Home Physical Sciences A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
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A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform

  • Maalee Almheidat , Humaira Yasmin EMAIL logo , Maryam Al Huwayz , Rasool Shah EMAIL logo and Samir A. El-Tantawy
Published/Copyright: October 16, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

This investigation explores the analytical solutions to the time-fractional multi-dimensional Navier–Stokes (NS) problem using advanced approaches, namely the Aboodh residual power series method and the Aboodh transform iteration method, within the context of the Caputo operator. The NS equation governs the motion of fluid flow and is essential in fluid dynamics, engineering, and atmospheric sciences. Given the equation’s extensive and diverse applicability across several disciplines, we are motivated to conduct a thorough analysis to understand the complex dynamics associated with the nonlinear events it describes. For this purpose, we effectively handle the challenges posed by fractional derivatives by utilizing the Aboodh approach. This will enable us to obtain accurate analytical approximations for the time fractional multi-dimensional NS equation. By conducting thorough analysis and computational simulations, we provide evidence of the efficiency and dependability of the suggested methodologies in accurately representing the dynamic behavior of fractional fluid flow systems. This work enhances our comprehension of the utilization of fractional calculus in fluid dynamics and provides valuable analytical instruments for examining intricate flow phenomena. Its interdisciplinary nature ensures that the findings are applicable to various scientific and engineering fields, making the research highly versatile and impactful.

Keywords: multi-dimensional fractional Navier–Stokes models; Aboodh residual power series method; Aboodh transform iteration method; Caputo operator

1 Introduction

In 1695, the concept of fractional derivative was established by the famous mathematician Leibnitz. Integral and non-integral differential operators are linked to fractional calculus. A system’s future state is determined by its current and preceding states since the fractional-order differential operator is non-local. A growing number of disciplines are realizing the non-linear complex problems that fractional calculus can solve, including mathematical biology, electrochemistry, fluid mechanics, viscoelasticity, and mathematical biology, among many others [1–3]. There has been a lot of focus on fractional differential equation (DE) advances in the last few years [4–9]. For the study of partial differential equations (PDEs), especially those equations derived from the Mathematics of Finance, symmetry analysis is a useful approach [10–12].

Mathematics and physics have recently been captivated by nonlinear fractional partial differential equations (NFPDEs) for the wide range of physical phenomena they can model. These phenomena can be found in fields as varied as mathematical physics, statistical mechanics, theoretical neuroscience, mathematical finance, electrochemistry, and cancer modeling, among many others [13–15]. Accordingly, NFPDE theory is effective for solving engineering issues (e.g., [16–22]). Modern approaches of complex systems and control theory propose the advancement in different fields such as neuroscience, network dynamics, and autonomous systems [23]. Studying the connectivity of the functional brain networks in boys with attention deficit hyperactivity disorder has questioned the degree of high-order interactions in these networks [24]. At the same time, multi-chimera states in higher-order networks of FHN oscillators have been studied and provided new findings about network dynamical activity [25]. Also, the modern fractional order control approach has improved self-driving cars, robustness, while the FOCNSTSMC has been implemented in the permanent magnet synchronous motor (PMSM) control system of electric vehicle [26]. Moreover, the investigation of the exact soliton solutions and wave structure for other DEs also remain a hot topic of research in mathematical physics today and also support the growth of theoretical and applied physics [27].

The flow of viscous fluids was governed by the Navier–Stokes (NS) model, which was discovered in 1822 [28–30]. One way to look at this equation is as a combination of Newton’s second rule of motion for fluids, the energy equation, the continuity equation, and the momentum equation. Fluids in pipes, airflow, ocean currents, and airflow across airplane wings are all possible representations of this equation [31–34]. It was not until 2005 that Salem and El-Shahed [35] did fractional modeling of NS equations. To generalize the classical NS equations, the authors [35] used different types of transformations. By combining the Laplace transform with the homotopy perturbation approach, Kumar et al. [36] could analytically solve a non-linear fractional model of the NS problem. Two previous studies that used HAM to resolve the non-linear fractional NS equation were Ganji et al. [37], and Ragab et al. [38]. Both Momani and Birajdar [39] and Maitama [40] used the Adomian decomposition approach to calculate the fractional NS model numerically. While Chaurasia and Kumar [41] solved the time-fractional NS equation by integrating the finite Hankel transform and Laplace transform, Kumar et al. [42] obtained an approximation of the issue by combining the Adomian decomposition approach with the Laplace transformation. In 2013, Arqub founded RPSM [43]. It arises from combining the Taylor series with the residual error function. The solution for DEs [44] was given by an infinite convergent series. Several DEs, including Boussinesq DEs, fuzzy DEs, Korteweg–de Vries (KdV) Burger’s equation, and many others [45–51], have driven the creation of new RPSM methods. These algorithms aim to produce accurate and quick approximations.

By integrating two effective methods, a new strategy for FPDEs was established. Some approaches that fall into these categories include those that use the Sumudu transform in conjunction with the Adomian decomposition method (ADM) and Shehu transformation [52], the RPSM with Laplace transformation [53–56], the natural transform [57], and the homotopy perturbation approach [58]. For further details on combining the two techniques, see [59–61]. This study used a novel combination method known as the ARPSM to discover approximation and precise solutions for time-fractional PDEs. This innovative method is significant because it combines the Aboodh transform (AT) technique with the RPSM [62,63].

The novel method transforms the initial problem into an AT space, finds several solutions to the modified equation, and then solves the original equation using the inverse AT. The novel method may be used to solve linear and nonlinear PDEs with power series expansion, and it does not need discretization, linearization, or perturbation. This approach is distinguished from the conventional power series technique because it is not dependent on recursion relations or matching coefficients of related words. Unlike the RPSM, which shows fractional derivatives, the suggested limit-based method shows series coefficients.

The computing effort and complexity needed are significant issues with the previously mentioned approaches. The unique contribution of this research is the Aboodh transform iterative method (ATIM) that we put forth as a solution to the NS model. Integrating the AT with the new iterative technique significantly reduces the computational complexity and labor required [64]. According to [65,66], the suggested approach yields a convergent series solution.

Based on the literature [62–66], the two most straightforward methods for solving fractional DEs are the Aboodh residual power series method (ARPSM) and the ATIM. These methods not only clarify all the symbolic notions utilized in analytical solutions immediately and comprehensively, but they also offer discretization-free and nonlinear numerical solutions to PDEs [67]. The purpose of this study is to evaluate ARPSM and ATIM in terms of how well they solve the NS model. These two methods have successfully resolved several linear and nonlinear fractional differential problems. For instance, the ARPSM and ATIM have been rigorously examined to obtain precise approximations for more complicated nonlinear fractional evolution equations, such as the fractional Hirota–Satsuma coupled KdV issue [68] and fractional damped Burger’s equation [69]. The researchers discovered that all derived approximations exhibit exceptional accuracy and enhanced stability, instilling confidence in the robustness of the ARPSM and ATIM.

The PDEs of the NS are solved in this article using ARPSM and ATIM. When compared to other numerical techniques, the results obtained by these methods are far more precise. Research comparing the numerical data is also included. Consistency in the outcomes produced by the proposed approaches is a powerful indicator of their efficacy and dependability. Visually, fractional-order derivatives get more eye-catching as their value increases. Because of this, the methods are efficient, precise, user-friendly, and robust against computational error phases. Due to this discovery, mathematicians will have an easier time solving many different types of PDEs.

The rest of the current work is introduced in the following form: Section 2 discusses the definitions, theorems, and lemmas of fractional calculus that will be utilized in analyzing the problem being examined. Section 3 covers both ARPSM and ATIM in great detail and provides an overview of their applications through the study of fractional DEs. In Section 4.1.2, two test problems concerning the NS model are discussed and analyzed using ARPSM and ATIM to show these approaches’ effectiveness and general applicability. In this section, all derived approximations are analyzed and discussed. Furthermore, the absolute error of these approximations is calculated and evaluated against one another to ascertain their effectiveness in handling the most challenging tasks. In the fifth and last part, we summarize our most significant findings. Furthermore, we included a substantial and supplementary part following the section on future activity connected to the present work.

