NEW APPROACH OF MHD BOUNDARY LAYER FLOW TOWARDS A POROUS STRETCHING SHEET VIA SYMMETRY ANALYSIS AND THE GENERALIZED EXP-FUNCTION METHOD

Due to importance of the slip effect on modeling the boundary layer flows, symmetries and exact solution investigations have been introduced in this paper for studying the effect of a slip boundary layer on the stretching sheet through a porous medium. The exact solution of the investigating model is obtained in term of exponential via the generalized Exp-Function method. This solution satisfies the boundary conditions. Finally, the effect of parameters on the velocity field is studied.


Introduction
Symmetry group analysis based on the transformation groups, now known as Lie groups, is the most important solution method for the nonlinear problems in the literature. This approach is used to analysis the symmetries of the differential equations. Then, the corresponding symmetry groups can be used to simplify the analysis of the problems governing by the differential equations in the engineering science, mathematical physics, and mechanics. Lie groups characterize the symmetry of the differential equations and may be a point, a contact, and a potential or a nonlocal symmetry. It has also been verified that these kinds of groups can be represented by their infinitesimals that contain dependent variables, independent variables and the derivatives of dependent variables as arguments. In the last century, the application of the Lie groups has been developed by a number of mathematicians. Ovsiannikov [1], Olver [2], Ibragimov [3], and Bluman and Kumei [4] are some of the mathematicians who have huge number of studies in that field [5][6][7][8][9].
The boundary layer [10][11][12][13] equations are especially interesting from a physical point of view because they have the capacity to admit a large number of invariant solutions i.e. basically closed-form solutions. In the present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differential equation (ODE). Prandtl's boundary layer equations admit more and different symmetry groups.
Symmetry groups or simply symmetries are invariant transformations which do not alter the structural form of the equation under investigation (Bluman and Kumei [1]).
This work is organized as follows. The problem is formulated in Section 2 and in Section 3 we calculate the symmetries of the thermal boundary layer equations. All invariant solutions of the thermal boundary layer equations in Section 4. Finally, we show the effect of parameters on the velocity field.

Formulation of the problem
We consider the steady state 2D magnetohydrodynamic (MHN) boundary layer, incompressible and viscous flow on stretching sheet through a porous medium, where M is the magnetic parameter, k p is the permeability parameter and f w is the mass transfer parameter, which is positive for suction and negative for injection.
In (1) u and υ are the components of velocity respectively in the x and y directions, k 0 is the permeability of the porous medium, B 0 is magnetic field of uniform strength and σ 0 is electrical conductivity, v = µ ρ is the kinematic viscosity, µ is the coefficient of fluid viscosity and ρ is the fluid density. By using the boundary layer approximations and neglecting viscous dissipation.
The appropriate boundary conditions for the problem are given by where B is the stretching rate, υ w is the wall velocity and the velocity components along x, y coordinates, respectively, are where ψ is the stream function.
Using the relations (3) in the boundary layer (2) and in the energy (1) we get the following equations ∂ψ ∂y The boundary conditions (2) then become

Symmetry analysis for the boundary layer equations
Firstly, we shall derive the similarity solutions using the Lie-group method [11] under which (1) is invariant.
Consider the one-parameter (ε) Lie group of infinitesimal transformations in (x, y, ψ) given by Lie point With associated infinitesimal form where "ε" is a small parameter.
If we set: The invariance conditions [1][2][3][4] where Γ (3) is given by where Equation (9) gives the following system of linear partial differential equations: Solving the system (13), after substitution from (12) into (13), and using the invariance of the boundary conditions (6), yields In order to study the group theoretic structure, the vector field operator V is written as where It is easy to verify, that the vector fields are closed under the Lie bracket as follows Further, from the symmetries given in (16) the following possibilities exist for the solution of (9).
Having determined the infinitessimals, the symmetry variables are found by solving the auxiliary equation

Reductions and exact solutions
Now we look the similarity solutions with respect to the generators V 1 The reduced system of ODEs is The boundary condition take the following forms We look for a similarity solution of (19) ,and boundary condition (20) as the following form: Using (21) we obtain the following self-similar equations subject to the boundary conditions where M =  [14][15][16], we assume that the solution of (22) can be expressed in the form where c, d, p and q are positive integers which are unknown to be further determined, a n and r m are unknown constants. In addition, φ(η) satisfies Riccati equation, In order to determine values of c and p, we balance the linear term of the highest order in Eq. (24) with the highest order nonlinear term f and f 2 , we have where a i and r i are determined coefficients only for simplicity. From balancing the lowest order and highest order of φ (26) and (27), we obtain −7d − c − 3 = −6d − 2c − 2, which leads to the limit c = d + 1,and 7q + p + 3 = 6q + 2p + 2,which leads to the limit p = q + 1, for simplicity d = q = 0, the function in Eq. (24), Substituting (28) into (22), equating to zero the coefficients of all powers of φ(η) yields a set of algebraic equations for γ 0 , γ 1 and γ −1 , we obtain the following system Solving the system of algebraic equations with the aid of Maple, we obtain the following results: Substituting (30) into (28), the solutions of (1) can be written as: where γ = a−1 r−1 . Now we have to apply the boundary conditions to the solution (19), noting that the third one is already satisfied. On using the first two boundary conditions we then need to solve the system: By solving Eq. (33) then substituting in Eq. (32), we obtain the closed form solution where

Conclusion
In this paper, the couple system of MHD boundary layer flow towards a porous stretching sheet have been reduced by symmetry method to ordinary differential equations. the exact solutions of ordinary differential equations is obtained by the generalized Exp-Function method. Finally, some plots have been given for study the effects of various parameters on velocity of fluid .

Acknowledgements
The authors would like to thank the deanship of scientific research of Majmaah niversity for the financial grant received for conducting this research.  Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper.