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

1  Introduction

Physical and technical workflows are recognized by fractional calculus (FC) and are succinctly explained by fractional differential equations (FDEs) [1–4]. Admittedly, classical simulation results of integer-order derivatives, which include empirical systems, do not underlie well in numerous situations [5–9]. 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. [10–14].

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 [15–19]. 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 [20–26]. 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 [39–41] 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={ℏ(ξ):∃ℳ,κ1,κ2>0;|ℏ(ξ)|<ℳe|ξ|κ;ξ∈(−1)ȷ×[0,+∞)}.

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

E[ℏ(ξ)]=ℋ(p)=p∫0+∞ℏ(ξ)e−ξpdξ,ξ>0,κ1<p<κ2.(1)

This transform has a stronger resemblance to the Laplace transform such as ℒ(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 ℏ be a mapping of three variables ϖ1,ϖ2 and ξ. Then the triple Elzaki transform of ℏ is stated as

Eϖ1Eϖ2Eξ[ℏ(ϖ1,ϖ2,ξ)]=ℋ(η,ζ,s)=ηζs∫0+∞∫0+∞∫0+∞ℏ(ϖ1,ϖ2,ξ)e−ϖ1η−ϖ2ζ−ξsdξdϖ2dϖ1,(2)

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

Eϖ1Eϖ2Eξ[ℏϖ1(ϖ1,ϖ2,ξ)]=1ηℋ(η,ζ,s)−ℋ(0,ζ,s),Eϖ1Eϖ2Eξ[ℏϖ2(ϖ1,ϖ2,ξ)]=1ζℋ(η,ζ,s)−ℋ(η,0,s),Eϖ1Eϖ2Eξ[ℏξ(ϖ1,ϖ2,ξ)]=1sℋ(η,ζ,s)−ℋ(η,ζ,0).(3)

Analogously, we have

Eϖ1Eϖ2Eξ[ℏϖ1ϖ1(ϖ1,ϖ2,ξ)]=1η2ℋ(η,ζ,s)−ℋ(0,ζ,s)−ηℏϖ1(0,ζ,s),Eϖ1Eϖ2Eξ[ℏϖ2ϖ2(ϖ1,ϖ2,ξ)]=1ζ2ℋ(η,ζ,s)−ℋ(η,0,s)−ζℏϖ1(η,0,s),Eϖ1Eϖ2Eξ[ℏξξ(ϖ1,ϖ2,ξ)]=1s2ℋ(η,ζ,s)−ℋ(η,ζ,0)−sℏϖ1(η,ζ,0).(4)

The inverse triple Elzaki transform Eη−1Eζ−1Es−1[ℋ(η,ζ,s)]=ℏ(ϖ1,ϖ2,ξ) is stated as follows:

ℏ(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[ℋ(η,ζ,s)]=ηζs∫0+∞∫0+∞∫0+∞eηϖ1+ζϖ2+sξE(1η)E(1ζ)E(1s)dηdζds.(5)

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

c𝒟ϕℏ(ϖ1)={1Γ(ℓ−ϕ)∫0ϖ1(ϖ1−ξ)ℓ−ϕ−1ℏ(ξ)dξ,ℓ−1<ϕ<ℓ,dℓdϖ1ℓℏ(ϖ1)ℓ=ϕ.

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

Theorem 2.1. For ϕ1,ϕ2,ϕ3>0,ℓ−1<ϕ1≤ℓ,n−1<ϕ2≤n,q−1<ϕ3≤q and ℓ,n,q∈N, so that f1∈Ci(R+×R+×R+),i=max{ℓ,n,q},f1(i)∈L1[(0,a1)×(0,b1)×(0,c1)] for any a1,b1,c1>0,|f1(ϖ1,ϖ2,q)|≤ℳeϖ1ξ1+ϖ2ξ2+ϖ3ξ3,0<a1<ϖ1,0<b1<ϖ2,0<c1<ξ, then the triple Elzaki transforms of Caputo’s fractional derivatives 𝒟ξϕ1ℏ(ϖ1,ϖ2,ξ),𝒟ξϕ2ℏ(ϖ1,ϖ2,ξ) and 𝒟ξϕ3ℏ(ϖ1,ϖ2,ξ) are described by

Eϖ1Eϖ2Eξ[𝒟ξϕ1ℏ(ϖ1,ϖ2,ξ)]=s−ϕ1ℋ(η,ζ,s)−∑i=0ℓ−1s2−ϕ1+iEϖ2Eϖ1[𝒟ξiℏ(ϖ1,ϖ2,0)],ℓ−1<ϕ1<ℓ,Eϖ1Eϖ2Eξ[𝒟ϖ2ϕ2ℏ(ϖ1,ϖ2,ξ)]=ζ−ϕ2ℋ(η,ζ,s)−∑j=0n−1ζ2−ϕ2+jEϖ1Eξ[𝒟ϖ2jℏ(ϖ1,0,ξ)],n−1<ϕ2<n

and

Eϖ1Eϖ2Eξ[𝒟ϖ1ϕ3ℏ(ϖ1,ϖ2,ξ)]=η−ϕ3ℋ(η,ζ,s)−∑κ=0q−1η2−ϕ3+κEϖ2Eξ[𝒟ϖ1κℏ(0,ϖ2,ξ)],q−1<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:

