Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative

Nasser Hassan Sweilam; Khloud Ramadan Khater; Zafer Mohamed Asker; Waleed Abdel Kareem

doi:10.1515/nleng-2024-0029

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Space–time variable-order carbon nanotube model using modified Atangana–Baleanu–Caputo derivative

Nasser Hassan Sweilam , Khloud Ramadan Khater , Zafer Mohamed Asker and Waleed Abdel Kareem

Published/Copyright: November 28, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

This study presents a space–time variable-order fractional carbon nanotube (CNT) mathematical model. Fractional differential equations successfully describe the model and its physical and biological properties. The mathematical model simulates the mixed free convection flow of a nanofluid in porous space using single- and multi-walled CNTs. The heat transfer characteristics of the base fluid (human blood) are studied. The numerical solutions for the temperature and velocity fields were derived from the modified Atangana–Baleanu–Caputo derivative, and the model was very effective. The higher compact finite difference method is used to study the numerical technique. Stabilization is construed as a technique related to John Neumann’s stabilization analysis. Additionally, the truncation error was studied, and various graphs were used to present numerical results. The results in the tables and the numerical figures emphasize that the schemes attained from applying the submitted numerical methods are completely compatible with the exact solution.

Keywords: carbon nanotube model; space–time variable-order fractional model; compact finite difference method; modified Atangana–Baleanu–Caputo derivative

1 Introduction

A carbon nanotube (CNT) is a tube made of carbon with a diameter in the nanometre range (nanoscale). They are one of the allotropes of carbon. Single-walled carbon nanotubes (SWCNTs) have diameters around 0.5–2.0 nm, about 1,00,000 times smaller than the width of a human hair [1]. Multi-walled carbon nanotubes (MWCNTs) consist of nested SWCNTs in a nested, tube-in-tube structure. Double- and triple-walled CNTs are special cases of MWCNT. CNTs can exhibit remarkable properties, such as exceptional tensile strength and thermal conductivity, because of their nanostructure and strength of the bonds between carbon atoms. Some SWCNT structures exhibit high electrical conductivity, while others are semiconductors. In addition, CNTs can be chemically modified. These properties are expected to be valuable in many areas of technology, such as electronics, optics, composite materials (replacing or complementing carbon fibres), nanotechnology (including nanomedicine), and other applications of materials science. Fractional calculus has been applied in modelling control systems, heat flux, temperature, entropy generation and diffusion, among others, because it is the popularization for integer-order calculus. Such operators reveal the distinguished characteristics of extended relationships, which cannot be proved by the criterion integer order differential equation, so it is a logical progression from the constant-order calculus, which is the variable-order calculus. The ordering may be seen in this way as functions in any variable, including time, space, and other variables. Therefore, some fractional derivatives such as Riemann–Liouville, Caputo, Caputo–Fabrizio and AtanganaBaleanu (AB), or Atangana–Baleanu–Caputo (ABC) derivatives were calculated numerically and theoretically for diverse fractional differential equations to explain such unusual behaviour for various mathematical problems. Later, the solutions of time-fractional differential equations were presented in previous studies [2–4], which were analyzed by RL derivative but it displays weakened singularity for the initial time t = 0 ; therefore, Caputo–Fabrizio fractional derivative by Caputo and Fabrizio was introduced [5], that comprises a nonsingular kernel that recognizes the material hesitations and variations with different scales. Then, some studied the ABC derivative, which comprises the nonsingular kernel and is more generalized than the CF definition [6]. Recently, Refai and Baleanu [7] submitted a modulation for the ABC derivative called the modified Atangana–Baleanu–Caputo (MABC) definition and showed that there are diverse fractional problems, which have the competence to be solved by the MABC derivative when they cannot be solved by the ABC derivative. Nowadays, the modelling of natural phenomena is gaining much interest among researchers. Such models give sufficient information about the different dynamics of infectious diseases [8]. Numerical treatments for the optimal control of two types of variable-order COVID-19 model and preventive inter ventions using fractional-calculus analysis has been introduced by Sweilam et al. [9]. Numerical solutions and geometric attractors of a fractional model of the cancer-immune based on the ABC derivative and the reproducing kernel scheme [10].

Nanofluid is one of the most important topics due to its boosted thermal properties compared to the base fluid [11–13]. Heat transfer generally counts on the thermophysical features for the nanoparticles, fractional volume, and thermophysical discriminators of the base fluid. The magnetic nanoparticle flow plays many real-world uses in engineering and industries such as hot rolling metal extrusion, extracting and producing energy, cooling frameworks, and fibreglass. The thermal shape of gamma alumina nanofluid for the Marangoni flow was addressed by Ganesh et al. [14]. Some more studies on the same topic are those of Pande et al. [15], Khalid et al. [16], and Khan et al. [17]. SWCNTs and MWCNTs for rotating plates were discussed by Alzahrani et al. [18]. Dimbeswar et al. [19] studied a few applications for a vertical tube of CNTs and the porosity of the medium (human blood) flow in the appearance of thermal irradiation and chemical response of first order. Inside this tube, SWCNT and MWCNT were replaced with blood as a base fluid. The simulation solutions that depend on the fractional derivatives notion are the preferable way for these classes of mathematics problems, which can illustrate numerical and analytical effective outcomes for some real-world problems. Thanaa et al. [20] investigated the influence of CNTs nanofluid on natural convection with Prabhakar-like thermal transport, near an infinite vertical heated plate. Ali et al. [21] explained the Casson nanofluid flow over an oscillating plate in the existence of a magnetic field and porosity. Sweilam et al. [22] investigated higher-order finite difference schemes to solve the time-fractional magnetohydrodynamic convection inflow of blood with CNT models. Basma et al. [23] studied a mathematical model of mixed convective flow based on CNTs suspended in ethylene glycol, which has been developed and derived by means of Fourier sine transform. As systems’ long variable memory evolves over time, variable-order fractional derivatives are an effective tool for characterising the impact of these changes. Systems’ lengthy memories are shown by the constant-order fractional derivative, but their short memories. Some studies and references mentioned therein examined the benefit of employing variable-order in fractional for ordinary differential equations or partial differential equations [24–29].

This article is organized in the following way. In Section 2, the model is presented together with its fractional representation. Section 3 contains the definitions for fractions. Additionally, Section 4 explains the variable-order fractional derivative as well as the higher-order finite difference approximations. The CNT model equations are discretized in Section 4.1. Section 6 presents the stability assessments and truncation errors for the proposed model. Section 7 contains the numerical graphical reports for the equations relating to nanofluid CNTs. Section 8 concludes everything.

2 Flow problem and its fractional form

Let us appreciate that the flow of a Casson-based incompressible liquid is unstable and absolute convective over a porous vibrating slab at an isothermal temperature T i n f . The behaviour of Casson-base fluid is presumed to denote the liquid (blood) properties and the analyses for transferring heat for SWCNTs and MWCNTs. The liquid is pretended to be well-conducting electrical, and B is a magnetic field with intensity B 0 utilized through the vertical way for the laminar when an angle of γ is used. In the beginning, the temperature is estimated as ω and the slab is not moving. Then, the heat for the liquid changes from T ω to T i n f , and the slab begins to shake with V 0 = U 0 H ( t ) cos ( ω t ) (Figure 1). The vibrating slab caused the flow. The tensor of Cauchy stress for Casson-base fluid is specified as [30]

(1) τ i j = 2 ( ψ B + P y 2 π ) e i j , π > π c , 2 ( ψ B + P y 2 π c ) e i j , π < π c . ,

where π = e i j e i j and ψ B , P y , π c represent the plastic dynamic viscosity, giving stress. The governing equations for the actual model are represented as [16,30]

