A Novel Approach to Solving Fractional Navier–Stokes Equations Using the Elzaki Transform

Abstract

A clear method is provided to explain a new approach for solving systems of fractional Navier–Stokes equations (SFNSEs) using initial conditions (ICs) that rely on the Elzaki transform (ET). A few steps show the technique’s validity and utility for handling SFNSE solutions. For fractional derivatives, the Caputo sense is used. This method does not need discretization or limiting assumptions and may be used to solve both linear and nonlinear SFNSEs. By eliminating round-off mistakes, the technique reduces the need for numerical calculations. Using examples, the new technique’s accuracy and efficacy are illustrated.

1. Introduction

In applied math and science, FPDEs are employed to investigate and illustrate a variety of practical problems, including diffusion, damping laws, mathematical biology, fluid mechanics, electrical circuits, diffusion, relaxation processes, etc. Therefore, finding a treatment for FPDEs is one of several essential objectives of scientific study. Although fractional derivatives (FDs) have an extensive history in mathematics, they were not employed extensively in research for an extended period of time. A fractional derivative’s unpopularity might be due in part to the ubiquity of non-equivalent meanings of the term [1]. Furthermore, because fractional derivatives are non-local, they cannot be precisely interpreted geometrically [2].

Numerous phenomena, including the oscillation of earthquakes [3], may be explained using FDs and faults in the fluid-dynamic traffic model brought on by the assumption of continuous traffic flow [4]. According to empirical data, fractional order phenomena are proposed in DEs and FPDEs for streams of seepage in porous media by [5,6], respectively. Mainardi [7] provides a summary of several uses of FDs in continuous mechanics and the mechanics of statistics. A new method for dynamic system parameter estimation using neural ordinary differential equations and the octonion linear canonical transform was studied by Yong Yang et al. [8] and Nan Jiang et al. [9].

It is thought to be advantageous to use Navier–Stokes equations (NSEs) to illustrate the physical aspects of scientific and engineering research. These equations are mostly used to control wind current around a wing, water streams in a line, sea flows, and climate estimation. Additionally, NSEs are closely linked to the fundamental design of automobiles and airplanes, bloodstream analysis, intensity station objectives, and contamination analysis. Furthermore, the coupling of Maxwell’s equation and NSEs is essential for the study of magnetohydrodynamics. Distinct models of reality have been used throughout the literature to control prepared material situations ever since they were first presented. In [10], RajaramaMohan Jenaand S. Chakraverty solved time-fractional Navier–Stokes equations using homotopy perturbation with ET, and the time-fractional Navier–Stokes equation was also solved in series by Burqan et al. [11] using a Laplace transform-based approach. The Laplace decomposition technique was used by Kumar et al. [12] to approach the analytical solution of the fractional Navier–Stokes equation.

NSEs are a well-established area of study in characterizing fluid motion. In certain straightforward situations, they can be resolved analytically, but because of their complexity, numerical techniques are typically needed. However, it becomes considerably more difficult to obtain numerical solutions when fractional derivatives are taken into account.

Fractional calculus’s application to the NSEs has drawn a lot of interest lately [10,13,14]. An FD is used in place of the time derivative in time-fractional NSEs, which are a fractional extension of the classical NSEs. Because of their broad use in fluid dynamics, geophysics, and several other scientific and technical domains, the resolution of time-fractional NSEs in n dimensions is a crucial area of study (see [15,16]); fractional equations were also studied in [17,18,19,20], and the solution of fractional differential equations using the integral transform was addressed in [21].

Because various initial approximations are used in successive iterations of the solution, the new procedure yields more accurate results.

The broad description of the proposed solution is applied to certain examples of the aforementioned problems. It is challenging to find analytical solutions for SFNSEs, including ICs. The current study uses an approach that is very simple to comprehend and use to produce closed-form analytical solutions to the SFNSEs.

In terms of physical interpretation, the FNSE solutions are highly helpful since they provide an explanation of the physics behind many important scientific and engineering phenomena, including airflow around a wing, water flow in pipes, ocean currents, and weather modeling. In their full and reduced versions, the FNSEs also aid in the design of automobiles and aero planes, research into blood flow, power plant design, pollution studies, and many other problems.

In this study, we solve SFNSEs using a specific analytical technique that makes use of the ET. Examples are given to illustrate this method’s efficacy and dependability. This method may be used to solve the functional equations that are produced when many processes are numerically represented; we also relied on the Caputo fractional derivatives in this study because of their simplicity and effectiveness.

We know that there is no specific method that can solve all systems of linear or nonlinear fractional differential equations, so many methods have been researched and discovered for this purpose, and among these methods is the method that has been presented. It will be developed and generalized in future studies to include a wider field of fractional differential equations.

This section covers some of the concepts and conclusions about fractional calculus and differential and integral calculus that are used in the current study.

Definition 1. 

The Riemann–Liouville fractional integral operator of order $\alpha \ge 0,$ of a function $\mathcal{Đ} \in C_{\mu }, \mu \ge -1,$ is defined as

$$J^{\alpha }\mathcal{Đ}\left(\varkappa \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle \underset{0}{\overset{\varkappa }{\int }}{\left(\varkappa -\upsilon \right)}^{\alpha - 1}\mathcal{Đ}\left(\upsilon \right) d\upsilon , \alpha >0},J^0 \mathcal{Đ}\left(\varkappa \right) = \mathcal{Đ}\left(\varkappa \right)$$

(1)

 where  $\Gamma$ is the gamma function and $\upsilon$ is a parameter.