∂ϕΦ∂ξϕ+Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2=ρ0(∂2Φ∂ϖ12+∂2Φ∂ϖ22)+1ρ∂σ∂ϖ1,ϖ1,ϖ2,ξ>0,∂ϕΨ∂ξϕ+Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2=ρ0(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22)−1ρ∂σ∂ϖ2,ϖ1,ϖ2,ξ>0,(6)

subject to

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

where ∂ϕ∂ξϕ is the Caputo fractional derivative and σ is the pressure. Also, if σ is known, then ℏ1=1ρ∂σ∂ϖ1 and ℏ2=1ρ∂σ∂ϖ2. To employ the TEADM, multiply (6) by ϖ1, which leads us

1sϕEϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]−∑κ=0ℓ−1Φ(κ)(η,ζ,0)s2−ϕ+κ=−Eϖ1Eϖ2Eξ(Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2)+Eϖ1Eϖ2Eξ(ρ0(∂2Φ∂ϖ12+∂2Φ∂ϖ22)+ℏ1),1sϕEϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]−∑κ=0ℓ−1Ψ(κ)(η,ζ,0)s2−ϕ+κ=−Eϖ1Eϖ2Eξ(Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2)+Eϖ1Eϖ2Eξ(ρ0(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22))−Eϖ1Eϖ2Eξ(ℏ2).(7)

By the virtue of Theorem 2.1, we obtain

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=s2ℱ1(η,ζ)−sϕEϖ1Eϖ2Eξ(Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(∂2Φ∂ϖ12+∂2Φ∂ϖ22))+sϕEϖ1Eϖ2Eξ(ℏ1),Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2𝒢1(η,ζ)−sϕEϖ1Eϖ2Eξ(Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22))−sϕEϖ1Eϖ2Eξ(ℏ2).(8)

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

Φ(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[s2ℱ1(η,ζ)−sϕEϖ1Eϖ2Eξ(Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(∂2Φ∂ϖ12+∂2Φ∂ϖ22))+sϕEϖ1Eϖ2Eξ(ℏ1)],Ψ(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[s2𝒢1(η,ζ)−sϕEϖ1Eϖ2Eξ(Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2)+sϕEϖ1Eϖ2Eξ(ρ0(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22))−sϕEϖ1Eϖ2Eξ(ℏ2)].(9)

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

Φ(ϖ1,ϖ2,ξ)=∑ℓ=0+∞Φℓ(ϖ1,ϖ2,ξ),Ψ(ϖ1,ϖ2,ξ)=∑ℓ=0+∞Ψℓ(ϖ1,ϖ2,ξ).(10)

Meanwhile, the non-linear expressions 𝒩1=Φ∂Φ∂ϖ1,𝒩2=Ψ∂Φ∂ϖ2,𝒩3=Φ∂Ψ∂ϖ1 and 𝒩4=Ψ∂Ψ∂ϖ2 are denoted by

𝒩1(Φ,Ψ)=∑ℓ=0+∞𝒜ℓ,𝒩2(Φ,Ψ)=∑ℓ=0+∞ℬℓ,𝒩3(Φ,Ψ)=∑ℓ=0+∞𝒞ℓ,𝒩4(Φ,Ψ)=∑ℓ=0+∞𝒟ℓ(11)

and are presented by the Adomian polynomials as

𝒜ℓ=1ℓ![∂ℓ∂θℓ{𝒩1(∑κ=0+∞θκΦκ,∑κ=0+∞θκΨκ)}]θ=0,ℬℓ=1ℓ![∂ℓ∂θℓ{𝒩2(∑κ=0+∞θκΦκ,∑κ=0+∞θκΨκ)}]θ=0,𝒞ℓ=1ℓ![∂ℓ∂θℓ{𝒩3(∑κ=0+∞θκΦκ,∑κ=0+∞θκΨκ)}]θ=0,𝒟ℓ=1ℓ![∂ℓ∂θℓ{𝒩4(∑κ=0+∞θκΦκ,∑κ=0+∞θκΨκ)}]θ=0.(12)

