Research article Topical Sections

Estimation techniques utilizing dual auxiliary variables in stratified two-phase sampling

  • In this research paper, an improved set of estimators for finding the finite population variance of a study variable under a stratified two-phase sampling design is introduced. These estimators rely on information about extreme values and the ranks of an auxiliary variable. We examined the properties of these estimators using first-order approximation, focusing on biases and mean squared errors (MSEs). Additionally, we conducted an extensive simulation study to evaluate their performance and validate our theoretical insights. Furthermore, in the application section, we employed some datasets to further assess the performances of our estimators as compared to other existing estimators. The results demonstrated that S2Q2 was the best-performing estimator, and significantly outperforms existing estimators, achieving a percent relative efficiency (PRE) in the exponential distribution as high as 385.467. The percent relative efficiency values were continuously higher than 100 in a variety of situations, with values as high as 353.129 in other distributions like the uniform and gamma. The suggested estimators are superior to the conventional estimators, as demonstrated by empirical assessments using datasets, where percent relative efficiency improvements ranged from 115.026 to 139.897. These results highlight the robustness and applicability of the proposed class of estimators in real-world sampling.

    Citation: Olayan Albalawi. Estimation techniques utilizing dual auxiliary variables in stratified two-phase sampling[J]. AIMS Mathematics, 2024, 9(11): 33139-33160. doi: 10.3934/math.20241582

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  • In this research paper, an improved set of estimators for finding the finite population variance of a study variable under a stratified two-phase sampling design is introduced. These estimators rely on information about extreme values and the ranks of an auxiliary variable. We examined the properties of these estimators using first-order approximation, focusing on biases and mean squared errors (MSEs). Additionally, we conducted an extensive simulation study to evaluate their performance and validate our theoretical insights. Furthermore, in the application section, we employed some datasets to further assess the performances of our estimators as compared to other existing estimators. The results demonstrated that S2Q2 was the best-performing estimator, and significantly outperforms existing estimators, achieving a percent relative efficiency (PRE) in the exponential distribution as high as 385.467. The percent relative efficiency values were continuously higher than 100 in a variety of situations, with values as high as 353.129 in other distributions like the uniform and gamma. The suggested estimators are superior to the conventional estimators, as demonstrated by empirical assessments using datasets, where percent relative efficiency improvements ranged from 115.026 to 139.897. These results highlight the robustness and applicability of the proposed class of estimators in real-world sampling.



    Fractional Calculus (FC) is the subject dealing with the derivative and integration of non-integer orders. In [1], the researchers generalize the classical diffusion and wave equations to different physical processes, such as slow diffusion, classical diffusion, diffusion-wave hybrid, and the classical wave equation. Differential equations (DEs) are valuable tools for describing critical natural phenomena such as phase transition, electrochemistry, electromagnetism, filtration, acoustics, cosmology, biochemistry, and the dynamics of biological groups [2]. The present idea of DEs has been further extended by using different derivatives operators with fractional orders. The mathematical models with fractional derivatives and integrations are more accurate and adequate than ordinary models[3]. There are various applications of fractional calculus in applied sciences like Poisson-Nerst Planck diffusion [4], earth quack nonlinear oscillation [5], air foil, chaos theory [6], fluid traffic [7], Financial [8], Zener [9], Electrodynamics [10], Cancer chemotherapy [11], Hepatitis B Virus [12], Tuberculosis [13], Pine wilt disease [14], Diabetes [15] and many other various applications in applied sciences [16,17]. The applications that have been mentioned above-attracted researchers to the subject and developed various sophisticated mathematical models in terms of fractional integrals and derivatives. These models are further analyzed and solved by using some numerical and analytical techniques such as the functional constraint's method [18], the iterated pseudo-spectral method [19], reduced differential transforms algorithm (RDTA)[20], q-homotopy analysis Shehu transform algorithm (q-HASTA) [20], predictor-corrector algorithm [21], Adams-Bashforth-Moulton algorithm [22], and the numerical method for DEs in fractional order: based on the definition of Grunwald-Letnikov (GL) fractional derivative [22].

    One of the most effective tools for researchers to simulate physical phenomena in nature, including fluid dynamics, mathematical biology, quantum physics, linear optics, and chemical kinetics, are considered to be fractional partial differential equations (FPDEs) [23]. Other physical phenomena are accurately modeled by FPDEs, such as the analytical solution of a coupled system of non-linear PDEs is presented in [24]; the solution of non-linear ODEs, which is previously achieved in [25]; non-linear PDEs, which is presented in[26]; fractional telegraph equations which are presented in [27], the unsteady fractional flow of a polytrophic gas model which is introduced in [28], and the fractional Fokker-Plank equation and the Schrödinger equation can be found in [29].

    Many researchers have tried hard to find the solutions of these FPDES through numerous techniques such as Homotopy perturbation transform method (HPTM) [30], Homotopy perturbation method (HPM) [31], Homotopy analysis method (HAM) [32], the Finite differences method[33], the multiple exponential function algorithms and variational iteration method (VIM) [34]. Moreover, Feng's first integral method[35], Abazari and Ganji provide the reduce differential transform method (RDTM) for PDEs [36], Mehshless method (MM) [37], Laplace Adomian decomposition method (LADM)[38], and modified homotopy perturbation method (MHPM) [39].

    The Riemann-Liouville (R-L) integral [40], Caputo [41,42], Caputo-Fabrizio[43] and Atangana-Baleanu[44] have been developed using the various fractional derivative operators (FDOs) to provide a precise meaning to the derivative with the optimal order of the derivative. These fractional derivative operators differ fundamentally from one another in that they each have a variety of kernels from which to choose depending on the demands of a given application. "Be aware that these three definitions of fractional derivatives have advantages and disadvantages. An arbitrary function does not need to be continuous at the origin or differentiable in order to take the RL fractional derivative. The fact that the RL derivative of a constant is not zero is one limitation in the RL's capacity for simulating real-world phenomena. On the other hand, the Caputo fractional derivative has the advantage of agreeing with the usual initial and boundary conditions included in the formulation of the equation. The disadvantage of Caputo's derivative is that we must first find the function's derivative in order to estimate the fractional derivative of a function in the Caputo sense. Atangana-Baleanu derivative tries to address some of the drawbacks from earlier. Besides, the fractional order derivative based on the AB operator is defined using limits rather than integrals. The details can be found in [41,42,43,44,45]."

    In this study, we used three different FDOs: the Caputo fractional differential operator (C-FDO), the Caputo-Fabrizio fractional differential operator (C-FFDO), and the Atangana-Baleanu fractional differential operator (A-BFDO) to achieve the analytical solutions of the non-linear fractional order 3D Navier Stokes equations with g1=g2=g3=0. The approximate analytical solution for time-fractional non-homogeneous Cauchy equation [20,46] are obtained with the help of a new iterative transform method (NITM)[47]. NITM is an analytical technique in which the solution is obtained by an infinite series with higher convergent components. The accuracy of the proposed method is shown by graphs and tables with various fractional operators. The structure of this research article is presented as follows: in Section 2, we discuss some preliminary concepts. In Sections 3 and 4, we discuss the procedure of NITM for FPDEs and the system of FPDEs restrictively. The numerical results are discussed in Section 5. Section 6 is the conclusion section.

    In this part of the paper, the preliminary concepts and important definitions are presented, which are very useful for the continuation of this research work.

    Definition 2.1. The Caputo fractional derivative operator (C-FDO) is given as [48]

    Dδˆtμ(ˆt)=1Γ(nδ)ˆt0(ˆtτ)nδ1f(n)(τ)dτ,   n1<δn  nN,  ˆt>0. (2.1)

    Definition 2.2. The Laplace transform (LT) of C-FDO is given as [48]

    Ł(Dδˆtμ(ˆt))=sδŁ[μ(ˆt)]n1k=0snk1Uk(0+),Ł(Dδˆtμ(ˆt))=sδU(s)sδ1U(0). (2.2)

    Definition 2.3. The formulation of the Caputo-Fabrizio fractional differential operator (C-FFDO) is [43]

    CFDδˆtμ(ˆt)=B(δ)1δˆt0eδ1δ(ˆtτ)τμ(τ)dτ (2.3)

    the function B(δ) is the normalization function depending on δB(0)=B(1)=1.

    Definition 2.4. The LT of the C-FFDO is define as [43]

    Ł(Dn+δˆtμ(ˆt))(s)=11δŁ(μn+1(ˆt))Ł(exp(δ1δˆt)),Ł(Dn+δˆtμ(ˆt))(s)=sn+1Ł(μ(ˆt))snμ(0)sn1μ(0)μ(n)(0)s+δ(1s).

    In particular, we have

    Ł(Dn+δˆtμ(ˆt))(s)=sŁ(μ(ˆt))s+δ(1s),    n=0,Ł(D1+δˆtμ(ˆt))(s)=s2Ł(μ(ˆt))sμ(0)μ(0)s+δ(1s),    n=1.

    Definition 2.5. The Mittag-leffler function was introduced in 1903[49], and is given as

    Eδ(Z)=n=0znΓ(δn+1),   where  z ϵ C.

    Definition 2.6. The Atangana-Baleanu fractional derivative operator (ABFDO) is defined as [50]

    ABCDδtμ(ˆt)=B(δ)1δˆt0Eδ(δ1δ(ˆtτ)δ)τμ(τ)dτ, (2.4)

    where B() is a normalization function such that B(0)=B(1)=1 and Eδ is the Mittag-leffler function defined in Definition 2.5.

    Definition 2.7. The LT of the ABFDO is define as[50]

    Ł(ABCDδˆtμ(ˆt))=sδ1B(δ)sδ(1δ)+δ(sμ(s)μ(0)). (2.5)

    Definition 2.8. The Aboodh transform (AT) is define by [51]

    A[u(ˆt)]=U(s)=1s0u(ˆt)esˆtdˆt,ˆt0,k1sk2. (2.6)

    Theorem 1. The AT of C-FDO is given as [51]

    A(Dδˆtμ(ˆt))=sδA[μ(ˆt)]n1k=0snk1Uk(0+). (2.7)

    Theorem 2. The AT of C-FFDO is defined as [52]

    A(CFCDδˆtμ(ˆt))=B(δ)(U(s)u(0)s). (2.8)

    Theorem 3. If U(s) is the AT of Atangana-Baleanu-Caputo operator then [53]

    U(s)=A(ABCDδˆtμ(ˆt))(s)=B(δ)s(1δ){sδŁ[u(ˆt)]sδ1u(0)sδ+δ1δ}. (2.9)

    Proofs. The proofs of these Theorems 1–3 can be found in [51,52,53].

    To understand the basic methodology of the NITM within three different fractional derivative operators (FDOs), we consider the following three different cases.

    Let us consider a general non-homogenous FPDEs within C-FDO of the form,

    Dδ+mˆtu(ξo,ˆt)=f(ξo,ˆt)+Lu(ξo,ˆt)+Nu(ξo,ˆt),    m1<δm,  mN, (3.1)

    having initial condition

    ku(ξo,0)ˆtk=θk(ξo),    k=0,1,,n1. (3.2)

    In Eq (3.1) L is linear and N is the non-linear operator, while f(ξo,ˆt) is a source term.

    Applying} the AT to Eq (3.1), we obtain

    A(u(ξo,ˆt))=ϑ(ξo,s)+(1sδ+m)A(Lu(ξo,ˆt)+Nu(ξo,ˆt)), (3.3)

    where

    ϑ(ξo,s)=1sm+1(smθ0(ξo)+sm1θ1(ξo)++θm(ξo))+1sδ+mf(ξo,s).

    Taking the inverse AT on Eq (3.3), we obtain

    u(ξo,ˆt)=ϑ(ξo,ˆt)+A(1sδ+m)A(Lu(ξo,ˆt)+Nu(ξo,ˆt)). (3.4)

    Next, we apply the New Iterative Method present in [54], we obtain an infinite series solution

    u(ξo,ˆt)==0u(ξo,ˆt). (3.5)

    Since L is linear

    L(=0u(ξo,ˆt))==0L(u(ξo,ˆt)). (3.6)

    The nonlinear operator N becomes

    N(=0u(ξo,ˆt))=N(u0(ξo,ˆt))+=0[N(i=0ui(ξo,ˆt))N(1i=0ui(ξo,ˆt))]. (3.7)

    In view of Eqs (3.3–3.5) and Eq (3.6) is equivalent to

    =0u(ξo,ˆt)=ϑ(ξo,ˆt)+A[(1sδ)A(=0Lu(ξo,ˆt))]+A[(1sδ)A(N(u0(ξo,ˆt)))+=0{N(i=0ui(ξo,ˆt))N(1i=0ui(ξo,ˆt))}]. (3.8)

    Next, we consider the following recurrence relation,

    u0(ξo,ˆt)=ϑ(ξo,ˆt),u1(ξo,ˆt)=A[(1sδ)A(L(u0(ξo,ˆt))+N(u0(ξo,ˆt)))], (3.9)
                                                   un+1(ξo,ˆt)=A(1sδ)A[(L(un(ξo,ˆt)))+j=1{N(ni=0ui(ξo,ˆt))N(n1i=0ui(ξo,ˆt))}]. (3.10)

    The NITM series form solution is given by

    u(ξo,ˆt)=u0+u1+u2++un. (3.11)

    We have a general non-homogenous AB-FDE of the form

    Dδ+mˆtu(ξo,ˆt)=f(ξo,ˆt)+Lu(ξo,ˆt)+Nu(ξo,ˆt),    m1<δm,  mN, (3.12)

    having initial condition

    ku(ξo,0)ˆtk=θk(ξo),    k=0,1,,n1. (3.13)

    In Eq (3.12) L is linear and N is the non-linear operator, while f(ξo,ˆt) is a source term.