(2) ρ n f ∂ u ( ξ , t ) ∂ t = ψ n f 1 + 1 B 1 ∂ 2 u ( ξ , t ) ∂ ξ 2 − σ n f B 0 2 sin γ + 1 + 1 B 1 ψ n f ϕ K u ( ξ , t ) + g ( ρ B T ) n f ( θ ( ξ , t ) − θ ∞ ) ,

(3) ( ρ C p ) n f ∂ θ ( ξ , t ) ∂ t = K n f ∂ 2 θ ( ξ , t ) ∂ ξ 2 , t > 0 , ξ ≥ 0 ,

with boundary and initial conditions,

u ( ξ , 0 ) = 0 , θ ( ξ , 0 ) = θ ∞ , ∀ ξ > 0 , u ( 0 , t ) = U 0 H ( t ) cos ( w t ) , θ ( 0 , t ) = θ w , for t ≥ 0 , u ( l , t ) = 0 , θ ( l , t ) = 0 , ∀ t > 0 ,

where ψ n f = ψ f ( 1 − ϕ ) 2.5 , ρ n f = ( 1 − ϕ ) ρ f + ϕ ρ s , ( ρ B T ) n f = ( 1 − ϕ ) ( ρ B T ) f + ϕ ( ρ B T ) s , ( ρ C p ) n f = ( 1 − ϕ ) ( ρ C p ) f + ϕ ( ρ C p ) s , σ n f = σ f 1 + 3 ϕ ( σ s σ f − 1 ) ( σ s σ f + 2 ) − ϕ ( σ s σ f − 1 ) . Assuming the non-dimensional variable equations in the following forms:

u * = u u o , ξ * = u o u f ξ , t * = u o 2 u f t , and T = θ − θ ∞ θ w − θ ∞ .

The aforementioned variables lead to the following set of equations:

(4) a 0 ∂ u ( ξ , t ) ∂ t = a 1 B ∂ 2 u ( ξ , t ) ∂ ξ 2 − a 2 M sin γ + a 1 K u ( ξ , t ) + a 3 G r T ( ξ , t ) ,

(5) a 4 Pr ∂ T ( ξ , t ) ∂ t = a 5 ∂ 2 T ( ξ , t ) ∂ ξ 2 , t > 0 , ξ ≥ 0 ,

where Eqs (4) and (5) represent the temperature and velocity fields with the initial and boundary conditions:

u ( ξ , 0 ) = 0 , T ( ξ , 0 ) = 0 , ∀ ξ > 0 , u ( 0 , t ) = H ( t ) cos ( w t ) , T ( 0 , t ) = 1 , f o r t ≥ 0 , u ( l , t ) = 0 , T ( l , t ) = 0 , ∀ t > 0 ,

where

M = σ f u f B 0 2 ρ f U 0 2 , B = B 1 1 + B 1 , K = k u 0 2 u f ϕ , Pr = ψ C p k f , and Gr = g ( β T u ) f ( θ w − θ ∞ ) U 0 3 , a 0 = ( 1 − ϕ f ) + ϕ ρ s ρ f , a 1 = 1 ( 1 − ϕ ) , a 2 = σ h n f σ f , a 3 = ( 1 − ϕ ) + ϕ ( ρ B T ) s ( ρ B T ) f , a 4 = ( 1 − ϕ ) + ϕ ( ρ C p ) s ( ρ C p ) f , a 5 = k n f k f = 1 − ϕ + 2 ϕ ( k s k s − k f ) ln k s + k f 2 k f 1 − ϕ + 2 ϕ ( k s k s − k f ) ln k s + k f 2 k f .

The magnetic number is M , B is the Casson fluid parameter, Pr is the Prandtl number, K is the permeability constant, and Gr is the Grashof number, while a x for x = 0 , 1 , … , 5 are the constants. The thermal attributes of the base fluids SWCNTs and MWCNTs are exhibited in Table 1.

Figure 1

Configuration and coordinate system.

Table 1

Thermophysical properties of SWCNTs and MWCNTs for base fluid [18]

Phsical parameters	Material	Base fluid	SWCNTs	MWCNTs
Density	ρ	1,053	2,600	1,600
Heat capacitance	C p	3,594	425	796
Electrical conductivity	σ	0.8	106–107	1.9 × 1 0 ‒ 4
Thermal conductivity	K	0.492	6,600	3,000
Casson fluid	B T × 1 0 ‒ 5	0.18	27	44

The time–space fractional for the presented model of variable-order fractions α ( ξ , t ) for time and β ( ξ , t ) for space to Eqs (4) and (5) is given

(6) D t α ( ξ , t ) MABC u t ( ξ , t ) = a 6 D ξ β ( ξ , t ) MABC u ξ ξ ( ξ , t ) − a 7 u ( ξ , t ) + a 8 T ( ξ , t ) , t > 0 , ξ ≥ 0 ,

(7) D t α ( ξ , t ) MABC T t ( ξ , t ) = a 9 D ξ β ( ξ , t ) MABC T ξ ξ ( ξ , t ) , 0 < α ( ξ , t ) < 1 , 1 < β ( ξ , t ) < 2 ,

where D t α ( ξ , t ) MABC and D ξ β ( ξ , t ) MABC are the MABC fractional operator for time and space [7] for the same boundary and initial conditions.

3 Preliminaries and mathematical details

The calculus of fractional derivatives is commonly studied through variable-order. Some applications of fractional calculus call for the integral or derivative order to depend on a system parameter rather than being kept constant. This is achieved with variable-order differential equations [31], which apply mostly to dynamic problems. There are many definitions of variable-order fractional derivatives of order 0 < α ( ξ , t ) < 1 [24,25] such as Grünwald–Letinkov’s definition (GL), RL definition, Caputo’s fractional derivative, AB fractional derivative in Caputo sense and the new definition of the modified ABC fractional operator in L 1 ( 0 , T ) in Caputo sense.

The RL and Caputo derivatives of fractional order of f ( t ) are, respectively, defined as
(8) D t α ( ξ , t ) R L f ( t ) = 1 Γ ( r − α ( ξ , t ) ) d r d t r ∫ 0 t ( t − τ ) ( r − α ( ξ , t ) − 1 ) f ( τ ) d τ ,

