A Fourth-Order Compact Finite Difference Scheme for Solving the Time Fractional Carbon Nanotubes Model

In this work, we deal with unsteady magnetohydrodynamic allowed convection inflow of blood with a carbon nanotubes model; the single and multiwalled carbon nanotubes of human blood are used as a based fluid. Two numerical methods used to study this model are the weighted average finite difference method and the nonstandard compact finite difference method. The proportional Caputo hybrid operator has been used to fractionalize the proposed model. Stability analysis has been construed by a kind of John von Neumann stability analysis. Numerical results are presented in diverse graphs, which manifest that the method is successful in solving the proposed model.


Introduction
Fractional calculus (FC) is a generalization of the integerorder calculus. In fractional calculus, researchers try to solve problems with α-order derivatives and integrals, where there are several definitions for derivatives of order α [1,2]. e most common derivatives are the Riemann-Liouville [3], Caputo [3], Caputo-Fabrizio [4], and the proportional Caputo hybrid [5] formulations. In scopes of fluid dynamic and engineering, most nanofluids waft issues are typically nonlinear character, and it is believed that the fractionalorder methods are the best suited models to act for such studies comparatively different conventional methods [6].
Oberlin et al. [7] were the first to initiate the carbon nanotubes (CNTs) as nanoparticles in 1976. CNTs are one of the nanomaterials that are vastly used in such parts in the last years. ey have got more attention because of their unrivalled advantages [8]. CNTs have extraordinary conductivity which helps them to form a network of conductive tubes. CNTs have also been used for thermal defence as thermal boundary materials. In 1995, a novel magnificence of warmth transferring fluids that may be engineered via placing metal nanoparticles in conventional heat transfer fluids became initiated by Choi [9]. With the expansion of nanotechnology, many nanomaterials were developed and utilized in the industry. Khan et al. [10] discussed the slip waft of Eyring-Powell nanoliquid film containing graphene nanoparticles. 3D nanofluid waft with heat and mass transferring analysis is over a linear straight floor with convective boundary conditions. Khalid et al. [11] investigated a case of effects of MHD human blood going with the flow in porosity of the waist CNTs and thermal evaluation. e case is stated and solved for an analytical solution by the usage of the Laplace transform method. Khan [12] investigated the Atangana-Baleanu fractional derivative to blood flow in nanofluids possessing without local and without singular kernels, and it was utilized to blood of nanofluid. Meyer et al. [13] discussed the convective warmth transferring fecundity experimentally of watery deferrals for the multiple-walled CNTs (MWCNTs) flowing horizontal straight tube.
Wang et al. [14,15] have paid vital interest to the CNTs with various consequences together with heat transfer, thermal conductivity, thermal radiation, the porosity of the medium, and so forth. Qureshi et al. [16] discussed a fractional model for the concentration system of blood ethanol with real data application, where they used the Atangana-Baleanu-Caputo and the Caputo-Fabrizio fractional operators. Saqib et al. [17] solved the fractional derivative nonsingular and local kernel to enhance heat transfer in CNT nanofluid over a sloping plate, and the exact solution expressed analytically for velocity and temperature profiles by the Laplace transformation technique.
Kalita et al. [18] 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, singlewalled CNT (SWCNT) and MWCNT were replaced with blood as a based fluid. e flow problem and its time-fractional form are given in Section 2. Preliminaries of the fractional derivatives definitions are given in Section 3. Moreover, the nonstandard finite difference method (NSFDM) and the nonstandard compact finite difference method (NSCFDM) are given in Section 4. We developed the weighted average nonstandard compact finite difference method (WANSCFDM) for the nanofluid CNTs equations in Section 5. Stability analyses of these schemes are given in Section 6. Numerical solutions for the nanofluid CNTs equations are graphically reported in Section 7. Finally, the conclusions are given in Section 8.