2 Terminologies of fractional calculus

Definition 2.1

[70] If the piecewise continuous function S ( w , Y ) is exponentially ordered, then the AT for Y ≥ 0 may be described in the following way:

A [ S ( w , Y ) ] = γ ( w , k ) = 1 k ∫ 0 ∞ S ( w , Y ) e − Y k d Y , r 1 ≤ k ≤ r 2 .

One of the many possible explanations for the iAT is as follows:

A − 1 [ γ ( w , k ) ] = S ( w , Y ) = 1 2 π i ∫ u − i ∞ u + i ∞ γ ( w , Y ) k e Y k d Y , w = ( w 1 , w 2 , … , w d ) ∈ R .

Lemma 2.2

[71,72] There are two piecewise continuous functions that are exponentially ordered, S 1 ( w , Y ) and S 2 ( w , Y ) , on the interval [ 0 , ∞ ] . Assuming λ 1 , λ 2 as constants and A [ S 1 ( w , Y ) ] = γ 1 ( w , Y ) , A [ S 2 ( w , Y ) ] = γ 2 ( w , Y ) . Therefore, the subsequent characteristics are true:

  1. A [ λ 1 S 1 ( w , Y ) + λ 2 S 2 ( w , Y ) ] = λ 1 γ 1 ( w , k ) + λ 2 γ 2 ( w , Y ) ,

  2. A − 1 [ λ 1 γ 1 ( w , Y ) + λ 2 γ 2 ( w , Y ) ] = λ 1 S 1 ( w , k ) + λ 2 S 2 ( w , Y ) ,

  3. A [ J Y d S ( w , Y ) ] = γ ( w , k ) k d ,

  4. A [ D Y d S ( w , Y ) ] = k d γ ( w , k ) − ∑ K = 0 r − 1 S K ( w , 0 ) k K − d + 2 , r − 1 < d ≤ r , r ∈ N .

Definition 2.3

[73] For the function S ( w , Y ) , the fractional derivative that Caputo defines as follows in relation to order d :

D Y d S ( w , Y ) = J Y m − d S ( m ) ( w , Y ) , r ≥ 0 , m − 1 < d ≤ m ,

where m , d ∈ R , J Y m − d is the integral of Liouville and Riemann of S ( w , Y ) , and w = ( w 1 , w 2 , … , w d ) ∈ R d .

Definition 2.4

[74] The power series expansion is written as

∑ r = 0 ∞ ℘ r ( w ) ( Y − Y 0 ) r d = ℘ 0 ( Y − Y 0 ) 0 + ℘ 1 ( Y − Y 0 ) d + ℘ 2 ( Y − Y 0 ) 2 d + ⋯ ,

where w = ( w 1 , w 2 , … , w d ) ∈ R d and d ∈ N .

Lemma 2.5

Consider the function S ( w , Y ) , which is ordered exponentially. Then, A [ S ( w , Y ) ] = γ ( w , k ) is the AT. Hence,

(1) A [ D Y r d S ( w , Y ) ] = k r d γ ( w , k ) − ∑ j = 0 r − 1 k d ( r − j ) − 2 D Y j d S ( w , 0 ) , 0 < d ≤ 1 ,

where D Y r d = D Y d . D Y d , ⋯ , D Y d ( r - t i m e s ) and w = ( w 1 , w 2 , … , w d ) ∈ R d and d ∈ N .

Proof

To show the validity of Eq. (1), we utilize the induction method.

Insert r = 1 in Eq. (1):

A [ D Y 2 d S ( w , Y ) ] = k 2 d γ ( w , k ) − k 2 d − 2 S ( w , 0 ) − k d − 2 D Y d S ( w , 0 ) .

For r = 1 , Eq. (1) holds on behalf of Lemma 2.2. In Eq. (1), we substitute r = 2

(2) A [ D r 2 d S ( w , Y ) ] = k 2 d γ ( w , k ) − k 2 d − 2 S ( w , 0 ) − k d − 2 D Y d S ( w , 0 ) .

The left-hand side (LHS) of Eq. (2) implies

(3) LHS = A [ D Y 2 d S ( w , Y ) ] .

Furthermore, Eq. (3) can be expressed subsequently

(4) LHS = A [ D Y d ( D Y d S ( w , Y ) ) ] .

By supposing the following formula:

(5) z ( w , Y ) = D Y d S ( w , Y ) ,

which leads to z ( w , Y ) into Eq. (4) yields

(6) LHS = A [ D Y d z ( w , Y ) ] .

Using definition (2.3) in Eq. (6) gives us

(7) LHS = A [ J 1 − d z ′ ( w , Y ) ] .

Take the Liouville and Riemann integral of Eq. (7) in the framework of AT

(8) LHS = A [ z ′ ( w , Y ) ] k 1 − d .

By the derivative characteristic of the AT, then Eq. (8) is modified to the following form:

(9) LHS = k d Z ( w , k ) − z ( w , 0 ) k 2 − d .

From Eq. (5), we obtain

Z ( w , k ) = k d γ ( w , k ) − S ( w , 0 ) k 2 − d ,

as Z ( w , k ) = A [ z ( w , Y ) ] , so Eq. (9) becomes

(10) LHS = k 2 d γ ( w , k ) − S ( w , 0 ) k 2 − 2 d − D Y d S ( w , 0 ) k 2 − d .

Now, assuming Eq. (1) holds for r = k , and then inserting r = k into Eq. (1) yields

(11) A [ D Y k d S ( w , Y ) ] = k k d γ ( w , k ) − ∑ j = 0 k − 1 k d ( k − j ) − 2 D Y j d D Y j d S ( w , 0 ) , 0 < d ≤ 1 .

Finally, we will have to show that Eq. (1) also holds for r = k + 1

(12) A [ D Y ( k + 1 ) d S ( w , Y ) ] = k ( k + 1 ) d γ ( w , k ) − ∑ j = 0 K k d ( ( k + 1 ) − j ) − 2 D Y j d S ( w , 0 ) .

The LHS of Eq. (12) implies

(13) LHS = A [ D Y k d ( D Y k d ) ] .

Let

D Y k d = g ( w , Y ) .

From Eq. (13), we obtain

(14) LHS = A [ D Y d g ( w , Y ) ] .

By the application of the Liouville and Riemann integral and derivative of Caputo, Eq. (14) becomes

(15) LHS = k d A [ D Y K d S ( w , Y ) ] − g ( w , 0 ) k 2 − d .

Using Eq. (73) yields

(16) LHS = k r d γ ( w , k ) − ∑ j = 0 r − 1 k d ( r − j ) − 2 D Y j d S ( w , 0 ) .

From Eq. (16), we derive this subsequent form generally

LHS = A [ D Y r d S ( w , 0 ) ] .

Using the mathematical induction method, Eq. (1) holds for r = k + 1 . Hence, it is proved that Eq. (1)  ∀ Z + holds.□

Lemma 2.6

Assume the exponential function S ( w , Y ) . If A [ S ( w , Y ) ] = γ ( w , k ) , the AT of S ( w , Y ) may be written in terms of multiple fractional Taylor s series (MFTS) as

(17) γ ( w , k ) = ∑ r = 0 ∞ ℘ r ( w ) k r d + 2 , k > 0 ,

where w = ( w 1 , w 2 , … , w d ) ∈ R d , d ∈ N .

Proof

Introducing the following Taylor’s series:

(18) S ( w , Y ) = ℘ 0 ( w ) + ℘ 1 ( w ) Y d Γ [ d + 1 ] + + ℘ 2 ( w ) Y 2 d Γ [ 2 d + 1 ] + ⋯ .

Applying AT yields

A [ S ( w , Y ) ] = A [ ℘ 0 ( w ) ] + A ℘ 1 ( w ) Y d Γ [ d + 1 ] + A ℘ 1 ( w ) Y 2 d Γ [ 2 d + 1 ] + ⋯

The feature of AT is then used to obtain the subsequent expression

A [ S ( w , Y ) ] = ℘ 0 ( w ) 1 k 2 + ℘ 1 ( w ) Γ [ d + 1 ] Γ [ d + 1 ] 1 k d + 2 + ℘ 2 ( w ) Γ [ 2 d + 1 ] Γ [ 2 d + 1 ] 1 k 2 d + 2 ⋯ .

Thus, we obtain a new variant of Taylor’s series in the AT framework.□

Lemma 2.7

Considering the new variant of Taylor’s series (17), we derive an multiple fractional power series (MFPS) form of the function A [ S ( w , Y ) ] = γ ( w , k )

(19) ℘ 0 ( w ) = lim k → ∞ k 2 γ ( w , k ) = S ( w , 0 ) .