    Taking the AT to Eq (3.12), we get

    A(u(ξo,ˆt))=χ(ξo,s)+((1δ)sδsδ)A(L(ξo,ˆt)+Nu(ξo,ˆt)), (3.14)

    where

    χ(ξo,s)=1sm+1(smθ0(ξo)+sm1θ1(ξo)++θm(ξo))+(1δ)sδ+δsδ+mf(ξo,s).

    Taking the inverse AT on Eq (3.14), we obtain

    u(ξo,ˆt)=ϑ(ξo,ˆt)+A((1δ)sδsδ)A(Lu(ξo,ˆt)+Nu(ξo,ˆt)), (3.15)

    where the source term is denoted by ϑ(ξo,ˆt). Next we apply New Iterative Method introduced in [54], we obtain an infinite series solution

    u(ξo,ˆt)==0u(ξo,ˆt). (3.16)

    Since L is linear

    L(=0u(ξo,ˆt))==0L(u(ξo,ˆt)). (3.17)

    The nonlinear term N is decomposed by

    N(=0u(ξo,ˆt))=N(u0(ξo,ˆt))+=0[N(i=0ui(ξo,ˆt))N(1i=0ui(ξo,ˆt))]. (3.18)

    In view of Eqs (3.15–3.17) and Eq (3.18) is equivalent to,

    =0u(ξo,ˆt)=ϑ(ξo,ˆt)+A((1δ)sδsδ)A[(L(un(ξo,ˆt)))+=0{N(i=0ui(ξo,ˆt))N(1i=0ui(ξo,ˆt))}], (3.19)

    furthermore, consider the recursive relation of the following form,

    u0(ξo,ˆt)=ϑ(ξo,ˆt),u1(ξo,ˆt)=A[((1δ)sδsδ)A(u(u0(ξo,ˆt))+N(u0(ξo,ˆt)))], (3.20)
                                                   un+1(ξo,ˆt)=A((1δ)sδsδ)A[(L(un(ξo,ˆt)))+j=1{N(ni=0ui(ξo,ˆt))N(n1i=0ui(ξo,ˆt))}]. (3.21)

    The NITM approximate solution is given by

    u(ξo,ˆt)=u0+u1+u2++un. (3.22)

    We have a general non-homogenous FPDEs of the form

    Dδ+mˆtu(ξo,ˆt)=f(ξo,ˆt)+Lu(ξo,ˆt)+Nu(ξo,ˆt),    m1<δm,  mN, (3.23)

    having initial condition

    ku(ξo,0)ˆtk=θk(ξo),    k=0,1,,m1, (3.24)

    where Dδ+mˆt is C-FFDO and L and N are the linear and non-linear operator respectively, while f(ξo,ˆt) is a source term.

    Taking the AT to Eq (3.23), we obtain

    A(u(ξo,ˆt))=ϑ(ξo,s)+(s+δ(1s)sm+1)A(Lu(ξo,ˆt)+Nu(ξo,ˆt)), (3.25)

    where

    ϑ(ξo,s)=1sm+1(smθ0(ξo)+sm1θ1(ξo)++θm(ξo)+s+δ(1s)sm+1f(ξo,s).

    Taking the inverse AT to Eq (3.25), we obtain

    u(ξo,ˆt)=ϑ(ξo,ˆt)+A(s+δ(1s)sm+1)A(Lu(ξo,ˆt)+Nu(ξo,ˆt)), (3.26)

    where the source term is denoted by ϑ(ξo,ˆt). Next, we apply the New Iterative Method introduced in [54], we obtain an infinite series solution

    u(ξo,ˆt)==0u(ξo,ˆt). (3.27)

    Since L is linear

    L(=0u(ξo,ˆt))==0L(u(ξo,ˆt)). (3.28)

    The nonlinear term N is decomposed by

    L(=0u(ξo,ˆt))=N(u0(ξo,ˆt))+=0[N(i=0ui(ξo,ˆt))N(1i=0ui(ξo,ˆt))]. (3.29)

    In view of Eqs (3.26–3.28) and Eq (3.29) is equivalent to

    =0u(ξo,ˆt)=ϑ(ξo,ˆt)+A(s+δ(1s)sm+1)A[(L(un(ξo,ˆt)))+=0{N(i=0ui(ξo,ˆt))N(1i=0ui(ξo,ˆt))}]. (3.30)

    Next, consider the recursive relation of the following form,

    u0(ξo,ˆt)=ϑ(ξo,ˆt),u1(ξo,ˆt)=A[(s+δ(1s)sm+1)A(L(u0(ξo,ˆt))+N(u0(ξo,ˆt)))], (3.31)
                                                   un+1(ξo,ˆt)=A(s+δ(1s)sm+1)A[(L(un(ξo,ˆt)))+j=1{N(ni=0ui(ξo,ˆt))N(n1i=0ui(ξo,ˆt))}]. (3.32)

    The NITM approximate solution is given by

    u(ξo,ˆt)=u0+u1+u2++un. (3.33)

    We give the following methods to study the NITM process.

    Consider a general system of non-homogenous Caputo type FPDEs of the form

    Dδ+mˆtu(ξo,ξ1,ξ2,ˆt)=f1(ξo,ξ1,ξ2,ˆt)+Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN,Dδ+mˆtμ(ξo,ξ1,ξ2,ˆt)=f2(ξo,ξ1,ξ2,ˆt)+Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN,Dδ+mˆtν(ξo,ξ1,ξ2,ˆt)=f3(ξo,ξ1,ξ2,ˆt)+Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN, (4.1)

    having initial condition

    ku(ξo,ξ1,ξ2,0)ˆtk=θk(ξo,ξ1,ξ2),    k=0,1,,1,kμ(ξo,ξ1,ξ2,0)ˆtk=k(ξo,ξ1,ξ2),    k=0,1,,1,kν(ξo,ξ1,ξ2,0)ˆtk=jk(ξo,ξ1,ξ2),    k=0,1,,1. (4.2)

    In Eq (4.1) f1(ξo,ξ1,ξ2,ˆt), f2(ξo,ξ1,ξ2,ˆt) and f3(ξo,ξ1,ξ2,ˆt), are the source terms and L and N are the linear and non-linear operator respectively, while Dδ+mˆt is a Caputo type fractional derivative operator.

    Taking the AT of Eq (4.1), we have

    A(u(ξo,ξ1,ξ2,ˆt))=ϑ1(ξo,ξ1,ξ2,s)+(1sδ)A(Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt)),A(μ(ξo,ξ1,ξ2,ˆt))=ϑ2(ξo,ξ1,ξ2,s)+(1sδ)A(Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt)),A(ν(ξo,ξ1,ξ2,ˆt))=ϑ3(ξo,ξ1,ξ2,s)+(1sδ)A(Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt)), (4.3)

    where

    ϑ1(ξo,ξ1,ξ2,s)=1sm+1(smθ0(ξo,ξ1,ξ2)+sm1θ1(ξo,ξ1,ξ2)++θm(ξo,ξ1,ξ2))+1sδ+mf1(ξo,ξ1,ξ2,s).
    ϑ2(ξo,ξ1,ξ2,s)=1sm+1(sm0(ξo,ξ1,ξ2)+sm11(ξo,ξ1,ξ2)++m(ξo,ξ1,ξ2))+1sδ+mf2(ξo,ξ1,ξ2,s).
    ϑ3(ξo,ξ1,ξ2,s)=1sm+1(smj0(ξo,ξ1,ξ2)+sm1j1(ξo,ξ1,ξ2)++jm(ξo,ξ1,ξ2))+1sδ+mf3(ξo,ξ1,ξ2,s).

    Taking the inverse AT on Eq (4.3), we obtain

    u(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt)+A(1sδ)A(Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt)),μ(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt)+A(1sδ)A(Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt)),ν(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt)+A(1sδ)A(Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt)). (4.4)

    Now, the source terms for the given system are presented as ϑ1(ξo,ξ1,ξ2,ˆt), ϑ2(ξo,ξ1,ξ2,ˆt) and ϑ3(ξo,ξ1,ξ2,ˆt) respectively.

    Next, applying the New Iterative Method introduced in [54] and considering the solution of an infinite series form,

    u(ξo,ξ1,ξ2,ˆt)==0u(ξo,ξ1,ξ2,ˆt),μ(ξo,ξ1,ξ2,ˆt)==0u(ξo,ξ1,ξ2,ˆt),ν(ξo,ξ1,ξ2,ˆt)==0u(ξo,ξ1,ξ2,ˆt). (4.5)

    Since L is linear

    u(=0u(ξo,ξ1,ξ2,ˆt))==0u(u(ξo,ξ1,ξ2,ˆt)),u(=0μ(ξo,ξ1,ξ2,ˆt))==0u(μ(ξo,ξ1,ξ2,ˆt)),u(=0ν(ξo,ξ1,ξ2,ˆt))==0u(ν(ξo,ξ1,ξ2,ˆt)), (4.6)

    and the nonlinear term N is decomposed as

    u(=0u(ξo,ξ1,ξ2,ˆt))=N(u0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0ui(ξo,ξ1,ξ2,ˆt))N(1i=0ui(ξo,ξ1,ξ2,ˆt))],u(=0μ(ξo,ξ1,ξ2,ˆt))=N(μ0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0μi(ξo,ξ1,ξ2,ˆt))N(1i=0μi(ξo,ξ1,ξ2,ˆt))],u(=0ν(ξo,ξ1,ξ2,ˆt))=N(ν0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0νi(ξo,ξ1,ξ2,ˆt))N(1i=0νi(ξo,ξ1,ξ2,ˆt))]. (4.7)

    In view of Eqs (4.4–4.6) and Eq (4.7), is equivalent to

    =0u(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt)+A[(1sδ)A(=0Lu(ξo,ξ1,ξ2,ˆt))]+A[(1sδ)×A(L(un(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0ui(ξo,ξ1,ξ2,ˆt))N(1i=0ui(ξo,ξ1,ξ2,ˆt))}],=0μ(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt)+A[(1sδ)A(=0Lμ(ξo,ξ1,ξ2,ˆt))]+A[(1sδ)×A(N(μ0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0μi(ξo,ξ1,ξ2,ˆt))N(1i=0μi(ξo,ξ1,ξ2,ˆt))}],=0ν(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt)+A[(1sδ)A(=0Lν(ξo,ξ1,ξ2,ˆt))]+A[(1sδ)×A(N(ν0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0νi(ξo,ξ1,ξ2,ˆt))N(1i=0νi(ξo,ξ1,ξ2,ˆt))}], (4.8)

    furthermore, we obtained a recursive relation of the following form

    u0(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt),μ0(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt),ν0(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt),u1(ξo,ξ1,ξ2,ˆt)=A[(1sδ)A(L(u0(ξo,ξ1,ξ2,ˆt))+L(un(ξo,ξ1,ξ2,ˆt)))],μ1(ξo,ξ1,ξ2,ˆt)=A[(1sδ)A(L(μ0(ξo,ξ1,ξ2,ˆt))+N(μ0(ξo,ξ1,ξ2,ˆt)))],ν1(ξo,ξ1,ξ2,ˆt)=A[(1sδ)A(L(ν0(ξo,ξ1,ξ2,ˆt))+N(ν0(ξo,ξ1,ξ2,ˆt)))], (4.9)
    un+1(ξo,ξ1,ξ2,ˆt)=A(1sδ)A[(L(un(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0ui(ξo,ξ1,ξ2,ˆt))N(n1i=0ui(ξo,ξ1,ξ2,ˆt))}],μn+1(ξo,ξ1,ξ2,ˆt)=A(1sδ)A[(L(μn(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0μi(ξo,ξ1,ξ2,ˆt))N(n1i=0μi(ξo,ξ1,ξ2,ˆt))}],νn+1(ξo,ξ1,ξ2,ˆt)=A(1sδ)A[(L(νn(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0νi(ξo,ξ1,ξ2,ˆt))N(n1i=0νi(ξo,ξ1,ξ2,ˆt))}]. (4.10)

    The NITM approximate solution is given by

    u(ξo,ξ1,ξ2,ˆt)=u0+u1+u2++un,μ(ξo,ξ1,ξ2,ˆt)=μ0+μ1+μ2++μn,ν(ξo,ξ1,ξ2,ˆt)=ν0+ν1+ν2++un. (4.11)

    We have a general non-homogenous system of C-FFDE of the form

    Dδ+mˆtu(ξo,ξ1,ξ2,ˆt)=f1(ξo,ξ1,ξ2,ˆt)+Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN,Dδ+mˆtμ(ξo,ξ1,ξ2,ˆt)=f2(ξo,ξ1,ξ2,ˆt)+Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN,Dδ+mˆtν(ξo,ξ1,ξ2,ˆt)=f3(ξo,ξ1,ξ2,ˆt)+Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN, (4.12)

    having initial condition

    ku(ξo,ξ1,ξ2,0)ˆtk=θk(ξo,ξ1,ξ2),    k=0,1,,1,kμ(ξo,ξ1,ξ2,0)ˆtk=k(ξo,ξ1,ξ2),    k=0,1,,1,kν(ξo,ξ1,ξ2,0)ˆtk=jk(ξo,ξ1,ξ2),    k=0,1,,1. (4.13)

    In Eq (4.12) f1(ξo,ξ1,ξ2,ˆt), f2(ξo,ξ1,ξ2,ˆt) and f3(ξo,ξ1,ξ2,ˆt), are the source terms, L and N are the linear and non-linear operator respectively, while Dδ+mˆt is Atangana-Baleanu type fractional derivative operator.

