Numerical analysis of multi-dimensional Navier–Stokes equation based on Yang–Abdel–Cattani fractional operator

1. Introduction

Nowadays, arbitrary order calculus is a hot study subject from a theoretical perspective and in a practical sense. It is well known that noninteger order calculus is a super set of an integer order calculus.¹ The concept of an arbitrary order calculus was born in 1695 by the legendary mathematicians in a letter written to L’Hopital from Leibniz.¹ The basic idea of the paper was about the derivative of order $n=0.5$. The world-class mind mathematicians Lagrange, Euler, Heaviside, Fourier, Riemann, Liouville, Abel, and others were contributed to the development of noninteger order calculus. After Caputo and Fabrizio introduced the first definition of nonsingular arbitrary order derivative, Atangana–Baleanu and Yang–Abdel–Aty–Cattani contributed on additional development of the new concept. Nonlocal operators are one of the key reasons why noninteger-order calculus is becoming increasingly popular. Very recently, the concept of arbitrary-order local and nonlocal operators has become a main study subject in science and technology, attracting a huge number of authors.² Few years ago, a lot of attempts on target were made to find more interesting, and fresh nonsingular arbitrary order derivatives depend on kernels. A novel noninteger order Caputo–Fabrizio operator derived in 2015 and this operator addressed several linear and nonlinear issues. In 2016, Atangana and Baleanu explored a famous nonsingular derivative within the context of the Mittag-Leffler kernel and applied it to a variety of issues.³ Yang et al.⁴ published a highly interesting noninteger-order derivative operator based on the fractional Rabotnov exponential function (FREF) four years ago in 2019. They used the proposed concept to solve an arbitrary-order heat transfer equation. Malyk et al.⁵ obtained numerical result of a nonlinear arbitrary order diffusion equation based on the Yang–Abdel–Cattani derivative operator.

Noninteger-order derivative equations have recently gained popularity due to their importance and applicability in mathematics and other natural and social sciences.⁶ In the last few years, noninteger differential equations have been used in almost all areas of science and technology. For example, authors in Ref. 7 have proposed HIV-1 infection model based on different fractional derivative operators. They solved the model numerically with different arbitrary order values. Srivastava et al.⁸ studied an arbitrary order mathematical model of diabetes and its resulting complications. In Ref. 9, authors have investigated SZIR Model of Zombies infection and the system was represented by fractional order derivatives. Numerical simulation-based study for prediction of COVID-19 spread in India was performed by the authors in Ref. 10. A SEIR model was created using fractional order nonlinear differential equations. Sharma et al.¹¹ comprehended the omicron variant model with the help of arbitrary order derivative operator. Authors in Ref. 12 investigated the effect of vaccination on COVID-19 transmission dynamics in Ethiopia. The model was represented by fractional order differential equations in frame of Atangana–Baleanu fractional derivative in Caputo sense. Habenom et al.¹³ performed analysis of fractional order COVID-19 pandemic model in Ethiopia. Fractional order hepatitis B virus model was solved with the help of Homotopy decomposition method by the authors of Ref. 14. With the help of Sumudu transform, Aychluh et al.¹⁵ solved an arbitrary order kinetic equation related to the Riemann xi of function. Dubey and Goswami¹⁶ studied a mathematical analysis of diabetes complication with the help of fractional order operator with no singular kernel.

As nonlinear problems exist anywhere, engineers and mathematicians have attempted to target nonlinear equations and their solutions.^17,18,19 The nonlinear arbitrary-order partial differential equations are the superset of nonlinear equations of positive integer order. Nonlinear equations describe the most significant processes in the planet. One of the most important nonlinear equations is the Navier–Stokes equation. For the first time in 1822, an integer order Navier–Stokes (N-S) equation has been originated.²⁰ It is a very famous and popular equation that plays a significant role in the study of the motion of viscous fluid flow. This equation describes many key phenomena, such as ocean circulation, liquid, blood, and air flows around the wings of an aircraft. For the first time in 2005, El-Shahed and Salem²¹ introduced the fractional N-S equation. In recent decades, the results of arbitrary order N-S equations have gained the considerable attention of several investigators in the science community. Arbitrary-order N-S equations have recently been studied and solved with the help of numerous analytical and numerical techniques. In 2006, Momani and Odibat²² solved a noninteger-order N-S equation using adomian decomposition method (ADM). The modified Laplace decomposition method²³ has been implemented efficiently to obtain analytic result for an arbitrary order N-S equation. In 2009, Khan²⁴ studied the arbitrary order N-S equation analytically with the help of He’s homotopy perturbation and variational iteration approaches. In 2016, Kangle and Sanyang²⁵ established Elzaki transform technique to derive the analytical results of the time-fractional N-S equation. Multi-dimensional N-S equation with arbitrary orders was investigated analytically by the authors of Ref. 26. Singh and Kumar²⁷ addressed the numerical simulation of time-fractional model of N-S equation. In 2022, Albalawi et al.²⁸ investigated multi-dimensional NS equation in Caputo sense. For additional reading on analytical and numerical schemes for solving N-S equation, an interesting reader is referred to Ref. 29 and the references cited in. In this paper, we implement the Laplace Adomian decomposition method (LADM) for the results of multidimensional nonlinear N-S equations in YAC sense. The computational process of the proposed approach is very easy and supported by the graphical plotting.

