ0 and Ψ0. Therefore, it includes initial condition, all of which are considered to be identified. Consequently, we devised

Φ0=sin(ϖ1+ϖ2)+ξϕΓ(ϕ+1),Ψ0=sin(ϖ1+ϖ2)ξϕΓ(ϕ+1).

Remember that ΦΦ(ϖ1,ξ)ϖ1==0+𝒜, ΨΦ(ϖ1,ξ)ϖ2==0+, ΦΨ(ϖ1,ξ)ϖ1==0+𝒞 and ΨΨ(ϖ1,ξ)ϖ2==0+𝒟 are the Adomian terms and nonlinear terms were characterized. Now, (26) can be expressed in an iterative way with the aid of (27), as follows:

=0+Φ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[=0+𝒜+=0+]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],=0+Ψ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[=0+𝒞+=0+𝒟]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].

Thanks to (11), the Adomian polynomials will express all forms of nonlinearity as

𝒜0=Φ0Φ0ϖ1,𝒜1=Φ0Φ1ϖ1+Φ1Φ0ϖ1,𝒜2=Φ0Φ2ϖ1+Φ1Φ1ϖ1+Φ2Φ0ϖ1,0=Ψ0Φ0ϖ2,1=Ψ0Φ1ϖ2+Ψ1Φ0ϖ2,2=Ψ0Φ2ϖ2+Ψ1Φ1ϖ2+Ψ2Φ0ϖ2,𝒞0=Φ0Ψ0ϖ1,𝒞1=Φ0Ψ1ϖ1+Φ1Ψ0ϖ1,𝒞2=Φ0Ψ2ϖ1+Φ1Ψ1ϖ1+Φ2Ψ0ϖ1,𝒟0=Ψ0Ψ0ϖ2,𝒟1=Ψ0Ψ1ϖ2+Ψ1Ψ0ϖ2,𝒟2=Ψ0Ψ2ϖ2+Ψ1Ψ1ϖ2+Ψ2Ψ0ϖ2.

For =0, we have

Φ1(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒜0+0]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φ0ϖ12+2Φ0ϖ22)]]=Eη1Eζ1Es1[2ρsϕ+2η2ζ2(η+ζ)(η2+1)(ζ2+1)]=2ρξϕΓ(ϕ+1)sin(ϖ1+ϖ2).

Similarly, we obtain

Ψ1(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒞0+𝒟0]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψ0ϖ12+2Ψ0ϖ22)]]=2ρξϕΓ(ϕ+1)sin(ϖ1+ϖ2)

Analogously, for =1,

Φ2(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒜1+1]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φ1ϖ12+2Φ1ϖ22)]]=Eη1Eζ1Es1[4ρ2s2ϕ+2η2ζ2(η+ζ)(η2+1)(ζ2+1)]=(2ρ)2ξ2ϕΓ(2ϕ+1)sin(ϖ1+ϖ2)

and

Ψ2(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒞1+𝒟1]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψ1ϖ12+2Ψ1ϖ22)]]=(2ρ)2ξ2ϕΓ(2ϕ+1)sin(ϖ1+ϖ2).

For =2,

Φ3(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒜2+2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φ2ϖ12+2Φ2ϖ22)]]=Eη1Eζ1Es1[8ρ3s3ϕ+2η2ζ2(η+ζ)(η2+1)(ζ2+1)]=(2ρ)3ξ3ϕΓ(3ϕ+1)sin(ϖ1+ϖ2)

and

Ψ3(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒞2+𝒟2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψ2ϖ12+2Ψ2ϖ22)]]=(2ρ)3ξ3ϕΓ(3ϕ+1)sin(ϖ1+ϖ2).

Continuing the same way, the iterative terms of Φ and Ψ,(>4) are presented as follows:

Φ(ϖ1,ϖ2,ξ)==0+Φ(ϖ1,ϖ2,ξ)=Φ0(ϖ1,ϖ2,ξ)+Φ1(ϖ1,ϖ2,ξ)+Φ2(ϖ1,ϖ2,ξ)+Φ3(ϖ1,ϖ2,ξ)+...,Ψ(ϖ1,ϖ2,ξ)==0+Ψ(ϖ1,ϖ2,ξ)=Ψ0(ϖ1,ϖ2,ξ)+Ψ1(ϖ1,ϖ2,ξ)+Ψ2(ϖ1,ϖ2,ξ)+Ψ3(ϖ1,ϖ2,ξ)+...,Φ(ϖ1,ϖ2,ξ)=ξϕΓ(ϕ+1)sin(ϖ1+ϖ2)[12ρξϕΓ(ϕ+1)+4ρ2ξ2ϕΓ(2ϕ+1)8ρ3ξ3ϕΓ(3ϕ+1)+...],Ψ(ϖ1,ϖ2,ξ)=ξϕΓ(ϕ+1)+sin(ϖ1+ϖ2)[12ρξϕΓ(ϕ+1)+4ρ2ξ2ϕΓ(2ϕ+1)8ρ3ξ3ϕΓ(3ϕ+1)+...].