(9) D t α ( ξ , t ) C f ( t ) = 1 Γ ( r − α ( ξ , t ) ) ∫ 0 t ( t − τ ) ( r − α ( ξ , t ) − 1 ) f r ( τ ) d τ , t > 0 , r − 1 < α ( ξ , t ) < r .
Let f ( x , t ) be a function in H 1 ( a ; b ) , b > a , and α ( ξ , t ) ∈ ( 0 , 1 ) , and then, the Caputo–Fabrizio derivative with fractional-order compact finite difference [4] and [5] will be defined as
(10) D t α ( ξ , t ) a CF f ( t ) = α ( ξ , t ) B ( α ( ξ , t ) ) ( 1 − α ( ξ , t ) ) ∫ 0 t ( f ( x , t ) − f ( x , τ ) ) × e − α ( ξ , t ) ( 1 − τ ) ( 1 − α ( ξ , t ) ) d τ , 0 < α ( ξ , t ) < 1 ,
where B ( α ( ξ , t ) ) = 1 − α ( ξ , t ) + α ( ξ , t ) Γ ( α ( ξ , t ) ) is the normalization function such that B ( 0 ) = B ( 1 ) = 1 .
The ABC fractional derivative is defined [6] as
(11) D t α ( ξ , t ) a A B C f ( t ) = B ( α ( ξ , t ) ) ( 1 − α ( ξ , t ) ) ∫ 0 t E α ( ξ , t ) ( − α ( ξ , t ) × ( t − s ) α ( ξ , t ) ( 1 − α ( ξ , t ) ) ) f ˙ ( s ) d s , 0 < α ( ξ , t ) < 1 ,
where E α ( ξ , t ) is the Mittag–Leffler function where E α ( ξ , t ) , β ( ξ , t ) ( K ) = ∑ r = 0 ∞ K r Γ ( r α ( ξ , t ) + β ( ξ , t ) ) , α ( ξ , t ) > 0 ; it is a modified form of Caputo–Fabrizio that presents the ideal properties of non-singularity and nonlocality of the kernel.
The MABC fractional operator in L 1 ( 0 , T ) in Caputo sense (follow [7]) was presented as
(12) D 0 α ( ξ , t ) MABC f ( t ) = B ( α ( ξ , t ) ) ( 1 − α ( ξ , t ) ) f ( t ) − E α ( ξ , t ) ( − μ α ( ξ , t ) t α ( ξ , t ) ) f ( 0 ) − μ α ( ξ , t ) ∫ 0 t ( t − s ) α ( ξ , t ) − 1 E α ( ξ , t ) , α ( ξ , t ) × ( − μ α ( ξ , t ) ( t − s ) α ( ξ , t ) ) f ( s ) d s , D 0 δ MABC = B ( α ( ξ , t ) ) ( 1 − α ( ξ , t ) ) f n − 1 ( t ) − E α ( ξ , t ) ( − μ α ( ξ , t ) t α ( ξ , t ) ) f n − 1 ( 0 ) − μ α ( ξ , t ) ∫ 0 t ( t − s ) α ( ξ , t ) − 1 E α ( ξ , t ) , α ( ξ , t ) × ( − μ α ( ξ , t ) ( t − s ) α ( ξ , t ) ) f n − 1 ( s ) d s ,
where μ α ( ξ , t ) = α ( ξ , t ) 1 − α ( ξ , t ) and B has the same properties as in Caputo–Fabrizio case. The derivative is defined for 0 < α ( ξ , t ) < 1 and n − 1 < δ < n . The MABC fractional operator that modulates the ABC fractional derivative includes the kernel with an integrable singularity originally, which leads to new solutions of several fractional differential equations and a description of the dynamics of complex processes that is better than the ABC fractional operator, for more information [7].

4 Numerical methods

4.1 High-order compact finite difference approximation (HOCFD)

HOCFD schemes are used for solving third-order differential equations created during the study of obstacle boundary value problems, which can enable obtaining of more terms in the Taylor series expansion. Taylor expansion is considered a handy tool for the derivation of order approximations to derivatives of any order. They have been shown to be highly accurate and efficient. They are constructed by modifying the second-order scheme that was developed by Noor et al. [32]. The convergence rate of the high-order compact scheme is third order, and the second-order scheme is fourth order. Our concern here is using an HOCFD formula for the spatial discretization of the problems to be considered later in this work.

Let us consider that the solution domain of our problem is n , m and l be positive integers when 0 = ξ 0 < ξ 1 < ξ 2 < … < ξ n = l and 0 = t o < t 1 < t 2 < … < t m = t max , the points have the coordinates ξ i + 1 = ξ i + d ξ , ( i = 0 , 1 , … , n ) and t j + 1 = t j + d t , ( j = 0 , 1 , … , m ) , where d ξ = l n , where d t = t max m , therefore, we need to create an estimate based on the step size of 2 ξ through the Taylor series expansion [4,22]

u ( ξ i + 2 d ξ ) = ∑ n = 0 ∞ ( 2 d ξ ) n n ! u n ( ξ i ) , u ( ξ i − 2 d ξ ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 d ξ ) n n ! u n ( ξ i ) .

A better approximation can be gained by combining these two assessments using the process called Richardson extrapolation. We will deduce the fourth-order centred standard finite difference scheme for the first derivative will be

(13) ∂ u ∂ ξ = ( u ( ξ i − 2 d ξ ) − 8 ( ξ i − d ξ ) + 8 ( ξ i + d ξ ) − u ( ξ i + 2 d ξ ) ) 12 d ξ + O ( ( d ξ ) 4 ) .

For problems (6) and (7), the value solutions are u ( ξ i , t j ) ≡ u i j and T ( ξ i , t j ) ≡ T i j , where d ξ = l n and d t = T i m e m .

4.2 Variable-order fractional derivatives for CNT model

In the following, we introduce the variable-order system for the mathematical models (6) and (7), where D t α ( ξ , t ) MABC is the time fractional operator and D ξ β ( ξ , t ) MABC is the space fractional operator:

(14) D t α ( ξ i , t j ) MABC u t ( ξ i , t j ) = a 6 D ξ β ( ξ i , t j ) MABC u ξ ξ ( ξ i , t j ) − a 7 u ( ξ i , t j ) + a 8 T ( ξ i , t j ) , t > 0 , ξ ≥ 0 ,

(15) D t α ( ξ i , t j ) MABC T t ( ξ i , t j ) = a 9 D ξ β ( ξ i , t j ) MABC T ξ ξ ( ξ i , t j ) , 0 < α ( ξ i , t j ) < 1 , 1 < β ( ξ i , t j ) < 2 ,

where

a 6 = a 1 B , a 7 = a 2 M sin γ + a 1 K , a 8 = a 3 G r , and a 9 = a 4 Pr a 5 .

The MABC time-space variable-order fractional derivatives for the CNT model of systems (14) and (15) will be derived as follows:

(16) B ( α ( j i ) ) ( 1 − α ( j i ) ) u ( j i ) − E α ( j i ) ( − μ α ( j i ) t α ( j i ) ) u ( 0 ) − μ α ( j i ) ∫ 0 t ( t − s ) α ( j i ) − 1 E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t − s ) α ( j i ) ) u ( ξ i , s j ) d s = a 6 B ( β ( j i ) ) ( 1 − β ( j i ) ) u ′ ( j i ) − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) u ′ ( 0 ) − μ β ( j i ) ∫ 0 ξ ( ξ − p ) β ( j i ) − 1 E β ( j i ) , β ( j i ) ( − μ β ( j i ) ( ξ − p ) β ( j i ) ) u ′ ( p i , t j ) d p − a 7 u i j + a 8 T i j , t > 0 , ξ ≥ 0 ,

(17) B ( α ( j i ) ) ( 1 − α ( j i ) ) T ( j i ) − E α ( j i ) ( − μ α ( j i ) t α ( j i ) ) T ( 0 ) − μ α ( j i ) ∫ 0 t ( t − s ) α ( j i ) − 1 E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t − s ) α ( j i ) ) T ( x i , s j ) d s = a 9 B ( β ( j i ) ) ( 1 − β ( j i ) ) T ′ ( j i ) − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) T ′ ( 0 ) − μ β ( j i ) ∫ 0 ξ ( ξ − p ) β ( j i ) − 1 E β ( j i ) , β ( j i ) ( − μ β ( j i ) ( ξ − p ) β ( j i ) ) T ′ ( p i , t j ) d p , 0 < α ( j i ) ≤ 1 , 1 < β ( j i ) ≤ 2 .