Definition 2. 

The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral, so the Caputo fractional derivative of $\mathcal{Đ}\left(\varkappa \right),$ is 

$$D^{\alpha }\mathcal{Đ}\left(\varkappa \right) = J^{m - \alpha }{D}^m\mathcal{Đ}\left(\varkappa \right)$$

(2)

For $m - 1 < \alpha \le m , m \in N, \varkappa > 0,$ and $\mathcal{Đ} \in C^m(0,\infty ).$

To determine the correct order of an FD, an ordinary derivative is calculated first, followed by a fractional integral.

Similar to the R-L fractional integral operator, the integer-order integration is a linear operation:

$$J^{\alpha } \left({\displaystyle \sum _{i = 1}^n{c}_i {\mathcal{Đ}}_i\left(\varkappa \right)}\right) = {\displaystyle \sum _{i = 1}^n{c}_i J^{\alpha }{\mathcal{Đ}}_i\left(\varkappa \right)}$$

(3)

where ${\left\{c_i\right\}}_{i= 1}^n$ are constants.

The use of the Caputo definition is supported by the belief that FD has a Caputo meaning in the setting of the current inquiry.

2. Elzaki Transform

The updated Sumudu transform (ET) is defined broadly in this article as follows:

$$E\left[\mathcal{Đ}(\varkappa )\right]=\zeta {\displaystyle \underset{0}{\overset{\infty }{\int }}\mathcal{Đ}(\varkappa ) e^{-\frac{\varkappa }{\zeta }}}d\varkappa =T(\zeta ), \varkappa >0,$$

(4)

where $\zeta$ is a complex value.

NPDEs [22,23] and PDEs with proportional delay in a single variable [24] are solved by ET. ET is a new integral transform which was introduced by Tarig ELzaki [23]. ET is a modified transform of Sumudu and Laplace transforms. It is worth mentioning that there are some differential equations with variable coefficients which may not be solved by Sumudu and Laplace transforms but may easily be solved with the aid of ET [25].

Lemma 1 

([21]). Let $\mathcal{Đ} (\varkappa ,\lambda ,\tau )$ be a function in which three variables have the ET. Then, we obtain

$$\begin{array}{l}E\left[{\displaystyle \frac{\mathcal{\partial } \mathcal{Đ} (\varkappa ,\lambda ,\tau )}{\mathcal{\partial }\tau }}\right]={\displaystyle \frac{1}{\zeta }}T(\varkappa ,\lambda ,\zeta )-\zeta \mathcal{Đ} (\varkappa ,\lambda ,0) , \\ E\left[{\displaystyle \frac{{\mathcal{\partial }}^2\mathcal{Đ} (\varkappa ,\lambda ,\tau )}{\mathcal{\partial }{\tau }^2}}\right]={\displaystyle \frac{1}{{\zeta }^2}}T(\varkappa ,\lambda ,\zeta )-\mathcal{Đ} (\varkappa ,\lambda ,0)-\zeta {\displaystyle \frac{\mathcal{\partial } \mathcal{Đ} (\varkappa ,\lambda ,0)}{\mathcal{\partial }\tau }} , \\ E\left[{\displaystyle \frac{\mathcal{\partial } \mathcal{Đ} (\varkappa ,\lambda ,\tau )}{\mathcal{\partial }\varkappa }}\right]={\displaystyle \frac{d}{d\varkappa }}\left[T(\varkappa ,\lambda ,\zeta )\right], E\left[{\displaystyle \frac{{\mathcal{\partial }}^2\mathcal{Đ} (\varkappa ,\lambda ,\tau )}{\mathcal{\partial }{\varkappa }^2}}\right]= {\displaystyle \frac{d^2}{d{\varkappa }^2}}\left[T(\varkappa ,\lambda ,\zeta )\right]\\ E\left[{\displaystyle \frac{\mathcal{\partial } \mathcal{Đ} (\varkappa ,\lambda ,\tau )}{\mathcal{\partial }\lambda }}\right]={\displaystyle \frac{d}{d\lambda }}\left[T(\varkappa ,\lambda ,\zeta )\right], E\left[{\displaystyle \frac{{\mathcal{\partial }}^2\mathcal{Đ} (\varkappa ,\lambda ,\tau )}{\mathcal{\partial }{\lambda }^2}}\right]= {\displaystyle \frac{d^2}{d{ƛ}^2}}\left[T(\varkappa ,\lambda ,\zeta )\right]\end{array}$$

ET of some functions:

$\mathcal{Đ}\left(\varkappa \right)$	$E\left[\mathcal{Đ}\left(\varkappa \right)\right]=T\left(\zeta \right)$
$1$	${\zeta }^2$
$\varkappa$	${\zeta }^3$
${\varkappa }^n$	$n! {\zeta }^{n+2}$
$e^{a\varkappa }$	${\displaystyle \frac{{\zeta }^2}{1-a\zeta }}$
$\sin a\varkappa$	${\displaystyle \frac{a{\zeta }^3}{1+a^2{\zeta }^2}}$
$\cos a\varkappa$	${\displaystyle \frac{{\zeta }^2}{1+a^2{\zeta }^2}}$

Mittag–LefflerFunctions (M-LFs)

In FDE solutions, the M-LFs are important and common. The scientific community is developing more interested in M-LFs as a result of the growing popularity of innovative models and pure and practical mathematics among academics and researchers. By focusing on the concept of M-LFs, we can clarify a variety of phenomena in various processes that evolve or degrade too rapidly to be sufficiently represented by traditional functions such as the exponential formula and its surrounds.