At ϕ=1 and =0, the actual solution of classical NSe is

Φ(ϖ1,ϖ2,ξ)=exp(2ρξ)sin(ϖ1+ϖ2),Ψ(ϖ1,ϖ2,ξ)=exp(2ρξ)sin(ϖ1+ϖ2).

Problem 4.2. We assume the two dimensional time-fractional NSe:

{ϕΦ(ϖ1,ξ)ξϕ+ΦΦ(ϖ1,ξ)ϖ1+ΨΦ(ϖ1,ξ)ϖ2=ρ(2Φϖ12+2Φϖ22)+,ϕΨ(ϖ1,ξ)ξϕ+ΦΨ(ϖ1,ξ)ϖ1+ΨΨ(ϖ1,ξ)ϖ2=ρ(2Ψϖ12+2Ψϖ22),(28)

subject to

{Φ(ϖ1,ϖ2,0)=exp(ϖ1+ϖ2),Ψ(ϖ1,ϖ2,0)=exp(ϖ1+ϖ2).(29)

Proof. By means of (22) and according to initial conditions given in (29), we have

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=s2Eϖ1Eϖ2[exp(ϖ1+ϖ2)]sϕEϖ1Eϖ2Eξ[ΦΦ(ϖ1,ξ)ϖ1+ΨΦ(ϖ1,ξ)ϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)+],Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2Eϖ1Eϖ2[exp(ϖ1+ϖ2)]sϕEϖ1Eϖ2Eξ[ΦΨ(ϖ1,ξ)ϖ1+ΨΨ(ϖ1,ξ)ϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)].

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=s2η2ζ2(1η)(1ζ)sϕEϖ1Eϖ2Eξ[ΦΦ(ϖ1,ξ)ϖ1+ΨΦ(ϖ1,ξ)ϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)+],Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2η2ζ2(1η)(1ζ)sϕEϖ1Eϖ2Eξ[ΦΨ(ϖ1,ξ)ϖ1+ΨΨ(ϖ1,ξ)ϖ2]+sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)].(30)

Employing the inverse triple Elzaki transform for (30)

Φ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2)+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ()]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΦ(ϖ1,ξ)ϖ1+ΨΦ(ϖ1,ξ)ϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],Ψ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2)Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ()]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΨ(ϖ1,ξ)ϖ1+ΨΨ(ϖ1,ξ)ϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].(31)

It follows that

Φ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2)+Eη1Eζ1Es1[η2ς2sϕ+2]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΦ(ϖ1,ξ)ϖ1+ΨΦ(ϖ1,ξ)ϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],Ψ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2)Eη1Eζ1Es1[η2ς2sϕ+2]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΨ(ϖ1,ξ)ϖ1+ΨΨ(ϖ1,ξ)ϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].(32)

Consequently, we have

Φ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2)+ξϕΓ(ϕ+1)Eη1Eζ1Es1[η2ς2sϕ+2]Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΦ(ϖ1,ξ)ϖ1+ΨΦ(ϖ1,ξ)ϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],Ψ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2)ξϕΓ(ϕ+1)+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ΦΨ(ϖ1,ξ)ϖ1+ΨΨ(ϖ1,ξ)ϖ2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].(33)

The infinite series solution for unknown function Φ(ϖ1,ϖ2,ξ) and Ψ(ϖ1,ϖ2,ξ) have the following form:

Φ(ϖ1,ϖ2,ξ)==0+Φ(ϖ1,ϖ2,ξ),Ψ(ϖ1,ϖ2,ξ)==0+Ψ(ϖ1,ϖ2,ξ).(34)

The Adomian approach is required to determine the zeroth elements Φ0 and Ψ0. Therefore, it includes initial condition, all of which are considered to be identified. Consequently, we devised

Φ0=exp(ϖ1+ϖ2)+ξϕΓ(ϕ+1),Ψ0=exp(ϖ1+ϖ2)ξϕΓ(ϕ+1).

