Applications of Magnetohydrodynamic Couple Stress Fluid Flow between Two Parallel Plates with Three Different Kernels

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 fractalfractional differential operators. We present the stability analysis and the error analysis.


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][2][3][4][5][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][12][13][14][15][16][17][18][19][20][21][22][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.

Preminaries
Definition 1. Assume that gðςÞ is a continuous function in the ðc 11 , d 11 Þ and fractal differentiable on ðc 11 , d 11 Þ with order η , then the fractal-fractional derivative of g of order λ in Riemann-Liouville sense with power law kernel is introduced as [ where dg x ð Þ Definition 2. Assume that gðςÞ is a continuous function in the ðc 11 , d 11 Þ and fractal differentiable on ðc 11 , d 11 Þ with order η , then the fractal-fractional derivative of g of order λ in Riemann-Liouville sense with the exponential decay kernel is introduced as [ Definition 3. Assume that gðςÞ is a continuous function in the ðc 11 , d 11 Þ and fractal differentiable on ðc 11 , d 11 Þ with order η , then the fractal-fractional derivative of g of order λ in Riemann-Liouville sense with the generalized Mittag-Leffler kernel is introduced as

Solution of the Problem with the Power Law Kernel
We take into consideration the Eq. (10) with fractalfractional differential operator using Definition 1 of power law kernel as The, we get v y, λ ð Þ= For simplicity, we take We discretize this equation at We apply the two-step Lagrange polynomial as Journal of Function Spaces We have Then, we will obtain We define Then, we get v n+1 We choose ε n i = δ n exp ðik m yÞ. Then, we have After simplification, we obtain

Journal of Function Spaces
We prove by induction. For n = 0, we obtain We should show jδ 1 /δ 0 j < 1: Therefore, we have Since we have for all m, we obtain When n > 1, we have Then, we get We assume that for all n ≥ 1, jδ 1 /δ 0 j < 1: We want to prove that jδ n+1 /δ 0 j < 1:. However, By induction hypothesis for all n ≥ 1,jδ n j < jδ 0 j, we have This inequality is true for all m: Thus, we reach We need to show that jδ n+1 /δ 0 j < 1: Thus, we reach

Solution of the Problem with the Generalized Mittag-Leffler Kernel
We take into consideration the Eq. (10) with fractalfractional differential operator using Definition 3 of Mittag-Leffler kernel as For simplicity, we define Then, we get v y, λ ð Þ= We discretize above Eq. (53) at ðy i , t n+1 Þ as v n+1 Then, we obtain v n+1 We have Journal of Function Spaces Then, we will obtain v n+! i = For simplicity, we let Then, we get v n+1 We choose ε n i = δ n exp ðik m yÞ:. Then, we acquire Then, we get We prove by induction. For n = 0, we have 7 Journal of Function Spaces Then, we get Thus, we reach After simplification, we obtain We should show jδ 1 /δ 0 j < 1:. Therefore, we have Since we have for all m, we obtain When n > 1, we have We suppose that for all n ≥ 1, jδ n /δ 0 j < 1:. We want to prove that jδ n+1 /δ 0 j < 1:. However,

Error Analysis
In this section, we will consider the error analysis.
Then, we get v y, t ð Þ= For simplicity, we take Then, we have v y, t ð Þ= At ðy i , t = t n+1 Þ, we get v n+1 Then, we have