M-LFs are defined as,

$${\in }_{\omega }\left(\varkappa \right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\varkappa }^n}{\Gamma \left(\omega n+1\right)}},{\in }_{\omega , \xi }\left(\varkappa \right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\varkappa }^n}{\Gamma \left(\omega n+\xi \right)}}.$$

(5)

where, $\Gamma$ is the Gamma function, and $\omega ,\xi$ are complex parameter with $\mathrm{Re}\omega >0, \mathrm{Re}\xi >0$.

For special values of $\omega ,\xi$, we have

$$\begin{array}{l}1. {\in }_{\omega , 1}\left(\varkappa \right) ={\in }_{\omega }\left(\varkappa \right) 2. {\in }_{0 , 1}\left(\varkappa \right)={\displaystyle \frac{1}{1-\varkappa }} 3. {\in }_{1 , 1}\left(\varkappa \right)=e^{\varkappa }\\ 4. {\in }_{2, 1}\left(-{\varkappa }^2\right)=\cos \varkappa 5. {\in }_{2, 2}\left(-{\varkappa }^2\right)={\displaystyle \frac{\sin \varkappa }{\varkappa }}\end{array}$$

We give certain lemmas here that allow one to deduce a function $\mathcal{Đ}(\varkappa )$ from its ET.

Lemma 2 

([21]). The ET of R-L operator of order $\alpha > 0$ is

$$E\left[J^{\alpha } \mathcal{Đ}\left(\varkappa \right)\right] = {\zeta }^{\alpha } T\left(\zeta \right).$$

(6)

where $\zeta$ is a complex value (ET parameter).

Proof. 

We begin with

$$E\left[J^{\alpha } \mathcal{Đ}\left(\varkappa \right)\right] = E\left[{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}} {\displaystyle \underset{0}{\overset{\varkappa }{\int }}{\left(\varkappa -\upsilon \right)}^{\alpha - 1}\mathcal{Đ}\left(\upsilon \right) d\upsilon }\right] = {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}} {\displaystyle \frac{1}{\zeta }} T\left(\zeta \right) G\left(\zeta \right)={\zeta }^{\alpha } T\left(\zeta \right),$$

where

$$G\left(\zeta \right) = E\left[{\varkappa }^{\alpha - 1}\right] = {\zeta }^{\alpha + 1}\Gamma \left(\alpha \right)$$

□

Lemma 3 

([21]). The ET of Caputo fractional derivative (CFD) for $\alpha >0, m - 1 < \alpha \le m, m \in N,$ is

$$\begin{array}{l}E\left[D_{\tau }^{\alpha }\mathcal{Đ} (\varkappa ,\lambda ,\tau )\right] = {\zeta }^{m-\alpha }\left[\begin{array}{l}{\displaystyle \frac{T(\varkappa ,\lambda ,\zeta )}{{\zeta }^m}} - {\displaystyle \frac{\mathcal{Đ} (\varkappa ,\lambda ,0) }{{\zeta }^{m-2}}} - {\displaystyle \frac{\frac{\mathcal{\partial } \mathcal{Đ} (\varkappa ,\lambda ,0)}{\mathcal{\partial }\tau }}{{\zeta }^{m - 3}}}\\ - \cdot \cdot \cdot - \zeta {\displaystyle \frac{{\mathcal{\partial }}^{m-1} \Phi (\varkappa ,\lambda ,0)}{\mathcal{\partial }{\tau }^{m-1}}}\end{array}\right]\\ m-1<\alpha \le m,\end{array}$$

$$Or, E\left[D_{\tau }^{\alpha }\mathcal{Đ} (\varkappa ,\lambda ,\tau )\right] ={\displaystyle \frac{1}{{\zeta }^{\alpha }}}E\left[\mathcal{Đ} (\varkappa ,\lambda ,\tau )\right]-{\displaystyle \sum _{k=0}^{m-1}\frac{{\mathcal{\partial }}^k \mathcal{Đ} (\varkappa ,\lambda ,0)}{\mathcal{\partial }{\tau }^k}}{\zeta }^{2-\alpha +k},$$

Lemma 4 

([21]). If the constants $\alpha , \beta > 0,$ $a \in C$ and ${\displaystyle \frac{1}{{\zeta }^{\alpha }}} > \left|a\right|,$ then the inverse of ET can be represented by the model below:

$$E^{- 1}\left[{\displaystyle \frac{{\zeta }^{\beta + 1}}{1 + a {\zeta }^{\alpha }}}\right] = {\varkappa }^{\beta - 1} {\in }_{\alpha , \beta }\left(- a {\varkappa }^{\alpha }\right)$$

3. The Method

In this section, we will study ET for solving SFNSEs where we can transform SFNSEs into a simple set of linear algebraic equations, and then these algebraic equations can be solved using the inverse of the ET.

The SFNSEs can be used in the following ways to clarify the core idea of the suggested study plan.