Remember that ΦΦ(ϖ1,ξ)ϖ1==0+𝒜, ΨΦ(ϖ1,ξ)ϖ2==0+, ΦΨ(ϖ1,ξ)ϖ1==0+𝒞 and ΨΨ(ϖ1,ξ)ϖ2==0+𝒟 are the Adomian terms and nonlinear terms were characterized. Now, (33) can be expressed in an iterative way with the aid of (34), as follows:

=0+Φ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[=0+𝒜+=0+]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φϖ12+2Φϖ22)]],=0+Ψ(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[=0+𝒞+=0+𝒟]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψϖ12+2Ψϖ22)]].

Thanks to (11), the Adomian polynomials will express all forms of nonlinearity as

𝒜0=Φ0Φ0ϖ1,𝒜1=Φ0Φ1ϖ1+Φ1Φ0ϖ1,𝒜2=Φ0Φ2ϖ1+Φ1Φ1ϖ1+Φ2Φ0ϖ1,0=Ψ0Φ0ϖ2,1=Ψ0Φ1ϖ2+Ψ1Φ0ϖ2,2=Ψ0Φ2ϖ2+Ψ1Φ1ϖ2+Ψ2Φ0ϖ2,𝒞0=Φ0Ψ0ϖ1,𝒞1=Φ0Ψ1ϖ1+Φ1Ψ0ϖ1,𝒞2=Φ0Ψ2ϖ1+Φ1Ψ1ϖ1+Φ2Ψ0ϖ1,𝒟0=Ψ0Ψ0ϖ2,𝒟1=Ψ0Ψ1ϖ2+Ψ1Ψ0ϖ2,𝒟2=Ψ0Ψ2ϖ2+Ψ1Ψ1ϖ2+Ψ2Ψ0ϖ2.

For =0, we have

Φ1(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒜0+0]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φ0ϖ12+2Φ0ϖ22)]]=Eη1Eζ1Es1[2ρsϕ+2η2ζ2(1η)(1ζ)]=2ρξϕΓ(ϕ+1)exp(ϖ1+ϖ2).

In a similar manner, we obtain

Ψ1(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒞0+𝒟0]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψ0ϖ12+2Ψ0ϖ22)]]=2ρξϕΓ(ϕ+1)exp(ϖ1+ϖ2).

Analogously, for =1,

Φ2(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒜1+1]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φ1ϖ12+2Φ1ϖ22)]]=Eη1Eζ1Es1[4ρ2s2ϕ+2η2ζ2(1η)(1ζ)]=(2ρ)2ξ2ϕΓ(2ϕ+1)exp(ϖ1+ϖ2)

and

Ψ2(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒞1+𝒟1]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψ1ϖ12+2Ψ1ϖ22)]]=(2ρ)2ξ2ϕΓ(2ϕ+1)exp(ϖ1+ϖ2).

For =2,

Φ3(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒜2+2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Φ2ϖ12+2Φ2ϖ22)]]=Eη1Eζ1Es1[8ρ3s3ϕ+2η2ζ2(1η)(1ζ)]=(2ρ)3ξ3ϕΓ(3ϕ+1)exp(ϖ1+ϖ2)

and

Ψ3(ϖ1,ϖ2,ξ)=Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[𝒞2+𝒟2]]+Eη1Eζ1Es1[sϕEϖ1Eϖ2Eξ[ρ(2Ψ2ϖ12+2Ψ2ϖ22)]]=(2ρ)3ξ3ϕΓ(3ϕ+1)exp(ϖ1+ϖ2).

Continuing the same way, the iterative terms of Φ and Ψ,(>4) are presented as follows:

Φ(ϖ1,ϖ2,ξ)==0+Φ(ϖ1,ϖ2,ξ)=Φ0(ϖ1,ϖ2,ξ)+Φ1(ϖ1,ϖ2,ξ)+Φ2(ϖ1,ϖ2,ξ)+Φ3(ϖ1,ϖ2,ξ)+...,Ψ(ϖ1,ϖ2,ξ)==0+Ψ(ϖ1,ϖ2,ξ)=Ψ0(ϖ1,ϖ2,ξ)+Ψ1(ϖ1,ϖ2,ξ)+Ψ2(ϖ1,ϖ2,ξ)+Ψ3(ϖ1,ϖ2,ξ)+...,Φ(ϖ1,ϖ2,ξ)=ξϕΓ(ϕ+1)exp(ϖ1+ϖ2)=0+(2ρ)ξϕΓ(ϕ+1),Ψ(ϖ1,ϖ2,ξ)=ξϕΓ(ϕ+1)+exp(ϖ1+ϖ2)=0+(2ρ)ξϕΓ(ϕ+1).