Proof

Assume the new variant of Taylor’s series

(20) ℘ 0 ( w ) = k 2 γ ( w , k ) − ℘ 1 ( w ) k d − ℘ 2 ( w ) k 2 d − ⋯

lim k → ∞ is applied on Eq. (19) and after simplification, we derive Eq. (20).□

Theorem 2.8

For the function A [ S ( w , Y ) ] = γ ( w , k ) have the MFPS form is given as follows:

γ ( w , k ) = ∑ 0 ∞ ℘ r ( w ) k r d + 2 , k > 0 ,

where w = ( w 1 , w 2 , … , w d ) ∈ R d and d ∈ N . Then, we have

℘ r ( w ) = D r r d S ( w , 0 ) ,

where D Y r d = D Y d . D Y d , … , D Y d ( r - t i m e s ) .

Proof

Consider Taylor’s series

(21) ℘ 1 ( w ) = k d + 2 γ ( w , k ) − k d ℘ 0 ( w ) − ℘ 2 ( w ) k d − ℘ 3 ( w ) k 2 d − ⋯ .

With the application of lim k → ∞ , Eq. (21) takes the following form:

℘ 1 ( w ) = lim k → ∞ ( k d + 2 γ ( w , k ) − k d ℘ 0 ( w ) ) − lim k → ∞ ℘ 2 ( w ) k d − lim k → ∞ ℘ 3 ( w ) k 2 d − ⋯ .

Computing limit, we derive

(22) ℘ 1 ( w ) = lim k → ∞ ( k d + 2 γ ( w , k ) − k d ℘ 0 ( w ) ) .

Equation (22) with the help of Lemma (2.5) becomes

(23) ℘ 1 ( w ) = lim k → ∞ ( k 2 A [ D Y d S ( w , Y ) ] ( k ) ) .

Applying Lemma (2.6) on Eq. (23), we deduce

℘ 1 ( w ) = D Y d S ( w , 0 ) .

As limit k → ∞ are applied, we obtain

℘ 2 ( w ) = k 2 d + 2 γ ( w , k ) − k 2 d ℘ 0 ( w ) − k d ℘ 1 ( w ) − ℘ 3 ( w ) k d − ⋯

From Lemma 2.6, we obtain

(24) ℘ 2 ( w ) = lim k → ∞ k 2 ( k 2 d γ ( w , k ) − k 2 d − 2 ℘ 0 ( w ) − k d − 2 ℘ 1 ( w ) ) .

Using Lemmas 2.5 and 2.7 again, Eq. (24) becomes

℘ 2 ( w ) = D Y 2 d S ( w , 0 ) .

Using the same process, we derive this subsequent expression

℘ 3 ( w ) = lim k → ∞ k 2 ( A [ D Y 2 d S ( w , d ) ] ( k ) ) .

Lemma 2.7 gives the following result

℘ 3 ( w ) = D Y 3 d S ( w , 0 ) .

Generally, we obtain

□ ℘ r ( w ) = D Y r d S ( w , 0 ) .

The convergence condition for the new variant of the Taylor series is given in the subsequent theorem.

Theorem 2.9

For ∣ k a A [ D Y ( k + 1 ) d S ( w , Y ) ] ∣ ≤ T , the residual R k ( w , k ) of the innovative form of MFTS validates the inequality

∣ R k ( w , k ) ∣ ≤ T k ( k + 1 ) d + 2 , 0 < k ≤ s .

Proof

Let A [ D Y r d S ( w , Y ) ] for each r = 0 , 1 , 2 , … , k + 1 , the function k is defined on 0 < k ≤ s . Assume that ∣ k 2 A [ D Y k + 1 S ( w , τ ) ] ∣ ≤ T on 0 < k ≤ s . Consider the subsequent expression which is obtained from Taylor’s series:

(25) R k ( w , k ) = γ ( w , k ) − ∑ r = 0 k ℘ r ( w ) k r d + 2 .

Theorem 2.8 allows one to write Eq. (25) in the following way:

(26) R k ( w , k ) = γ ( w , k ) − ∑ r = 0 k D Y r d S ( w , 0 ) k r d + 2 .

Multiplying k ( k + 1 ) d + 2 with Eq. (26), we deduce

(27) k ( k + 1 ) d + 2 R k ( w , k ) = k 2 ( k ( k + 1 ) d γ ( w , k ) − ∑ r = 0 k k ( k + 1 − r ) d − 2 D Y r d S ( w , 0 ) ) .

Implement Lemma 2.5 on Eq. (27)

(28) k ( k + 1 ) d + 2 R k ( w , k ) = k 2 A [ D Y ( k + 1 ) d S ( w , Y ) ] .

Take the absolute

(29) ∣ k ( k + 1 ) d + 2 R k ( w , k ) ∣ = ∣ k 2 A [ D Y ( k + 1 ) d S ( w , Y ) ] ∣ .

Using Eq. (29), we obtain

(30) − T k ( k + 1 ) d + 2 ≤ R k ( w , k ) ≤ T k ( k + 1 ) d + 2 .

From Eq. (30), we obtain this final result

□ ∣ R k ( w , k ) ∣ ≤ T k ( k + 1 ) d + 2 .

3 Methodologies

3.1 ARPSM

Here, we introduce the framework of the proposed method and how this method is utilized to find the solution for the linear and nonlinear PDEs.

Step 1: Suppose the following form of general PDE:

(31) D Y q d S ( w , Y ) + κ ( w ) N ( S ) − ϖ ( w , S ) = 0 .

Step 2: Applying AT to Eq. (31) as follows:

(32) A [ D Y q d S ( w , Y ) + κ ( w ) N ( S ) − ϖ ( w , S ) ] = 0 .

Eq. (32) after the utilization of Lemma 2.5 becomes

(33) γ ( w , s ) = ∑ j = 0 q − 1 D Y j S ( w , 0 ) s q d + 2 − κ ( w ) Y ( s ) s q d + F ( w , s ) s q d ,

where A [ ϖ ( w , S ) ] = F ( w , s ) , A [ N ( S ) ] = Y ( s ) .

Step 3: Rewrite Eq. (33) as follows:

γ ( w , s ) = ∑ r = 0 ∞ ℘ r ( w ) s r d + 2 , s > 0 .

Step 4: Use the process below for the solution:

℘ 0 ( w ) = lim s → ∞ s 2 γ ( w , s ) = S ( w , 0 ) .

Implement Theorem 2.9

℘ 1 ( w ) = D Y d S ( w , 0 ) , ℘ 2 ( w ) = D Y 2 d S ( w , 0 ) , ⋮ ℘ b ( w ) = D Y b d S ( w , 0 ) ,

Step 5: To obtain γ ( w , s ) , use the K th -truncate series

γ K ( w , s ) = ∑ r = 0 K ℘ r ( w ) s r d + 2 , s > 0 , γ K ( w , s ) = ℘ 0 ( w ) s 2 + ℘ 1 ( w ) s d + 2 + ⋯ + ℘ b ( w ) s b d + 2 + ∑ r = b + 1 K ℘ r ( w ) s r d + 2 .

Step 6: The Kth-truncated function of Aboodh residual and the function of Aboodh residual (FAR) of Eq. (33) are taken independently

ARes ( w , s ) = γ ( w , s ) − ∑ j = 0 q − 1 D Y j S ( w , 0 ) s j d + 2 + κ ( w ) Y ( s ) s j d − F ( w , s ) s j d

and

(34) ARes K ( w , s ) = γ K ( w , s ) − ∑ j = 0 q − 1 D Y j S ( w , 0 ) s j d + 2 + κ ( w ) Y ( s ) s j d − F ( w , s ) s j d .

Step 7: In Eq. (34), put γ K ( w , s )

(35) ARes K ( w , s ) = ℘ 0 ( w ) s 2 + ℘ 1 ( w ) s d + 2 + ⋯ + ℘ b ( w ) s b d + 2 + ∑ r = b + 1 K ℘ r ( w ) s r d + 2 − ∑ j = 0 q − 1 D Y j S ( w , 0 ) s j d + 2 + κ ( w ) Y ( s ) s j d − F ( w , s ) s j d .