Consider the system of FPDE [25]:

$$\begin{array}{l}{D}_{\tau }^{\alpha }\mathcal{Đ}(\varkappa ,\lambda ,\tau )+R_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]=g_1(\varkappa ,\lambda ,\tau ) , \\ D_{\tau }^{\alpha } Ŧ(\varkappa ,\lambda ,\tau )+R_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]=g_2(\varkappa ,\lambda ,\tau ) , \\ 0<\alpha \le 1, \varkappa ,\lambda ,\tau >0, \end{array}$$

(7)

With the ICs,

$$\mathcal{Đ}(\varkappa ,\lambda ,0)=h_1(\varkappa ,\lambda ) , Ŧ(\varkappa ,\lambda ,0)=h_2(\varkappa ,\lambda ) ,$$

(8)

where $D_{\tau }^{\alpha }(\varkappa ,\lambda ,\tau )$ is a Caputo fractional derivative, $R_1,R_2$,$N_1, N_2$ are linear and nonlinear operators, $h_1(\varkappa ,\lambda ) , h_2(\varkappa ,\lambda )$ are functions, and $g_1(\varkappa ,\lambda ,\tau ), g_2(\varkappa ,\lambda ,\tau )$ are inhomogeneous terms.

We can find the following equations by applying the ET to Equation (7) and using the ICs in Equation (8).

$$\begin{array}{l}E\left[D_{\tau }^{\alpha }\mathcal{Đ}(\varkappa ,\lambda ,\tau )\right]+E\left\{R_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]\right\}=E\left[g_1(\varkappa ,\lambda ,\tau )\right] , \\ E\left[D_{\tau }^{\alpha } Ŧ(\varkappa ,\lambda ,\tau )\right]+E\left\{R_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]\right\}=E\left[g_2(\varkappa ,\lambda ,\tau )\right] , \\ E\left(\mathcal{Đ}(\varkappa ,\lambda ,\tau )\right)={\zeta }^2{h}_1(\varkappa ,\lambda )-{\zeta }^{\alpha }E\left\{R_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]-g_1(\varkappa ,\lambda ,\tau )\right\}\\ E\left(Ŧ(\varkappa ,\lambda ,\tau )\right)={\zeta }^2{h}_2(\varkappa ,\lambda )-{\zeta }^{\alpha }E\left\{R_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]-g_2(\varkappa ,\lambda ,\tau )\right\},\end{array}$$

(9)

The solutions of Equation (7) should be in series form:

$$\mathcal{Đ}(\varkappa ,\lambda ,\tau ) ={\displaystyle \sum _{n=0}^{\infty }{\mathcal{Đ}}_n} , Ŧ(\varkappa ,\lambda ,\tau ) ={\displaystyle \sum _{n=0}^{\infty }{Ŧ}_n}$$

(10)

Apply the inverse of ET to Equation (9), and use Equation (10) to obtain

$$\begin{array}{l}{\displaystyle \sum _{n=0}^{\infty }\mathcal{Đ}(\varkappa ,\lambda ,\tau )}=G_1(\varkappa ,\lambda ,\tau )-E^{-1}\left\{{\zeta }^{\alpha }E\left[R_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_1\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]\right]\right\}\\ {\displaystyle \sum _{n=0}^{\infty }Ŧ(\varkappa ,\lambda ,\tau )}=G_2(\varkappa ,\lambda ,\tau )-E^{-1}\left\{{\zeta }^{\alpha }E\left[R_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]+N_2\left[\mathcal{Đ},Ŧ,{\mathcal{Đ}}_{\varkappa },{Ŧ}_{\varkappa }\right]\right]\right\}\end{array}$$

(11)

$G_1(\varkappa ,\lambda ,\tau ), G_2(\varkappa ,\lambda ,\tau ),$ are terms derived from the source terms and the ICs.

This approach relies on our ability to elect the initial iterations that yield exact results in a single step.

We can write Equation (11) in the following relations that can be used iteratively to obtain the solutions.

$$\begin{array}{l}{\mathcal{Đ}}_{n+1}(\varkappa ,\lambda ,\tau )=E^{-1}\left\{{\zeta }^{\alpha }E\left[R_1\left[\left({\mathcal{Đ}}_n\right),\left({Ŧ}_n\right),{\left({\mathcal{Đ}}_n\right)}_{\varkappa },{\left({Ŧ}_n\right)}_{\varkappa }\right]+N_1\left[\left({\mathcal{Đ}}_n\right),\left({Ŧ}_n\right),{\left({\mathcal{Đ}}_n\right)}_{\varkappa },{\left({Ŧ}_n\right)}_{\varkappa }\right]\right]\right\} ,\\ {\mathcal{Đ}}_0(\varkappa ,\lambda ,\tau )=G_1(\varkappa ,\lambda ,\tau ),\\ {Ŧ}_{n+1}(\varkappa ,\lambda ,\tau )=E^{-1}\left\{{\zeta }^{\alpha }E\left[R_2\left[\left({\mathcal{Đ}}_n\right),\left({Ŧ}_n\right),{\left({\mathcal{Đ}}_n\right)}_{\varkappa },{\left({Ŧ}_n\right)}_{\varkappa }\right]+N_2\left[\left({\mathcal{Đ}}_n\right),\left({Ŧ}_n\right),{\left({\mathcal{Đ}}_n\right)}_{\varkappa },{\left({Ŧ}_n\right)}_{\varkappa }\right]\right]\right\} ,\\ {Ŧ}_0(\varkappa ,\lambda ,\tau )=G_2(\varkappa ,\lambda ,\tau ),\end{array}$$

(12)

Equation (12) allowed us to determine the following: ${\mathcal{Đ}}_0, {\mathcal{Đ}}_1, {\mathcal{Đ}}_2, \dots , {Ŧ}_0, {Ŧ}_1, {Ŧ}_2, \dots ,$

Equation (10) can then be used to find the result.