The Flow Problem and Its Fractional Form
Consider the unstable transportation inflow of human blood CNT-based nanofluid through a columnar platelet with isothermal heat T inf (ambient heat). e nanofluid is supposed to be with electric carrying with a regular magnetic domain B with intensity B 0 utilized in the way vertical to the laminar [11]. e half-space laminar is included in the porosity of the medium satiated with human blood as base nanofluids containing both SWCNT and MWCNT. Let the based fluid and CNTs be in thermal balance, and no slip happens between them; at first, at time t � 0, the nanofluid and the lamina are stable with constant heat T inf . As in [11], after a small interval of time t > 0, the plate vibrates with V � U 0 H(t)cos(Qt), and the ambient field of temperature of the plate T inf rises to T w . e temperature and velocity fields equations are with the following boundary and initial conditions: where where ϕ r , (r � 1, 2, . . . , 5) are the constant terms: 2 e Scientific World Journal e time-fractional forms of equations (1) and (2) are given: where CPC D α t (ξ, t) is the constant proportional Caputo timefractional operator [19], for the same boundary and initial condition (3), where A, C, F, and L are the constants: e following analytical solutions are a special case for the temperature and velocity fields by taking Qt � 0 which agree with the impulsive motion of the plate [11], e parameters that appeared in the model are given in Table 1.
e function erfc is a complementary error function; it is widely used in statistical computations, for instance, where it is also known as the standard normal cumulative probability. e complementary error function is defined as [20]

Basic Preliminaries
We are going to reminiscence several important definitions for fractional derivatives. Caputo's fractional-order derivative for 0 < α < 1 and Γ be the Euler gamma function of a differentiable function f(t) and is defined as follows [3]: e Riemann-Liouville integral, where α > 0 and f(t) is an integrable function, is defined by [3] as e hybrid fractional operator is a new fractional operator that is defined by combining the proportional e Scientific World Journal 3 definition, Caputo, and Riemann-Liouville definitions (Baleanu et al. [19]): where K 0 (α), K 1 (α) are the constants ( [21]) defined with respect to time and depending only on the parameter α. As in [19], consider the kernels as follows:

NSFDM.
In this part, we introduce several comments related to the NSFDM, first proposed by Mickens [22]. e derivative term of the forward method du/dt is substituted by u(t + k) − u(t)/k, where k � Δt is the step size and k ⟶ 0; in the Mickens schemes, this term is substituted by is a continuous function of step size k, where the function φ(k) satisfies the following conditions: In the centered method, the derivative term Δξ is the step size; in the schemes of Mickens, this term is . Let Z and N are the two positive integers, the mesh points have the coordinates ξ i+1 � ξ i + h, (i � 0, 1, . . . , N) and t j+1 � t j + k, (j � 0, 1, . . . , Z), and the values of the solution u(ξ, t) on these grid points are e forward NSFD formula for the first order of the time and the centered NSFD formula for the second order of space will be for more details ( [14]).

NSCFDM.
e expansion of Taylor is considered a very helpful tool for the derivation of higher-order approximation to derivatives of all orders. Our advantage in this work is A better approximation can be gained by combining these two assessments using the process called Richardson extrapolation. We will deduce that the fourth-order centered nonstandard finite difference scheme for the second derivative will be