At ϕ=1 and =0, the actual solution of classical NSe is

Φ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2+2ρξ),Ψ(ϖ1,ϖ2,ξ)=exp(ϖ1+ϖ2+2ρξ).

Problem 4.3. We assume the system of time-fractional NSe with 1=1=3=0

{ϕΦξϕ+ΦΦϖ1+ΨΦϖ2+ΥΦϖ3=ρ(2Φϖ12+2Φϖ22+2Φϖ32),ϕΨξϕ+ΦΨϖ1+ΨΨϖ2+ΥΨϖ3=ρ(2Ψϖ12+2Ψϖ22+2Ψϖ32),ϕΥξϕ+ΦΥϖ1+ΨΥϖ2+ΥΥϖ3=ρ(2Υϖ12+2Υϖ22+2Υϖ32),(35)

subject to

{Φ(ϖ1,ϖ2,ϖ3,0)=0.5ϖ1+ϖ2+ϖ3,Ψ(ϖ1,ϖ2,ϖ3,0)=ϖ10.5ϖ2+ϖ3,Υ(ϖ1,ϖ2,ϖ3,0)=ϖ1+ϖ20.5ϖ3.(36)

Proof. By applying the quadruple Elzaki transform on (35), we have

Eϖ1Eϖ2Eϖ3Eξ[ϕΦξϕ+ΦΦϖ1+ΨΦϖ2+ΥΦϖ3]=Eϖ1Eϖ2Eϖ3Eξ[ρ(2Φϖ12+2Φϖ22+2Φϖ32)],Eϖ1Eϖ2Eϖ3Eξ[ϕΨξϕ+ΦΨϖ1+ΨΨϖ2+ΥΨϖ3]=Eϖ1Eϖ2Eϖ3Eξ[ρ(2Ψϖ12+2Ψϖ22+2Ψϖ32)],Eϖ1Eϖ2Eϖ3Eξ[ϕΥξϕ+ΦΥϖ1+ΨΦϖ2+ΥΥϖ3]=Eϖ1Eϖ2Eϖ3Eξ[ρ(2Υϖ12+2Υϖ22+2Υϖ32)].

On making the use of differentiation property of the Elzaki transform, we obtain

1sϕEϖ1Eϖ2Eϖ3Eξ[Φ(ϖ1,ϖ2,ϖ3,ξ)]κ=01Φ(κ)(η,ζ,r1,0)s2ϕ+κ=Eϖ1Eϖ2Eϖ3Eξ[ΦΦϖ1+ΨΦϖ2+ΥΦϖ3]+Eϖ1Eϖ2Eϖ3Eξ[ρ(2Φϖ12+2Φϖ22+2Φϖ32)],1sϕEϖ1Eϖ2Eϖ3Eξ[Ψ(ϖ1,ϖ2,ϖ3,ξ)]κ=01Ψ(κ)(η,ζ,r1,0)s2ϕ+κ=Eϖ1Eϖ2Eϖ3Eξ[ΦΨϖ1+ΨΨϖ2+ΥΨϖ3]+Eϖ1Eϖ2Eϖ3Eξ[ρ(2Ψϖ12+2Ψϖ22+2Ψϖ32)],1sϕEϖ1Eϖ2Eϖ3Eξ[Υ(ϖ1,ϖ2,ϖ3,ξ)]κ=01Υ(κ)(η,ζ,r1,0)s2ϕ+κ=Eϖ1Eϖ2Eϖ3Eξ[ΦΥϖ1+ΨΥϖ2+ΥΥϖ3]+Eϖ1Eϖ2Eϖ3Eξ[ρ(2Υϖ12+2Υϖ22+2Υϖ32)].(37)

According to initial conditions and simple computations yields

Eϖ1Eϖ2Eϖ3Eξ[Φ(ϖ1,ϖ2,ϖ3,ξ)]=s2[0.5η3ζ2r12+η2ζ3r22+η2ζ2r13]sϕEϖ1Eϖ2Eϖ3Eξ[ΦΦϖ1+ΨΦϖ2+ΥΦϖ3]+sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Φϖ12+2Φϖ22+2Φϖ32)],Eϖ1Eϖ2Eϖ3Eξ[Ψ(ϖ1,ϖ2,ϖ3,ξ)]=s2[η3ζ2r120.5η2ζ3r22+η2ζ2r13]sϕEϖ1Eϖ2Eϖ3Eξ[ΦΨϖ1+ΨΨϖ2+ΥΨϖ3]+sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Ψϖ12+2Ψϖ22+2Ψϖ32)],Eϖ1Eϖ2Eϖ3Eξ[Υ(ϖ1,ϖ2,ϖ3,ξ)]=s2[η3ζ2r12+η2ζ3r220.5η2ζ2r13]sϕEϖ1Eϖ2Eϖ3Eξ[ΦΥϖ1+ΨΥϖ2+ΥΥϖ3]+sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Υϖ12+2Υϖ22+2Υϖ32)].(38)