We show that the provided method can effectively handle SFNSEs involving ICs and ETs.

Note1: The choices of the initial iterations are very important as they can activate the convergence of the method and succeed in calculating the exact solutions in a few steps. This approach is dependent on how we select the initial iterations that result in exact solutions in a few steps. We can select initial iterations of all or a portion of the functions $g_1(\varkappa ,\lambda ,\tau ) , g_2(\varkappa ,\lambda ,\tau ) ,$ if the ICs are equal to zero.

4. Illustrative Examples

Since fractional Navier–Stokes equation systems are very complicated and difficult to solve manually, some simple examples are discussed and the remaining complicated equations will be studied in the near future.

We used the method described in this study to obtain exact solutions for SFNSE.

Example 1. 

Consider the SFNSE:

$$\begin{array}{l}{D}_{\tau }^{\alpha }\mathcal{Đ}+\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }+Ŧ{\mathcal{Đ}}_{\lambda }={\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right) , \\ D_{\tau }^{\alpha }Ŧ+\mathcal{Đ}{Ŧ}_{\varkappa }+Ŧ{Ŧ}_{\lambda }={\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right) , 0<\alpha \le 1 , \varkappa ,\lambda ,\tau >0\end{array}$$

(13)

With the ICs,

$$\mathcal{Đ}(\varkappa ,\lambda ,0)=-e^{\varkappa +\lambda }, Ŧ(\varkappa ,\lambda ,0)=e^{\varkappa +\lambda },$$

(14)

where $\mathcal{Đ}, Ŧ$ represent fluid vector and ${\sigma }_0$ is the density.

We shall deduct the quantities $2{\sigma }_0\mathcal{Đ}, 2{\sigma }_0Ŧ$ from either side of Equation (13) in order to expedite the solution process and implement the effective solution approach.

As for adding certain quantities to both sides of the equations, this was based on the solution method after many attempts, and the goal was to reach the Mittag–Leffler functions or the exponential function that facilitates calculations, and these additions and subtractions did not affect the original equations.

As a result, Equation (13) becomes

$$\begin{array}{l}{D}_{\tau }^{\alpha }\mathcal{Đ}+\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }+Ŧ{\mathcal{Đ}}_{\lambda }-2{\sigma }_0\mathcal{Đ}={\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right)-2{\sigma }_0\mathcal{Đ} , \\ D_{\tau }^{\alpha }Ŧ+\mathcal{Đ}{Ŧ}_{\varkappa }+Ŧ{Ŧ}_{\lambda }-2{\sigma }_0Ŧ={\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right) -2{\sigma }_0Ŧ, 0<\alpha \le 1 , \varkappa ,\lambda ,\tau >0\end{array}$$

ICs and ET can be used to obtain

$$\begin{array}{l}{\displaystyle \frac{1}{{\zeta }^{\alpha }}}E\left[\mathcal{Đ}\right]-\mathcal{Đ}(\varkappa ,\lambda ,0){\zeta }^{2-\alpha }-2{\sigma }_{0}E\left[\mathcal{Đ}\right]=E\left[{\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right) -2{\sigma }_0\mathcal{Đ}-\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }-Ŧ{\mathcal{Đ}}_{\lambda }\right] , \\ {\displaystyle \frac{1}{{\zeta }^{\alpha }}}E\left[Ŧ\right]-Ŧ(\varkappa ,\lambda ,0){\zeta }^{2-\alpha }-2{\sigma }_{0}E\left[Ŧ\right]=E\left[{\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right)-2{\sigma }_0Ŧ-\mathcal{Đ}{Ŧ}_{\varkappa }-Ŧ{Ŧ}_{\lambda }\right],\\ E\left[\mathcal{Đ}\right]=-{\displaystyle \frac{{\zeta }^2}{1-2{\sigma }_0{\zeta }^{\alpha }}}{e}^{\varkappa +\lambda }+{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right) -2{\sigma }_0\mathcal{Đ}-\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }-Ŧ{\mathcal{Đ}}_{\lambda }\right] , \\ E\left[Ŧ\right]={\displaystyle \frac{{\zeta }^2}{1-2{\sigma }_0{\zeta }^{\alpha }}}{e}^{\varkappa +\lambda }+{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right)-2{\sigma }_0Ŧ-\mathcal{Đ}{Ŧ}_{\varkappa }-Ŧ{Ŧ}_{\lambda }\right],\end{array}$$

The inverse ET indicates the following:

$$\begin{array}{l}\mathcal{Đ}=-{\in }_{\alpha ,1}\left(2{\sigma }_0{\tau }^{\alpha }\right)e^{\varkappa +\lambda }+E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right) -2{\sigma }_0\mathcal{Đ}-\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }-Ŧ{\mathcal{Đ}}_{\lambda }\right]\right\}, \\ Ŧ={\in }_{\alpha ,1}\left(2{\sigma }_0{\tau }^{\alpha }\right)e^{\varkappa +\lambda }+E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right)-2{\sigma }_0Ŧ-\mathcal{Đ}{Ŧ}_{\varkappa }-Ŧ{Ŧ}_{\lambda }\right]\right\},\end{array}$$