WANSCFDM
Now, we will use the WANSCFD scheme to obtain the discretization formulas for the temperature and velocity equations. For getting the discretization formulas of equations (4) and (5), we need to substitute the WANSCFD method of the centered formula for space (20) into equations (6) and (7), where ω is the weighting factor: Equations (13) and (14) are the WANSCFD schemes for the temperature and velocity fields. At the case of ω � 1, we have the forward Euler fractional quadrature scheme, and if we put ω � 1/2, we get Crank-Nicholson fractional quadrature scheme, but at ω � 0, the scheme is called totally implicit, which have been studied, e.g., in [23].
Our aim in the current study is to introduce numerical solutions of time-order fractional for equations (6) and (7) with the new derivative operator (hybrid operator) (Baleanu et al. [24]), which is discretized as follows: Here, CPC stands for constant proportional Caputo derivative [5]. e discretization of time-order fractional for the Riemann-Liouville operator is given by where e Scientific World Journal e fraction [t j /k] means the integer part of t j /k and the parameter p represents the order of approximation which are dependent on the choice of W (1− α) k . e above expression is not the only one because there are different expressions of the weights W (α) k [25]. e coefficients W (α) k can be evaluated by the recursive formula: where the discretization of time-fractional order of Caputo derivative is given by where 0 < α < 1; by substituting equations (25), (26), and (28) into equations (21) and (22), we will get the time-order fractional discretization of hybrid derivative for temperature and velocity equations: where T j i is the truncating error. More details for discretization in fractional calculus can be found in previous studies such as [24,25].

Stability Analysis
To check the stability of schemes (29) and (30), we applying a kind of the Jon von Neumann method [22,24] by considering systems (4) and (5) can be written in the following form: Writing this system in a matrix form is as follows: 6 e Scientific World Journal and the above system (31) can be formed using the WANSCFD method as follows: Applying the mathematical required steps system (32) will take the following form: where , are the constants, where k � 1, 2, . . . , m − 1. Applying the von Neumann stability analysis by assuming that X j i � χ j e niqk into system (33), where n � �� � − 1 √ and q ∈ R, divide the deduced equation by χ j e niqk and put every By using the Euler formulas e nϑ � cos(ϑ) + nsin(ϑ) and e − nϑ � cos(ϑ) − nsin(ϑ) and making some necessary arrangements, we will have that e mode will be stable as long as ‖η‖ ≤ 1.
e Scientific World Journal 7

Results and Discussion
To clarify the performance of the proposed method for solving the suggested model, we will study the effects of various flow parameters (α, ϕ, c, M, K, Pr, and Gr) that are distinguished in multifigures identifying the temperature and velocity profiles for blood. e influences of all the above parameters are displayed for human blood (SWCNTs and MWCNTs); the Pr is taken 21 and 25, respectively.
(i) e desirable results in Figure 1 show the behavior of the stable and unstable solutions for the velocity field (22) of CNTs using the WANSCFDM (ii) e results in Figure 2 can be carried out from different values for ϕ into temperature, and velocity profiles (22) and (23) are reported for SWCNTs and MWCNTs; it is obtained from the time-fractional type and WANSCFD schemes discussed above at α � 0.5. of ω, such that sometimes in using the WASCFD M, we get best results from the Crank-Nicholson fractional scheme at ω � 0.5, but for the WANS CFD method, we get the best results from the

Conclusions
e proportional Caputo hybrid operator is used to fractionalize the model of the nanofluid flow of human blood CNTs over a vertical plate. e effects of the magnetic area and the porosity medium are taken into consideration. e numerical results for temperature and velocity fields are calculated by the method of WANSCFD. Numerical results are presented in diverse graphs and mentioned with physical reasonings, and all computations had been run with the use of Matlab programming. e main findings extracted are as follows [27][28][29]: (i) e velocity of nanofluid decreases with the increase in ϕ, magnetic parameter, and permeability parameter (ii) ere is an inverse relationship between the volume fraction parameter, magnetic parameter, and Casson fluid parameter (iii) e Casson nanofluid flow has the same influence on temperature and velocity profiles for both single and multiwalled CNTs (iv) CPC fractional derivative model can be qualified to solve the biological properties than the integer order model (v) e numerical methods, highly accurate WANSFD, WASCFD, and WANSCFD, are used to study the presented model as shown in tables, so we can conclude by comparative results that the WANSCFDM was more accurate. (vi) e stability analysis of the proposed WANSCFDM is construed by a kind of the standard John von Neumann stability analysis technique (vii) e numerical solutions in this study are in good agreement with the exact solutions

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.