5 Discreitization

5.1 Time discreitization

Using the definition given by Eq. (12) and applying the definition of the Mittag–Leffler function and applying the Taylor series expansion to discretize the function u ( t ) , we obtain

u ( t ) = u ( t k ) + ( t − t k ) u ′ ( t k ) + ( t − t k ) 2 2 u ″ ( t k ) + ( O ( t − t k ) 3 ) , u ′ ( t k ) = u ( t k + 1 ) − u ( t k − 1 ) 2 Δ t − u ( 3 ) ( t k ) 6 ( Δ t ) 2 + O ( ( Δ t ) 4 )

so,

(18) u ( t ) = u ( t k ) + ( t − t k ) u ( t k + 1 ) − u ( t k − 1 ) 2 Δ t − ( t − t k ) u ( 3 ) ( t k ) 6 ( Δ t ) 2 + O ( ( t − t k ) 2 ) ,

we have that,

D t α ( j i ) MABC u ( j i ) = B ( α ( j i ) ) ( 1 − α ( j i ) ) u ( j i ) − E α ( j i ) ( − μ α ( j i ) t α ( j i ) ) u ( i , 0 ) − μ α ( j i ) ∫ 0 t ( t − s ) α ( j i ) − 1 E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t − s ) α ( j i ) ) u ( ξ i , s j ) d s , 0 < α ( j i ) ≤ 1 , = B ( α ( j i ) ) ( 1 − α ( j i ) ) u ( j i ) − E α ( j i ) ( − μ α ( j i ) t α ( j i ) ) u ( i , 0 ) − μ α ( j i ) ∫ 0 t ( t − s ) α ( j i ) − 1 E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t − s ) α ( j i ) ) × u ( ξ , t k ) + ( s − t k ) u ( i , t k + 1 ) − u ( i , t k − 1 ) 2 Δ t d s + R k

(19) = B ( α ( j i ) ) ( 1 − α ( j i ) ) u ( j i ) − B ( α ( j i ) ) ( 1 − α ( j i ) ) × μ α ( j i ) ∫ 0 t ( t − s ) α ( j i ) − 1 E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t − s ) α ( j i ) ) u ( ξ , t k ) d s + μ α ( j i ) ∫ 0 t ( t − s ) α ( j i ) − 1 E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t − s ) α ( j i ) ) ( s − t k ) u ( i , t k + 1 ) − u ( i , t k − 1 ) 2 Δ t d s + R k , 0 < α ( j i ) ≤ 1 .

Now, for simplification, we consider that

∑ k = 0 m ∫ ( k − 1 ) Δ t ( k ) Δ t ( t k − s ) ( α − 1 ) E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t k − s ) α ( j i ) ) u ( ξ , t k ) d s = ∑ k = 1 m u ( ξ i , t k ) ( t m − t k ) α i j E α ( j i ) , α ( j i + 1 ) ( − μ α ( j i ) ( t m − t k ) α ( j i ) ) − ( t m − t k + 1 ) α i j E α ( j i ) , α ( j i + 1 ) ( − μ α ( j i ) ( t m − t k + 1 ) α ( j i ) ) = ∑ k = 1 m u ( ξ i , t k ) δ α i j

and

∑ k = 0 m ∫ ( k − 1 ) Δ t ( k ) Δ t ( t k − s ) ( α − 1 ) E α ( j i ) , α ( j i ) ( − μ α ( j i ) ( t k − s ) α ( j i ) ) × ( s − t k ) u ( i , t k + 1 ) − u ( i , t k − 1 ) 2 Δ t d s = ∑ k = 1 m u ( i , t k + 1 ) − u ( i , t k − 1 ) 2 Δ t × ( t m − t k ) α i j + 1 E α ( j i ) , α ( j i + 2 ) ( − μ α ( j i ) ( t m − t k ) α ( j i ) ) − Δ t ( t m − t k + 1 ) α i j × E α ( j i ) , α ( j i + 1 ) ( − μ α ( j i ) ( t m − t k + 1 ) α ( j i ) ) − ( t m − t k + 1 ) α i j + 1 E α ( j i ) , α ( j i + 2 ) ( − μ α ( j i ) ( t m − t k + 1 ) α ( j i ) ) = ∑ k = 1 m u ( i , t k + 1 ) − u ( i , t k − 1 ) 2 Δ t δ α i j * .

After making some necessary mathematical arguments in Eq. (19),

(20) D t α ( j i ) MABC u ( j i ) = B ( α ( j i ) ) ( 1 − α ( j i ) ) u ( j i ) − B ( α ( j i ) ) ( 1 − α ( j i ) ) × ∑ k = 1 m [ C k + 1 u ( ξ , t k + 1 ) + C k u ( ξ , t k ) + C k − 1 u ( ξ , t k − 1 ) ] + R k ,

where

C k + 1 = μ α ( j i ) 2 Δ t δ α i j * , C k = 2 Δ t μ α ( j i ) δ α i j , C k − 1 = − C k + 1 .

5.2 Space discreitization

(21) D ξ β ( j i ) MABC u ξ ξ ( j i ) = B ( β ( j i ) ) ( 1 − β ( j i ) ) u ′ ( j i ) − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) u ′ ( t ) − μ β ( j i ) ∫ 0 ξ ( ξ − p ) β ( j i ) − 1 E β ( j i ) , β ( j i ) ( − μ β ( j i ) ( ξ − p ) β ( j i ) ) u ′ ( p i , t j ) d p .

For simplification, we consider that

∫ 0 ξ ( ξ − p ) β ( j i ) − 1 E β ( j i ) , β ( j i ) ( − μ β ( j i ) ( ξ − p ) β ( j i ) ) u ( p i , t j ) d p = ∑ q = 0 n ∫ ( q − 1 ) Δ ξ ( q ) Δ ξ E β ( j i ) , β ( j i ) ( − μ β ( j i ) ( ξ − p ) β ( i j ) ) d p = ∑ q = 0 n u ′ ( ξ i − q , t j ) ( ξ n − ξ q ) E β ( j i ) , β ( j i + 1 ) ( − μ β ( j i ) ( ξ n − ξ q ) β ( j i ) ) − ( ξ n − ξ q + 1 ) E β ( j i ) , β ( j i + 1 ) ( − μ β ( j i ) ( ξ n − ξ q + 1 ) β ( j i ) ) = ∑ q = 0 n u ′ ( ξ i − q , t j ) δ β i j .

By substituting Eq. (13) into Eq. (21), we obtain

(22) D ξ β ( j i ) MABC u ξ ξ ( j i ) = B ( β ( j i ) ) ( 1 − β ( j i ) ) u i − 2 j + 1 − 8 u i − 1 j + 1 + 8 u i + 1 j + 1 − u i + 2 j + 1 12 Δ ξ − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) u ′ ( t ) − μ β i j B ( β ( j i ) ) ( 1 − β ( j i ) ) ∑ q = 0 n × u i − 2 − q j + 1 − 8 u i − 1 − q j + 1 + 8 u i + 1 − q j + 1 − u i + 2 − q j + 1 12 Δ ξ δ β i j + R q , 1 < β ( j i ) ≤ 2 .

For simplicity, we will write the system equation in the form:

(23) D ξ β ( j i ) MABC u ξ ξ ( j i ) = B ( β ( j i ) ) ( 1 − β ( j i ) ) [ − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) u ′ ( 0 , t ) ] + ( C q − 2 − C q − 1 + C q + 1 − C q + 2 ) + R q , 1 < β ( j i ) ≤ 2 ,

where

C q − 2 = B ( β i j ) ( 1 − β i j ) u i − 2 j + 1 12 Δ ξ − μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j u i − 2 − q j + 1 12 Δ ξ , C q − 1 = − B ( β i j ) ( 1 − β i j ) u i − 1 j + 1 12 Δ ξ + μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j u i − 1 − q j + 1 12 Δ ξ , C q + 1 = B ( β i j ) ( 1 − β i j ) u i + 1 j + 1 12 Δ ξ − μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j u i + 1 − q j + 1 12 Δ ξ , C q + 2 = B ( β i j ) ( 1 − β i j ) u i + 2 j + 1 12 Δ ξ − μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j u i + 2 − q j + 1 12 Δ ξ ,

HOCFD discretization for the velocity and temperature fields is obtained by substituting Eqs (20) and (23) into schemes (16) and (17) as follows:

(24) B ( α ( j i ) ) ( 1 − α ( j i ) ) u ( j i ) − B ( α ( j i ) ) ( 1 − α ( j i ) ) ∑ k = 1 m × [ C k + 1 u ( ξ , t k + 1 ) + C k u ( ξ , t k ) + C k − 1 u ( ξ , t k − 1 ) ] − a 6 B ( β ( j i ) ) ( 1 − β ( j i ) ) [ − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) u ′ ( 0 , t ) ] + ( C q − 2 − C q − 1 + C q + 1 − C q + 2 ) + a 7 u i j − a 8 T i j = R 1 , t > 0 , ξ ≥ 0