The iteration formulae that employ an initial approximation are as follows:

$$\begin{array}{l}{\mathcal{Đ}}_{n+1}=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\left({\mathcal{Đ}}_n\right)}_{\varkappa \varkappa }+{\left({\mathcal{Đ}}_n\right)}_{\lambda \lambda }\right) -2{\sigma }_0\left({\mathcal{Đ}}_n\right)-\left({\mathcal{Đ}}_n\right){\left({\mathcal{Đ}}_n\right)}_{\varkappa }-\left({Ŧ}_n\right){\left({\mathcal{Đ}}_n\right)}_{\lambda }\right]\right\},\\ {Ŧ}_{n+1}=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\left({Ŧ}_n\right)}_{\varkappa \varkappa }+{\left({Ŧ}_n\right)}_{\lambda \lambda }\right)-2{\sigma }_0\left({Ŧ}_n\right)-\left({\mathcal{Đ}}_n\right){\left({Ŧ}_n\right)}_{\varkappa }-\left({Ŧ}_n\right){\left({Ŧ}_n\right)}_{\lambda }\right]\right\},\\ {\mathcal{Đ}}_0=-{\in }_{\alpha ,1}\left(2{\sigma }_0{\tau }^{\alpha }\right)e^{\varkappa +\lambda } , {Ŧ}_0={\in }_{\alpha ,1}\left(2{\sigma }_0{\tau }^{\alpha }\right)e^{\varkappa +\lambda }\end{array}$$

(15)

Equation (15) gives

$${\mathcal{Đ}}_1=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[0\right]\right\}=0, {Ŧ}_1=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1-2{\sigma }_0{\zeta }^{\alpha }}}E\left[0\right]\right\}=0$$

From Equation (15), the following values can be found easily: ${\mathcal{Đ}}_2=0, {\mathcal{Đ}}_3=0, \dots , {Ŧ}_2=0, {Ŧ}_3=0, \dots ,$

Therefore, the following are the solutions to Equation (13):

$$\mathcal{Đ}(\varkappa ,\lambda ,\tau )=-{\in }_{\alpha ,1}\left(2{\sigma }_0{\tau }^{\alpha }\right)e^{\varkappa +\lambda } , Ŧ(\varkappa ,\lambda ,\tau )={\in }_{\alpha ,1}\left(2{\sigma }_0{\tau }^{\alpha }\right)e^{\varkappa +\lambda }$$

If $\alpha =1,$ then $\mathcal{Đ}(\varkappa ,\lambda ,\tau )=-e^{\varkappa +\lambda +2{\sigma }_0 \tau } , Ŧ(\varkappa ,\lambda ,\tau )=e^{\varkappa +\lambda +2{\sigma }_0 \tau }$.

The solutions obtained clearly show a strong agreement with FRDTM [26] and HPETM [17].

Example 2. 

Considering the SFNSE,

$$\begin{array}{l}{D}_{\tau }^{\alpha }\mathcal{Đ}+\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }+Ŧ{\mathcal{Đ}}_{\lambda }={\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right) , \\ D_{\tau }^{\alpha }Ŧ+\mathcal{Đ}{Ŧ}_{\varkappa }+Ŧ{Ŧ}_{\lambda }={\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right) , 0<\alpha \le 1 , \varkappa ,\lambda ,\tau >0\end{array}$$

(16)

With the IC,

$$\mathcal{Đ}(\varkappa ,\lambda ,0)=-\sin (\varkappa +\lambda ), Ŧ(\varkappa ,\lambda ,0)=\sin (\varkappa +\lambda ),$$

(17)

As in the first example, we add $2{\sigma }_0\mathcal{Đ}, 2{\sigma }_0Ŧ$ to both sides of Equation (16), respectively, and then Equation (16) becomes

$$\begin{array}{l}{D}_{\tau }^{\alpha }\mathcal{Đ}+\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }+Ŧ{\mathcal{Đ}}_{\lambda }+2{\sigma }_0\mathcal{Đ}={\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right)+2{\sigma }_0\mathcal{Đ} , \\ D_{\tau }^{\alpha }Ŧ+\mathcal{Đ}{Ŧ}_{\varkappa }+Ŧ{Ŧ}_{\lambda }+2{\sigma }_0Ŧ={\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right) +2{\sigma }_0Ŧ, 0<\alpha \le 1 , \varkappa ,\lambda ,\tau >0\end{array}$$

The ICs and ET can be used to obtain

$$\begin{array}{l}E\left[\mathcal{Đ}\right]=-{\displaystyle \frac{{\zeta }^2}{1+2{\sigma }_0{\zeta }^{\alpha }}}\sin (\varkappa +\lambda )+{\displaystyle \frac{{\zeta }^{\alpha }}{1+2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\mathcal{Đ}}_{\varkappa \varkappa }+{\mathcal{Đ}}_{\lambda \lambda }\right) +2{\sigma }_0\mathcal{Đ}-\mathcal{Đ}{\mathcal{Đ}}_{\varkappa }-Ŧ{\mathcal{Đ}}_{\lambda }\right],\\ E\left[Ŧ\right]={\displaystyle \frac{{\zeta }^2}{1+2{\sigma }_0{\zeta }^{\alpha }}}\sin (\varkappa +\lambda )+{\displaystyle \frac{{\zeta }^{\alpha }}{1+2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({Ŧ}_{\varkappa \varkappa }+{Ŧ}_{\lambda \lambda }\right)+2{\sigma }_0Ŧ-\mathcal{Đ}{Ŧ}_{\varkappa }-Ŧ{Ŧ}_{\lambda }\right],\end{array}$$