    Applying the AT to Eq (4.12), we have

    A(u(ξo,ξ1,ξ2,ˆt))=ϑ1(ξo,ξ1,ξ2,s)+((1δ)sδ+δsδ)A(Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt)),A(μ(ξo,ξ1,ξ2,ˆt))=ϑ2(ξo,ξ1,ξ2,s)+((1δ)sδ+δsδ)A(Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt)),A(ν(ξo,ξ1,ξ2,ˆt))=ϑ3(ξo,ξ1,ξ2,s)+((1δ)sδ+δsδ)A(Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt)), (4.14)

    where

    ϑ1(ξo,ξ1,ξ2,s)=1sm+1(smθ0(ξo,ξ1,ξ2)+sm1θ1(ξo,ξ1,ξ2)++θm(ξo,ξ1,ξ2))+(1δ)sδ+δsδ+mf1(ξo,ξ1,ξ2,s).
    ϑ2(ξo,ξ1,ξ2,s)=1sm+1(sm0(ξo,ξ1,ξ2)+sm11(ξo,ξ1,ξ2)++m(ξo,ξ1,ξ2))+(1δ)sδ+δsδ+mf2(ξo,ξ1,ξ2,s).
    ϑ3(ξo,ξ1,ξ2,s)=1sm+1(smg0(ξo,ξ1,ξ2)+sm1g1(ξo,ξ1,ξ2)++gm(ξo,ξ1,ξ2))+(1δ)sδ+δsδ+mf3(ξo,ξ1,ξ2,s).

    Taking the inverse AT to Eq (4.14), we obtain

    u(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt)+A((1δ)sδ+δsδ)A(Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt)),μ(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt)+A((1δ)sδ+δsδ)A(Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt)),ν(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt)+A((1δ)sδ+δsδ)A(Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt)), (4.15)

    where the source term is denoted by ϑ(ξo,ξ1,ξ2,ˆt). Further we apply NIM introduced in [54]. We consider the solution as an infinite series given as

    u(ξo,ξ1,ξ2,ˆt)==0u(ξo,ξ1,ξ2,ˆt),μ(ξo,ξ1,ξ2,ˆt)==0μ(ξo,ξ1,ξ2,ˆt),ν(ξo,ξ1,ξ2,ˆt)==0ν(ξo,ξ1,ξ2,ˆt). (4.16)

    Since L is linear

    L(=0u(ξo,ξ1,ξ2,ˆt))==0L(u(ξo,ξ1,ξ2,ˆt)),L(=0μ(ξo,ξ1,ξ2,ˆt))==0L(μ(ξo,ξ1,ξ2,ˆt)),L(=0ν(ξo,ξ1,ξ2,ˆt))==0L(ν(ξo,ξ1,ξ2,ˆt)). (4.17)

    The nonlinear operator N is decomposed as

    N(=0u(ξo,ξ1,ξ2,ˆt))=N(u0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0ui(ξo,ξ1,ξ2,ˆt))N(1i=0ui(ξo,ξ1,ξ2,ˆt))],N(=0μ(ξo,ξ1,ξ2,ˆt))=N(μ0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0μi(ξo,ξ1,ξ2,ˆt))N(1i=0μi(ξo,ξ1,ξ2,ˆt))],N(=0ν(ξo,ξ1,ξ2,ˆt))=N(ν0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0νi(ξo,ξ1,ξ2,ˆt))N(1i=0νi(ξo,ξ1,ξ2,ˆt))]. (4.18)

    In view of Eqs (4.15–4.17) and Eq (4.18) is equivalent to

    =0u(ξo,ξ1,ξ2,ˆt)=ϑ(ξo,ξ1,ξ2,ˆt)+A[((1δ)sδ+δsδ)A(N(u0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0ui(ξo,ξ1,ξ2,ˆt))N(1i=0ui(ξo,ξ1,ξ2,ˆt))}],=0μ(ξo,ξ1,ξ2,ˆt)=ϑ(ξo,ξ1,ξ2,ˆt)+A[((1δ)sδ+δsδ)A(=0Lμ(ξo,ξ1,ξ2,ˆt))]+A[((1δ)sδ+δsδ)A(N(μ0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0μi(ξo,ξ1,ξ2,ˆt))N(1i=0μi(ξo,ξ1,ξ2,ˆt))}],=0ν(ξo,ξ1,ξ2,ˆt)=ϑ(ξo,ξ1,ξ2,ˆt)+A[((1δ)sδ+δsδ)A(=0Lν(ξo,ξ1,ξ2,ˆt))]+A[((1δ)sδ+δsδ)A(N(ν0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0νi(ξo,ξ1,ξ2,ˆt))N(1i=0νi(ξo,ξ1,ξ2,ˆt))}], (4.19)

    considering the following recursive relation

    u0(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt),μ0(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt),ν0(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt),u1(ξo,ξ1,ξ2,ˆt)=A[((1δ)sδ+δsδ)A(L(u0(ξo,ξ1,ξ2,ˆt))+N(u0(ξo,ξ1,ξ2,ˆt)))],μ1(ξo,ξ1,ξ2,ˆt)=A[((1δ)sδ+δsδ)A(L(μ0(ξo,ξ1,ξ2,ˆt))+N(μ0(ξo,ξ1,ξ2,ˆt)))],ν1(ξo,ξ1,ξ2,ˆt)=A[((1δ)sδ+δsδ)A(L(ν0(ξo,ξ1,ξ2,ˆt))+N(ν0(ξo,ξ1,ξ2,ˆt)))], (4.20)
                                                                                                                                              un+1(ξo,ξ1,ξ2,ˆt)=A((1δ)sδ+δsδ)A[(L(un(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0ui(ξo,ξ1,ξ2,ˆt))N(n1i=0ui(ξo,ξ1,ξ2,ˆt))}],μn+1(ξo,ξ1,ξ2,ˆt)=A((1δ)sδ+δsδ)A[(N(μ0(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0μi(ξo,ξ1,ξ2,ˆt))N(n1i=0μi(ξo,ξ1,ξ2,ˆt))}],νn+1(ξo,ξ1,ξ2,ˆt)=A((1δ)sδ+δsδ)A[(N(ν0(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0νi(ξo,ξ1,ξ2,ˆt))N(n1i=0νi(ξo,ξ1,ξ2,ˆt))}]. (4.21)

    The NITM approximate solution is given by

    u(ξo,ξ1,ξ2,ˆt)=u0+u1+u2++un,μ(ξo,ξ1,ξ2,ˆt)=μ0+μ1+μ2++μn,ν(ξo,ξ1,ξ2,ˆt)=ν0+ν1+ν2++νn. (4.22)

    Let us consider a general non-homogenous system of FPDEs in terms of C-FPDO,

    Dδ+mˆtu(ξo,ξ1,ξ2,ˆt)=f1(ξo,ξ1,ξ2,ˆt)+Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN,Dδ+mˆtμ(ξo,ξ1,ξ2,ˆt)=f2(ξo,ξ1,ξ2,ˆt)+Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN,Dδ+mˆtν(ξo,ξ1,ξ2,ˆt)=f3(ξo,ξ1,ξ2,ˆt)+Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt),    m1<δm,  mN, (4.23)

    having initial condition

    ku(ξo,ξ1,ξ2,0)ˆtk=gk(ξo,ξ1,ξ2),    k=0,1,,1,kμ(ξo,ξ1,ξ2,0)ˆtk=hk(ξo,ξ1,ξ2),    k=0,1,,1,kν(ξo,ξ1,ξ2,0)ˆtk=jk(ξo,ξ1,ξ2),    k=0,1,,1. (4.24)

    In Eq (4.23) f1(ξo,ξ1,ξ2,ˆt), f2(ξo,ξ1,ξ2,ˆt) and f3(ξo,ξ1,ξ2,ˆt), are the source terms, L and N are the linear and non-linear operator respectively, while Dδ+mˆt is the Caputo-Fabrizio type fractional derivative operator.

    Applying the AT to Eq (4.23), we have

    A(u(ξo,ξ1,ξ2,ˆt))=ϑ1(ξo,ξ1,ξ2,s)+(s+δ(1s)sm+1)A(Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt)),A(μ(ξo,ξ1,ξ2,ˆt))=ϑ2(ξo,ξ1,ξ2,s)+(s+δ(1s)sm+1)A(Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt)),A(ν(ξo,ξ1,ξ2,ˆt))=ϑ2(ξo,ξ1,ξ2,s)+(s+δ(1s)sm+1)A(Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt)), (4.25)

    where

    ϑ1(ξo,ξ1,ξ2,s)=1sm+1(smθ0(ξo,ξ1,ξ2)+sm1θ1(ξo,ξ1,ξ2)++θm(ξo,ξ1,ξ2)+s+δ(1s)sm+1f1(ξo,ξ1,ξ2,s).
    ϑ2(ξo,ξ1,ξ2,s)=1sm+1(smθ0(ξo,ξ1,ξ2)+sm1θ1(ξo,ξ1,ξ2)++θm(ξo,ξ1,ξ2)+s+δ(1s)sm+1f2(ξo,ξ1,ξ2,s).
    ϑ3(ξo,ξ1,ξ2,s)=1sm+1(smθ0(ξo,ξ1,ξ2)+sm1θ1(ξo,ξ1,ξ2)++θm(ξo,ξ1,ξ2)+s+δ(1s)sm+1f3(ξo,ξ1,ξ2,s).

    Taking the inverse AT on Eq (4.25), we obtain

    u(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt)+A(s+δ(1s)sm+1)A(Lu(ξo,ξ1,ξ2,ˆt)+Nu(ξo,ξ1,ξ2,ˆt)),μ(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt)+A(s+δ(1s)sm+1)A(Lμ(ξo,ξ1,ξ2,ˆt)+Nμ(ξo,ξ1,ξ2,ˆt)),ν(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt)+A(s+δ(1s)sm+1)A(Lν(ξo,ξ1,ξ2,ˆt)+Nν(ξo,ξ1,ξ2,ˆt)), (4.26)

    where the source terms are denoted by ϑ1(ξo,ξ1,ξ2,ˆt), ϑ2(ξo,ξ1,ξ2,ˆt) and ϑ3(ξo,ξ1,ξ2,ˆt). Further we apply NIM introduced in [48].We consider the solution as an infinite series given as

    u(ξo,ξ1,ξ2,ˆt)==0u(ξo,ξ1,ξ2,ˆt),μ(ξo,ξ1,ξ2,ˆt)==0μ(ξo,ξ1,ξ2,ˆt),ν(ξo,ξ1,ξ2,ˆt)==0ν(ξo,ξ1,ξ2,ˆt). (4.27)

    Since L is linear

    L(=0u(ξo,ξ1,ξ2,ˆt))==0L(u(ξo,ξ1,ξ2,ˆt)),L(=0μ(ξo,ξ1,ξ2,ˆt))==0L(μ(ξo,ξ1,ξ2,ˆt)),L(=0ν(ξo,ξ1,ξ2,ˆt))==0L(ν(ξo,ξ1,ξ2,ˆt)). (4.28)

    The nonlinear operator N is decomposed as

    N(=0u(ξo,ξ1,ξ2,ˆt))=N(u0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0ui(ξo,ξ1,ξ2,ˆt))N(1i=0ui(ξo,ξ1,ξ2,ˆt))],N(=0μ(ξo,ξ1ν,ξ2,ˆt))=N(μ0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0μi(ξo,ξ1,ξ2,ˆt))N(1i=0μi(ξo,ξ1,ξ2,ˆt))],N(=0ν(ξo,ξ1,ξ2,ˆt))=N(ν0(ξo,ξ1,ξ2,ˆt))+=0[N(i=0νi(ξo,ξ1,ξ2,ˆt))N(1i=0νi(ξo,ξ1,ξ2,ˆt))]. (4.29)