(25) B ( α ( j i ) ) ( 1 − α ( j i ) ) T ( j i ) − B ( α ( j i ) ) ( 1 − α ( j i ) ) ∑ k = 1 m × [ C k + 1 T ( ξ , t k + 1 ) + C k T ( ξ , t k ) + C k − 1 T ( ξ , t k − 1 ) ] − a 9 B ( β ( j i ) ) ( 1 − β ( j i ) ) [ − E β ( j i ) ( − μ β ( j i ) t β ( j i ) ) T ′ ( 0 , t ) ] + ( D q − 2 − D q − 1 + D q + 1 − D q + 2 ) ] = R 2 ,

where

D q − 2 = B ( β i j ) ( 1 − β i j ) T i − 2 j + 1 12 Δ ξ − μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j T i − 2 − q j + 1 12 Δ ξ , D q − 1 = − B ( β i j ) ( 1 − β i j ) T i − 1 j + 1 12 Δ ξ + μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j T i − 1 − q j + 1 12 Δ ξ , D q + 1 = B ( β i j ) ( 1 − β i j ) T i + 1 j + 1 12 Δ ξ − μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j T i + 1 − q j + 1 12 Δ ξ , D q + 2 = B ( β i j ) ( 1 − β i j ) T i + 2 j + 1 12 Δ ξ − μ β i j B ( β i j ) ( 1 − β i j ) ∑ q = 0 n δ β i j T i + 2 − q j + 1 12 Δ ξ .

5.3 Algorithm to solve (6) and (7)

This algorithm consists of the following steps:

Step 1. Let k = ( t max − t 1 ) m > 0 , m = 20 if 0 ≤ t 1 < t max = 1 ; start with the initial conditions ( u ( 0 , t ) = H ( t ) cos ( w t ) and T ( 0 , t ) = 1 ).
Step 2. Let h = ( l − ξ 1 ) n > 0 , n = 20 if 0 ≤ ξ 1 < l = 4 ; start with the boundary conditions ( u i ( ξ , 0 ) = 0 and T ( ξ , 0 ) = 0 ).
Step 3. Solve systems (20) and (23) using the MABC method (12) for m = n every step to obtain the stable state.
Step 4. Solve Steps 1, 2, and 3 using Matlab code and the fsolve system of nonlinear equations for 0 < α < 1 and 1 < β < 2 .
Step 5. Insert the mlf.m file to solve the Mittag–Leffler function in the MABC method in Step 4.
Step 6. The program will make loops for every iteration interval ( n , m ), comparing the numerical results of Step 4 with the exact solution at α = 1 and β = 2 .
Step 7. Output the current values as solutions for the values of the variables and physics constants for the tables and graphics.

6 Stability and error estimates

6.1 Stability analysis

The stability of schemes (24) and (25) was examined using a technique related to the Jon Von Neumann method [22] by considering R 1 = R 2 = 0 for systems (24) and (25), which can be written in the form

D t α i j MABC u t − a 6 D t β i j MABC u ξ ξ + a 7 u i j − a 8 T i j = 0 , D t α i j MABC T t − a 9 D t β i j MABC T ξ ξ = 0 , t > 0 , ξ ≥ 0 .

By writing this system in a matrix form as follows:

(26) Y 1 D t α i j MABC X t + Y 2 D t β i j MABC X ξ ξ + Y 3 X = 0 ,

X = u T , Y 1 = 1 0 0 1 , Y 2 = − a 6 0 0 − a 9 , Y 3 = a 7 − a 8 0 0 ,

system (26) can be formed as follows:

(27) Y 1 B ( α i j ) ( 1 − α i j ) X i j − μ α i j ∑ k = 1 m X i j − k δ α i j + Y 2 B ( β i j ) ( 1 − β i j ) X i − 2 j + 1 − 8 X i − 1 j + 1 + 8 X i + 1 j + 1 − X i + 2 j + 1 12 Δ ξ − μ β i j ∑ q = 1 n X i − 2 − q j + 1 − 8 X i − 1 − q j + 1 + 8 X i + 1 − q j + 1 − X i + 2 − q j + 1 12 Δ ξ δ β i j + Y 3 X i j = 0 .

By applying the mathematical required steps, system (27) will take the form,

(28) A 1 X i j − A 2 ∑ k = 1 m X i j − k δ α i j + A 3 X i − 2 j + 1 − A 4 X i − 1 j + 1 + A 5 X i + 1 j + 1 − A 6 X i + 2 j + 1 − A 7 ∑ q = 0 n X i − 2 − q j + 1 δ β i j + A 8 ∑ q = 0 n X i − 1 − q j + 1 δ β i j − A 9 ∑ q = 0 n X i + 1 − q j + 1 δ β i j + A 10 ∑ q = 0 n X i + 2 − q j + 1 δ β i j = 0 ,

where A d ( d = 1 , 2 , 3 , … , 10 ) are the constants, where k = 1 , 2 , … , m − 1 , and q = 0 , 1 , 2 , … , n − 1 ,

A 1 = Y 1 B ( α i j ) ( 1 − α i j ) + Y 3 , A 2 = Y 1 B ( α i j ) ( 1 − α i j ) μ α i j , A 3 = Y 2 B ( β i j ) ( 1 − β i j ) 1 12 Δ ξ , A 4 = Y 2 B ( β i j ) ( 1 − β i j ) 8 12 Δ ξ , A 5 = A 4 , A 6 = A 3 , A 7 = Y 2 B ( β i j ) ( 1 − β i j ) μ α i j 12 Δ ξ , A 8 = Y 2 B ( β i j ) ( 1 − β i j ) 8 μ α i j 12 Δ ξ , A 9 = A 8 , A 10 = A 7 .

Applying the von Neumann stability analysis by assuming that X i j = ℵ j e I ϒ k i into system (28), where I = − 1 and g ∈ R as follows:

A 1 ℵ j e I ϒ k i − A 2 ∑ k = 1 m ℵ j − k e I ϒ k i δ α i j + A 3 ℵ j + 1 e I ϒ k ( i − 2 ) − A 4 ℵ j + 1 e I ϒ k ( i − 1 ) + A 5 ℵ j + 1 e I ϒ k ( i + 1 ) − A 6 ℵ j + 1 e I ϒ k ( i + 2 ) + ∑ q = 0 n ( − A 7 ℵ j + 1 e I ϒ k ( i − 2 − q ) + A 8 ℵ j + 1 e I ϒ k ( i − 1 − q ) − A 9 ℵ j + 1 e I ϒ k ( i + 1 − q ) + A 10 ℵ j + 1 e I ϒ k ( i + 2 − q ) ) δ β i j = 0 .

Divide the deduced equation by ℵ j e I i ϒ k and put every ℵ j + 1 ⁄ ℵ j = η ; using the Euler formulas e I ϑ − e − I ϑ = 2 I sin ( ϑ ) [22] and making some necessary arrangements, we will have that

A 1 − A 2 ∑ k = 1 m ℵ − k δ α i j + A 3 η e − 2 I ϒ k − A 4 η e − I ϒ k + A 5 η e I ϒ k − A 6 η e 2 I ϒ k + ∑ q = 0 n ( − A 7 η e − I ϒ k ( 2 + q ) + A 8 η e − I ϒ ( 1 + q ) k − A 9 η e I ϒ k ( 1 − q ) + A 10 η e I ϒ k ( 2 − q ) ) δ β i j = 0 , η = A 1 − A 2 ∑ k = 1 m ℵ − k δ α i j − 2 I A 3 sin ( 2 ϒ k ) + 2 I A 4 sin ( ϒ k ) + ∑ q = 0 n ( 2 I A 7 sin 2 ϒ k − 2 I A 8 sin ϒ k ) δ β i j .

Therefore, given the condition, schemes (24) and (25) are stable if

∣ η ∣ ≤ 1 .

