Abstract
In this paper, we investigate the implementations of newly introduced nonlocal differential operators as convolution of power law, exponential decay law, and the generalized Mittag-Leffler law with fractal derivative in fluid dynamics. The new operators are referred as fractal-fractional differential operators. The governing equations for the problem are constructed with the fractal-fractional differential operators. We present the stability analysis and the error analysis.
1. Introduction
Magnetohydrodynamics (MHD) deals with the study of the motion of electrically conducting fluids in the presence of the magnetic field. MHD flow has significant importance applications between infinite parallel plates in various areas such as geophysical, astrophysical, and metallurgical processing, MHD generators, pumps, geothermal reservoirs, polymer technology, and mineral industries [1–6]. In last few decades, fractional calculus has taken much interest in many fields [7, 8]. There are many definitions for the fractional derivative operators, and among them are Caputo-Fabrizio (CF) [9] and Atangana and Baleanu (AB) [10] definitions of fractional derivatives with a nonlocal and nonsingular kernels having all the characteristics of the old definitions [7, 11–23]. Farman et al. [24] have analyzed the numerical solution of SEIR Epidemic model of measles with noninteger time fractional derivatives by using the Laplace Adomian decomposition method. Ghanbari and Djilali [25] have taken mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population. Ghanbari and Atangana [26, 27] have given the new edge detecting techniques based on fractional derivatives with nonlocal and nonsingular kernels. Recently, another idea of differentiation has been proposed by Atanagna [28].
We organize our manuscript as follows. We present the main definitions in Section 2. We construct the problem formulation in Section 3. We present the analysis of the model with the power law kernel in Section 4. We give the analysis of the model with the exponential decay kernel in Section 5. We discuss the analysis of the model with the Mittag-Leffler kernel in Section 6. We present the error analysis in Section 7. We give the conclusion in the last section.
2. Preminaries
Definition 1. Assume that is a continuous function in the and fractal differentiable on with order , then the fractal-fractional derivative of of order in Riemann-Liouville sense with power law kernel is introduced as [29] where
Definition 2. Assume that is a continuous function in the and fractal differentiable on with order , then the fractal-fractional derivative of of order in Riemann-Liouville sense with the exponential decay kernel is introduced as [29]
Definition 3. Assume that is a continuous function in the and fractal differentiable on with order , then the fractal-fractional derivative of of order in Riemann-Liouville sense with the generalized Mittag-Leffler kernel is introduced as [29]
3. Problem Formulation
We consider into Eqs. (5)-(8), and we obtain where is the magnetic field parameter, and is the Reynold number and We demonstrate the geometry of the physical model in Figure 1.
4. Solution of the Problem with the Power Law Kernel
We take into consideration the Eq. (10) with fractal-fractional differential operator using Definition 1 of power law kernel as
The, we get
For simplicity, we take
We discretize this equation at and get
We apply the two-step Lagrange polynomial as
Thus, we will get
We have
Then, we will obtain
We define
Then, we get
We choose . Then, we have
After simplification, we obtain
We prove by induction. For we obtain
We should show Therefore, we have
Since we have for all we obtain
When we have
Then, we get
We assume that for all , We want to prove that . However,
By induction hypothesis for all , we have
This inequality is true for all Thus, we reach
We need to show that Thus, we reach
5. Solution of the Problem with the Exponential Decay Kernel
We consider Eq. (10) with fractal-fractional differential operator using Definition 2 of exponential decay kernel as
For simplicity, we define
Then, we reach
We discretize Eq. (36) at and as
Then, we obtain where
Thus, we acquire
For simplicity, we let
We choose . Therefore, we reach
After simplification, we obtain
Thus, we have
For we get
The implies
This is true for all . Thus, we get
We assume that . Thus, we need to show
Thus, we obtain
This inequality is true for all . Thus, we get
6. Solution of the Problem with the Generalized Mittag-Leffler Kernel
We take into consideration the Eq. (10) with fractal-fractional differential operator using Definition 3 of Mittag-Leffler kernel as
For simplicity, we define
Then, we get
We discretize above Eq. (53) at as
Then, we obtain
We have
Then, we will obtain
For simplicity, we let
Then, we get
We choose . Then, we acquire
Then, we get
We prove by induction. For , we have
Then, we get
Thus, we reach
After simplification, we obtain
We should show . Therefore, we have
Since we have for all , we obtain
When we have
We suppose that for all , . We want to prove that . However,
By induction hypothesis for all , , we have
This inequality is true for all Thus, we reach
We need to show that . Thus, we get
7. Error Analysis
In this section, we will consider the error analysis.
Then, we get
For simplicity, we take
Then, we have
At , we get
Then, we have
We have
Therefore, we acquire where
Therefore, we obtain
Remark 4. Error analysis with exponential decay kernel and Mittag-Leffler kernel can be obtained likewise. Therefore, we misplaced the error analysis for them.
8. Conclusion
In this paper, we investigated the fractional MHD incompressible couple stress fluid flow between two parallel plates. We discussed the discretization and the stability analysis for three different kernels. Additionally, we discussed the error analysis of the model in details.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflict of interest.