    In view of Eqs (4.26–4.28) and Eq (4.29) is equivalent to

    =0u(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt)+A[(s+δ(1s)sm+1)A(=0Lu(ξo,ξ1,ξ2,ˆt))]+A[(s+δ(1s)sm+1)A(N(u0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0ui(ξo,ξ1,ξ2,ˆt))N(1i=0ui(ξo,ξ1,ξ2,ˆt))}],=0μ(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt)+A[(s+δ(1s)sm+1)A(=0Lμ(ξo,ξ1,ξ2,ˆt))]+A[(s+δ(1s)sm+1)A(N(μ0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0μi(ξo,ξ1,ξ2,ˆt))N(1i=0μi(ξo,ξ1,ξ2,ˆt))}],=0ν(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt)+A[(s+δ(1s)sm+1)A(=0Lν(ξo,ξ1,ξ2,ˆt))]+A[(s+δ(1s)sm+1)A(N(ν0(ξo,ξ1,ξ2,ˆt)))+=0{N(i=0νi(ξo,ξ1,ξ2,ˆt))N(1i=0νi(ξo,ξ1,ξ2,ˆt))}], (4.30)

    considering the following recursive relation

    u0(ξo,ξ1,ξ2,ˆt)=ϑ1(ξo,ξ1,ξ2,ˆt),μ0(ξo,ξ1,ξ2,ˆt)=ϑ2(ξo,ξ1,ξ2,ˆt),ν0(ξo,ξ1,ξ2,ˆt)=ϑ3(ξo,ξ1,ξ2,ˆt),u1(ξo,ξ1,ξ2,ˆt)=A[(s+δ(1s)sm+1)A(L(u0(ξo,ξ1,ξ2,ˆt))+N(u0(ξo,ξ1,ξ2,ˆt)))],μ1(ξo,ξ1,ξ2,ˆt)=A[(s+δ(1s)sm+1)A(L(μ0(ξo,ξ1,ξ2,ˆt))+N(μ0(ξo,ξ1,ξ2,ˆt)))],ν1(ξo,ξ1,ξ2,ˆt)=A[(s+δ(1s)sm+1)A(L(ν0(ξo,ξ1,ξ2,ˆt))+N(ν0(ξo,ξ1,ξ2,ˆt)))], (4.31)
                                                   un+1(ξo,ξ1,ξ2,ˆt)=A(s+δ(1s)sm+1)A[(L(un(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0ui(ξo,ξ1,ξ2,ˆt))N(n1i=0ui(ξo,ξ1,ξ2,ˆt))}],μn+1(ξo,ξ1,ξ2,ˆt)=A(s+δ(1s)sm+1)A[(L(μn(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0μi(ξo,ξ1,ξ2,ˆt))N(n1i=0μi(ξo,ξ1,ξ2,ˆt))}],νn+1(ξo,ξ1,ξ2,ˆt)=A(s+δ(1s)sm+1)A[(L(νn(ξo,ξ1,ξ2,ˆt)))+j=1{N(ni=0νi(ξo,ξ1,ξ2,ˆt))N(n1i=0νi(ξo,ξ1,ξ2,ˆt))}]. (4.32)

    The NITM approximate solution is given by

    u(ξo,ξ1,ξ2,ˆt)=u0+u1+u2++un,μ(ξo,ξ1,ξ2,ˆt)=μ0+μ1+μ2++μn,ν(ξo,ξ1,ξ2,ˆt)=ν0+ν1+ν2++νn. (4.33)

    In this section, we will find the analytical solutions of a non-homogeneous, time-fractional Cauchy equation with different fractional derivative operators.

    The non-homogeneous time-fractional Cauchy equation is [20,46]

    0Dδˆtu+uξo=ξo,ˆt>0,ξoR,0<δ1, (5.1)

    having initial condition

    u(ξo,0)=eξo.

    The exact solution is given as [20,46]

    u(ξo,ˆt)=eξoˆt+ˆt(ξoˆt2).

    Applying AT to Eq (5.1), we obtain

    A(u(ξo,ˆt))=u(ξo,0)s+1sδA(ξouξo), (5.2)

    applying the inverse AT on Eq (5.2), we have

    u(ξo,ˆt)=u(ξo,0)+A[1sδA(ξouξo)], (5.3)

    so the iterative scheme is

    u0(ξo,ˆt)=u(ξo,0)=eξo,un+1(ξo,ˆt)=A[1sδA(ξounξo)],    n=0,1,, (5.4)

    put n = 0 in Eq (5.4), we have

    u1(ξo,ˆt)=A[1sδA(ξou0ξo)],
    u1(ξo,ˆt)=ξoˆtδΓ(δ+1)eξoˆtδΓ(δ+1).

    Put n = 1 in Eq (5.4), we have

    u2(ξo,ˆt)=A[1sδA(ξou1ξo)],u2(ξo,ˆt)=ξoˆtδΓ(δ+1)ˆt2δΓ(2δ+1)+eξoˆt2δΓ(2δ+1).

    The approximate NITM solution using the Caputo operator with three terms iterations

    u(ξo,ˆt)=u0(ξo,ˆt)+u1(ξo,ˆt)+u2(ξo,ˆt),
    u(ξo,ˆt)=eξo+ξoˆtδΓ(δ+1)eξoˆtδΓ(δ+1)+ξoˆtδΓ(δ+1)ˆt2δΓ(2δ+1)+eξoˆt2δΓ(2δ+1).

    When the series goes to infinity term then we obtain the following solution

    u(ξo,ˆt)=eξoˆt+ˆt(ξoˆt2),

    which is the the exact solution of the Eq (5.1). The exact solution are also yield in VIM [46], q-HASTA and RDTA [20].

    Applying AT to Eq (5.1), we obtain

    A(u(ξo,ˆt))=u(ξo,0)s+((1δ)sδ+δsδ)A(ξouξo), (5.5)

    applying the inverse AT on Eq (5.5), we have

    u(ξo,ˆt)=u(ξo,0)+A[((1δ)sδ+δsδ)A(ξouξo)], (5.6)

    so the iterative scheme of Eq (5.6), is

    u0(ξo,ˆt)=u(ξo,0)=eξo,un+1(ξo,ˆt)=A[((1δ)sδ+δsδ)A(ξounξo)],    n=0,1,, (5.7)

    put n = 0 in Eq (5.7)

    u1(ξo,ˆt)=A[((1δ)sδ+δsδ)A(ξou0ξo)],
    u1(ξo,ˆt)=(1δ)ξo(1δ)eξo+δξoˆtδΓ(δ+1)δeξoˆtδΓ(δ+1).

    Put n = 1 in Eq (5.7), we have

    u2(ξo,ˆt)=A[((1δ)sδ+δsδ)A(ξou1ξo)],
    u2(ξo,ˆt)={(1δ)ξo(1δ)2+(1δ)2eξo(1δ)δˆtδΓ(δ+1)+(1δ)δeξoˆtδΓ(δ+1)+δξoˆtδΓ(δ+1)δ(1δ)ˆtδΓ(δ+1)+δ(1δ)eξoˆtδΓ(δ+1)δ2ˆt2δΓ(2δ+1)+δ2eξoˆt2δΓ(2δ+1).

    The NITM solution with three terms approximation within Atangana-Baleanu operator is

    u(ξo,ˆt)=u0(ξo,ˆt)+u1(ξo,ˆt)+u2(ξo,ˆt),
    u(ξo,ˆt)={eξo+(1δ)ξo(1δ)eξo+δˆtδΓ(δ+1)δeξoˆtδΓ(δ+1)(1δ)(1δ)2+(1δ)2eξo(1δ)δˆtδΓ(δ+1)+(1δ)δeξoˆtδΓ(δ+1)+δξoˆtδΓ(δ+1)δ(1δ)ˆtδΓ(δ+1)+δ(1δ)eξoˆtδΓ(δ+1)δ2ˆt2δΓ(2δ+1)+δ2eξoˆt2δΓ(2δ+1).

    Applying AT to Eq (5.1), we obtain

    A(u(ξo,ˆt))=u(ξo,0)s+(s+(1s)δs)A(ξouξo), (5.8)

    taking the inverse AT on Eq (5.8), we have

    u(ξo,ˆt)=u(ξo,0)+A[(s+(1s)δs)A(ξouξo)], (5.9)

    so the iterative scheme of Eq (5.9), is

    u0(ξo,ˆt)=u(ξo,0)=eξo,un+1(ξo,ˆt)=A[(s+(1s)δs)A(ξounξo)],   n=0,1,, (5.10)

    put n = 0 in Eq (5.10)

    u1(ξo,ˆt)=A[(s+(1s)δs)A(ξou0ξo)],
    u1(ξo,ˆt)=ξoeξo+δξoˆtΓ(2)eξoˆtΓ(2)δξo+δeξo.

    Put n = 1 in Eq (5.10),

    u2(ξo,ˆt)=A[(s+(1s)δs)A(ξou1ξo)],
    u2(ξo,ˆt)={ξo1+eξoδˆtΓ(2)+δeξoˆtΓ(2)+δδeξo+δξoˆtΓ(2)δˆtΓ(2)+δeξoˆtΓ(2)δ2ˆt2Γ(3)+δ2eξoˆt2Γ(3)+δ2ˆtΓ(2)δ2eξoˆtΓ(2)δξo+δδeξo+δ2ˆtΓ(2)δ2eξoˆtΓ(2)δ2+δ2eξo.

    The NITM solution with three terms approximation within Caputo Fabrizio operator is

    u(ξo,ˆt)=u0(ξo,ˆt)+u1(ξo,ˆt)+u2(ξo,ˆt),

    the series form approximate solution to three term iterations is given as

    u(ξo,ˆt)={ξo+δξoˆtΓ(2)eξoˆtΓ(2)δξo+δeξo+ξo1+eξoδˆtΓ(2)+δeξoˆtΓ(2)+δδeξo+δξoˆtΓ(2)δˆtΓ(2)+δeξoˆtΓ(2)δ2ˆt2Γ(3)+δ2eξoˆt2Γ(3)+δ2ˆtΓ(2)δ2eξoˆtΓ(2)δξo+δδeξo+δ2ˆtΓ(2)δ2eξoˆtΓ(2)δ2+δ2eξo, (5.11)

    Consider fractional order 3D Navier Stokes equation of the form[38]

    Dδˆtu+uuξo+μuξ1+νuξ2=ρ0(2uξo2+2uξ12+2uξ22)1ρρξo,Dδˆtμ+uμξo+μμξ1+νμξ2=ρ0(2μξo2+2μξ12+2μξ22)1ρρξ1,Dδˆtν+uνξo+μνξ1+ννξ2=ρ0(2νξo2+2νξ12+2νξ22)1ρρξ2, (5.12)

    having initial condition

    u(ξo,ξ1,ξ2,0)=0.5ξo+ξ1+ξ2,μ(ξo,ξ1,ξ2,0)=ξo0.5ξ1+ξ2,ν(ξo,ξ1,ξ2,0)=ξo+ξ10.5ξ2.

    The exact solution of the Eq (5.12), is

    u(ξo,ξ1,ξ2,ˆt)=0.5ξo+ξ1+ξ22.25ξoˆt12.25ˆt2,μ(ξo,ξ1,ξ2,ˆt)=ξo0.5ξ1+ξ22.25ξ1ˆt12.25ˆt2,ν(ξo,ξ1,ξ2,ˆt)=ξo+ξ10.5ξ22.25ξ2ˆt12.25ˆt2,

    where

    g1=1ρρξo,    g2=1ρρξ1,    g3=1ρρξ2.

    Case 1: The solution of system (5.12), using NITM within C-FDO.

    Applying AT on both sides of Eq (5.12), we obtain

    A(u(ξo,ξ1,ξ2,ˆt))=u(ξo,ξ1,ξ2,0)s+1sδA[ρ0(2uξo2+2uξ12+2uξ22)+g1(uuξo+μuξ1+νuξ2)],A(μ(ξo,ξ1,ξ2,ˆt))=μ(ξo,ξ1,ξ2,0)s+1sδA[ρ0(2μξo2+2μξ12+2μξ22)+g2(uμξo+μμξ1+νμξ2)],A(ν(ξo,ξ1,ξ2,ˆt))=ν(ξo,ξ1,ξ2,0)s+1sδA[ρ0(2νξo2+2νξ12+2νξ22)+g3(uνξo+μνξ1+ννξ2)], (5.13)

    applying the inverse AT on Eq (5.13), we have

    u(ξo,ξ1,ξ2,ˆt)=u(ξo,ξ1,ξ2,0)+A[1sδA{ρ0(2uξo2+2uξ12+2uξ22)+g1(uuξo+μuξ1+νuξ2)}],μ(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)+A[1sδA{ρ0(2μξo2+2μξ12+2μξ22)+g2(uμξo+μμξ1+νμξ2)}],ν(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)+A[1sδA{ρ0(2νξo2+2νξ12+2νξ22)+g3(uνξo+μνξ1+ννξ2)}], (5.14)

    so the iterative scheme is

    u0(ξo,ξ1,ξ2,ˆt)=u(ξo,ξ1,ξ2,0)=0.5ξo+ξ1+ξ2,μ0(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)=ξo0.5ξ1+ξ2,ν0(ξo,ξ1,ξ2,ˆt)=ν(ξo,ξ1,ξ2,0)=ξo+ξ10.5ξ2,un+1(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2unξo2+2unξ12+2unξ22)+g1(ununξo+μnunξ1+νnunξ2)}],μn+1(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2μnξo2+2μnξ12+2μnξ22)+g2(unμnξo+μnμnξ1+νnμnξ2)}],νn+1(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2νnξo2+2νnξ12+2νnξ22)+g3(unνnξo+μnνnξ1+νnνnξ2)}],  n=0,1,, (5.15)