6.2 Truncation error

We will estimate the truncation error for the proposed numerical method in the following. From the definition of truncation error given by (24) and (25), the description of HOCFD scheme (13) also depending on Taylor series expansion for variable time and space, we can claim the following:

u i j + k = u ( ξ i , t j + Δ t ) = u i j + ( Δ t ) ∂ u ∂ t ( i , j ) + 1 2 ( Δ t ) 2 ∂ 2 u ∂ t 2 ( i , j ) + 1 6 ( Δ t ) 3 ∂ 3 u ∂ t 3 ( i , j ) + … ,

u i − 2 j + 1 = u ( ξ i − 2 Δ ξ , t j + Δ t ) = u i j + 1 − ( 2 Δ ξ ) ∂ u ∂ ξ ( i , j + 1 ) + 1 2 ( 2 Δ ξ ) 2 ∂ 2 u ∂ ξ 2 ( i , j + 1 ) − 1 6 ( 2 Δ ξ ) 3 × ∂ 3 u ∂ ξ 3 ( i , j + 1 ) + 1 24 ( 2 Δ ξ ) 4 ∂ 4 u ∂ ξ 4 ( i , j + 1 ) − 1 120 ( 2 Δ ξ ) 5 ∂ 5 u ∂ ξ 5 ( i , j + 1 ) + … ,

u i + 2 j + 1 = u ( ξ i + 2 Δ ξ , t j + Δ t ) = u i j + 1 + ( 2 Δ ξ ) ∂ u ∂ ξ ( i , j + 1 ) + 1 2 ( 2 Δ ξ ) 2 ∂ 2 u ∂ ξ 2 ( i , j + 1 ) + 1 6 ( 2 Δ ξ ) 3 × ∂ 3 u ∂ ξ 3 ( i , j + 1 ) + 1 24 ( 2 Δ ξ ) 4 ∂ 4 u ∂ ξ 4 ( i , j + 1 ) + 1 120 ( 2 Δ ξ ) 5 ∂ 5 u ∂ ξ 5 ( i , j + 1 ) + … ,

u i − 1 j + 1 = u ( ξ i − Δ ξ , t j + Δ t ) = u i j + 1 − ( Δ ξ ) ∂ u ∂ ξ ( i , j + 1 ) + 1 2 ( Δ ξ ) 2 ∂ 2 u ∂ ξ 2 ( i , j + 1 ) − 1 6 ( Δ ξ ) 3 × ∂ 3 u ∂ ξ 3 ( i , j + 1 ) + 1 24 ( Δ ξ ) 4 ∂ 4 u ∂ ξ 4 ( i , j + 1 ) − 1 120 ( Δ ξ ) 5 ∂ 5 u ∂ ξ 5 ( i , j + 1 ) + … ,

u i + 1 j + 1 = u ( ξ i + Δ ξ , t j + Δ t ) = u i j + 1 + ( Δ ξ ) ∂ u ∂ ξ ( i , j + 1 ) + 1 2 ( Δ ξ ) 2 ∂ 2 u ∂ ξ 2 ( i , j + 1 ) + 1 6

(29) ( Δ ξ ) 3 ∂ 3 u ∂ ξ 3 ( i , j + 1 ) + 1 24 ( Δ ξ ) 4 ∂ 4 u ∂ ξ 4 ( i , j + 1 ) + 1 120 ( Δ ξ ) 5 ∂ 5 u ∂ ξ 5 ( i , j + 1 ) + … ,

then create an expansion for the same Taylor series for u i − 2 − q j + 1 , u i − 1 − q j + 1 , u i + 1 − q j + 1 , u i + 2 − q j + 1 , and create the same for temperature field equation T ( ξ , t ) . Enter all in Eqs (24) and (25), and we can obtain the following,

(30) R 1 i j = O ( Δ t ) 2 + S ( Δ ξ ) 4 , R 2 i j = O ( Δ t ) 2 + S ( Δ ξ ) 4 .

7 Numerical discussions

Numerous numerical examples of the CNT model are taken for this part as a variable-order with different values is necessary to demonstrate the viability of the proposed model. We can show that the variable-order for considering the stated model is genuine by assessing the impact of different flow parameters ( α ( ξ , t ) , β ( ξ , t ) , ϕ , γ , M , K ). With the use of all the aforementioned variables, the model for SWCNTs and MWCNTs has been demonstrated, and they can be easily differentiated in several graphs displaying the velocity and temperature profiles for the proposed model. After solving the SWCNTs and MWCNTs numerically and using the MABC fractional operator to manage the variable-order, the numerical results were compared to the exact answer and the accuracy of this model is demonstrated by comparing the findings to those of Khalid et al. [16], Dimbeswar et al. [19], and Sweilam et al. [22] for the velocity field.

The results in Figure 2 show the HOCFD method results of the numerical solutions for the velocity field for SWCNTs and MWCNTs where α ( ξ , t ) = β ( ξ , t ) = e ( ( − ξ t ) ⁄ 300 ) .
Figure 3 shows up the FDM and variable-order results for SWCNTs and MWCNTs velocity fields where α ( ξ , t ) = β ( ξ , t ) = e ( ( − ξ t ) ⁄ 300 ) .
Figure 4 shows up the VOFDM and HOCFD method results for SWCNTs and MWCNTs velocity fields where α ( ξ , t ) = 0.5 + 0.5 e ( − ( t . 2 ) − 1 ) and β ( ξ , t ) = 0.97 − 0.5 ( cos ( ξ ) sin ( t ) ) 2 .
Figure 5 shows up the VOFDM and HOCFD method results for SWCNTs and MWCNTs velocity fields where α ( ξ , t ) = 0.05 + 0.04 cos ( p i ( ξ t ) ) + 0.01 t and β ( ξ , t ) = 0.01 * exp ( − t ξ ) .
Figure 6 shows up the behaviour of velocity field for SWCNT and MWCNT using the VOFDM and HOCFD methods where α ( ξ , t ) = exp ( ( − ξ t ) ⁄ 300 ) and β ( ξ , t ) = 0.01 * exp ( − t ξ ) .
Figure 7 shows the errors that were analyzed for SWCNT and MWCNT using the FDM and HOCFD methods where α ( ξ , t ) = e ( ( − ξ t ) ⁄ 300 ) and β ( ξ , t ) = 0.01 * exp ( − t ξ ) .
Table 2 discuss the errors for FDM and HOCFD method results for SWCNT and MWCNT velocity fields where α ( ξ , t ) = β ( ξ , t ) = 0.99876 , e ( ( − ξ t ) ⁄ 300 ) , 0.97 − 0.5 ( ( cos ( ξ ) sin ( t ) ) 2 ) , 0.5 + 0.5 e ( − ( t 2 ) − 1 ) .
Table 3 shows the errors for FDM and HOCFD method results for SWCNT and MWCNT velocity fields, where α ( ξ , t ) = 0.99876 , e ( ( − ξ t ) ⁄ 300 ) , 0.97 − 0.5 ( ( cos ( ξ ) sin ( t ) ) 2 ) , α = e ( ( − ξ t ) ⁄ 300 ) and β ( ξ , t ) = 0.99876 , e ( ( − ξ t ) ⁄ 300 ) , 0.97 − 0.5 ( ( cos ( ξ ) sin ( t ) ) 2 ) , 0.5 + 0.5 e ( − ( t 2 ) − 1 ) .
Table 4 shows a comparison of the numerical scheme (24) with the exact solution for SWCNTs at α ( ξ , t ) = 1 and the maximum error at different functions α ( ξ , t ) = e ( ( − ξ t ) ⁄ 300 ) , α ( ξ , t ) = 0.01 exp ( − t ξ ) , α ( ξ , t ) = 0.5 + 0.5 exp ( − ( t 2 ) − 1 ) , α ( ξ , t ) = 0.05 + 0.04 cos ( π ξ t ) + 0.01 , and α ( ξ , t ) = 0.97 − 0.5 ( cos ( ξ ) sin ( t ) ) 2 .
Table 5 shows the numerical results for MWCNTs at a l p h a ( ξ , t ) = 1 and the maximum error at the variable-order α ( ξ , t ) = e ( ( − ξ t ) ⁄ 300 ) , α ( ξ , t ) = 0.01 exp ( − t ξ ) , α ( ξ , t ) = 0.5 + 0.5 exp ( − ( t 2 ) − 1 ) , α ( ξ , t ) = 0.05 + 0.04 cos ( π ξ t ) + 0.01 , and α ( ξ , t ) = 0.97 − 0.5 ( cos ( ξ ) sin ( t ) ) 2 .
Table 6 shows the comparison of the numerical results between the present work and similar works at Q = 0.05 , Gr = 0 , 5, and γ = 0.5 , 1 .