The recurrence connections will parse similarly.

$$\begin{array}{l}{\mathcal{Đ}}_{n+1}=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1+2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\left({\mathcal{Đ}}_n\right)}_{\varkappa \varkappa }+{\left({\mathcal{Đ}}_n\right)}_{\lambda \lambda }\right) +2{\sigma }_0\left({\mathcal{Đ}}_n\right)-\left({\mathcal{Đ}}_n\right){\left({\mathcal{Đ}}_n\right)}_{\varkappa }-\left({Ŧ}_n\right){\left({\mathcal{Đ}}_n\right)}_{\lambda }\right]\right\},\\ {Ŧ}_{n+1}=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1+2{\sigma }_0{\zeta }^{\alpha }}}E\left[{\sigma }_0\left({\left({Ŧ}_n\right)}_{\varkappa \varkappa }+{\left({Ŧ}_n\right)}_{\lambda \lambda }\right)+2{\sigma }_0\left({Ŧ}_n\right)-\left({\mathcal{Đ}}_n\right){\left({Ŧ}_n\right)}_{\varkappa }-\left({Ŧ}_n\right){\left({Ŧ}_n\right)}_{\lambda }\right]\right\},\\ {\mathcal{Đ}}_0=-{\in }_{\alpha ,1}\left(-2{\sigma }_0{\tau }^{\alpha }\right)\sin (\varkappa +\lambda ) , {Ŧ}_0={\in }_{\alpha ,1}\left(-2{\sigma }_0{\tau }^{\alpha }\right)\sin (\varkappa +\lambda )\end{array}$$

(18)

Equation (13) gives

$${\mathcal{Đ}}_1=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1+2{\sigma }_0{\zeta }^{\alpha }}}E\left[0\right]\right\}=0, {Ŧ}_1=E^{-1}\left\{{\displaystyle \frac{{\zeta }^{\alpha }}{1+2{\sigma }_0{\zeta }^{\alpha }}}E\left[0\right]\right\}=0$$

and ${\mathcal{Đ}}_2=0, {\mathcal{Đ}}_3=0, \dots , {Ŧ}_2=0, {Ŧ}_3=0, \dots ,$

Therefore, the following are the solutions to Equation (16):

$$\mathcal{Đ}(\varkappa ,\lambda ,\tau )=-{\in }_{\alpha ,1}\left(-2{\sigma }_0{\tau }^{\alpha }\right)\sin (\varkappa +\lambda ) , Ŧ(\varkappa ,\lambda ,\tau )={\in }_{\alpha ,1}\left(-2{\sigma }_0{\tau }^{\alpha }\right)\sin (\varkappa +\lambda )$$

If $\alpha =1,$ then $\mathcal{Đ}(\varkappa ,\lambda ,\tau )=-e^{\varkappa +\lambda -2{\sigma }_0 \tau } , Ŧ(\varkappa ,\lambda ,\tau )=e^{\varkappa +\lambda -2{\sigma }_0 \tau }.$

Once more, the results clearly demonstrate a high degree of agreement with HPETM [10] and FRDTM [26].

Note2: Although one study solely evaluates the method’s effectiveness in a specific type of SFNSE, I believe it will pave the way for further research and its extension to all fractional equations in future studies.

5. Numerical Analysis

This section compares the approximate and exact solutions in a table and graph to demonstrate the accuracy and utility of the proposed method. In Figure 1a–e, Figure 2a–e, Figure 3a–e and Figure 4a–e, the 3-D and 2-D graph solutions of Examples 1 and 2 generated using the present approaches are compared with the solutions at these charts. This comparison demonstrates how close to exact results the approximate solutions determined by this method are.

Table 1. Errors in ET solution for different values of the function $\mathcal{Đ}(\varkappa ,\lambda ,\tau )$ with $\lambda =\varkappa =\pi$ and ${\sigma }_0=1$, $\lambda \in [0,2\pi ]$, $\tau \in [0,5]$ in Example 1.

$\mathit{\tau }$	ET at $\mathit{\alpha }\mathbf{=}\mathbf{0.75}$	ET at $\mathit{\alpha }\mathbf{=}\mathbf{0.85}$	ET at $\mathit{\alpha }\mathbf{=}\mathbf{0.95}$	Exact at $\mathit{\alpha }\mathbf{=}\mathbf{1}$	Error at $\mathit{\alpha }\mathbf{=}\mathbf{0.75}$	Error at $\mathit{\alpha }\mathbf{=}\mathbf{0.85}$	Error at $\mathit{\alpha }\mathbf{=}\mathbf{0.95}$
0	0	0	0	0	0	0	0
0.25	−352.0455600	−340.4036197	−329.8347830	−328.8159866	23.229573	11.5876	1.0187964
0.5	−217.3828893	−207.7083154	−200.1793427	−199.5158089	17.86708	8.19251	0.6635338
0.75	−142.4530514	−131.5726108	−123.0691661	−122.3207488	20.132303	9.25186	0.7484173
1	−96.62788655	−85.25989990	−76.28517361	−75.49053641	21.13735	9.76936	0.7946372

,

Table 2. Errors in ET solution for different values of the function for $Ŧ(\varkappa ,\lambda ,\tau )$ with $\lambda =\varkappa =\pi$ and ${\sigma }_0=1$ in Example 1.