    put n = 0 in Eq (5.15), we have

    u1(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2u0ξo2+2u0ξ12+2u0ξ22)+g1(u0u0ξo+μ0u0ξ1+ν0u0ξ2)}],μ1(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2μ0ξo2+2μ0ξ12+2μ0ξ22)+g2(u0μ0ξo+μ0μ0ξ1+ν0μ0ξ2)}],ν1(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2ν0ξo2+2ν0ξ12+2ν0ξ22)+g3(u0ν0ξo+μ0ν0ξ1+ν0ν0ξ2)}],
    u1(ξo,ξ1,ξ2,ˆt)=A[1sδA{g1((0.5ξo+ξ1+ξ2)(0.5)+(ξo0.5ξ1+ξ2)(1)+(ξo+ξ10.5ξ2)(1))}],μ1(ξo,ξ1,ξ2,ˆt)=A[1sδA{g2((0.5ξo+ξ1+ξ2)(1)+(ξo0.5ξ1+ξ2)(0.5)+(ξo+ξ10.5ξ2)(1))}],ν1(ξo,ξ1,ξ2,ˆt)=A[1sδA{g3((0.5ξo+ξ1+ξ2)(1)+(ξo0.5ξ1+ξ2)(1)+(ξo+ξ10.5ξ2)(0.5))}],
    u1(ξo,ξ1,ξ2,ˆt)=A[1sδ{g1s2.25ξos}]=A[g1sδ+12.25ξosδ+1],μ1(ξo,ξ1,ξ2,ˆt)=A[1sδ{g2s2.25ξ1s}]=A[g2sδ+12.25ξ1sδ+1],ν1(ξo,ξ1,ξ2,ˆt)=A[1sδ{g3s2.25ξ2s}]=A[g3sδ+12.25ξ2sδ+1],
    u1(ξo,ξ1,ξ2,ˆt)=g1ˆtδΓ(δ+1)2.25ξoˆtδΓ(δ+1),μ1(ξo,ξ1,ξ2,ˆt)=g2ˆtδΓ(δ+1)2.25ξ1ˆtδΓ(δ+1),ν1(ξo,ξ1,ξ2,ˆt)=g3ˆtδΓ(δ+1)2.25ξ2ˆtδΓ(δ+1).

    Put n = 1 in Eq (5.15), we have

    u2(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2u1ξo2+2u1ξ12+2u1ξ22)+g1(u1u1ξo+μ1u1ξ1+ν1u1ξ2)}],μ2(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2μ1ξo2+2μ1ξ12+2μ1ξ22)+g2(u1μ1ξo+μ1μ1ξ1+ν1μ1ξ2)}],ν2(ξo,ξ1,ξ2,ˆt)=A[1sδA{ρ0(2ν1ξo2+2ν1ξ12+2ν1ξ22)+g3(u1ν1ξo+μ1ν1ξ1+ν1ν1ξ2)}],

    after simplification we get,

    u2(ξo,ξ1,ξ2,ˆt)=g1ˆtδΓ(δ+1)+2.25g1ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1)2.252ξoˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1),μ2(ξo,ξ1,ξ2,ˆt)=g2ˆtδΓ(δ+1)+2.25g2ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1)2.252ξ1ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1),ν2(ξo,ξ1,ξ2,ˆt)=g3ˆtδΓ(δ+1)+2.25g3ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1)2.252ξ2ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1).

    The approximate NITM solution using the Caputo operator with three terms iterations

    u(ξo,ξ1,ξ2,ˆt)=u0(ξo,ξ1,ξ2,ˆt)+u1(ξo,ξ1,ξ2,ˆt)+u2(ξo,ξ1,ξ2,ˆt),μ(ξo,ξ1,ξ2,ˆt)=μ0(ξo,ξ1,ξ2,ˆt)+μ1(ξo,ξ1,ξ2,ˆt)+μ2(ξo,ξ1,ξ2,ˆt),ν(ξo,ξ1,ξ2,ˆt)=ν0(ξo,ξ1,ξ2,ˆt)+ν1(ξo,ξ1,ξ2,ˆt)+ν2(ξo,ξ1,ξ2,ˆt),
    u(ξo,ξ1,ξ2,ˆt)=0.5ξo+ξ1+ξ2+g1ˆtδΓ(δ+1)2.25ξoˆtδΓ(δ+1)+g1ˆtδΓ(δ+1)+2.25g1ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1)2.252ξoˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1),μ(ξo,ξ1,ξ2,ˆt)=ξo0.5ξ1+ξ2+g2ˆtδΓ(δ+1)2.25ξ1ˆtδΓ(δ+1)+g2ˆtδΓ(δ+1)+2.25g2ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1)2.252ξ1ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1),ν(ξo,ξ1,ξ2,ˆt)=ξo+ξ10.5ξ2+g3ˆtδΓ(δ+1)2.25ξ2ˆtδΓ(δ+1)+g3ˆtδΓ(δ+1)+2.25g3ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1)2.252ξ2ˆt3δΓ(2δ+1)(Γ(δ+1))2Γ(3δ+1).

    Case 2: NITM solution of system (5.12) within AB operator.

    ApplyAT to both sides of Eq (5.12), we get

    A(u(ξo,ξ1,ξ2,ˆt))=u(ξo,ξ1,ξ2,0)s+((1δ)sδ+δsδ)A[ρ0(2uξo2+2uξ12+2uξ22)+g1(uuξo+μuξ1+νuξ2)],A(μ(ξo,ξ1,ξ2,ˆt))=μ(ξo,ξ1,ξ2,0)s+((1δ)sδ+δsδ)A[ρ0(2μξo2+2μξ12+2μξ22)+g2(uμξo+μμξ1+νμξ2)],A(ν(ξo,ξ1,ξ2,ˆt))=ν(ξo,ξ1,ξ2,0)s+((1δ)sδ+δsδ)A[ρ0(2νξo2+2νξ12+2νξ22)+g3(uνξo+μνξ1+ννξ2)], (5.16)

    applying the inverse AT on Eq (5.16), we have

    u(ξo,ξ1,ξ2,ˆt)=u(ξo,ξ1,ξ2,0)+A[((1δ)+δsδ)A{ρ0(2uξo2+2uξ12+2uξ22)+g1(uuξo+μuξ1+νuξ2)}],μ(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)+A[((1δ)+δsδ)A{ρ0(2μξo2+2μξ12+2μξ22)+g2(uμξo+μμξ1+νμξ2)}],ν(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)+A[((1δ)+δsδ)A{ρ0(2νξo2+2νξ12+2νξ22)+g3(uνξo+μνξ1+ννξ2)}], (5.17)

    so the iterative scheme is

    u0(ξo,ξ1,ξ2,ˆt)=u(ξo,ξ1,ξ2,0)=0.5ξo+ξ1+ξ2,μ0(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)=ξo0.5ξ1+ξ2,ν0(ξo,ξ1,ξ2,ˆt)=ν(ξo,ξ1,ξ2,0)=ξo+ξ10.5ξ2,un+1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2unξo2+2unξ12+2unξ22)+g1(ununξo+μnunξ1+νnunξ2)}],μn+1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2μnξo2+2μnξ12+2μnξ22)+g2(unμnξo+μnμnξ1+νnμnξ2)}],νn+1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2νnξo2+2νnξ12+2νnξ22)+g3(unνnξo+μnνnξ1+νnνnξ2)}],  n=0,1,..., (5.18)

    put n = 0 in Eq (5.18), we have

    u1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2u0ξo2+2u0ξ12+2u0ξ22)+g1(u0u0ξo+μ0u0ξ1+ν0u0ξ2)}],μ1(ξo,ξ1,ξ2,t)=A[((1δ)+δsδ)A{ρ0(2μ0ξo2+2μ0ξ12+2μ0ξ22)+g2(u0μ0ξo+μ0μ0ξ1+ν0μ0ξ2)}],ν1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2ν0ξo2+2ν0ξ12+2ν0ξ22)+g3(u0ν0ξo+μ0ν0ξ1+ν0ν0ξ2)}],
    u1(ξo,ξ1,ξ2,ˆt)=(1δ)g12.25(1δ)ξo+δg1ˆtδΓ(δ+1)2.25δξoˆtδΓ(δ+1),μ1(ξo,ξ1,ξ2,ˆt)=(1δ)g22.25(1δ)ξ1+δg2ˆtδΓ(δ+1)2.25δξ1ˆtδΓ(δ+1),ν1(ξo,ξ1,ξ2,ˆt)=(1δ)g32.25(1δ)ξ2+δg3ˆtδΓ(δ+1)2.25δξ2ˆtδΓ(δ+1).

    Put n = 1 in Eq (5.18), we have

    u2(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2u1ξo2+2u1ξ12+2u1ξ22)+g1(u1u1ξo+μ1u1ξ1+ν1u1ξ2)}],μ2(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2μ1ξo2+2μ1ξ12+2μ1ξ22)+g2(u1μ1ξo+μ1μ1ξ1+ν1μ1ξ2)}],ν2(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δsδ)A{ρ0(2ν1ξo2+2ν1ξ12+2ν1ξ22)+g3(u1ν1ξo+μ1ν1ξ1+ν1ν1ξ2)}],
    u2(ξo,ξ1,ξ2,ˆt)={g1(1δ)+2.25(1δ)3g1+2.25δ(1δ)2g1ˆtδΓ(δ+1)2.25(1δ)3ξo2.25(1δ)2ξoˆtδΓ(δ+1)+2.25δ(1δ)2g1ˆtδΓ(δ+1)+2.252δ2(1δ)g1ˆt2δ+1Γ(δ+1)22.252δ(1δ)2ξoˆtδΓ(δ+1)2.252δ2(1δ)ξoˆt2δ+1Γ(δ+1)2+δg1ˆtδΓ(δ+1)+2.25δ(1δ)2g1ˆtδΓ(δ+1)+2.25δ2(1δ)g1ˆt2δΓ(2δ+1)2.25δ(1δ)2ξoˆtδΓ(δ+1)2.25δ(1δ)ξoˆt2δΓ(2δ+1)+2.25δ2(1δ)g1ˆt2δΓ(2δ+1)+2.252δ3g1ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1)2.252δ2(1δ)ξoˆt2δΓ(2δ+1)2.252δ3ξoˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1),μ2(ξo,ξ1,ξ2,ˆt)={g2(1δ)+2.25(1δ)3g2+2.25δ(1δ)2g2ˆtδΓ(δ+1)2.25(1δ)3ξ12.25(1δ)2ξ1ˆtδΓ(δ+1)+2.25δ(1δ)2g2ˆtδΓ(δ+1)+2.252δ2(1δ)g2ˆt2δ+1Γ(δ+1)22.252δ(1δ)2ξ1ˆtδΓ(δ+1)2.252δ2(1δ)ξ1ˆt2δ+1Γ(δ+1)2+δg2ˆtδΓ(δ+1)+2.25δ(1δ)2g2ˆtδΓ(δ+1)+2.25δ2(1δ)g2ˆt2δΓ(2δ+1)2.25δ(1δ)2ξ1ˆtδΓ(δ+1)2.25δ(1δ)ξ1ˆt2δΓ(2δ+1)+2.25δ2(1δ)g2ˆt2δΓ(2δ+1)+2.252δ3g2ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1)2.252δ2(1δ)ξ1ˆt2δΓ(2δ+1)2.252δ3ξ1ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1),ν2(ξo,ξ1,ξ2,ˆt)={g3(1δ)+2.25(1δ)3g3+2.25δ(1δ)2g3ˆtδΓ(δ+1)2.25(1δ)3ξ22.25(1δ)2ξ2ˆtδΓ(δ+1)+2.25δ(1δ)2g3ˆtδΓ(δ+1)+2.252δ2(1δ)g3ˆt2δ+1Γ(δ+1)22.252δ(1δ)2ξ2ˆtδΓ(δ+1)2.252δ2(1δ)ξ2ˆt2δ+1Γ(δ+1)2+δg3ˆtδΓ(δ+1)+2.25δ(1δ)2g3ˆtδΓ(δ+1)+2.25δ2(1δ)g3ˆt2δΓ(2δ+1)2.25δ(1δ)2ξ2ˆtδΓ(δ+1)2.25δ(1δ)ξ2ˆt2δΓ(2δ+1)+2.25δ2(1δ)g3ˆt2δΓ(2δ+1)+2.252δ3g3ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1)2.252δ2(1δ)ξ2ˆt2δΓ(2δ+1)2.252δ3ξ2ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1).