Step 8: Multiply Eq. (35) with s K d + 2

(36) s K d + 2 ARes K ( w , s ) = s K d + 2 ℘ 0 ( w ) s 2 + ℘ 1 ( w ) s d + 2 + ⋯ + ℘ b ( w ) s b d + 2 + ∑ r = b + 1 K ℘ r ( w ) s r d + 2 − ∑ j = 0 q − 1 D Y j S ( w , 0 ) s j d + 2 + κ ( w ) Y ( s ) s j d − F ( w , s ) s j d .

Step 9: Take lim s → ∞ of Eq. (36) to obtain the following result:

lim s → ∞ s K d + 2 A Res K ( w , s ) = lim s → ∞ s K d + 2 ℘ 0 ( w ) s 2 + ℘ 1 ( w ) s d + 2 + ⋯ + ℘ b ( w ) s b d + 2 + ∑ r = b + 1 K ℘ r ( w ) s r d + 2 − ∑ j = 0 q − 1 × D Y j S ( w , 0 ) s j d + 2 + κ ( w ) Y ( s ) s j d − F ( w , s ) s j d .

Step 10: For ℘ K ( w ) , we can solve the following equation:

lim s → ∞ ( s K d + 2 A Res K ( w , s ) ) = 0 ,

where K = b + 1 , b + 2 , … .

Step 11: Insert ℘ K ( w ) in Eq. (33).

Step 12: S K ( w , Y ) is obtained as a solution by applying iAT.

3.2 ATIM methodology

Here, we assume the following time fractional general PDE:

(37) D Y d S ( w , Y ) = Φ ( S ( w , Y ) , D w β S ( w , Y ) , D w 2 β S ( w , Y ) , D w 3 β S ( w , Y ) ) , 0 < d , β ≤ 1 ,

with the following IC:

(38) S ( k ) ( w , 0 ) = h k , k = 0 , 1 , 2 , … , m − 1 ,

where Φ are operators of S ( w , Y ) , D w β S ( w , Y ) , D w 2 β S ( w , Y ) , and D w 3 β S ( w , Y ) , and S ( Y , w ) , is the unknown function. Applying AT on Eq. (37) yields

(39) A [ S ( w , Y ) ] = 1 s d ∑ k = 0 m − 1 S ( k ) ( w , 0 ) s 2 − d + k + A [ Φ ( S ( w , Y ) , D w β S ( w , Y ) , D w 2 β S × ( w , Y ) , D w 3 β S ( w , Y ) ) ] ) .

The iAT of Eq. (39) yields

(40) S ( w , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S ( k ) ( w , 0 ) s 2 − d + k + A [ Φ ( S ( w , Y ) , D w β S × ( w , Y ) , D w 2 β S ( w , Y ) , D w 3 β S ( w , Y ) ) ] ) ] .

Using ATIM, we obtain an infinite series as a solution

(41) S ( w , Y ) = ∑ i = 0 ∞ S i .

The operator Φ ( S , D w β S , D w 2 β S , D w 3 β S ) is decomposed as follows:

(42) Φ ( S , D w β S , D w 2 β S , D w 3 β S ) = Φ ( S 0 , D w β S 0 , D w 2 β S 0 , D w 3 β S 0 ) + ∑ i = 0 ∞ Φ ∑ k = 0 i ( S k , D w β S k , D w 2 β S k , D w 3 β S k ) − Φ ∑ k = 1 i − 1 ( S k , D w β S k , D w 2 β S k , D w 3 β S k ) .

We obtain the following expression by inserting Eqs (42) and (41) into Eq. (40):

(43) ∑ i = 0 ∞ S i ( w , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S ( k ) ( w , 0 ) s 2 − d + k + A [ Φ ( S 0 , D w β S 0 , D w 2 β S 0 , D w 3 β S 0 ) ] + A − 1 1 s d A ∑ i = 0 ∞ Φ ∑ k = 0 i ( S k , D w β S k , D w 2 β S k , D w 3 β S k ) − A − 1 1 s d A Φ ∑ k = 1 i − 1 ( S k , D w β S k , D w 2 β S k , D w 3 β S k )

S 0 ( w , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S ( k ) ( w , 0 ) s 2 − d + k , S 1 ( w , Y ) = A − 1 1 s d ( A [ Φ ( S 0 , D w β S 0 , D w 2 β S 0 , D w 3 β S 0 ) ] ) , ⋮

(44) S m + 1 ( w , Y ) = A − 1 1 s d A ∑ i = 0 ∞ Φ ∑ k = 0 i ( S k , D w β S k , D w 2 β S k , D w 3 β S k ) − A − 1 1 s d A Φ ∑ k = 1 i − 1 ( S k , D w β S k , D w 2 β S k , D w 3 β S k ) , m = 1 , 2 , … .

Generally, we obtain the solution for the given PDE (37)

(45) S ( w , Y ) = ∑ i = 0 m − 1 S i .

4 Application of ARPSM and ATIM

Here, we consider two test problems concerning the NS model to show the effectiveness and general applicability of the two suggested approaches.

4.1 Example 1

Here, we consider the two-dimensional incompressible time-fractional NS model [75,76]

(46) D Y d S 1 ( w , G , Y ) − a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) − a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) + S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) + S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) − q = 0 , D Y d S 2 ( w , G , Y ) − a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) − a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) + S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) + S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) + q = 0 , where 0 < d ≤ 1 ,

which is subjected to the initial conditions (ICs)

(47) S 1 ( w , G , 0 ) = − sin ( w + G ) , S 2 ( w , G , 0 ) = sin ( w + G ) .

For the integer case, the exact solutions to model (46) read

(48) S 1 ( w , G , Y ) = − e − 2 a Y sin ( w + G ) , S 2 ( w , G , Y ) = e − 2 a Y sin ( w + G ) .

4.1.1 Analyzing example 1 via ARPSM

By applying AT on Eq. (46) and then using Eq. 47, we obtain

(49) S 1 ( w , G , s ) + sin ( w + G ) s 2 − a s d ∂ 2 ∂ w 2 S 1 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 1 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 1 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 1 ( w , G , s ) − 1 s d + 1 [ q ] = 0 , S 2 ( w , G , s ) − sin ( w + G ) s 2 − a s d ∂ 2 ∂ w 2 S 2 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 2 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 2 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 2 ( w , G , s ) + 1 s d + 1 [ q ] = 0 .

Therefore, the necessary term series is stated as

(50) S 1 ( w , G , s ) = − sin ( w + G ) s 2 + ∑ r = 1 k f r ( w , G , s ) s r d + 1 , S 2 ( w , G , s ) = sin ( w + G ) s 2 + ∑ r = 1 k g r ( w , G , s ) s r d + 1 , r = 1 , 2 , 3 , 4 , … .

The Aboodh residual functions (ARFs) are represented as

(51) A Y Res ( w , G , s ) = S 1 ( w , G , s ) + sin ( w + G ) s 2 − a s d ∂ 2 ∂ w 2 S 1 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 1 ( w , G , s ) + 1 s d A Y A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 1 ( w , G , s ) + 1 s d A Y A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 1 ( w , G , s ) − 1 s d + 1 [ q ] = 0 , A Y Res ( w , G , s ) = S 2 ( w , G , s ) − sin ( w + G ) s 2 − a s d ∂ 2 ∂ w 2 S 2 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 2 ( w , G , s ) + 1 s d A Y A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 2 ( w , G , s ) + 1 s d A Y A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 2 ( w , G , s ) + 1 s d + 1 [ q ] = 0 ,

and the k th -ARFs read as

(52) A Y Res k ( w , G , s ) = S 1 k ( w , G , s ) + sin ( w + G ) s 2 − a s d ∂ 2 ∂ w 2 S 1 k ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 1 k ( w , G , s ) + 1 s d A Y A Y − 1 S 1 k ( w , G , s ) × ∂ ∂ w A Y − 1 S 1 k ( w , G , s ) + 1 s d A Y A Y − 1 S 2 k ( w , G , s ) × ∂ ∂ G A Y − 1 S 1 k ( w , G , s ) − 1 s d + 1 [ q ] = 0 , A Y Res k ( w , G , s ) = S 2 k ( w , G , s ) − sin ( w + G ) s 2 − a s d ∂ 2 ∂ w 2 S 2 k ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 2 k ( w , G , s ) + 1 s d A Y A Y − 1 S 1 k ( w , G , s ) × ∂ ∂ w A Y − 1 S 2 k ( w , G , s ) + 1 s d A Y A Y − 1 S 2 k ( w , G , s ) × ∂ ∂ G A Y − 1 S 2 k ( w , G , s ) + 1 s d + 1 [ q ] = 0 .