Employing the inverse quadruple Elzaki transform for (38)

Φ(ϖ1,ϖ2,ϖ3,ξ)=0.5ϖ1+ϖ2+ϖ3Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ΦΦϖ1+ΨΦϖ2+ΥΦϖ3]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Φϖ12+2Φϖ22+2Φϖ32)]],Ψ(ϖ1,ϖ2,ϖ3,ξ)=ϖ10.5ϖ2+ϖ3Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ΦΨϖ1+ΨΨϖ2+ΥΨϖ3]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Ψϖ12+2Ψϖ22+2Ψϖ32)]],Υ(ϖ1,ϖ2,ϖ3,ξ)=ϖ1+ϖ20.5ϖ3Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ΦΥϖ1+ΨΥϖ2+ΥΥϖ3]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Υϖ12+2Υϖ22+2Υϖ32)]].(39)

The infinite series solution for unknown function Φ(ϖ1,ϖ2,ϖ3,ξ),Ψ(ϖ1,ϖ2,ϖ3,ξ) and Υ(ϖ1,ϖ2,ϖ3,ξ) have the following form:

Φ(ϖ1,ϖ2,ϖ3,ξ)==0+Φ(ϖ1,ϖ2,ϖ3,ξ),Ψ(ϖ1,ϖ2,ϖ3,ξ)==0+Ψ(ϖ1,ϖ2,ϖ3,ξ),Υ(ϖ1,ϖ2,ϖ3,ξ)==0+Υ(ϖ1,ϖ2,ϖ3,ξ).(40)

The Adomian approach is required to determine the zeroth elements Φ0,Ψ0 and Υ0. Therefore, it includes initial condition, all of which are considered to be identified. Consequently, we devised

Φ0=0.5ϖ1+ϖ2+ϖ3,Ψ0=ϖ10.5ϖ2+ϖ3,Υ0=ϖ1+ϖ20.5ϖ3.

=0+Φ(ϖ1,ϖ2,ϖ3,ξ)=Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[=0+𝒜+=0++=0+𝒞]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Φϖ12+2Φϖ22+2Φϖ32)]],=0+Ψ(ϖ1,ϖ2,ϖ3,ξ)=Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[=0+𝒟+=0++=0+]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Ψϖ12+2Ψϖ22+2Ψϖ32)]],=0+Υ(ϖ1,ϖ2,ϖ3,ξ)=Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[=0+𝒢+=0++=0+]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Υϖ12+2Υϖ22+2Υϖ32)]].

Thanks to (11), the Adomian polynomials will express all forms of nonlinearity as

𝒜0=Φ0Φ0ϖ1,𝒜1=Φ0Φ1ϖ1+Φ1Φ0ϖ1,𝒜2=Φ0Φ2ϖ1+Φ1Φ1ϖ1+Φ2Φ0ϖ1,0=Ψ0Φ0ϖ2,1=Ψ0Φ1ϖ2+Ψ1Φ0ϖ2,2=Ψ0Φ2ϖ2+Ψ1Φ1ϖ2+Ψ2Φ0ϖ2,𝒞0=Υ0Φ0ϖ3,𝒞1=Υ0Φ1ϖ3+Υ1Φ0ϖ3,𝒞2=Υ0Φ2ϖ3+Υ1Φ1ϖ1+Υ2Φ0ϖ3,𝒟0=Φ0Ψ0ϖ1,𝒟1=Φ0Ψ1ϖ1+Φ1Ψ0ϖ1,𝒟2=Φ0Ψ2ϖ1+Φ1Ψ1ϖ1+Φ2Ψ0ϖ1,0=Ψ0Ψ0ϖ2,1=Ψ0Ψ1ϖ2+Ψ1Ψ0ϖ2,2=Ψ0Ψ2ϖ2+Ψ1Ψ1ϖ1+Ψ2Ψ0ϖ2,0=Υ0Ψ0ϖ3,1=Υ0Ψ1ϖ3+Υ1Ψ0ϖ3,2=Υ0Ψ2ϖ3+Υ1Ψ1ϖ1+Υ2Ψ0ϖ3,𝒢0=Φ0Υ0ϖ1,𝒢1=Φ0Υ1ϖ1+Φ1Υ0ϖ1,𝒢2=Φ0Υ2ϖ1+Φ1Υ1ϖ1+Φ2Υ0ϖ1,0=Ψ0Υ0ϖ2,1=Ψ0Υ1ϖ2+Ψ1Υ0ϖ2,2=Ψ0Υ2ϖ2+Ψ1Υ1ϖ1+Ψ2Υ0ϖ2,0=Υ0Υ0ϖ3,1=Υ0Υ1ϖ3+Υ1Υ0ϖ3,2=Υ0Υ2ϖ3+Υ1Υ1ϖ1+Υ2Υ0ϖ3,