    The NITM solution with three terms approximation within the Atangana-Baleanu operator is

    u(ξo,ξ1,ξ2,ˆt)=u0(ξo,ξ1,ξ2,ˆt)+u1(ξo,ξ1,ξ2,ˆt)+u2(ξo,ξ1,ξ2,ˆt),μ(ξo,ξ1,ξ2,ˆt)=μ0(ξo,ξ1,ξ2,ˆt)+μ1(ξo,ξ1,ξ2,ˆt)+μ2(ξo,ξ1,ξ2,ˆt),ν(ξo,ξ1,ξ2,ˆt)=ν0(ξo,ξ1,ξ2,ˆt)+ν1(ξo,ξ1,ξ2,ˆt)+ν2(ξo,ξ1,ξ2,ˆt),
    u(ξo,ξ1,ξ2,ˆt)={0.5ξo+ξ1+ξ2+(1δ)g12.25(1δ)ξo+δg1ˆtδΓ(δ+1)2.25δξoˆtδΓ(δ+1)+g1(1δ)+2.25(1δ)3g1+2.25δ(1δ)2g1ˆtδΓ(δ+1)2.25(1δ)3ξo2.25(1δ)2ξoˆtδΓ(δ+1)+2.25δ(1δ)2g1ˆtδΓ(δ+1)+2.252δ2(1δ)g1ˆt2δ+1Γ(δ+1)22.252δ(1δ)2ξoˆtδΓ(δ+1)2.252δ2(1δ)ξoˆt2δ+1Γ(δ+1)2+δg1ˆtδΓ(δ+1)+2.25δ(1δ)2g1ˆtδΓ(δ+1)+2.25δ2(1δ)g1ˆt2δΓ(2δ+1)2.25δ(1δ)2ξoˆtδΓ(δ+1)2.25δ(1δ)ξoˆt2δΓ(2δ+1)+2.25δ2(1δ)g1ˆt2δΓ(2δ+1)+2.252δ3g1ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1)2.252δ2(1δ)ξoˆt2δΓ(2δ+1)2.252δ3ξoˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1),μ(ξo,ξ1,ξ2,ˆt)={ξo0.5ξ1+ξ2+(1δ)g22.25(1δ)ξ1+δg2ˆtδΓ(δ+1)2.25δξ1ˆtδΓ(δ+1)+g2(1δ)+2.25(1δ)3g2+2.25δ(1δ)2g2ˆtδΓ(δ+1)2.25(1δ)3ξ12.25(1δ)2ξ1ˆtδΓ(δ+1)+2.25δ(1δ)2g2ˆtδΓ(δ+1)+2.252δ2(1δ)g2ˆt2δ+1Γ(δ+1)22.252δ(1δ)2ξ1ˆtδΓ(δ+1)2.252δ2(1δ)ξ1ˆt2δ+1Γ(δ+1)2+δg2ˆtδΓ(δ+1)+2.25δ(1δ)2g2ˆtδΓ(δ+1)+2.25δ2(1δ)g2ˆt2δΓ(2δ+1)2.25δ(1δ)2ξ1ˆtδΓ(δ+1)2.25δ(1δ)ξ1ˆt2δΓ(2δ+1)+2.25δ2(1δ)g2ˆt2δΓ(2δ+1)+2.252δ3g2ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1)2.252δ2(1δ)ξ1ˆt2δΓ(2δ+1)2.252δ3ξ1ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1),ν(ξo,ξ1,ξ2,ˆt)={ξo+ξ10.5ξ2+(1δ)g32.25(1δ)ξ2+δg3ˆtδΓ(δ+1)2.25δξ2ˆtδΓ(δ+1)+g3(1δ)+2.25(1δ)3g3+2.25δ(1δ)2g3ˆtδΓ(δ+1)2.25(1δ)3ξ22.25(1δ)2ξ2ˆtδΓ(δ+1)+2.25δ(1δ)2g3ˆtδΓ(δ+1)+2.252δ2(1δ)g3ˆt2δ+1Γ(δ+1)22.252δ(1δ)2ξ2ˆtδΓ(δ+1)2.252δ2(1δ)ξ2ˆt2δ+1Γ(δ+1)2+δg3ˆtδΓ(δ+1)+2.25δ(1δ)2g3ˆtδΓ(δ+1)+2.25δ2(1δ)g3ˆt2δΓ(2δ+1)2.25δ(1δ)2ξ2ˆtδΓ(δ+1)2.25δ(1δ)ξ2ˆt2δΓ(2δ+1)+2.25δ2(1δ)g3ˆt2δΓ(2δ+1)+2.252δ3g3ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1)2.252δ2(1δ)ξ2ˆt2δΓ(2δ+1)2.252δ3ξ2ˆt3δΓ(2δ+1)Γ(δ+1)2Γ(3δ+1).

    Case 3: The solution of system(5.12), using NITM with a Caputo Fabrizio operator.

    Applying AT to Eq (5.12), we obtain

    A(u(ξo,ξ1,ξ2,ˆt))=u(ξo,ξ1,ξ2,0)s+(s+(1s)δs)A[ρ0(2uξo2+2uξ12+2uξ22)+g1(uuξo+μuξ1+νuξ2)],A(μ(ξo,ξ1,ξ2,ˆt))=μ(ξo,ξ1,ξ2,0)s+(s+(1s)δs)A[ρ0(2μξo2+2μξ12+2μξ22)+g2(uμξo+μμξ1+νμξ2)],A(ν(ξo,ξ1,ξ2,ˆt))=ν(ξo,ξ1,ξ2,0)s+(s+(1s)δs)A[ρ0(2νξo2+2νξ12+2νξ22)+g3(uνξo+μνξ1+ννξ2)], (5.19)

    applying the inverse AT on Eq (5.19), we have

    u(ξo,ξ1,ξ2,ˆt)=u(ξo,ξ1,ξ2,0)+A[((1δ)+δs)A{ρ0(2uξo2+2uξ12+2uξ22)+g1(uuξo+μuξ1+νuξ2)}],μ(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)+A[((1δ)+δs)A{ρ0(2μξo2+2μξ12+2μξ22)+g2(uμξo+μμξ1+νμξ2)}],ν(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)+A[((1δ)+δs)A{ρ0(2νξo2+2νξ12+2νξ22)+g3(uνξo+μνξ1+ννξ2)}], (5.20)

    so the iterative scheme is

    u0(ξo,ξ1,ξ2,ˆt)=u(ξo,ξ1,ξ2,0)=0.5ξo+ξ1+ξ2,μ0(ξo,ξ1,ξ2,ˆt)=μ(ξo,ξ1,ξ2,0)=ξo0.5ξ1+ξ2,ν0(ξo,ξ1,ξ2,ˆt)=ν(ξo,ξ1,ξ2,0)=ξo+ξ10.5ξ2,un+1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2unξo2+2unξ12+2unξ22)+g1(ununξo+μnunξ1+νnunξ2)}],μn+1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2μnξo2+2μnξ12+2μnξ22)+g2(unμnξo+μnμnξ1+νnμnξ2)}],νn+1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2νnξo2+2νnξ12+2νnξ22)+g3(unνnξo+μnνnξ1+νnνnξ2)}],  n=0,1,, (5.21)

    put n = 0 in Eq (5.21) we have

    u1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2u0ξo2+2u0ξ12+2u0ξ22)+g1(u0u0ξo+μ0u0ξ1+ν0u0ξ2)}],μ1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2μ0ξo2+2μ0ξ12+2μ0ξ22)+g2(u0μ0ξo+μ0μ0ξ1+ν0μ0ξ2)}],
    ν1(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2ν0ξo2+2ν0ξ12+2ν0ξ22)+g3(u0ν0ξo+μ0ν0ξ1+ν0ν0ξ2)}],
    u1(ξo,ξ1,ξ2,ˆt)=(1δ)g12.25(1δ)ξo+δg1ˆt2.25δξoˆt,μ1(ξo,ξ1,ξ2,ˆt)=(1δ)g22.25(1δ)ξ1+δg2ˆt2.25δξ1ˆt,ν1(ξo,ξ1,ξ2,ˆt)=(1δ)g32.25(1δ)ξ2+δg3ˆt2.25δξ2ˆt.

    Put n = 1 in Eq (5.21), we have

    u2(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2u1ξo2+2u1ξ12+2u1ξ22)+g1(u1u1ξo+μ1u1ξ1+ν1u1ξ2)}],μ2(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2μ1ξo2+2μ1ξ12+2μ1ξ22)+g2(u1μ1ξo+μ1μ1ξ1+ν1μ1ξ2)}],ν2(ξo,ξ1,ξ2,ˆt)=A[((1δ)+δs)A{ρ0(2ν1ξo2+2ν1ξ12+2ν1ξ22)+g3(u1ν1ξo+μ1ν1ξ1+ν1ν1ξ2)}],
    u2(ξo,ξ1,ξ2,ˆt)={(1δ)g1+2.25(1δ)3g12.252(1δ)3ξo+2.25δ(1δ)2g1ˆt2.252δ(1δ)2ξoˆt+2.25δ(1δ)2g1ˆt2.252δ(1δ)2ξoˆt+2.25δ2(1δ)g1ˆt22.252δ2(1δ)ξoˆt2+δg1ˆt+2.25δ(1δ)2g1ˆt2.252δ(1δ)2ξoˆt+2.25δ2(1δ)g1ˆt2Γ(3)2.252δ2(1δ)ξoˆt2Γ(3)+2.25δ2(1δ)g1ˆt2Γ(3)2.252δ2(1δ)ξoˆt2Γ(3)+2.25δ3g1ˆt3Γ(3)Γ(4)2.252δ3ξoˆt3Γ(3)Γ(4),μ2(ξo,ξ1,ξ2,ˆt)={(1δ)g2+2.25(1δ)3g22.252(1δ)3ξ1+2.25δ(1δ)2g2ˆt2.252δ(1δ)2ξ1ˆt+2.25δ(1δ)2g2ˆt2.252δ(1δ)2ξ1ˆt+2.25δ2(1δ)g2ˆt22.252δ2(1δ)ξ1ˆt2+δg2ˆt+2.25δ(1δ)2g2ˆt2.252δ(1δ)2ξ1ˆt+2.25δ2(1δ)g2ˆt2Γ(3)2.252δ2(1δ)ξ1ˆt2Γ(3)+2.25δ2(1δ)g2ˆt2Γ(3)2.252δ2(1δ)ξ1ˆt2Γ(3)+2.25δ3g2ˆt3Γ(3)Γ(4)2.252δ3ξ1ˆt3Γ(3)Γ(4),ν2(ξo,ξ1,ξ2,ˆt)={(1δ)g3+2.25(1δ)3g32.252(1δ)3ξ2+2.25δ(1δ)2g3ˆt2.252δ(1δ)2ξ2ˆt+2.25δ(1δ)2g3ˆt2.252δ(1δ)2ξ2ˆt+2.25δ2(1δ)g3ˆt22.252δ2(1δ)ξ2ˆt2+δg3ˆt+2.25δ(1δ)2g3ˆt2.252δ(1δ)2ξ2ˆt+2.25δ2(1δ)g3ˆt2Γ(3)2.252δ2(1δ)ξ2ˆt2Γ(3)+2.25δ2(1δ)g3ˆt2Γ(3)2.252δ2(1δ)ξ2ˆt2Γ(3)+2.25δ3g3ˆt3Γ(3)Γ(4)2.252δ3ξ2ˆt3Γ(3)Γ(4).

    The NITM solution with three terms approximation within Caputo Fabrizio operator is

    u(ξo,ξ1,ξ2,ˆt)=u0(ξo,ξ1,ξ2,ˆt)+u1(ξo,ξ1,ξ2,ˆt)+u2(ξo,ξ1,ξ2,ˆt),μ(ξo,ξ1,ξ2,ˆt)=μ0(ξo,ξ1,ξ2,ˆt)+μ1(ξo,ξ1,ξ2,ˆt)+μ2(ξo,ξ1,ξ2,ˆt),ν(ξo,ξ1,ξ2,ˆt)=ν0(ξo,ξ1,ξ2,ˆt)+ν1(ξo,ξ1,ξ2,ˆt)+ν2(ξo,ξ1,ξ2,ˆt),

    the series form solution is obtain by

    u(ξo,ξ1,ξ2,ˆt)={0.5ξo+ξ1+ξ2+(1δ)g12.25(1δ)ξo+δg1ˆt2.25δξoˆt+(1δ)g1+2.25(1δ)3g12.252(1δ)3ξo+2.25δ(1δ)2g1ˆt2.252δ(1δ)2ξoˆt+2.25δ(1δ)2g1ˆt2.252δ(1δ)2ξoˆt+2.25δ2(1δ)g1ˆt22.252δ2(1δ)ξoˆt2+δg1ˆt+2.25δ(1δ)2g1ˆt2.252δ(1δ)2ξoˆt+2.25δ2(1δ)g1ˆt2Γ(3)2.252δ2(1δ)ξoˆt2Γ(3)+2.25δ2(1δ)g1ˆt2Γ(3)2.252δ2(1δ)ξoˆt2Γ(3)+2.25δ3g1ˆt3Γ(3)Γ(4)2.252δ3ξoˆt3Γ(3)Γ(4),μ(ξo,ξ1,ξ2,ˆt)={ξo0.5ξ1+ξ2+(1δ)g22.25(1δ)ξ1+δg2ˆt2.25δξ1ˆt+(1δ)g2+2.25(1δ)3g22.252(1δ)3ξ1+2.25δ(1δ)2g2ˆt2.252δ(1δ)2ξ1ˆt+2.25δ(1δ)2g2ˆt2.252δ(1δ)2ξ1ˆt+2.25δ2(1δ)g2ˆt22.252δ2(1δ)ξ1ˆt2+δg2ˆt+2.25δ(1δ)2g2ˆt2.252δ(1δ)2ξ1ˆt+2.25δ2(1δ)g2ˆt2Γ(3)2.252δ2(1δ)ξ1ˆt2Γ(3)+2.25δ2(1δ)g2ˆt2Γ(3)2.252δ2(1δ)ξ1ˆt2Γ(3)+2.25δ3g2ˆt3Γ(3)Γ(4)2.252δ3ξ1ˆt3Γ(3)Γ(4),ν(ξo,ξ1,ξ2,ˆt)={ξo+ξ10.5ξ2+(1δ)g32.25(1δ)ξ2+δg3ˆt2.25δξ2ˆt+(1δ)g3+2.25(1δ)3g32.252(1δ)3ξ2+2.25δ(1δ)2g3ˆt2.252δ(1δ)2ξ2ˆt+2.25δ(1δ)2g3ˆt2.252δ(1δ)2ξ2ˆt+2.25δ2(1δ)g3ˆt22.252δ2(1δ)ξ2ˆt2+δg3ˆt+2.25δ(1δ)2g3ˆt2.252δ(1δ)2ξ2ˆt+2.25δ2(1δ)g3ˆt2Γ(3)2.252δ2(1δ)ξ2ˆt2Γ(3)+2.25δ2(1δ)g3ˆt2Γ(3)2.252δ2(1δ)ξ2ˆt2Γ(3)+2.25δ3g3ˆt3Γ(3)Γ(4)2.252δ3ξ2ˆt3Γ(3)Γ(4).

