Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a Powell–Eyring non-Newtonian fluid

ABSTRACT In this work, the boundary layer flow of a Powell–Eyring non-Newtonian fluid over a stretching sheet has been investigated by a reproducing kernel method. Reproducing kernel functions are used to obtain the solutions. The approximate solutions are demonstrated, and the proposed technique is compared with some well-known methods. Convergence analysis of the technique is presented. The accuracy of the reproducing kernel method has been proved.


Introduction
The investigation of flow and transport operation in non-Newtonian fluids have taken very important interest with the significance of different such fluids in the industry, chemical engineering and biological processes [1]. Additionally, the problem of boundary layer flow over a stretching sheet has many industrial implementations [2]. The connection between the shear stress and the rate of strain in such fluids are very complex. The viscoelastic properties in non-Newtonian fluids perform more strain in the resulting equations than Navier-Stokes equations. Many authors have been captivated by the flow analysis of non-Newtonian fluids [3]. A valuable and complex non-Newtonian fluid is the Powell-Eyring fluid, which has some advances over other non-Newtonian fluid models such as Powell and Eyring in some aspects [4]. Zaman et al. [5] have applied the homotopy analysis method to incompressible Powell-Eyring flow in a pipe with porous walls. Nadeem and Saleem [6] have studied on the series of solutions of an unsteady Powell-Eyring nanofluid flow about a rotating cone. Malik et al. [7] have researched mixed convection in a magnetohydrodynamic (MHD) Powell-Eyring nanofluid over a stretching sheet. Nadeem et al. [8] have studied the MHD flow of a Powell-Eyring fluid between parallel heated plates. Parand et al. [9] have investigated the laminar two-dimensional flow of an incompressible Powell-Eyring non-Newtonian fluid over a linearly stretching sheet with the indirect radial basis function (IRBF) method.
Many nonlinear differential equations do not have analytical solutions. Therefore, scientists have used numerical methods such as finite-difference method [10], finite-element method [11], spectral methods [12], and meshless methods [13] to approximate the solutions of these problems.
The main goal of this work is to apply the reproducing kernel method (RKM) using reproducing kernel functions for investigating the nonlinear differential equation of the Powell-Eyring problem, in an unbounded domain. This method has been applied to many problems successfully [14][15][16][17][18].
We consider a stretching sheet with a linear velocity V τ t. V τ = α is the linear stretching velocity and t is the distance from the slit. The shear tensor in a Powell-Eyring model is given as [4] The second-order approximation of the function is presented as [4] sin h −1 1 P The boundary layer problems for an incompressible fluid based on the Powell-Eyring model is given as [4] ∂v ∂t The kinematic viscosity is given as μ = η/ρ. For Equations (3) and (4), the boundary conditions are given as [4]: The following equations are acquired by dimensionless stream function z(ζ ), where ζ is the similarity variable: where Then, we obtain [4] ( in which ε and δ are the material fluid parameters. These quantities have the following definitions: The boundary conditions for Equation (8) are acquired by Equation (5) as [4] z(0) = 0, We investigate Equation (8) with its boundary conditions (10) in the reproducing kernel Hilbert space in this paper.

Reproducing kernel Hilbert spaces
We should construct two useful reproducing kernel Hilbert spaces to investigate our problem. We will obtain reproducing kernel functions in these spaces. We will use these functions to get approximate solutions of the problem.

Definition 2.1:
We firstly need to construct V 4 2 [0, ∞) for our problem. We define it as follow: We have the inner product and norm in this space as and Reproducing kernel function W p of the reproducing kernel Hilbert space V 4 2 [0, ∞) is obtained by reproducing property as (14) where by the inner product of reproducing kernel Hilbert We get by integrations by parts. We acquire by reproducing property. We choose Therefore, we obtain by (17). When ζ = p, we have Thus, we get Since we obtain and W p (ζ ) is the reproducing kernel function of reproducing kernel Hilbert space V 4 2 [0, ∞). Therefore, this function satisfies the boundary conditions as We can obtain the unknown coefficients A i (p) and B i (p) (i = 1, 2, . . . , 8) by (19)- (26). Then, the reproducing kernel function W p (ζ ) is obtained as

Approximate solutions in V 4 2 [0, ∞)
We use the following transformation: to homogenize the boundary conditions. We can check our transformation as Then, we have We put these functions into the following equation: Then, we obtain If we make necessary operations, we can acquire We give the solutions of the problem in the V 4 2 [0, ∞). We define the bounded linear operator J : Then the problem gets the form: is the exact solution of (54), then we have where from (55) and uniqueness of the solution of (54).
The approximate solution u n (ζ ) can be written as

Numerical results
The absolute errors obtained from the approximation of the solution of Equation (8) was given in this section. We solved Equation (8) applying the presented technique. All computations were implemented by Maple 18. We demonstrated our results by Table 1.

Conclusion
We constructed the reproducing kernel Hilbert space method for the numerical solution of the nonlinear Powell-Eyring equation. The solution of this problem has implementations in many fields of sciences. We also determined that this method can be useful in dealing with other nonlinear differential equations, which utilized as governing dynamical models in nonlinear science problems and in studies of various physical fields. The principal benefit of this method is that highly accurate solutions were obtained using reproducing kernel functions.