For =0, we have

Φ1(ϖ1,ϖ2,ϖ3,ξ)=Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[𝒜0+0+𝒞0]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Φ0ϖ12+2Φ0ϖ22+2Φ0ϖ32)]]=Eη1Eζ1Er11Es1[2.25η3ζ2r12sϕ+2]=2.25ϖ1ξϕΓ(ϕ+1).

In a similar way, we obtain

Ψ1(ϖ1,ϖ2,ϖ3,ξ)=2.25ϖ2ξϕΓ(ϕ+1),Υ1(ϖ1,ϖ2,ϖ3,ξ)=2.25ϖ3ξϕΓ(ϕ+1).

Analogously, for =1,

Φ2(ϖ1,ϖ2,ϖ3,ξ)=Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[𝒜1+1+𝒞1]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Φ1ϖ12+2Φ1ϖ22+2Φ1ϖ32)]]=Eη1Eζ1Er11Es1[2.25η3ζ2r12s2ϕ+24.5η2ζ3r12s2ϕ+24.5η2ζ2r13s2ϕ+2]=2(2.25)ξ2ϕΓ(2ϕ+1)(0.5ϖ1+ϖ2+ϖ3).

In a similar manner, we obtain

Ψ2(ϖ1,ϖ2,ϖ3,ξ)=2(2.25)ξ2ϕΓ(2ϕ+1)(ϖ10.5ϖ2+ϖ3),Υ2(ϖ1,ϖ2,ϖ3,ξ)=2(2.25)ξ2ϕΓ(2ϕ+1)(ϖ1+ϖ20.5ϖ3).

For =2,

Φ3(ϖ1,ϖ2,ϖ3,ξ)=Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[𝒜2+2+𝒞2]]+Eη1Eζ1Er11Es1[sϕEϖ1Eϖ2Eϖ3Eξ[ρ(2Φ2ϖ12+2Φ2ϖ22+2Φ2ϖ32)]]=Eη1Eζ1Er11Es1[(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2s3ϕ+2η3ζ2r12]=(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2ϖ1ξ3ϕΓ(3ϕ+1),Ψ3(ϖ1,ϖ2,ϖ3,ξ)=(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2ϖ2ξ3ϕΓ(3ϕ+1),

and

Υ1(ϖ1,ϖ2,ϖ3,ξ)=(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2ϖ3ξ3ϕΓ(3ϕ+1).

Continuing the same way, the iterative terms of Φ,Ψ and Υ (>4) are presented as follows:

Φ(ϖ1,ϖ2,ϖ3,ξ)==0+Φ(ϖ1,ϖ2,ϖ3,ξ)=Φ0(ϖ1,ϖ2,ϖ3,ξ)+Φ1(ϖ1,ϖ2,ϖ3,ξ)+Φ2(ϖ1,ϖ2,ϖ3,ξ)+Φ3(ϖ1,ϖ2,ϖ3,ξ)+...,Ψ(ϖ1,ϖ2,ϖ3,ξ)==0+Ψ(ϖ1,ϖ2,ϖ3,ξ)=Ψ0(ϖ1,ϖ2,ϖ3,ξ)+Ψ1(ϖ1,ϖ2,ϖ3,ξ)+Ψ2(ϖ1,ϖ2,ϖ3,ξ)+Ψ2(ϖ1,ϖ2,ϖ3,ξ)+...,Υ(ϖ1,ϖ2,ϖ3,ξ)==0+Υ(ϖ1,ϖ2,ϖ3,ξ)=Υ0(ϖ1,ϖ2,ϖ3,ξ)+Υ1(ϖ1,ϖ2,ϖ3,ξ)+Υ2(ϖ1,ϖ2,ϖ3,ξ)+Υ3(ϖ1,ϖ2,ϖ3,ξ)+...,

