Some of Semi Analytical Methods for Blasius Problem

In this paper, the Adomian methods, differential transform methods, and Taylor series methods are applied to non-linear differential equations which is called Blasius problem in fluid mechanics. The solutions of the Blasius problem for two cases are obtained by using these methods and their results are shown in table.

In generally, it is considered the two-dimensional flow over a semi-infinite flat plain, is governed by where the prime denotes the derivatives with respect to a non-dimensional variable ,


, where U is the velocity at infinity;  is the kinematic viscosity coefficient; x and y are the two independent coordinates.In this work, when with the boundary conditions as The physical character of boundary layer apparently needs close and far away solution to match as was done by Blasius.So, to obtain the solution of this problem, we consider the boundary conditions as: For inner case 5.5 For inner-outer case 4  5.5 For outer case 5.5 8

  
; boundary conditions are; The paper is organized as follows: Blasius problem is solved by using ADM in &2; by using DTM in &3 and also, by using Taylor Series Method in &4, then it is given some concluding remarks in &5.

Solution of Blasius Problem by Adomian's Decomposition Method (ADM)
For solving the following equation of the form where is non-linear mapping, X is Banach space, f is known function, by using Adomian's decomposition method is taken that the solution u can be following convergent series form: In putting the Equations (2.2) and ( 2.3) into the Equation (2.1), it gives 0 0 where to determine the so-called Adomian's polynomials A n from u n , where  is a scalar parameter Operating with N −1 on Equation (1.3), then it gives: For inner case: f 0 determined from the boundary conditions (1.5) and than the other components determined from Equation (2.11) as follows [16][17][18]: For outer case: Similarly, f 0 determined from Equations (1.7) and than f 1 determined from Equation (2.11), so on: (2.13) 

Solution of Blasius Problem by Differential Transform Method (DTM)
where a and b are any arbitrary constants.Then, we obtain the following equation: In this study, similarly [19], it is applied DTM for the Equation (1.3): where   F k shows the differential transform of   f  .For inner case: (1.5) boundary conditions (BCs) are transformed where c is an arbitrary constant.Then, we get the following equation: For inner-outer case: From (1.6) boundary conditions, 4 <  0 < 5 taking the interval of  0 is found a, b, and c constants, is obtained similar result as in [19].

Solution of Blasius Problem by Taylor Series (TS)
For outer case: (1.7) boundary conditions (BCs) are transformed Taylor's series method is used for solving Blasius problem.This method assumes that the solution   f  and derivative of   f  can be taken power series as where . c = 0.332 [1], c = 0.332057 [21], c = 0.333338 (this paper).
For outer case: Under the boundary conditions (1.7), it gives

Conclusion
In this study, some of the semi analytical methods were applied by author in the non-linear Blasius problem which names are Adomian decomposition method, dif-ferential transform method, and Taylor series method.It was obtained their results for two cases by using these methods.Their results were presented in the Table 1.
The results are shown that all of these methods are powerful and efficient technique for finding semi analytical solutions for Blasius problem in the fluid mechanics.
means no slip at the wall 1 which means layer solutions merges into the inviscid solution

0
9) where shows the k th Fréchet derivative of N at .To demonstrate Adomian's solution of the Blasius problem, the differential operator