    We used an approximate analytical scheme, and the obtained solution is displayed through graphs and tables. Figure 1 shows the comparison 2D solution plots of the example (5.1) for different fractional order δ and with different fractional operators. Figures 24 are the solution plots for the N.S. system (5.2) at different fractional operators and various fractional order δ respectively. Figure 5 is the compression 2D-solution plots of exact and approximate solutions for the N.S. system (5.2). Table 1 is the comparison table of example (5.1) between the exact solution and approximate solution at δ=1 and with three different operators. Table 2 shows the numerical simulation of example (5.1) at different fractional orders, space and time levels while the derivative is used in the Caputo sense. Tables 35 represent a numerical simulation of example (5.2) at different fractional operators, orders, space, and time levels for u, μ, and ν respectively. Graphs and tables show the accuracy of the proposed techniques. All the numerical simulations are done by Maple code on maple 2023.

    Figure 1.  Comparison 2D solution plots of the example (5.1) for different fractional order δ and with different fractional operators.
    Figure 2.  u-solution plots of example (5.2) at different fractional operators and various fractional order δ.
    Figure 3.  μ-solution plots of example (5.2) at different fractional operators and various fractional order δ.
    Figure 4.  ν-solution plots of example (5.2) at different fractional operators and various fractional order δ.
    Figure 5.  Compression 2D-solution plots of exact and approximate solution for the N.S. system (5.2).
    Table 1.  The comparison table of example (5.1) between exact solution and approximate solution.
    CO ABO CFO Exact AE AE AE
    ξo ˆt CO ABO CFO
    δ=1 δ=1 δ=1 δ=1 δ=1 δ=1 δ=1
    0.25 1.2832415 1.2839915 1.2832415 1.2837420 5.0050×104 2.4950×104 5.0050×104
    0.50 0.001 1.6480728 1.6485728 1.6480728 1.6480733 5.0000×104 4.9950×104 5.0000×104
    0.75 2.1163835 2.1166335 2.1163835 2.1158840 4.9950×104 7.4950×104 4.9950×104
    1 2.7175644 2.7175644 2.7175644 2.7165649 9.9950×104 9.9950×104 9.9950×104

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical simulation of u-solution of example (5.1) at different fractional orders, space and time levels.
    ξo t Exact solution Approximate AE at δ=0.99 AE at δ=0.90 AE at δ=0.80
    0.001 1.283181953 1.282991533 0.000190420 0.000591967 0.002314176
    0.002 1.282350466 1.281957932 0.000392534 0.000965369 0.003757763
    0.003 1.281523640 1.280924613 0.000599027 0.001266220 0.004951399
    0.004 1.280699938 1.279891574 0.000808364 0.001521869 0.005996186
    0.25 0.005 1.279878630 1.278858813 0.001019817 0.001744804 0.006936485
    0.006 1.279059290 1.277826330 0.001232960 0.001942193 0.007797048
    0.007 1.278241634 1.276794124 0.001447510 0.002118675 0.008593574
    0.008 1.277425463 1.275762193 0.001663270 0.002277489 0.009336846
    0.009 1.276610626 1.274730535 0.001880091 0.002421028 0.010034701
    0.010 1.275797006 1.273699150 0.002097856 0.002551126 0.010693077
    0.001 1.648072874 1.647572874 0.000500000 0.000450736 0.000195889
    0.002 1.647425125 1.646425123 0.001000002 0.000911208 0.000209887
    0.003 1.646778026 1.645278019 0.001500007 0.001375353 0.000163031
    0.004 1.646131576 1.644131558 0.002000018 0.001841897 0.000078045
    0.5 0.005 1.645485774 1.642985739 0.002500035 0.002310239 0.000034670
    0.006 1.644840620 1.641840561 0.003000059 0.002780021 0.000169136
    0.007 1.644196116 1.640696021 0.003500095 0.003251011 0.000321484
    0.008 1.643552260 1.639552119 0.004000141 0.003723045 0.000488997
    0.009 1.642909053 1.638408853 0.004500200 0.004195995 0.000669674
    0.010 1.642266494 1.637266220 0.005000274 0.004669771 0.000861976
    0.001 2.338959359 2.338157875 0.000801484 0.000165158 0.001230254
    0.002 2.338282899 2.336670234 0.001612665 0.000511426 0.001738675
    0.003 2.337611483 2.335183929 0.002427554 0.000919095 0.002035230
    0.004 2.336943842 2.333698957 0.003244885 0.001365587 0.002203814
    0.85 0.005 2.336279381 2.332215315 0.004064066 0.001840538 0.002281462
    0.006 2.335617740 2.330733000 0.004884740 0.002337980 0.002288952
    0.007 2.334958687 2.329252012 0.005706675 0.002854027 0.002239532
    0.008 2.334302054 2.327772347 0.006529707 0.003385957 0.002142342
    0.009 2.333647715 2.326294002 0.007353713 0.003931753 0.002004063
    0.010 2.332995574 2.324816977 0.008178597 0.004489858 0.001829773
    0.001 2.717509945 2.716564405 0.000945540 0.000231370 0.001333724
    0.002 2.716750692 2.714848698 0.001901994 0.000666649 0.001854647
    0.003 2.715997341 2.713134703 0.002862638 0.001171307 0.002136123
    0.004 2.715248472 2.711422418 0.003826054 0.001719936 0.002272635
    1 0.005 2.714503404 2.709711841 0.004791563 0.002300863 0.002305932
    0.006 2.713761742 2.708002969 0.005758773 0.002907364 0.002259461
    0.007 2.713023220 2.706295798 0.006727422 0.003535055 0.002148184
    0.008 2.712287647 2.704590327 0.007697320 0.004180866 0.001982435
    0.009 2.711554884 2.702886553 0.008668331 0.004842513 0.001769765
    0.010 2.710824822 2.701184472 0.009640350 0.005518245 0.001515921

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical simulation of u-solution of example (5.2) at different fractional operators, orders, space and time levels.
    Operators ξo,ξ1,ξ2 t Approximate Exact solution AE at δ=0.90 AE at δ=0.95 AE at δ=0.99 AE at δ=1
    0.001 0.374437500 0.374438342 0.000605299 0.000249210 0.000043607 0.000000843
    0.003 0.373312489 0.373320060 0.001456767 0.000622670 0.000116025 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.002176140 0.000949425 0.000186502 0.000020989
    0.007 0.371062355 0.371103414 0.002828462 0.001253588 0.000258784 0.000041059
    0.009 0.369937192 0.370004933 0.003437432 0.001544171 0.000334285 0.000067741
    0.001 0.748874999 0.748876685 0.001210599 0.000498421 0.000087214 0.000001686
    0.003 0.746624977 0.746640120 0.002913535 0.001245340 0.000232051 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.004352280 0.001898850 0.000373004 0.000041979
    0.007 0.742124711 0.742206828 0.005656925 0.002507176 0.000517569 0.000082118
    C-FDO 0.009 0.739874385 0.740009867 0.006874865 0.003088343 0.000668570 0.000135482
    0.001 1.123312499 1.123315027 0.501815898 0.000747630 0.000130820 0.000002528
    0.003 1.119937466 1.119960179 0.504370302 0.001868010 0.000348076 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.506528420 0.002848275 0.000559506 0.000062968
    0.007 1.113187066 1.113310242 0.508485386 0.003760763 0.000776353 0.000123176
    0.009 1.109811577 1.110014800 0.510312297 0.004632514 0.001002855 0.000203223
    0.001 1.497749998 1.874441717 0.000608674 0.000252584 0.000046982 0.376691719
    0.003 1.493249954 1.873350435 0.001487143 0.000653045 0.000146401 0.380100481
    1 0.005 1.488749789 1.872292816 0.002260519 0.001033805 0.000270882 0.383543027
    0.007 1.484249421 1.871268807 0.002993855 0.001418981 0.000424178 0.387019386
    0.009 1.479748770 1.870278358 0.003710857 0.001817596 0.000607710 0.390529588
    0.001 0.374437500 0.374438342 0.058076048 0.028505113 0.005664253 0.000000843
    0.003 0.373312489 0.373320060 0.058855962 0.028829615 0.005725785 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.059513521 0.029110717 0.005785674 0.000020989
    0.007 0.371062355 0.371103414 0.060110751 0.029371626 0.005847630 0.000041059
    0.009 0.369937192 0.370004933 0.060670143 0.029620993 0.005913049 0.000067741
    0.001 0.748874999 0.748876685 0.116152097 0.057010225 0.011328506 0.000001686
    0.003 0.746624977 0.746640120 0.117711924 0.057659231 0.011451571 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.119027042 0.058221435 0.011571349 0.000041979
    0.007 0.742124711 0.742206828 0.120221502 0.058743252 0.011695260 0.000082118
    CF-FDO 0.009 0.739874385 0.740009867 0.121340285 0.059241987 0.011826098 0.000135482
    0.001 1.123312499 1.123315027 0.674228145 0.085515337 0.016992759 0.000002528
    0.003 1.119937466 1.119960179 0.676567886 0.086488846 0.017177357 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.678540563 0.087332152 0.017357024 0.000062968
    0.007 1.113187066 1.113310242 0.680332252 0.088114877 0.017542890 0.000123176
    0.009 1.109811577 1.110014800 0.682010428 0.088862980 0.017739146 0.000203223
    0.001 1.497749998 0.747751682 0.232302506 0.114018763 0.022655325 0.749998316
    0.003 1.493249954 0.743265051 0.235408661 0.115303274 0.022887954 0.749984903
    1 0.005 1.488749789 0.738791557 0.238011895 0.116400680 0.023100508 0.749958232
    0.007 1.484249421 0.734330960 0.240360307 0.117403807 0.023307824 0.749918461
    0.009 1.479748770 0.729883021 0.242543858 0.118347262 0.023515483 0.749865749
    0.001 0.374437500 0.374438342 0.057494595 0.028265053 0.005621884 0.000000843
    0.003 0.373312489 0.373320060 0.057458804 0.028234478 0.005618312 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.057431335 0.028211504 0.005621628 0.000020989
    0.007 0.371062355 0.371103414 0.057412144 0.028196086 0.005631792 0.000041059
    0.009 0.369937192 0.370004933 0.057401187 0.028188186 0.005648765 0.000067741
    0.001 0.748874999 0.748876685 0.114989190 0.056530106 0.011243769 0.000001686
    0.003 0.746624977 0.746640120 0.114917608 0.056468957 0.011236624 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.114862670 0.056423007 0.011243256 0.000041979
    0.007 0.742124711 0.742206828 0.114824288 0.056392173 0.011263584 0.000082118
    A-BFDO 0.009 0.739874385 0.740009867 0.114802374 0.056376371 0.011297530 0.000135482
    0.001 1.123312499 1.123315027 0.672483784 0.084795158 0.016865652 0.000002528
    0.003 1.119937466 1.119960179 0.672376412 0.084703435 0.016854936 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.672294006 0.084634510 0.016864884 0.000062968
    0.007 1.113187066 1.113310242 0.672236432 0.084588259 0.016895375 0.000123176
    0.009 1.109811577 1.110014800 0.672203560 0.084564556 0.016946295 0.000203223
    0.001 1.497749998 0.747751682 0.229976692 0.113058524 0.022485850 0.749998316
    0.003 1.493249954 0.743265051 0.229820028 0.112922726 0.022458061 0.749984903
    1 0.005 1.488749789 0.738791557 0.229683151 0.112803824 0.022444322 0.749958232
    0.007 1.484249421 0.734330960 0.229565880 0.112701649 0.022444471 0.749918461
    0.009 1.479748770 0.729883021 0.229468035 0.112616030 0.022458348 0.749865749