Φ(ϖ1,ϖ2,ϖ3,ξ)=0.5ϖ1+ϖ2+ϖ32.25ϖ1ξϕΓ(ϕ+1)+2(2.25)ξ2ϕΓ(2ϕ+1)(0.5ϖ1+ϖ2+ϖ3)(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2ϖ1ξ3ϕΓ(3ϕ+1)+...,Ψ(ϖ1,ϖ2,ϖ3,ξ)=ϖ10.5ϖ2+ϖ32.25ϖ2ξϕΓ(ϕ+1)+2(2.25)ξ2ϕΓ(2ϕ+1)(ϖ10.5ϖ2+ϖ3)(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2ϖ2ξ3ϕΓ(3ϕ+1)+...,Υ(ϖ1,ϖ2,ϖ3,ξ)=ϖ1+ϖ20.5ϖ32.25ϖ3ξϕΓ(ϕ+1)+2(2.25)ξ2ϕΓ(2ϕ+1)(ϖ1+ϖ20.5ϖ3)(2.25)2(4(Γ(ϕ+1))2+Γ(2ϕ+1))(Γ(ϕ+1))2ϖ3ξ3ϕΓ(3ϕ+1)+....

At ϕ=1, the actual solution of system of NSe is

Φ(ϖ1,ϖ2,ϖ3,ξ)=(0.5ϖ1+ϖ2+ϖ32.25ϖ1ξ)(12.25ξ2)1,Ψ(ϖ1,ϖ2,ϖ3,ξ)=(ϖ10.5ϖ2+ϖ32.25ϖ2ξ)(12.25ξ2)1,Υ(ϖ1,ϖ2,ϖ3,ξ)=(ϖ1+ϖ20.5ϖ32.25ϖ3ξ)(12.25ξ2)1.

5  Results and Discussion

•   In Figs. 1 and 2, the 3D surfaces with respect to the Caputo fractional derivative operator at ϕ=1,ρ=0.3 and =0 are presented. Furthermore, we have demonstrated the graphs of the approximate solutions of functional values Φ(ϖ1,ϖ2,ξ) and Ψ(ϖ1,ϖ2,ξ) of the NSe for Caputo fractional derivative operator, respectively. The Figs. 1a and 2b depicted the approximate solutions, because the TEADM proved to be relatively compelling for solving nonlinear PDEs without linearization, perturbation or discretization. In Figs. 3 and 4, we have described the approximate solution of the problem with respect to the proposed method and exact solution, respectively. One can deduce from the figures that approximate solution approaches the exact solution as the fractional parameter changes from integer-order ϕ=1.

•   In Figs. 58, the 3D surfaces with respect to the Caputo fractional derivative operator at ϕ=1,ρ=0.3 and =0 are presented. Furthermore, we have demonstrated the graphs of the approximate solutions of functional values Φ(ϖ1,ϖ2,ξ) and Ψ(ϖ1,ϖ2,ξ) of the NSe for Caputo fractional derivative operator, respectively. The Figs. 5b and 6b depicted the approximate solutions, because the TEADM proved to be relatively promising for solving nonlinear PDEs without linearization, perturbation or discretization. Also, in Fig. 7, the comparison simulation for both Φ and Ψ are also presented to demonstrate the strong correlation between the exact and approximate solution of the proposed technique.

•   In Figs. 911, we have given the comparison of the exact, ϖ1ϖ2-slice of exact solution given in (35) and in Figs. 1214, it can be seen that the solution to the problem which is given by (35) with respect to integer-order parameter in the framework of Caputo fractional derivative. Taking into consideration the results of the article, we can view that only a few components of the series derived by the Adomian decomposition method coupled with the Elzaki transform provide the exact solution. Additionally, the comparison simulation for Φ,Ψ and Υ are also presented to demonstrate the strong correlation between the exact and approximate solution by implementing QEADM.

•   Tables 17 presented a comparison analysis of the findings shown in [31]. The analysis predicts that our projected scheme is close to harmony with the exact solutions.

images

Figure 1: Simulation of the (a) exact and (b) TEADM solutions of Φ(ϖ1,ϖ2,ξ) at ξ=3 of Problem 4.1 when ϕ=1, ρ=0.3, and =0

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Figure 2: Simulation of the (a) exact and (b) TEADM solutions of Ψ(ϖ1,ϖ2,ξ) of Problem 4.1 with the parameters when ϕ=1, ρ=0.3, and =0

