The Application of Homotopy Perturbation Method to Blasius Equations

Blasius equation is one of the fundamental and basic equations of fluid dynamics, which described the velocity profile of the fluid in the boundary layer theory on a half infinite interval (Belhachmi et al., 2000; Datta, 2003). Kuiken (1981) and He (1998) have investigated analytical and numerical solution methods to handle this problem. Two forms of Blasius equation appear in the fluid mechanic theory, where each is subjected to specific physical conditions. The equation has the forms as follows:


INTRODUCTION
Blasius equation is one of the fundamental and basic equations of fluid dynamics, which described the velocity profile of the fluid in the boundary layer theory on a half infinite interval (Belhachmi et al., 2000;Datta, 2003).Kuiken (1981) and He (1998) have investigated analytical and numerical solution methods to handle this problem.Two forms of Blasius equation appear in the fluid mechanic theory, where each is subjected to specific physical conditions.The equation has the forms as follows: It is well known that the Blasius equation is the mother of all boundary layer equations in fluid mechanics.Obviously, it is difficult to solve these differential equations analytically.Perturbation method is one of the well-known methods to solve nonlinear problems; it is based on the existence of small/large parameters, the so called perturbation quantity (Nayfeh, 2000).Many nonlinear problems do not contain such kind of the perturbation quantity and we can use non perturbation methods such as the Adomain's decomposition method (Adomian, 1994).In this study, the Homotopy Perturbation Method (HPM) has been applied to achieve the solution of Blasius equation which is one of the basic equations of fluid dynamics.
In this study, the basic idea of the HPM is introduced and the Homotopy Perturbation Method (HPM) has been applied to achieve the solution of Blasius equation which is one of the basic equations of fluid dynamics.Thus, two forms of Blasius equation are solved through HPM, then compare is made with exact solution of Variation Iteration Method (VIM) achieved by Abdul-Majid (2007).

BASIC IDEA OF HOMOTOPY PERTURBATION METHOD (HPM)
To illustrate the basic ideas of this method, we consider the following equation: With the boundary condition of: where, A : A general differential operator B : A boundary operator f (r) : A known analytical function Γ : The boundary of the domain Ω A can be divided into two parts which are L and N, where L is linear and N is nonlinear.Equation (3) can therefore be rewritten as follows: Homotopy perturbation structure is shown as follows: In Eq. ( 6), P ∈ [0, 1] is an embedding parameter and u 0 is the first approximation that satisfies the boundary condition.We can assume that the solution of Eq. ( 6) can be written as a power series in p, as following: ....
And the best approximation for solution is: .... lim 3 2 1 0 1 Interested readers may refer to works of Ganji and Sadighi (2007) for a detailed discussion on the above convergence and the principle of the HPM.

APPLICATION OF HPM TO BLASIUS EQUATION
The first form of Blasius equation: We first consider the first form of Blasius equation: Now we construct the following homotopy for Eq.(1):  8) into (10) and rearranging based on powers of p-terms, we have: 0 ) ( : Solving Eq. ( 11) to (13) subject to initial condition given by Eq. ( 1), we have: And when P →1, then we can obtain: It can be concluded that the curves are satisfactorily feet.Since Blasius equation cannot be easily solved by analytical methods, therefore, the perturbation method is valid for first form of Blasius equation.
The second form of Blasius equation: We first consider the second form of Blasius equation: The exact solution of VIM obtained by Abdul-Majid ( 2007) is compared with HPM in Fig. 2.
According to Fig. 2, it can be seen that the curves are approximately agreed.

CONCLUSION
In this study, the Homotopy Perturbation Method (HPM) has been applied to achieve the solution of Blasius equation which is one of the basic equations of fluid dynamics.It has been shown that HPM provides very accurate numerical solutions for non-liner problems in comparison with Variation Iteration Method (VIM).The numerical results we achieved here justify the advantage of this method and it can be concluded that HPM can successfully solve the two forms of Blasius equations; however, the results are more accurate for first form of Blasius equation.
the same as that obtained by Abdul-Majid (2007) through VIM.Figure1presents a comparison between VIM and HPM solution.

Fig. 1 :
Fig. 1: The obtained solution by HPM in comparison with VIM Continuous line: HPM; Dashed line: VIM