Here, we consider the following time-fractional model of N-S equation in YAC sense :

where $F=(\psi ,\vartheta ,\eta )$, $𝔱,p$ denote the fluid vector, time and the pressure, respectively. $(\psi ,\vartheta ,\eta )$ are spatial components in $\mathrm{\Phi }$ and $\partial \mathrm{\Phi }$ is the boundary of $\mathrm{\Phi }$, $\gamma$ denotes dynamic viscosity and $\rho$ the density while ${\rho }_0=\gamma ∕\rho$ is the kinematic viscosity of the flow. ${}_0^{YAC}{𝔇}_{𝔱}^{\kappa }$ is the Yang–Abel–Cattani fractional derivative operator of order $\kappa$. In standard form, Eq. (1.1) becomes with starting values where ${\nabla }^2=\frac{{\partial }^2}{\partial \psi }+\frac{{\partial }^2}{\partial \vartheta }+\frac{{\partial }^2}{\partial \eta }$.

This paper is arranged as follows. In Sec. 1, introduction of noninteger order operators and history of N-S equation and fractional calculus are written here. We next recall some fundamental definitions and properties about the fresh arbitrary order YAC derivative and integral in Sec. 2. The solution procedures, existence and oneness of the results and the 3D plots are found in Sec. 3. Finally, in Sec. 4, the total work of our paper is concluded.

2. The Yang–Abdel–Cattani Fractional Calculus

Here, we introduce some key definitions and properties of FREF and an arbitrary order YAC derivative and integral operators.

Definition 2.1. The Rabotnov exponential function of noninteger order $\kappa$ is defined as follows^4,5 :

⁴ and its $𝔏$-transform is where $𝔏$ is the Laplace transform.

Definition 2.2. The generalized YAC arbitrary-order derivative in terms of the FREF is defined as follows^4,5:

⁴ and its $𝔏$-transform is

Definition 2.3. For $0<𝔴\in ℝ$ and $\kappa \in (0,1]$, the following defines the fractional integral in terms of the FREF of order $\kappa$^4,5 :

The $𝔏$-transform of Eq. (2.5) is⁴

Definition 2.4. Prabhakar generalized Mittag-Leffler function and is defined as follows:

where ${(\varpi )}_i$ is a Pochhammer notation³⁰ and has the following form : The following are special functions, which are connected with the Prabhakar function (see Ref. 30). where ${𝔈}_{\varpi }^{(\upsilon )}(u,\kappa )$ is a polynomial of degree $\varpi$ in $u^{\kappa }$ studied in Ref. 30. where ${\mathrm{\text{L}}}_{\varpi }^{(\upsilon )}$ is the well-known Laguerre polynomial. In this paper, the results of an arbitrary-order N-S equation in YAC sense will be stated in terms of the Prabhakar functions defined in Eqs. (2.9) and (2.10).

3. Laplace Adomian Decomposition Method

Considering the following arbitrary order partial differential equation in the YAC sense and taking $x=(q,r,s)$.

The linear and nonlinear terms of the first equation of the system (1.2) are denoted by ${\mathrm{\text{L}}}_1$ and ${\mathrm{\text{N}}}_1$, respectively, and $S_1(x,t)$ is the source term with the following beginning value : Applying the $𝔏$-transform on left and right sides of Eq. (3.1), we get The ADM solution is The nonlinear term in the problem is expressed as where are called Adomian polynomials. Putting Eqs. (3.5) and (3.6) in Eq. (3.4), we get Applying the decomposition method, we get and Using ${𝔏}^{-1}$-transform to Eqs. (3.9) and (3.10), we have and For the second and third equations of (1.2), we have the following expressions : and where $B_n$ and $C_n$ are Adomian polynomials corresponding to the nonlinear terms of the second and third equations of a system (1.2), respectively.