$\mathit{\tau }$	ET at $\mathit{\alpha }\mathbf{=}\mathbf{0.75}$	ET at $\mathit{\alpha }\mathbf{=}\mathbf{0.85}$	ET at $\mathit{\alpha }\mathbf{=}\mathbf{0.95}$	Exact at $\mathit{\alpha }\mathbf{=}\mathbf{1}$	Error at $\mathit{\alpha }\mathbf{=}\mathbf{0.75}$	Error at $\mathit{\alpha }\mathbf{=}\mathbf{0.85}$	Error at $\mathit{\alpha }\mathbf{=}\mathbf{0.95}$
0	0	0	0	0	0	0	0
0.25	352.0455600	340.4036197	329.8347830	328.8159866	−23.22957	−11.58763	−1.018796
0.5	217.3828893	207.7083154	200.1793427	199.5158089	−17.86708	−8.192507	−0.663534
0.75	142.4530514	131.5726108	123.0691661	122.3207488	−20.1323	−9.251862	−0.748417
1	96.62788655	85.25989990	76.28517361	75.49053641	−21.13735	−9.769363	−0.794637

,

Table 3. Errors in ET solution for different values of the function $\mathcal{Đ}(\varkappa ,\lambda ,\tau )$ with $\varkappa =\pi$, $\lambda = \frac{\pi }{2}$ and ${\sigma }_0=1$ in Example 2.

τ	ET at $\mathit{\alpha }\mathbf{=}\dots$	ET at $\mathit{\alpha }\mathbf{=}\dots$	ET at $\mathit{\alpha }\mathbf{=}\dots$	Exact at $\mathit{\alpha }\mathbf{=}1$	Error at $\mathit{\alpha }\mathbf{=}\dots$	Error at $\mathit{\alpha }\mathbf{=}\dots$	Error at $\mathit{\alpha }\mathbf{=}\dots$
0	0	0	0	0	0	0	0
0.25	0.6574249216	0.6356842653	0.6159475676	0.6140450236	−0.04338	−0.021639	−0.001903
0.5	0.4059500962	0.3878833835	0.3738234582	0.3725843470	−0.033366	−0.015299	−0.001239
0.75	0.2660229154	0.2457043155	0.2298246196	0.2284269930	−0.037596	−0.017277	−0.001398
1	0.1804470442	0.1592179802	0.1424581928	0.1409742534	−0.039473	−0.018244	−0.001484

and

Table 4. Errors in ET solution for different values of the function for $Ŧ(\varkappa ,\lambda ,\tau )$ with $\varkappa =\pi$, $\lambda = \frac{\pi }{2}$ and ${\sigma }_0=1$ in Example 2.

$\mathit{\tau }$	ET at $\mathit{\alpha }\mathbf{=}\dots$	ET at $\mathit{\alpha }\mathbf{=}\dots$	ET at $\mathit{\alpha }\mathbf{=}\dots$	Exact at $\mathit{\alpha }\mathbf{=}1$	Error at $\mathit{\alpha }\mathbf{=}\dots$	Error at $\mathit{\alpha }\mathbf{=}\dots$	Error at $\mathit{\alpha }\mathbf{=}\dots$
0	0	0	0	0	0	0	0
0.25	−0.6574249216	−0.6356842653	−0.6159475676	−0.6140450236	0.04338	0.021639242	0.001902544
0.5	−0.4059500962	−0.3878833835	−0.3738234582	−0.3725843470	0.033366	0.015299037	0.001239111
0.75	−0.2660229154	−0.2457043155	−0.2298246196	−0.2284269930	0.037596	0.017277323	0.001397627
1	−0.1804470442	−0.1592179802	−0.1424581928	−0.1409742534	0.039473	0.018243727	0.001483939

present a comparison of the exact and approximate solutions for each example for $\alpha =1, 0.95, 0.85, 0.75,$ and varying values of $\tau$; these Table 1, Table 2, Table 3 and Table 4 also represent the approximate solution residual error of the SFNSEs with ICs at distinct values of $\alpha$. As the fractional order $\alpha$ increases, the error approaches small values. This indicates that the solution approach employed in this work is effective and produces accurate results. Furthermore, we will compare the numerical behavior of the solutions to SFNSEs with equations with integer derivatives. When $\alpha =1$, the closed-form solutions for Examples 1and 2 are simply calculated. The numbers and tables provide evidence of this strategy’s efficacy. It is very clear from these figures that the approximate solution approaches the exact solution when $\alpha$ reaches one.

6. Conclusions

In this work, analytical investigations are conducted for the two SFNSEs with ICs. A novel ET-based method is proposed for the rapid and effective resolution of SFNSEs. Two examples are provided to illustrate the usefulness of the proposed approach. The results obtained clearly show a strong agreement with FRDTM [26] and HPETM [10]. Future research will use this approach and its remarkable ability to use integrated circuits to solve several linear and nonlinear SFNSE problems to solve a wide range of initial value problems because it is straightforward, computationally inexpensive, and easy to use, as demonstrated by the solutions provided in this paper. In addition to what was mentioned in Section 4 above, we will try to solve the problems related to nonlinear boundary conditions or nonlinear source term to demonstrate the effectiveness in future research. We will then evaluate the accuracy and compare the results with standard numerical methods such as finite-difference or spectral methods.