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

Eϖ1Eϖ2Eξ[∑ℓ=0+∞Φℓ(ϖ1,ϖ2,ξ)]=s2ℱ1(η,ζ)−sϕEϖ1Eϖ2Eξ(∑ℓ=0+∞(𝒜ℓ+ℬℓ))+sϕEϖ1Eϖ2Eξ(ρ0(∑ℓ=0+∞∂2Φℓ∂ϖ12+∑ℓ=0+∞∂2Φℓ∂ϖ22))+sϕEϖ1Eϖ2Eξ(ℏ1)(13)

and

Eϖ1Eϖ2Eξ[∑ℓ=0+∞Ψℓ(ϖ1,ϖ2,ξ)]=s2𝒢1(η,ζ)−sϕEϖ1Eϖ2Eξ(∑ℓ=0+∞(𝒞ℓ+𝒟ℓ))+sϕEϖ1Eϖ2Eξ(ρ0(∑ℓ=0+∞∂2Ψℓ∂ϖ12+∑ℓ=0+∞∂2Ψℓ∂ϖ22))−sϕEϖ1Eϖ2Eξ(ℏ2).(14)

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

∑ℓ=0+∞Φℓ(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[s2ℱ1(η,ζ)]−Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(∑ℓ=0+∞(𝒜ℓ+ℬℓ))]+Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(ρ0(∑ℓ=0+∞∂2Φℓ∂ϖ12+∑ℓ=0+∞∂2Φℓ∂ϖ22))+sϕEϖ1Eϖ2Eξ(ℏ1)](15)

and

∑ℓ=0+∞Ψℓ(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[s2𝒢1(η,ζ)]−Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(∑ℓ=0+∞(𝒞ℓ+𝒟ℓ))]+Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(ρ0(∑ℓ=0+∞∂2Ψℓ∂ϖ12+∑ℓ=0+∞∂2Ψℓ∂ϖ22))−sϕEϖ1Eϖ2Eξ(ℏ2)].(16)

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

Φ0(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[s2ℱ1(η,ζ)],Ψ0(ϖ1,ϖ2,ξ)=Eη−1Eζ−1Es−1[s2𝒢1(η,ζ)],(17)

and the rest of the components Φℓ+1,ℓ≥0 are presented by

Φℓ+1(ϖ1,ϖ2,ξ)=−Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(𝒜ℓ+ℬℓ)]+Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(ρ0(∂2Φℓ∂ϖ12+∂2Φℓ∂ϖ22))+sϕEϖ1Eϖ2Eξ(ℏ1)].(18)

and

Ψℓ+1(ϖ1,ϖ2,ξ)=−Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(𝒞ℓ+𝒟ℓ)]+Eη−1Eζ−1Es−1[sϕEϖ1Eϖ2Eξ(ρ0(∂2Ψℓ∂ϖ12+∂2Ψℓ∂ϖ22))−sϕEϖ1Eϖ2Eξ(ℏ2)].(19)

where Eϖ1Eϖ2Eξ is the triple E-transform in respecting ϖ1,ϖ2,ξ1 and Eη−1Eζ−1Es−1 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:

{∂ϕΦ∂ξϕ+Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2=ρ(∂2Φ∂ϖ12+∂2Φ∂ϖ22)+ℏ,ϖ1,ϖ2,ξ>0,∂ϕΨ∂ξϕ+Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2=ρ(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22)−ℏ,ϖ1,ϖ2,ξ>0,ℓ−1<ϕ<ℓ;(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[∂ϕΦ∂ξϕ+Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2]=Eϖ1Eϖ2Eξ[ρ(∂2Φ∂ϖ12+∂2Φ∂ϖ22)+ℏ],Eϖ1Eϖ2Eξ[∂ϕΨ∂ξϕ+Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2]=Eϖ1Eϖ2Eξ[ρ(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22)−ℏ].

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

1sϕEϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]−∑κ=0ℓ−1Φ(κ)(η,ζ,0)s2−ϕ+κ=−Eϖ1Eϖ2Eξ[Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2]+Eϖ1Eϖ2Eξ[ρ(∂2Φ∂ϖ12+∂2Φ∂ϖ22)+ℏ],1sϕEϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]−∑κ=0ℓ−1Ψ(κ)(η,ζ,0)s2−ϕ+κ=−Eϖ1Eϖ2Eξ[Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2]+Eϖ1Eϖ2Eξ[ρ(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22)−ℏ].(22)

According to initial conditions and simple computations yields

Eϖ1Eϖ2Eξ[Φ(ϖ1,ϖ2,ξ)]=−s2Eϖ1Eϖ2[sin⁡(ϖ1+ϖ2)]−sϕEϖ1Eϖ2Eξ[Φ∂Φ∂ϖ1+Ψ∂Φ∂ϖ2]+sϕEϖ1Eϖ2Eξ[ρ(∂2Φ∂ϖ12+∂2Φ∂ϖ22)+ℏ],Eϖ1Eϖ2Eξ[Ψ(ϖ1,ϖ2,ξ)]=s2Eϖ1Eϖ2[sin⁡(ϖ1+ϖ2)]−sϕEϖ1Eϖ2Eξ[Φ∂Ψ∂ϖ1+Ψ∂Ψ∂ϖ2]+sϕEϖ1Eϖ2Eξ[ρ(∂2Ψ∂ϖ12+∂2Ψ∂ϖ22)−ℏ].