3.1. Existence and oneness of the result

Lemma 3.1. The YAC operator of order $\kappa$ possesses the Lipschitz condition with d as a Lipschitz constant. That is

Proof. Using the definition of YAC fractional derivative of order $\kappa$, we have

Now, applying the first-order derivative’s Lipschitz condition, then, we can find a small constant such as Finally, we get the following result: Hence, the proof. □

Theorem 3.1. Let us assume that the function $\mathrm{\Psi }(x,𝔱,\psi ,{\psi }_q,{\psi }_r,{\psi }_s,{\psi }_{qq},{\psi }_{rr},{\psi }_{ss})$ satisfies the Lipschitz condition as

We also assume that

where ${\theta }_1,{\theta }_2,{\theta }_3,{\beta }_1,{\beta }_2,{\beta }_3\in {ℝ}^{+}$ then the system (1.2) has a unique solution.

Proof. We define

We first show that $\mathrm{\Psi }$ satisfies Lipschitz condition. Consider

where $\mathrm{\text{N}}=\mathrm{\text{M}}+{\mathrm{\text{A}}}_1{\theta }_1+{\mathrm{\text{A}}}_2{\theta }_2+{\mathrm{\text{A}}}_3{\theta }_3+{\mathrm{\text{B}}}_1{\beta }_1+{\mathrm{\text{B}}}_2{\beta }_2+{\mathrm{\text{B}}}_3{\beta }_3\in {ℝ}^{+}$.

Applying the Yang–Abdel–Cattani fractional integral to the first equation of (1.2), we have

For convenience, we write Finally, we have Now, we consider For the above map to be a contraction, we must have Hence the existence and the oneness of the result follows as a consequence of the Banach fixed point theory. □

Example 1. Consider the two-dimensional time-fractional N-S equation with zero source terms

with the starting values Taking the $𝔏$-transform of left and right sides of Eq. (3.18) and after some simplification, we have Now, taking ${𝔏}^{-1}$-transform on two sides of Eq. (3.20), we arrived to the following form: Using the ADM procedure, we get where Adomian polynomial components $A_n(\psi ,\eta )$ are given as follows :

and so on. For $n=0,1,2,\dots$ and Following the same processes above, one can obtain the subsequent terms as follows :

Therefore, the general solution is and

Example 2. Consider an arbitrary order Yang–Abdel–Cattani N-S equation (3.18) with respect to the beginning values given here

We follow similar processes above and obtain the following successive terms: The next few terms have the following expressions :

Therefore, the general solution is

and

Example 3. Here, we will address the solution for time-fractional-ordered ($2+1$)-layered N-S condition (1.2) subject to the following starting values :

Following the same procedures of the above two examples, the successive terms can be found as follows:

The solution has the following form:

4. Conclusion

An arbitrary order N-S equation is one of the most significant equations used to govern the motion of viscous fluid flow. The first aim of the current task is to obtain the results of noninteger order multi-dimensional N-S equations, where noninteger operator is taken in a new YAC sense. The Laplace adomian decomposition approach is applied for the numerical analysis of time-fractional model of N-S equations with beginning values. With the consideration of different fractional order values in YAC fractional operator, new results of the nonlinear N-S models have been obtained. Existence and oneness of the technique for the proposed new fractional order N-S model in YAC sense have been successfully shown. The novelty and advantage of this work were that we suggested a new extension of an arbitrary order multi-dimensional N-S equation in YAC sense and discussed the numerical results in graphical form, which were very important to investigating the motion of viscous fluid flow. Three test examples are used to support our theoretical procedures. The outcomes of the system are in terms of three parametric Prabhakar function. The graphical results have been found with the use of MATLAB R2016a mathematical tool. Figures 1–6 are plotted with different fractional orders and these graphs support our theoretical procedures. Future work will further investigate the multi-dimensional N-S equation in the updated version of the fractional operator as it is still open.

Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

Acknowledgment

The authors would like to extend their sincere gratitude to the editor and reviewers for their insightful comments and suggestions that raised the paper’s quality.

Conflicts of Interest

No potential conflict of interest was reported by the authors.

Funding

This research is not funded by any agency.

ORCID

Mulualem Aychluh  https://orcid.org/0000-0002-5295-1559

Minilik Ayalew  https://orcid.org/0000-0002-5060-246X

D. L. Suthar  https://orcid.org/0000-0001-9978-2177

S. D. Purohit  https://orcid.org/0000-0002-1098-5961