     | Show Table
    DownLoad: CSV
    Table 4.  Numerical simulation of μ-soltion of example (5.2) at different fractional operators, orders, space and time levels.
    Operators ξo,ξ1,ξ2 t Approximate Exact solution AE at δ=0.90 AE at δ=0.95 AE at δ=0.99 AE at δ=1
    0.001 0.374437500 0.374438342 0.000605299 0.000249210 0.000043607 0.000000843
    0.003 0.373312489 0.373320060 0.001456767 0.000622670 0.000116025 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.002176140 0.000949425 0.000186502 0.000020989
    0.007 0.371062355 0.371103414 0.002828462 0.001253588 0.000258784 0.000041059
    0.009 0.369937192 0.370004933 0.003437432 0.001544171 0.000334285 0.000067741
    0.001 0.748874999 0.748876685 0.001210599 0.000498421 0.000087214 0.000001686
    0.003 0.746624977 0.746640120 0.002913535 0.001245340 0.000232051 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.004352280 0.001898850 0.000373004 0.000041979
    0.007 0.742124711 0.742206828 0.005656925 0.002507176 0.000517569 0.000082118
    C-FDO 0.009 0.739874385 0.740009867 0.006874865 0.003088343 0.000668570 0.000135482
    0.001 1.123312499 1.123315027 0.501815898 0.000747630 0.000130820 0.000002528
    0.003 1.119937466 1.119960179 0.504370302 0.001868010 0.000348076 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.506528420 0.002848275 0.000559506 0.000062968
    0.007 1.113187066 1.113310242 0.508485386 0.003760763 0.000776353 0.000123176
    0.009 1.109811577 1.110014800 0.510312297 0.004632514 0.001002855 0.000203223
    0.001 1.497749998 1.874441717 0.000608674 0.000252584 0.000046982 0.376691719
    0.003 1.493249954 1.873350435 0.001487143 0.000653045 0.000146401 0.380100481
    1 0.005 1.488749789 1.872292816 0.002260519 0.001033805 0.000270882 0.383543027
    0.007 1.484249421 1.871268807 0.002993855 0.001418981 0.000424178 0.387019386
    0.009 1.479748770 1.870278358 0.003710857 0.001817596 0.000607710 0.390529588
    0.001 0.374437500 0.374438342 0.058076048 0.028505113 0.005664253 0.000000843
    0.003 0.373312489 0.373320060 0.058855962 0.028829615 0.005725785 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.059513521 0.029110717 0.005785674 0.000020989
    0.007 0.371062355 0.371103414 0.060110751 0.029371626 0.005847630 0.000041059
    0.009 0.369937192 0.370004933 0.060670143 0.029620993 0.005913049 0.000067741
    0.001 0.748874999 0.748876685 0.116152097 0.057010225 0.011328506 0.000001686
    0.003 0.746624977 0.746640120 0.117711924 0.057659231 0.011451571 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.119027042 0.058221435 0.011571349 0.000041979
    0.007 0.742124711 0.742206828 0.120221502 0.058743252 0.011695260 0.000082118
    CF-FDO 0.009 0.739874385 0.740009867 0.121340285 0.059241987 0.011826098 0.000135482
    0.001 1.123312499 1.123315027 0.674228145 0.085515337 0.016992759 0.000002528
    0.003 1.119937466 1.119960179 0.676567886 0.086488846 0.017177357 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.678540563 0.087332152 0.017357024 0.000062968
    0.007 1.113187066 1.113310242 0.680332252 0.088114877 0.017542890 0.000123176
    0.009 1.109811577 1.110014800 0.682010428 0.088862980 0.017739146 0.000203223
    0.001 1.497749998 0.747751682 0.232302506 0.114018763 0.022655325 0.749998316
    0.003 1.493249954 0.743265051 0.235408661 0.115303274 0.022887954 0.749984903
    1 0.005 1.488749789 0.738791557 0.238011895 0.116400680 0.023100508 0.749958232
    0.007 1.484249421 0.734330960 0.240360307 0.117403807 0.023307824 0.749918461
    0.009 1.479748770 0.729883021 0.242543858 0.118347262 0.023515483 0.749865749
    0.001 0.374437500 0.374438342 0.057494595 0.028265053 0.005621884 0.000000843
    0.003 0.373312489 0.373320060 0.057458804 0.028234478 0.005618312 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.057431335 0.028211504 0.005621628 0.000020989
    0.007 0.371062355 0.371103414 0.057412144 0.028196086 0.005631792 0.000041059
    0.009 0.369937192 0.370004933 0.057401187 0.028188186 0.005648765 0.000067741
    0.001 0.748874999 0.748876685 0.114989190 0.056530106 0.011243769 0.000001686
    0.003 0.746624977 0.746640120 0.114917608 0.056468957 0.011236624 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.114862670 0.056423007 0.011243256 0.000041979
    0.007 0.742124711 0.742206828 0.114824288 0.056392173 0.011263584 0.000082118
    A-BFDO 0.009 0.739874385 0.740009867 0.114802374 0.056376371 0.011297530 0.000135482
    0.001 1.123312499 1.123315027 0.672483784 0.084795158 0.016865652 0.000002528
    0.003 1.119937466 1.119960179 0.672376412 0.084703435 0.016854936 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.672294006 0.084634510 0.016864884 0.000062968
    0.007 1.113187066 1.113310242 0.672236432 0.084588259 0.016895375 0.000123176
    0.009 1.109811577 1.110014800 0.672203560 0.084564556 0.016946295 0.000203223
    0.001 1.497749998 0.747751682 0.229976692 0.113058524 0.022485850 0.749998316
    0.003 1.493249954 0.743265051 0.229820028 0.112922726 0.022458061 0.749984903
    1 0.005 1.488749789 0.738791557 0.229683151 0.112803824 0.022444322 0.749958232
    0.007 1.484249421 0.734330960 0.229565880 0.112701649 0.022444471 0.749918461
    0.009 1.479748770 0.729883021 0.229468035 0.112616030 0.022458348 0.749865749

     | Show Table
    DownLoad: CSV
    Table 5.  Numerical simulation of ν-soltion of example (5.2) at different fractional operators, orders, space and time levels.
    Operators ξo,ξ1,ξ2 t Approximate Exact solution AE at δ=0.90 AE at δ=0.95 AE at δ=0.99 AE at δ=1
    0.001 0.374437500 0.374438342 0.000605299 0.000249210 0.000043607 0.000000843
    0.003 0.373312489 0.373320060 0.001456767 0.000622670 0.000116025 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.002176140 0.000949425 0.000186502 0.000020989
    0.007 0.371062355 0.371103414 0.002828462 0.001253588 0.000258784 0.000041059
    0.009 0.369937192 0.370004933 0.003437432 0.001544171 0.000334285 0.000067741
    0.001 0.748874999 0.748876685 0.001210599 0.000498421 0.000087214 0.000001686
    0.003 0.746624977 0.746640120 0.002913535 0.001245340 0.000232051 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.004352280 0.001898850 0.000373004 0.000041979
    0.007 0.742124711 0.742206828 0.005656925 0.002507176 0.000517569 0.000082118
    C-FDO 0.009 0.739874385 0.740009867 0.006874865 0.003088343 0.000668570 0.000135482
    0.001 1.123312499 1.123315027 0.501815898 0.000747630 0.000130820 0.000002528
    0.003 1.119937466 1.119960179 0.504370302 0.001868010 0.000348076 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.506528420 0.002848275 0.000559506 0.000062968
    0.007 1.113187066 1.113310242 0.508485386 0.003760763 0.000776353 0.000123176
    0.009 1.109811577 1.110014800 0.510312297 0.004632514 0.001002855 0.000203223
    0.001 1.497749998 1.874441717 0.000608674 0.000252584 0.000046982 0.376691719
    0.003 1.493249954 1.873350435 0.001487143 0.000653045 0.000146401 0.380100481
    1 0.005 1.488749789 1.872292816 0.002260519 0.001033805 0.000270882 0.383543027
    0.007 1.484249421 1.871268807 0.002993855 0.001418981 0.000424178 0.387019386
    0.009 1.479748770 1.870278358 0.003710857 0.001817596 0.000607710 0.390529588
    0.001 0.374437500 0.374438342 0.058076048 0.028505113 0.005664253 0.000000843
    0.003 0.373312489 0.373320060 0.058855962 0.028829615 0.005725785 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.059513521 0.029110717 0.005785674 0.000020989
    0.007 0.371062355 0.371103414 0.060110751 0.029371626 0.005847630 0.000041059
    0.009 0.369937192 0.370004933 0.060670143 0.029620993 0.005913049 0.000067741
    0.001 0.748874999 0.748876685 0.116152097 0.057010225 0.011328506 0.000001686
    0.003 0.746624977 0.746640120 0.117711924 0.057659231 0.011451571 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.119027042 0.058221435 0.011571349 0.000041979
    0.007 0.742124711 0.742206828 0.120221502 0.058743252 0.011695260 0.000082118
    CF-FDO 0.009 0.739874385 0.740009867 0.121340285 0.059241987 0.011826098 0.000135482
    0.001 1.123312499 1.123315027 0.674228145 0.085515337 0.016992759 0.000002528
    0.003 1.119937466 1.119960179 0.676567886 0.086488846 0.017177357 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.678540563 0.087332152 0.017357024 0.000062968
    0.007 1.113187066 1.113310242 0.680332252 0.088114877 0.017542890 0.000123176
    0.009 1.109811577 1.110014800 0.682010428 0.088862980 0.017739146 0.000203223
    0.001 1.497749998 0.747751682 0.232302506 0.114018763 0.022655325 0.749998316
    0.003 1.493249954 0.743265051 0.235408661 0.115303274 0.022887954 0.749984903
    1 0.005 1.488749789 0.738791557 0.238011895 0.116400680 0.023100508 0.749958232
    0.007 1.484249421 0.734330960 0.240360307 0.117403807 0.023307824 0.749918461
    0.009 1.479748770 0.729883021 0.242543858 0.118347262 0.023515483 0.749865749
    0.001 0.374437500 0.374438342 0.057494595 0.028265053 0.005621884 0.000000843
    0.003 0.373312489 0.373320060 0.057458804 0.028234478 0.005618312 0.000007571
    0.25 0.005 0.372187447 0.372208437 0.057431335 0.028211504 0.005621628 0.000020989
    0.007 0.371062355 0.371103414 0.057412144 0.028196086 0.005631792 0.000041059
    0.009 0.369937192 0.370004933 0.057401187 0.028188186 0.005648765 0.000067741
    0.001 0.748874999 0.748876685 0.114989190 0.056530106 0.011243769 0.000001686
    0.003 0.746624977 0.746640120 0.114917608 0.056468957 0.011236624 0.000015142
    0.50 0.005 0.744374894 0.744416873 0.114862670 0.056423007 0.011243256 0.000041979
    0.007 0.742124711 0.742206828 0.114824288 0.056392173 0.011263584 0.000082118
    A-BFDO 0.009 0.739874385 0.740009867 0.114802374 0.056376371 0.011297530 0.000135482
    0.001 1.123312499 1.123315027 0.672483784 0.084795158 0.016865652 0.000002528
    0.003 1.119937466 1.119960179 0.672376412 0.084703435 0.016854936 0.000022713
    0.75 0.005 1.116562342 1.116625310 0.672294006 0.084634510 0.016864884 0.000062968
    0.007 1.113187066 1.113310242 0.672236432 0.084588259 0.016895375 0.000123176
    0.009 1.109811577 1.110014800 0.672203560 0.084564556 0.016946295 0.000203223
    0.001 1.497749998 0.747751682 0.229976692 0.113058524 0.022485850 0.749998316
    0.003 1.493249954 0.743265051 0.229820028 0.112922726 0.022458061 0.749984903
    1 0.005 1.488749789 0.738791557 0.229683151 0.112803824 0.022444322 0.749958232
    0.007 1.484249421 0.734330960 0.229565880 0.112701649 0.022444471 0.749918461
    0.009 1.479748770 0.729883021 0.229468035 0.112616030 0.022458348 0.749865749

     | Show Table
    DownLoad: CSV

    In the present paper, the fractional views of the three-dimensional Navier-Stokes and non-homogeneous equations are examined by using the new iterative transform method (NITM). The fractional derivatives are replaced with different operators such as Atangana Baleanu, Caputo, and Caputo Fabrizio operators. The nonlinear terms in each problem are represented by the Jafaari polynomial, which has direct implementation over the entire problem. The solutions are obtained with the singular and non-singular kernels of the operators and confirm their effectiveness in the simulation of every problem. It is observed that the solutions under different operators are identical and verify the valuable dynamics of the suggested problems. Considering the benefits of the present operators, the expansion will be greatly appreciated in order to add new operators and approaches in the future. The current approach can be extended to solve other fractional problems due to its simple and straightforward implementation.

    FC Fractional calculus
    DEs Differential equations
    NITM New iterative transform method
    FPDEs fractional partial differentia equations
    C-FDO Caputo fractional differential operator
    C-FFDO Caputo-Fabrizio fractional differential operators
    A-BFDO Atangana-Baleanu fractional differential operator
    FDOs fractional differential operators
    AT Aboodh transform
    LT Laplace transform
    AE Absolute error

    The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2023 under project number FRB660073/0164.

    Qasim khan (Methodology, Software, Conceptualization, & Writing original draft);

    Anthony Suen (Supervision, Writing-review);

    Hassan Khan (Conceptualization, Draft Writing);

    Poom Kummam (Funding, Draft Writing).

    The authors declare there are no conflicts of interest.



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