Constructing Uniform Approximate Analytical Solutions for the Blasius Problem

and Applied Analysis 3

In this paper, we introduce a weight function   (; ) in (8) whose values cluster to 0 for  < /2 and to 1 for  > /2 when  is large enough.Then, employing a series approximate solutions   () for the Blasius function () near the origin  = 0 and a reference solution () away from the origin, we propose a weighted averaging method (11) based on the function   (; ).The presented analytical solution  , (; ), a smooth function on interval [0, ], highly reflects the near origin solution   () for  < /2 and the faraway solution () for  > /2.Furthermore, the solution  , (; ) can be continuously extended to the semi-infinite interval [0, ∞).For practical performance, a procedure to choose appropriate parameters (, , ) in  , (; ) is included.In addition, to improve the accuracy of  , (; ), we propose a corrected approximation formula including an auxiliary term which properly reflects the behavior of the deviation  , (; ) − ().Results of numerical experiments, compared with the aforementioned existing method [27], illustrate availability of the proposed method.

Series Solutions and a Reference Solution
For simplicity we consider the case of  = 1/2 and  = 1.The power series of the Blasius stream function () for this case is known as Abstract and Applied Analysis where  =  0 / √ 2 and the coefficients   are computed from the recurrence [1] In fact, the series becomes This series, however, converges for || <  = 5.6900380545.
In this paper, we will use a partial sum with an integer  ≥ 0, for an approximate solution to the Blasius function () near the origin.On the other hand, for a reference solution approximating () far from the origin, we consider the following linear function: based on the property (ii) in the previous section.Figure 1(a) illustrates graphs of the series approximate solutions   () with  = 0, 1, 2, 3, 4 on the interval [0, 8].Therein, the dotted line indicates the numerical solution for the Blasius function ().It is observed that   () overshoots () when  is even and undershoots when  is odd.In addition, Figure 1(b) shows the graph of the reference solution () which undershoots ().To illustrate motivation of the main idea proposed in the next section, graphs of the differences   () − () with  = 0, 2, 4 and () − () are included in Figure 2, where () is replaced by the numerical solution.

Uniform Approximate Analytical Solutions
For some  > 1 and  > 0 we introduce a weight function   (; ) defined as It should be noted that 0 ≤   (; ) ≤ 1 and it is strictly increasing on the interval [0, ] with   (; /2) = 1/2 for any .In addition, for a large  it follows that This implies that the value of   (; ) goes close to 0 for  < /2 and to 1 for  > /2 as  increases.Figure 3 shows the graphs of   (; ) with  = 10 and  = 2, 4, 8, for example.Moreover, we can find that the inverse function of   (; ) =  takes a form of In order to improve the accuracy of the approximate solutions for the Blasius function, we propose a weighted average of the series solution   () and the reference solution () as Therein, for given  and , we may take the optimal value of , denoted by  * , which minimizes the  2 -norm of the residual function  , (; ) defined as From the property (9) of the weight function   (; ), it follows that for  large enough with This implies that the point  = /2 is a threshold between the near origin series solution   () and the faraway reference solution ().
We now summarize the procedure to choose the parameters , , and  in the proposed solution  , (; ) in (11) as follows.
(S1) Considering the undershoot of the reference solution (), take an even integer  ≥ 0 in the series solution    which overshoots the Blasius function () (see Figure 1).
Abstract and Applied Analysis  (S3) Find the optimal exponent  =  * of   (; ) which minimizes ‖ , (; )‖ 2 defined in (12), that is, satisfies As a result, we may expect that the presented approximate solution  , (; ) with the parameters (, , ) determined by the procedure (S1)-(S3) will become a corrected approximate solution which improves accuracy of both the series solution   () and the reference solution () over the interval [0, ].
For example, when we take  = 0, from Figure 2, we can find  ≈ 3 and thus we may set  = 2 = 6.The optimal exponent is  * ≈ 4.31 which is obtained by the software, Mathematica V.9.By the similar way, we can choose the values of  and  * for other cases of .Table 1 includes the results for the some small values,  = 0, 2, 4, where   indicates the nearest integer to the optimal exponent  * .Figure 4 illustrates the availability of the presented approximate solution  , (; ) with (, , ) = (4, 8.5, 11) given in Table 1.Additionally, numerical results for the  2norm errors of the approximate solution  , (; ) and its derivatives are given in Table 2.

Further Improvement of the Approximate Solution
In a particular case of (, , ) = (4, 8.5, 11.49), observing the behavior of the difference error  , (; )−(), we propose a ( Comparing the results with those in Table 2, one can find that the corrected approximation f, (; ) and its first derivative f , (; ) reasonably improve the accuracy of  , (; ) and   , (; ).
For comparison with the existing approximation method, we consider the modified generalized Laguerre function Tau method introduced in the literature [27] such as based on the generalized Laguerre polynomials    () for  = 0.5, 0.8, 1, 1.3, 1.5 and a scaling parameter  > 0. For the unknown coefficients   's the Tau method [28,29] is used, which generates a nonlinear system of algebraic equations.Thus a Newton-like iterative method is required to determine the coefficients   's as a result.
To improve the accuracy, we introduced a correction method f, (; ) in (17) which includes an additional term reflecting the deviation of  , (; ) from the Blasius function ().As a result we can observe that the presented method is available for approximation to () and   () while the approximation to the second derivative   () is not so effective.

Conclusions
For the Blasius problem on the semi-infinite interval we proposed a uniformly accurate approximation formula  , (; ) in (11).The proposed method employs the weight function   (; ) in ( 8) to combine a near origin series solution and a faraway reference solution.
Comparing the presented solutions  , (; ) and f, (; ) with the existing solution  Par  (  ), a solution from the generalized Laguerre spectral approach [27] based on Tau method, we summarize advantages of the presented method with discussions as follows.
(i) The presented solution  , (; ) is composed of simple forms of known solutions, that is, a series solution   () and a reference solution () =  + √ 2 0 , while the spectral method requires solving a nonlinear system of algebraic equations.This implies that the presented method will save number of evaluations in numerical implementation.
(ii) The corrected solution f, (; ) highly improves accuracy of  , (; ) with a small number of terms  = 4, and numerical results show that it is comparable to the spectral solution  Par  (  ) with  = 21.
(iii) There is a room for further improvement of the present method, for example, by replacing the weight function   (, ) by some more appropriate one or employing other partial solutions instead of   () or ().
To conclude, though the presented method is limitedly applicable to the Blasius problem unlike the spectral methods,   we may expect to develop an extensive method for other nonlinear differential equations motivated by the advantages above.