$Figure 2 Behaviour of the stable solutions for the velocity field of CNTs using the VOFDM and HOCFD methods at the variable-orders α ( ξ , t ) = β ( ξ , t ) = e ( ( ‒ ξ t ) ⁄ 300 ) \alpha \left(\xi ,t)=\beta \left(\xi ,t)={e}^{\left(\left(‒\xi t)/300)} .$

Figure 2

Behaviour of the stable solutions for the velocity field of CNTs using the VOFDM and HOCFD methods at the variable-orders α ( ξ , t ) = β ( ξ , t ) = e ( ( ‒ ξ t ) ⁄ 300 ) .

$Figure 3 Solutions of velocity field for SWCNTs and MWCNTs at t = 1 t=1 , M = 0.5 M=0.5 , γ = 1 , ϕ = 0.7 \gamma =1,\phi =0.7 , Gr = 0, Pr = 25, K = 0.3 K=0.3 , and B = 0.5 B=0.5 , where α ( ξ , t ) = β ( ξ , t ) = e ( ( ‒ ξ t ) ⁄ 300 ) \alpha \left(\xi ,t)=\beta \left(\xi ,t)={e}^{\left(\left(‒\xi t)/300)} .$

Figure 3

Solutions of velocity field for SWCNTs and MWCNTs at t = 1 , M = 0.5 , γ = 1 , ϕ = 0.7 , Gr = 0, Pr = 25, K = 0.3 , and B = 0.5 , where α ( ξ , t ) = β ( ξ , t ) = e ( ( ‒ ξ t ) ⁄ 300 ) .

$Figure 4 Solutions of velocity field for SWCNTs and MWCNTs using the VOFDM and HOCFD methods at t = 1 t=1 , M = 0.5 M=0.5 , γ = 0.05 , 1 , ϕ = 0.2 , 0.5 \gamma =0.05,1,\phi =0.2,0.5 , Gr = 0,0.1, Pr = 25, K = 0.5 K=0.5 , 2, and B = 0.5 B=0.5 , where α ( ξ , t ) = 0.5 + 0.5 e ( ‒ ( t . 2 ) ‒ 1 ) \alpha \left(\xi ,t)=0.5+0.5{e}^{\left(‒\left(t{.}^{2})‒1)} and β ( ξ , t ) = 0.97 ‒ 0.5 ( cos ( ξ ) s i n ( t ) ) 2 \beta \left(\xi ,t)=0.97‒0.5{\left(\cos \left(\xi )sin\left(t))}^{2} .$

Figure 4

Solutions of velocity field for SWCNTs and MWCNTs using the VOFDM and HOCFD methods at t = 1 , M = 0.5 , γ = 0.05 , 1 , ϕ = 0.2 , 0.5 , Gr = 0,0.1, Pr = 25, K = 0.5 , 2, and B = 0.5 , where α ( ξ , t ) = 0.5 + 0.5 e ( ‒ ( t . 2 ) ‒ 1 ) and β ( ξ , t ) = 0.97 ‒ 0.5 ( cos ( ξ ) s i n ( t ) ) 2 .

$Figure 5 Solutions of velocity field for SWCNTs and MWCNTs using the VOFDM and HOCFD methods at t = 1 t=1 , M = 0.05 M=0.05 , γ = 0.5 \gamma =0.5 , ϕ = 0.5 \phi =0.5 , Gr = 0.1,0.5, Pr = 21, 25, K = 0.5 K=0.5 , 5, and B = 0.5 B=0.5 , where α ( ξ , t ) = 0.05 + 0.04 cos ( π ( ξ t ) ) + 0.01 t \alpha \left(\xi ,t)=0.05+0.04\cos \left(\pi \left(\xi t))+0.01t and β ( ξ , t ) = 0.01 * exp ( ‒ t ξ ) \beta \left(\xi ,t)=0.01* \exp \left(‒t\xi ) .$

Figure 5

Solutions of velocity field for SWCNTs and MWCNTs using the VOFDM and HOCFD methods at t = 1 , M = 0.05 , γ = 0.5 , ϕ = 0.5 , Gr = 0.1,0.5, Pr = 21, 25, K = 0.5 , 5, and B = 0.5 , where α ( ξ , t ) = 0.05 + 0.04 cos ( π ( ξ t ) ) + 0.01 t and β ( ξ , t ) = 0.01 * exp ( ‒ t ξ ) .

$Figure 6 Behaviour of velocity field for SWCNTs and MWCNTs using the VOFDM and HOCFD methods at t = 1 t=1 , M = 0.5 M=0.5 , 2.5, γ = 0.05 \gamma =0.05 , 0.1, ϕ = 0.5 \phi =0.5 , Gr = 0.7, Pr = 21, 25, K = 0.4 K=0.4 , 3 and B = 0.5 B=0.5 , where α ( ξ , t ) = exp ( ( ‒ ξ t ) ⁄ 300 ) \alpha \left(\xi ,t)=\exp \left(\left(‒\xi t)/300) and β ( ξ , t ) = 0.01 * exp ( ‒ t ξ ) \beta \left(\xi ,t)=0.01* \exp \left(‒t\xi ) .$

Figure 6

Behaviour of velocity field for SWCNTs and MWCNTs using the VOFDM and HOCFD methods at t = 1 , M = 0.5 , 2.5, γ = 0.05 , 0.1, ϕ = 0.5 , Gr = 0.7, Pr = 21, 25, K = 0.4 , 3 and B = 0.5 , where α ( ξ , t ) = exp ( ( ‒ ξ t ) ⁄ 300 ) and β ( ξ , t ) = 0.01 * exp ( ‒ t ξ ) .

$Figure 7 Error analysis for SWCNTs and MWCNTs using FDM and HOCFD methods at γ = 0.5 , ϕ = 0.5 \gamma =0.5,\phi =0.5 , Gr = 0.5, Pr = 21, K = 0.5 K=0.5 , and B = 0.5 B=0.5 , where α ( ξ , t ) = exp ( ( ‒ ξ t ) ⁄ 300 ) \alpha \left(\xi ,t)=\exp \left(\left(‒\xi t)/300) and β ( ξ , t ) = 0.01 * exp ( ‒ t ξ ) \beta \left(\xi ,t)=0.01* \exp \left(‒t\xi ) .$

Figure 7

Error analysis for SWCNTs and MWCNTs using FDM and HOCFD methods at γ = 0.5 , ϕ = 0.5 , Gr = 0.5, Pr = 21, K = 0.5 , and B = 0.5 , where α ( ξ , t ) = exp ( ( ‒ ξ t ) ⁄ 300 ) and β ( ξ , t ) = 0.01 * exp ( ‒ t ξ ) .