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Figure 3: Simulation of the exact and TEADM solutions of Φ(ϖ1,ϖ2,ξ) of Problem 4.1 when ϖ1=1.2 to 2.4, ϖ2=0.2 to 1.4, at ϕ=1, ρ=0.3, and =0, for various values of ξ

images

Figure 4: Simulation of the exact and TEADM solutions of Ψ(ϖ1,ϖ2,ξ) of Problem 4.1 when ϖ1=1.2 to 2.4, ϖ2=0.2 to 1.4, at ϕ=1, ρ=0.3, and =0, for various values of ξ

images

Figure 5: Simulation of the (a) exact and (b) TEADM solutions of Φ(ϖ1,ϖ2,ξ) at ξ=3 of Problem 4.2 with the parameters when ϕ=1, ρ=0.3, and =0

images

Figure 6: Simulation of the (a) exact and (b) TEADM solutions of Ψ(ϖ1,ϖ2,ξ) at ξ=3 of Problem 4.2 with the parameters ϕ=1, ρ=0.3, and =0

images

Figure 7: Simulation of the exact and TEADM solutions of Φ(ϖ1,ϖ2,ξ) of Problem 4.2 for ϖ1=1.2 to 2.4, ϖ2=0.2 to 1.4, at ϕ=1, ρ=0.3, and =0, for various values of ξ

images

Figure 8: Simulation of the exact and TEADM solutions of Ψ(ϖ1,ϖ2,ξ) of Problem 4.2 when ϖ1=1.2 to 2.4, ϖ2=0.2 to 1.4, at ϕ=1, ρ=0.3, and =0, for various values of ξ

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Figure 9: Simulation of the (a) exact solution of Problem 4.3 (b) ϖ1ϖ2-slice of exact solution (c) QEADM solutions of Φ(ϖ1,ϖ2,ϖ3,ξ) of Problem 4.3 when ϕ=1, ρ=0.3, ξ=0.01 and =0, (d) ϖ1ϖ2-slice of solution of (c)

images

Figure 10: Simulation of the (a) exact solution of Problem 4.3 (b) ϖ1ϖ2-slice of exact solution (c) QEADM solutions of Ψ(ϖ1,ϖ2,ϖ3,ξ) of problem 4.3, at ϕ=1, ρ=0.3, ξ=0.01 and =0, (d) ϖ1ϖ2-slice of solution of (c)

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Figure 11: Simulation of the (a) exact solution of Problem 4.3 (b) ϖ1ϖ2-slice of exact solution (c) QEADM solutions of Υ(ϖ1,ϖ2,ϖ3,ξ) of Problem 4.3 when ϕ=1, ρ=0.3, ξ=0.01 and =0, (d) ϖ1ϖ2-slice of solution of (c)

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Figure 12: Comparison between the exact and QEADM solutions of Φ(ϖ1,ϖ2,ϖ3,ξ) of Problem 4.3 for ϖ1,ϖ2,ϖ3= 0.4 to 1.0 at ϕ=1, ρ=0.3, and =0, for various values of ξ

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Figure 13: Comparison between the exact and QEADM solutions of Ψ(ϖ1,ϖ2,ϖ3,ξ) of Problem 4.3 for ϖ1,ϖ2,ϖ3= 0.4 to 1.0 at ϕ=1, ρ=0.3, and =0, for various values of ξ

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Figure 14: Comparison between the exact and QEADM solutions of Υ(ϖ1,ϖ2,ϖ3,ξ) of Problem 4.3 for ϖ1,ϖ2,ϖ3= 0.4 to 1.0 at ϕ=1, ρ=0.3, and =0, for various values of ξ

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6  Conclusion

The exploration of theoretical models designed to highlight physical process manifestations has indeed drawn researchers’ consideration to their potential to generate provocative results with adequate techniques. In the present investigation, we succeeded in finding analytical solutions to certain nonlinear fractional Navier-Stokes equations employing TEADM and QEADM. More precisely, the hybrid method reported the series of solutions by considering a new iterative approach. The high accuracy of the obtained results with the anticipated technique is expounded in the context of numerical simulation, and the highly complicated behavior has been captured in terms of surface plots. These kinds of investigations can help us examine more intriguing system results, and they widen the window for advancements and reformation in the notion of examining and predicting the more dynamic behavior of the commensurate models that illustrate physical processes. For future research, we extend this study by using the time delays and white noise environments to incorporate the revolutionary techniques of fractional calculus.

Funding Statement: The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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Cite This Article

Chu, Y., Rashid, S., Kubra, K. T., Inc, M., Hammouch, Z. et al. (2023). Analysis and Numerical Computations of the Multi-Dimensional, Time-Fractional Model of Navier-Stokes Equation with a New Integral Transformation. CMES-Computer Modeling in Engineering & Sciences, 136(3), 3025–3060.


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