Follow this process step by step to find the values of f r ( w , G , s ) and g r ( w , G , s ) . Substitute the r th residual function of Aboodh Eq. (52) in the truncated r th series Eq. (50) and take lim s → ∞ ( s r d + 1 ) on both sides of the expression. Then, evaluate the limit to solve A Y Res S 1 , r ( w , G , s ) = 0 and A Y Res S 2 , r ( w , G , s ) = 0 , for r = 1 , 2 , 3 , … . By using this process, we obtain some terms, which are given as follows:

(53) f 1 ( w , G , s ) = 2 a sin ( w + G ) + q , g 1 ( w , G , s ) = − 2 a sin ( w + G ) − q ,

(54) f 2 ( w , G , s ) = − 4 a 2 sin ( w + G ) , g 2 ( w , G , s ) = 4 a 2 sin ( w + G ) ,

and so on.

Now, by putting the values of f r ( w , G , s ) and g r ( w , G , s ) for r = 1 , 2 , 3 , … , in Eq. (50), we finally obtain

(55) S 1 ( w , G , s ) = − sin ( w + G ) s 2 + 2 a sin ( w + G ) + q s d + 1 − 4 a 2 sin ( w + G ) s 2 d + 1 + ⋯ , S 2 ( w , G , s ) = sin ( w + G ) s 2 − 2 a sin ( w + G ) + q s d + 1 + 4 a 2 sin ( w + G ) s 2 d + 1 + ⋯ .

By applying iAT, the following approximation is obtained:

(56) S 1 ( w , G , Y ) = − sin ( w + G ) + Y d ( 2 a sin ( w + G ) + q ) Γ ( d + 1 ) − 4 a 2 Y 2 d sin ( w + G ) Γ ( 2 d + 1 ) + ⋯

and

(57) S 2 ( w , G , Y ) = sin ( w + G ) − Y d ( 2 a sin ( w + G ) + q ) Γ ( d + 1 ) + 4 a 2 Y 2 d sin ( w + G ) Γ ( 2 d + 1 ) + ⋯ .

4.1.2 Analyzing example 1 via ATIM

Here, we apply ATIM to analyze model (46). For this purpose, we apply AT to model (46), which leads to

(58) A [ D Y d S 1 ( w , G , Y ) ] = 1 s d ∑ k = 0 m − 1 S 1 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) + q , A [ D Y d S 2 ( w , G , Y ) ] = 1 s d ∑ k = 0 m − 1 S 2 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) − q .

After applying the iAT to both sides of (58), the following result is obtained:

(59) S 1 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 1 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) + q , S 2 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 2 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) − q .

By applying AT, the following equation is obtained in an iterative manner:

S 1 0 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 1 ( k ) ( w , G , 0 ) s 2 − d + k = A − 1 S 1 ( w , G , 0 ) s 2 = − sin ( w + G ) ,

S 2 0 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 2 ( k ) ( w , G , 0 ) s 2 − d + k = A − 1 S 2 ( w , G , 0 ) s 2 = sin ( w + G ) .

By applying the RL integral into Eq. (46), the following equivalent form is obtained:

(60) S 1 ( w , G , Y ) = − sin ( w + G ) − A a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) + q , S 2 ( w , G , Y ) = sin ( w + G ) − A a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) − q .

By using the ATIM method, the following terms are achieved:

(61) S 1 0 ( w , G , Y ) = − sin ( w + G ) , S 1 1 ( w , G , Y ) = Y d ( 2 a sin ( w + G ) + q ) Γ ( d + 1 ) , S 1 2 ( w , G , Y ) = − 4 a 2 Y 2 d sin ( w + G ) Γ ( 2 d + 1 ) .

(62) S 2 0 ( w , G , Y ) = sin ( w + G ) , S 2 1 ( w , G , Y ) = − Y d ( 2 a sin ( w + G ) + q ) Γ ( d + 1 ) , S 2 2 ( w , G , Y ) = 4 a 2 Y 2 d sin ( w + G ) Γ ( 2 d + 1 ) .

The ATIM method yields the following final approximation:

(63) S 1 ( w , G , Y ) = S 1 0 ( w , G , Y ) + S 1 1 ( w , G , Y ) + S 1 2 ( w , G , Y ) + ⋯ = − sin ( w + G ) + Y d ( 2 a sin ( w + G ) + q ) Γ ( d + 1 ) − 4 a 2 Y 2 d sin ( w + G ) Γ ( 2 d + 1 ) + ⋯

and

(64) S 2 ( w , G , Y ) = S 2 0 ( w , G , Y ) + S 2 1 ( w , G , Y ) + S 2 2 ( w , G , Y ) + ⋯ = sin ( w + G ) − Y d ( 2 a sin ( w + G ) + q ) Γ ( d + 1 ) + 4 a 2 Y 2 d sin ( w + G ) Γ ( 2 d + 1 ) + ⋯ .

For simplicity, from now on, in all figures, we will use G 1 ( x , y , t ) ≡ G 1 instead of S 1 ( w , G , Y ) ≡ G 1 and p ≡ d . Also, G 2 ( x , y , t ) ≡ G 2 is used instead of S 2 ( w , G , Y ) .

It is clear from the approximations (56) and (57) that S 1 ( w , G , Y ) = − S 2 ( w , G , Y ) . Moreover, it is observed that the derived approximations (56) and (57) using the ARPSM are equivalent to the derived approximations (63) and (64) using the ATIM. Thus, in the current analysis, we are sufficient only to analyze the derived approximations (56) and (57). The approximations (56) and (57) are graphically investigated as illustrated in Figures 1 and 2, respectively. In these figures, we studied the effect of the fractional parameter p ≡ d on the profile of the approximations (56) and (57). Additionally, the absolute errors of the two approximations (56) and (57) are estimated as compared to the exact solutions (48) of the integer cases as evident in Tables 1 and 2, respectively. The numerical findings clearly demonstrate that the derived approximations exhibit exceptional accuracy and stability throughout the entire study domain. This, in turn, enhances the effectiveness of the used techniques in analyzing the most complex issues.

Figure 1 The approximation (56), S 1 ( w , G , Y ) ≡ G 1 ( x , y , t ) ≡ G 1 {{\mathfrak{S}}}_{1}\left({\mathfrak{w}},{\mathfrak{G}},{\mathfrak{Y}})\equiv {G}_{1}\left(x,y,t)\equiv {G}_{1} is plotted vs p ≡ d p\equiv {\mathfrak{d}} : (a) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.4 p=0.4 , (b) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.7 p=0.7 , (c) 3D graph in the ( x , y ) \left(x,y) -plane for p = 1 p=1 , and (d) 2D graph at different values of p p for y = 0.1 y=0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) \left(a,q,t)=\left(1,0,1) .
Figure 1

The approximation (56), S 1 ( w , G , Y ) ≡ G 1 ( x , y , t ) ≡ G 1 is plotted vs p ≡ d : (a) 3D graph in the ( x , y ) -plane for p = 0.4 , (b) 3D graph in the ( x , y ) -plane for p = 0.7 , (c) 3D graph in the ( x , y ) -plane for p = 1 , and (d) 2D graph at different values of p for y = 0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) .

Figure 2 The approximation (57), S 2 ( w , G , Y ) ≡ G 2 ( x , y , t ) ≡ G 2 {{\mathfrak{S}}}_{2}\left({\mathfrak{w}},{\mathfrak{G}},{\mathfrak{Y}})\equiv {G}_{2}\left(x,y,t)\equiv {G}_{2} is plotted vs p ≡ d p\equiv {\mathfrak{d}} : (a) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.4 p=0.4 , (b) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.7 p=0.7 , (c) 3D graph in the ( x , y ) \left(x,y) -plane for p = 1 p=1 , and (d) 2D graph at different values of p p for y = 0.1 y=0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) \left(a,q,t)=\left(1,0,1) .
Figure 2

The approximation (57), S 2 ( w , G , Y ) ≡ G 2 ( x , y , t ) ≡ G 2 is plotted vs p ≡ d : (a) 3D graph in the ( x , y ) -plane for p = 0.4 , (b) 3D graph in the ( x , y ) -plane for p = 0.7 , (c) 3D graph in the ( x , y ) -plane for p = 1 , and (d) 2D graph at different values of p for y = 0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) .