Table 2

Define the errors for SWCNTs using FDM and HOCFD method at ξ = 0.01 ‒ 2 , t = 0.5 , M = 0.5 , γ = 0.05 , Q = 0.5 , Gr = 0.1, Pr = 25, B = 0.5

	FD	HOCFD
t = 0.5 ,	α = 0.99876	α ( ξ , t ) = 0.5 + 0.5 e ( ‒ ( t 2 ) ‒ 1 )
ξ =	β = 0.99876	β ( ξ , t ) = 0.5 + 0.5 e ( ‒ ( t 2 ) ‒ 1 )
0.01	1.737 × 1 0 ‒ 2	3.386 × 1 0 ‒ 3
0.2	4.373 × 1 0 ‒ 3	7.737 × 1 0 ‒ 4
0.4	1.101 × 1 0 ‒ 3	1.768 × 1 0 ‒ 4
0.6	2.771 × 1 0 ‒ 4	4.040 × 1 0 ‒ 5
0.8	6.977 × 1 0 ‒ 5	9.231 × 1 0 ‒ 6
1	1.756 × 1 0 ‒ 5	2.109 × 1 0 ‒ 6
1.2	4.421 × 1 0 ‒ 6	4.820 × 1 0 ‒ 7
1.4	1.113 × 1 0 ‒ 6	1.101 × 1 0 ‒ 7
1.6	2.802 × 1 0 ‒ 7	2.517 × 1 0 ‒ 8
1.8	7.054 × 1 0 ‒ 8	5.751 × 1 0 ‒ 9
2	1.775 × 1 0 ‒ 8	1.314 × 1 0 ‒ 9

Table 3

Define the errors for MWCNTs using FDM and HOCFD method at ξ = 0.01 ‒ 2 , t = 0.5 , M = 0.5 , γ = 1 , Q = 0.2 , Gr = 0, Pr = 25, K = 0.5 , B = 0.5

	FD	HOCFD
t = 0.5 ,	α = 0.99876	α ( ξ , t ) = e ( ( ‒ ξ t ) ⁄ 300 )
ξ =	β = 0.99876	β ( ξ , t ) = 0.5 + 0.5 e ( ‒ ( t 2 ) ‒ 1 )
0.01	1.587 × 1 0 ‒ 2	1.403 × 1 0 ‒ 2
0.2	3.978 × 1 0 ‒ 3	3.665 × 1 0 ‒ 3
0.4	1.085 × 1 0 ‒ 3	9.532 × 1 0 ‒ 4
0.6	2.959 × 1 0 ‒ 4	2.468 × 1 0 ‒ 4
0.8	8.073 × 1 0 ‒ 5	6.368 × 1 0 ‒ 5
1	2.202 × 1 0 ‒ 5	1.636 × 1 0 ‒ 5
1.2	6.006 × 1 0 ‒ 6	4.188 × 1 0 ‒ 6
1.4	1.638 × 1 0 ‒ 6	2.714 × 1 0 ‒ 7
1.6	4.468 × 1 0 ‒ 7	6.873 × 1 0 ‒ 8
1.8	1.218 × 1 0 ‒ 7	1.734 × 1 0 ‒ 8
2	3.324 × 1 0 ‒ 8	4.363 × 1 0 ‒ 9

Table 4

Comparison with exact solution at the variable-order α ( ξ , t ) = 1 ?A3B2 tlsb -.1pt?>and the maximum error at α ( ξ , t ) where M = 0.5 , γ = 0.05 , Q = 0.5 , Gr = 0.1, Pr = 25, B = 0.5

α ( ξ , t ) at t = 0.6 , y = 2.4	SWCNTs
1	1.7458 × 1 0 ‒ 2
0.98765	1.7374 × 1 0 ‒ 2
0.97 ‒ 0.5 ( cos ( ξ ) sin ( t ) ) 2	1.7024 × 1 0 ‒ 2
0.05 + 0.04 cos ( π ( ξ t ) ) + 0.01	1.3961 × 1 0 ‒ 2
0.5 + 0.5 exp ( ‒ ( t 2 ) ‒ 1 )	1.4817 × 1 0 ‒ 2
exp ( ( ‒ ξ t ) ⁄ 300 )	1.6326 × 1 0 ‒ 2
0.01 exp ( ‒ t ξ )	1.3928 × 1 0 ‒ 2

Table 5

Comparison with exact solution at α ( ξ , t ) = 1 and the maximum error at α ( ξ , t ) where M = 0.5 , γ = 0.5 , Q = 0.5 , Gr = 0, Pr = 25, B = 0.7

α ( ξ , t ) at t = 0.6 , y = 2.4	MWCNTs
1	1.9752 × 1 0 ‒ 2
0.98765	1.97003 × 1 0 ‒ 2
0.97 ‒ 0.5 ( cos ( ξ ) sin ( t ) ) 2	1.9478 × 1 0 ‒ 2
0.05 + 0.04 cos ( π ( ξ t ) ) + 0.01	1.7139 × 1 0 ‒ 2
0.5 + 0.5 exp ( ‒ ( t 2 ) ‒ 1 )	1.7853 × 1 0 ‒ 2
exp ( ( ‒ ξ t ) ⁄ 300 )	1.8950 × 1 0 ‒ 2
0.01 exp ( ‒ t ξ )	1.7088 × 1 0 ‒ 2

Table 6

Velocity field for SWCNTs human blood model at M = 0.5 , Pr = 25, K = 1 , and t = 1

ξ	Khalid et al. [16]	Dimbeswar et al. [19]	Sweilam et al. [22]	Present work
	ϕ = 0.05 , Gr = 0 ,	ϕ = 0.05 , Gr = 5 ,	ϕ = 0.05 , Gr = 0 , γ = 1	ϕ = Q = 0.05 , Gr = 0
0.0	0	0	0	0
0.2	0.49671	1.33792	0.18033	0.31299
0.4	0.65312	1.67091	0.13376	6.0549 × 1 0 ‒ 2
0.6	0.56651	1.39606	8.8443 × 1 0 ‒ 2	1.1167 × 1 0 ‒ 2
0.8	0.32218	0.77614	4.4049 × 1 0 ‒ 2	1.9699 × 1 0 ‒ 3
1	0	0	0	0

8 Conclusions

In this study, we successfully developed the numerical approximation for the MABC operator for the proposed CNT model. The model of CNT nanofluid flow across a vertical plate was fractionalized using the MABC operator. The mentioned model was very effective, and the velocity field numerical data were derived using the variable-order approach. Numerical results, which were displayed in a variety of graphs and tables, validated theoretical analyses such as variable-order fractional derivatives and higher-order finite difference approximations. The following are the key conclusions:

The thermal buoyancy force’s proportionate impact on the viscous hydrodynamic force in the boundary layer is shown by the Grashof number. As the Grashof number rises, we see an increase in the blood flow velocity as a result of the thermal buoyancy force becoming stronger.
Notes that the effects of nanoparticle volume fraction Q and magnetic number ( M ) on the velocity field. The resistive behaviour of M causes the flow velocity to slow down because, when blood flows in the presence of a magnetic field, the action of magnetization causes the charged particles in the blood flow to rotate.
The velocity is accelerated for Gr and Q , whereas MWCNTs are more dominant over SWCNTs as shown in the previous tables.
Using the John Von Neumann stability analysis approach, the constancy analysis of the referred-to variable-order was put to the test.
The results in the tables and the numerical figures display that the schemes attained from applying the submitted numerical methods are completely compatible with the exact solution.
We can apply the MABC operator to an enormous ambit of problems defined encountered in technology and science.
The comparison in table shows that there is great agreement among the investigations regarding the velocity of human blood in CNTs.
Truncation error was calculated.

Funding information: This research received no external funding.
Author contributions: Formal analysis, software, writing, K.-R.K.; manuscript administration, N.-H.S.; writing-review editing, N.-H.S., W.-A.K. and Z.-M.A. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2023-05-18

Revised: 2024-06-26

Accepted: 2024-08-19

Published Online: 2024-11-28

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0029

Keywords for this article

carbon nanotube model; space–time variable-order fractional model; compact finite difference method; modified Atangana–Baleanu–Caputo derivative

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