Table 1

Absolute error of the approximation S 1 ( w , G , Y )

Y w ARPSM/ATIM d = 1 Exact ARPSM/ATIM Error d = 1.00
0.1 0.1 ‒ 0.198272 ‒ 0.198272 2.647600 × 1 0 ‒ 10
0.2 ‒ 0.294930 ‒ 0.294930 3.938300 × 1 0 ‒ 10
0.3 ‒ 0.388640 ‒ 0.388640 5.189649 × 1 0 ‒ 10
0.4 ‒ 0.478468 ‒ 0.478468 6.389145 × 1 0 ‒ 10
0.5 ‒ 0.563514 ‒ 0.563514 7.524803 × 1 0 ‒ 10
0.3 0.1 ‒ 0.197481 ‒ 0.197481 7.141380 × 1 0 ‒ 9
0.2 ‒ 0.293752 ‒ 0.293752 1.062278 × 1 0 ‒ 8
0.3 ‒ 0.387089 ‒ 0.387089 1.399805 × 1 0 ‒ 8
0.4 ‒ 0.476558 ‒ 0.476558 1.723346 × 1 0 ‒ 8
0.5 ‒ 0.561265 ‒ 0.561265 2.029667 × 1 0 ‒ 8
0.5 0.1 ‒ 0.196693 ‒ 0.196693 3.302894 × 1 0 ‒ 8
0.2 ‒ 0.292580 ‒ 0.292580 4.913048 × 1 0 ‒ 8
0.3 ‒ 0.385544 ‒ 0.385544 6.474112 × 1 0 ‒ 8
0.4 ‒ 0.474655 ‒ 0.474655 7.970489 × 1 0 ‒ 8
0.5 ‒ 0.559024 ‒ 0.559024 9.387228 × 1 0 ‒ 8
0.7 0.1 ‒ 0.195907 ‒ 0.195907 9.054099 × 1 0 ‒ 8
0.2 ‒ 0.291412 ‒ 0.291412 1.346795 × 1 0 ‒ 7
0.3 ‒ 0.384005 ‒ 0.384004 1.774724 × 1 0 ‒ 7
0.4 ‒ 0.472761 ‒ 0.472760 2.184920 × 1 0 ‒ 7
0.5 ‒ 0.556793 ‒ 0.556793 2.573285 × 1 0 ‒ 7
1.0 0.1 ‒ 0.194736 ‒ 0.194735 2.635732 × 1 0 ‒ 7
0.2 ‒ 0.289669 ‒ 0.289669 3.920646 × 1 0 ‒ 7
0.3 ‒ 0.381708 ‒ 0.381707 5.166386 × 1 0 ‒ 7
0.4 ‒ 0.469933 ‒ 0.469932 6.360506 × 1 0 ‒ 7
0.5 ‒ 0.553463 ‒ 0.553462 7.491073 × 1 0 ‒ 7
Table 2

Absolute error of the approximation S 2 ( w , G , Y )

Y w ARPSM/ATIM d = 1 Exact ARPSM/ATIM Error d = 1.00
0.1 0.1 0.198272 0.198272 2.647600 × 1 0 ‒ 10
0.2 0.294930 0.294930 3.938300 × 1 0 ‒ 10
0.3 0.388640 0.388640 5.189649 × 1 0 ‒ 10
0.4 0.478468 0.478468 6.389145 × 1 0 ‒ 10
0.5 0.563514 0.563514 7.524803 × 1 0 ‒ 10
0.3 0.1 0.197481 0.197481 7.141380 × 1 0 ‒ 9
0.2 0.293752 0.293752 1.062278 × 1 0 ‒ 8
0.3 0.387089 0.387089 1.399805 × 1 0 ‒ 8
0.4 0.476558 0.476558 1.723346 × 1 0 ‒ 8
0.5 0.561265 0.561265 2.029667 × 1 0 ‒ 8
0.5 0.1 0.196693 0.196693 3.302894 × 1 0 ‒ 8
0.2 0.292580 0.292580 4.913048 × 1 0 ‒ 8
0.3 0.385544 0.385544 6.474112 × 1 0 ‒ 8
0.4 0.474655 0.474655 7.970489 × 1 0 ‒ 8
0.5 0.559024 0.559024 9.387228 × 1 0 ‒ 8
0.7 0.1 0.195907 0.195907 9.054099 × 1 0 ‒ 8
0.2 0.291412 0.291412 1.346795 × 1 0 ‒ 7
0.3 0.384005 0.384004 1.774724 × 1 0 ‒ 7
0.4 0.472761 0.472760 2.184920 × 1 0 ‒ 7
0.5 0.556793 0.556793 2.573285 × 1 0 ‒ 7
1.0 0.1 0.194736 0.194735 2.635732 × 1 0 ‒ 7
0.2 0.289669 0.289669 3.920646 × 1 0 ‒ 7
0.3 0.381708 0.381707 5.166386 × 1 0 ‒ 7
0.4 0.469933 0.469932 6.360506 × 1 0 ‒ 7
0.5 0.553463 0.553462 7.491073 × 1 0 ‒ 7

4.2 Example 2

Here, we consider the same time-fractional NS model given in Eq. (46) but with different ICs as follows:

(65) S 1 ( w , G , 0 ) = − e w + G , S 2 ( w , G , 0 ) = e w + G .

According to this case and for the integer case, the exact solutions to model (46) read as follows:

(66) S 1 ( w , G , Y ) = − e w + G + 2 a Y , S 2 ( w , G , Y ) = e w + G + 2 a Y .

4.2.1 Analyzing example 2 via ARPSM

Here, we apply ARPSM to analyze model (46). For this purpose, we apply AT on model (46) and with the help of ICs (65), we obtain

(67) S 1 ( w , G , s ) + e w + G s 2 − a s d ∂ 2 ∂ w 2 S 1 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 1 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 1 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 1 ( w , G , s ) − 1 s d + 1 [ q ] = 0 ,

S 2 ( w , G , s ) − e w + G s 2 − a s d ∂ 2 ∂ w 2 S 2 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 2 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 2 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 2 ( w , G , s ) + 1 s d + 1 [ q ] = 0 .

Therefore, the necessary term series is stated as

(68) S 1 ( w , G , s ) = − e w + G s 2 + ∑ r = 1 k f r ( w , G , s ) s r d + 1 , S 2 ( w , G , s ) = e w + G s 2 + ∑ r = 1 k g r ( w , G , s ) s r d + 1 , r = 1 , 2 , 3 , 4 ⋯ .

The ARFs are represented as

(69) A Y Res ( w , G , s ) = S 1 ( w , G , s ) + e w + G s 2 − a s d ∂ 2 ∂ w 2 S 1 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 1 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 1 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 1 ( w , G , s ) − 1 s d + 1 [ q ] = 0 ,

A Y Res ( w , G , s ) = S 2 ( w , G , s ) − e w + G s 2 − a s d ∂ 2 ∂ w 2 S 2 ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 2 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 ( w , G , s ) × ∂ ∂ w A Y − 1 S 2 ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 ( w , G , s ) × ∂ ∂ G A Y − 1 S 2 ( w , G , s ) + 1 s d + 1 [ q ] = 0 ,

and the k th -ARFs as

(70) A Y Res k ( w , G , s ) = S 1 k ( w , G , s ) + e w + G s 2 − a s d ∂ 2 ∂ w 2 S 1 k ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 1 k ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 k ( w , G , s ) × ∂ ∂ w A Y − 1 S 1 k ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 k ( w , G , s ) × ∂ ∂ G A Y − 1 S 1 k ( w , G , s ) − 1 s d + 1 [ q ] = 0 ,

A Y Res k ( w , G , s ) = S 2 k ( w , G , s ) − e w + G s 2 − a s d ∂ 2 ∂ w 2 S 2 k ( w , G , s ) − a s d ∂ 2 ∂ G 2 S 2 k ( w , G , s ) + 1 s d A Y [ A Y − 1 S 1 k ( w , G , s ) × ∂ ∂ w A Y − 1 S 2 k ( w , G , s ) + 1 s d A Y [ A Y − 1 S 2 k ( w , G , s ) × ∂ ∂ G A Y − 1 S 2 k ( w , G , s ) + 1 s d + 1 [ q ] = 0 .

Follow this process step by step to find the values of f r ( w , G , s ) and g r ( w , G , s ) . Substitute the r th residual function of Aboodh Eq. (70) in the truncated r th series Eq. (68) and take lim s → ∞ ( s r d + 1 ) on both sides of the expression. Then evaluate the limit to solve A Y Res S 1 , r ( w , G , s ) = 0 and A Y Res S 2 , r ( w , G , s ) = 0 , for r = 1 , 2 , 3 , … . By using this process, we obtain some terms, which are given as follows:

(71) f 1 ( w , G , s ) = q − 2 a e w + G , g 1 ( w , G , s ) = 2 a e w + G − q ,

(72) f 2 ( w , G , s ) = − 4 a 2 e w + G , g 2 ( w , G , s ) = 4 a 2 e w + G ,

and so on.

Now, by putting the values of f r ( w , G , s ) and g r ( w , G , s ) , for r = 1 , 2 , 3 , … , , in Eq. (68), we obtain

(73) S 1 ( w , G , s ) = − e w + G s 2 + q − 2 a e w + G s d + 1 − 4 a 2 e w + G s 2 d + 1 + ⋯ , S 2 ( w , G , s ) = e w + G s 2 + 2 a e w + G − q s d + 1 + 4 a 2 e w + G s 2 d + 1 + ⋯ .

Applying iAT on model (73) yields the following approximations

(74) S 1 ( w , G , s ) = − e w + G + Y d ( q − 2 a e w + G ) Γ ( d + 1 ) − 4 a 2 Y 2 d e w + G Γ ( 2 d + 1 ) + ⋯

and

(75) S 2 ( w , G , s ) = e w + G + Y d ( 2 a e w + G − q ) Γ ( d + 1 ) + 4 a 2 Y 2 d e w + G Γ ( 2 d + 1 ) + ⋯ .

4.2.2 Analyzing example 2 via ATIM

Here, we proceed to apply ATIM for analyzing model (46). For this purpose, we apply AT to model (46), which leads to

(76) A [ D Y d S 1 ( w , G , Y ) ] = 1 s d ∑ k = 0 m − 1 S 1 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) + q , A [ D Y d S 2 ( w , G , Y ) ] = 1 s d ∑ k = 0 m − 1 S 2 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) − q .

After applying the iAT to both sides of model (76), the following result is obtained:

(77) S 1 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 1 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) + q , S 2 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 2 ( k ) ( w , G , 0 ) s 2 − d + k + A a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) − q .

Using the AT in an iterative manner, we obtain

S 1 0 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 1 ( k ) ( w , G , 0 ) s 2 − d + k = A − 1 S 1 ( w , G , 0 ) s 2 = − e w + G ,

S 2 0 ( w , G , Y ) = A − 1 1 s d ∑ k = 0 m − 1 S 2 ( k ) ( w , G , 0 ) s 2 − d + k = A − 1 S 2 ( w , G , 0 ) s 2 = e w + G .

By applying the RL integral into Eq. (46), the following equivalent form is obtained:

(78) S 1 ( w , G , Y ) = − e w + G − A a ∂ 2 ∂ w 2 S 1 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 1 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 1 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 1 ( w , G , Y ) + q ,

S 2 ( w , G , Y ) = e w + G − A a ∂ 2 ∂ w 2 S 2 ( w , G , Y ) + a ∂ 2 ∂ G 2 S 2 ( w , G , Y ) − S 1 ( w , G , Y ) ∂ ∂ w S 2 ( w , G , Y ) − S 2 ( w , G , Y ) ∂ ∂ G S 2 ( w , G , Y ) − q .

Using the ATIM method, the following terms are achieved:

(79) S 1 0 ( w , G , Y ) = − e w + G , S 1 1 ( w , G , Y ) = Y d ( q − 2 a e w + G ) Γ ( d + 1 ) , S 1 2 ( w , G , Y ) = − 4 a 2 Y 2 d e w + G Γ ( 2 d + 1 ) .

(80) S 2 0 ( w , G , Y ) = e w + G , S 2 1 ( w , G , Y ) = Y d ( 2 a e w + G − q ) Γ ( d + 1 ) , S 2 2 ( w , G , Y ) = 4 a 2 Y 2 d e w + G Γ ( 2 d + 1 ) .

The following approximations using the ATIM are obtained:

(81) S 1 ( w , G , Y ) = S 1 0 ( w , G , Y ) + S 1 1 ( w , G , Y ) + S 1 2 ( w , G , Y ) + ⋯ = − e w + G + Y d ( q − 2 a e w + G ) Γ ( d + 1 ) − 4 a 2 Y 2 d e w + G Γ ( 2 d + 1 ) + ⋯ ,

(82) S 2 ( w , G , Y ) = S 2 0 ( w , G , Y ) + S 2 1 ( w , G , Y ) + S 2 2 ( w , G , Y ) + ⋯ = e w + G + Y d ( 2 a e w + G − q ) Γ ( d + 1 ) + 4 a 2 Y 2 d e w + G Γ ( 2 d + 1 ) + ⋯ .

In this example, one can see from the approximations (74) and (75) that S 1 ( w , G , Y ) = − S 2 ( w , G , Y ) . Furthermore, it is shown that the derived approximations (74) and (75) obtained through the ARPSM are similar to the derived approximations (81) and (82) obtained through the ATIM. Hence, it is just necessary for us to analyze the generated approximations (74) and (75). Thus, the impact of the fractional parameter p ≡ d on the profile of the approximations (74) and (75) is examined as shown in Figures 3 and 4, respectively. Moreover, the absolute errors of the two approximations (74) and (75) are estimated as compared to the exact solutions (66) of the integer cases as shown in Tables 3 and 4, respectively. The analysis results demonstrate that the generated approximations exhibit excellent accuracy and increased stability across the entire study domain. This validates the precision and effectiveness of the techniques employed in the analysis.

Figure 3 The approximation (74), S 1 ( w , G , Y ) ≡ G 1 ( x , y , t ) ≡ G 1 {{\mathfrak{S}}}_{1}\left({\mathfrak{w}},{\mathfrak{G}},{\mathfrak{Y}})\equiv {G}_{1}\left(x,y,t)\equiv {G}_{1} is plotted vs p ≡ d p\equiv {\mathfrak{d}} : (a) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.4 p=0.4 , (b) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.7 p=0.7 , (c) 3D graph in the ( x , y ) \left(x,y) -plane for p = 1 p=1 , and (d) 2D graph at different values of p p for y = 0.1 y=0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) \left(a,q,t)=\left(1,0,1) .
Figure 3

The approximation (74), S 1 ( w , G , Y ) ≡ G 1 ( x , y , t ) ≡ G 1 is plotted vs p ≡ d : (a) 3D graph in the ( x , y ) -plane for p = 0.4 , (b) 3D graph in the ( x , y ) -plane for p = 0.7 , (c) 3D graph in the ( x , y ) -plane for p = 1 , and (d) 2D graph at different values of p for y = 0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) .

Figure 4 The approximation (75), S 2 ( w , G , Y ) ≡ G 2 ( x , y , t ) ≡ G 2 {{\mathfrak{S}}}_{2}\left({\mathfrak{w}},{\mathfrak{G}},{\mathfrak{Y}})\equiv {G}_{2}\left(x,y,t)\equiv {G}_{2} is plotted vs p ≡ d p\equiv {\mathfrak{d}} : (a) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.4 p=0.4 , (b) 3D graph in the ( x , y ) \left(x,y) -plane for p = 0.7 p=0.7 , (c) 3D graph in the ( x , y ) \left(x,y) -plane for p = 1 p=1 , and (d) 2D graph at different values of p p for y = 0.1 y=0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) \left(a,q,t)=\left(1,0,1) .
Figure 4

The approximation (75), S 2 ( w , G , Y ) ≡ G 2 ( x , y , t ) ≡ G 2 is plotted vs p ≡ d : (a) 3D graph in the ( x , y ) -plane for p = 0.4 , (b) 3D graph in the ( x , y ) -plane for p = 0.7 , (c) 3D graph in the ( x , y ) -plane for p = 1 , and (d) 2D graph at different values of p for y = 0.1 . Here, ( a , q , t ) = ( 1 , 0 , 1 ) .

Table 3

Absolute error of the approximation S 1 ( w , G , Y )

Y w ARPSM/ATIM d = 1 Exact ARPSM/ATIM Error d = 1.00
0.1 0.1 ‒ 1.22385 ‒ 1.22385 1.629351 × 1 0 ‒ 9
0.2 ‒ 1.35256 ‒ 1.35256 1.800712 × 1 0 ‒ 9
0.3 ‒ 1.49481 ‒ 1.49481 1.990094 × 1 0 ‒ 9
0.4 ‒ 1.65202 ‒ 1.65202 2.199394 × 1 0 ‒ 9
0.5 ‒ 1.82577 ‒ 1.82577 2.430706 × 1 0 ‒ 9
0.3 0.1 ‒ 1.22875 ‒ 1.22875 4.403653 × 1 0 ‒ 8
0.2 ‒ 1.35798 ‒ 1.35798 4.866789 × 1 0 ‒ 8
0.3 ‒ 1.50080 ‒ 1.50080 5.378634 × 1 0 ‒ 8
0.4 ‒ 1.65864 ‒ 1.65864 5.944310 × 1 0 ‒ 8
0.5 ‒ 1.83308 ‒ 1.83308 6.569478 × 1 0 ‒ 8
0.5 0.1 ‒ 1.23368 ‒ 1.23368 2.040770 × 1 0 ‒ 7
0.2 ‒ 1.36342 ‒ 1.36343 2.255400 × 1 0 ‒ 7
0.3 ‒ 1.50682 ‒ 1.50682 2.492602 × 1 0 ‒ 7
0.4 ‒ 1.66529 ‒ 1.66529 2.754752 × 1 0 ‒ 7
0.5 ‒ 1.84043 ‒ 1.84043 3.044472 × 1 0 ‒ 7
0.7 0.1 ‒ 1.23862 ‒ 1.23862 5.605487 × 1 0 ‒ 7
0.2 ‒ 1.36889 ‒ 1.36889 6.195021 × 1 0 ‒ 7
0.3 ‒ 1.51286 ‒ 1.51286 6.846557 × 1 0 ‒ 7
0.4 ‒ 1.67196 ‒ 1.67197 7.566616 × 1 0 ‒ 7
0.5 ‒ 1.84781 ‒ 1.84781 8.362404 × 1 0 ‒ 7
1.0 0.1 ‒ 1.24608 ‒ 1.24608 1.636712 × 1 0 ‒ 6
0.2 ‒ 1.37713 ‒ 1.37713 1.808846 × 1 0 ‒ 6
0.3 ‒ 1.52196 ‒ 1.52196 1.999085 × 1 0 ‒ 6
0.4 ‒ 1.68203 ‒ 1.68203 2.209330 × 1 0 ‒ 6
0.5 ‒ 1.85893 ‒ 1.85893 2.441687 × 1 0 ‒ 6
Table 4

Absolute error of the approximation S 2 ( w , G , Y )

Y w ARPSM/ATIM d = 1 Exact ARPSM/ATIM Error d = 1.00
0.1 0.1 1.22385 1.22385 1.629351 × 1 0 ‒ 9
0.2 1.35256 1.35256 1.800712 × 1 0 ‒ 9
0.3 1.49481 1.49481 1.990094 × 1 0 ‒ 9
0.4 1.65202 1.65202 2.199394 × 1 0 ‒ 9
0.5 1.82577 1.82577 2.430706 × 1 0 ‒ 9
0.3 0.1 1.22875 1.22875 4.403653 × 1 0 ‒ 8
0.2 1.35798 1.35798 4.866789 × 1 0 ‒ 8
0.3 1.50080 1.50080 5.378634 × 1 0 ‒ 8
0.4 1.65864 1.65864 5.944310 × 1 0 ‒ 8
0.5 1.83308 1.83308 6.569478 × 1 0 ‒ 8
0.5 0.1 1.23368 1.23368 2.040770 × 1 0 ‒ 7
0.2 1.36342 1.36343 2.255400 × 1 0 ‒ 7
0.3 1.50682 1.50682 2.492602 × 1 0 ‒ 7
0.4 1.66529 1.66529 2.754752 × 1 0 ‒ 7
0.5 1.84043 1.84043 3.044472 × 1 0 ‒ 7
0.7 0.1 1.23862 1.23862 5.605487 × 1 0 ‒ 7
0.2 1.36889 1.36889 6.195021 × 1 0 ‒ 7
0.3 1.51286 1.51286 6.846557 × 1 0 ‒ 7
0.4 1.67196 1.67197 7.566616 × 1 0 ‒ 7
0.5 1.84781 1.84781 8.362404 × 1 0 ‒ 7
1.0 0.1 1.24608 1.24608 1.636712 × 1 0 ‒ 6
0.2 1.37713 1.37713 1.808846 × 1 0 ‒ 6
0.3 1.52196 1.52196 1.999085 × 1 0 ‒ 6
0.4 1.68203 1.68203 2.209330 × 1 0 ‒ 6
0.5 1.85893 1.85893 2.441687 × 1 0 ‒ 6

5 Conclusion

Our investigation has yielded valuable insights into the behavior of fractional fluid flow systems by examining the analytical solutions of the time-fractional multi-dimensional NS equation. In our analysis, the ARPSM and the ATIM, operating within the Caputo operator’s framework, have been implemented to analyze the test problems. We have analyzed four different models related to the NS equation to verify the accuracy and efficiency of the used approaches. During the analysis, we discussed all derived approximations numerically and graphically through a number of tables and graphs for these approximations against the fractional parameter. In addition, we examined the properties of the nonlinear phenomena described by these approximations and analyzed how the fractional parameter influences each profile of these phenomena to understand the propagation dynamics comprehensively and uncover any ambiguity that may arise in the practical results described by this family. Furthermore, we calculated the absolute error of all derived approximations compared to the exact solutions for the integer case. By addressing the complexities introduced by fractional derivatives, we have demonstrated the efficacy and reliability of the proposed techniques for capturing the dynamic behavior of the multi-dimensional NS equation. The results of our research enhance the comprehension of how fractional calculus may be applied in fluid dynamics and provide essential analytical instruments for investigating intricate flow phenomena in different scientific and technical fields. Further research could explore additional applications of the Aboodh methods in addressing other complex PDEs and their fractional counterparts. This could significantly extend our capabilities in modeling and understanding intricate physical phenomena, potentially reshaping our understanding of many physical and engineering fields and their applications.

6 Future work

Due to the positive results obtained from the utilization of Aboodh’s techniques, it is possible to apply these techniques to analyze different types of fractional wave equations that arise in various plasma models to comprehend the ambiguity surrounding certain behaviors that accompany some nonlinear phenomena (e.g., solitary, shock, cnoidal waves, etc.) occurring in different plasma models. Accordingly, these approaches could be applied to solve and examine the third-order dispersion KdV-type equations [77–79] and fifth-order dispersion KdV-type equations (Kawahara-types equations) [80–82] and many other higher-order nonlinearity evolution equations. In addition, these approaches can be employed to analyze the nonlinear Schrödinger-type equations and examine how fractional parameters affect the properties of various nonlinear phenomena, such as rogue waves, envelope dark solitons, envelope bright solitons, envelope gray-solitons, modulated breathers, and many others [83–85].

,

Acknowledgments

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R439), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. KFU241716].

  1. Funding information: The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R439), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. KFU241716].

  2. Author contribution: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Received: 2024-03-31
Revised: 2024-07-19
Accepted: 2024-07-25
Published Online: 2024-10-16

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2024-0081
Keywords for this article
multi-dimensional fractional Navier–Stokes models; Aboodh residual power series method; Aboodh transform iteration method; Caputo operator
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
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  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
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  39. A comparative analysis of shielding effectiveness in glass and concrete containers
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  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
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  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
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  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
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  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
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  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
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  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
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  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
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  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
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  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
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  107. Special Issue on Predicting pattern alterations in nature - Part II
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  109. On the existence and numerical simulation of Cholera epidemic model
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  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
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  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
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  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
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  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
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  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
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