AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 495734 10.1155/2014/495734 495734 Research Article Constructing Uniform Approximate Analytical Solutions for the Blasius Problem http://orcid.org/0000-0003-2889-4459 Yun Beong In Jodar Lucas Department of Statistics and Computer Science Kunsan National University Gunsan 573-701 Republic of Korea kunsan.ac.kr 2014 1232014 2014 17 11 2013 05 02 2014 06 02 2014 12 3 2014 2014 Copyright © 2014 Beong In Yun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a simple constructive method which assures uniform accuracy of the approximate analytical solutions for the Blasius problem on the semi-infinite interval 0,. The method is based on a weight function having an S-shape to reflect a series solution near the origin x=0 and a reference solution far from the origin. Numerical results show the efficiency of the proposed method.

1. Introduction

For the Blasius problem (1)Nf(x):=f′′′(x)+αf(x)f′′(x)=0,0x<, subject to the boundary conditions (2)f(0)=f(0)=0,f()=β, we recall the well-known properties  of the so-called Blasius function f(x) as follows:

f′′(0)=κ=αβ3κ0 with κ0=0.4695999883

limx{f(x)-βx}=(β/α)B0 with B0=-1.2167806216.

Though the Blasius problem looks simple, search for an approximate analytical solution is known to be quite difficult. Until now, in the literature , lots of analytical methods have been proposed. Recently, in approximation of the solutions of nonlinear differential equations in unbounded domain, several efficient spectral methods  have been proposed. These methods reduce solving the nonlinear equation to solving a system of nonlinear algebraic equations.

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

2. Series Solutions and a Reference Solution

For simplicity we consider the case of α=1/2 and β=1. The power series of the Blasius stream function f(x) for this case is known as (3)Sf(x)=j=0(-12)jajκj+1(3j+2)!x3j+2, where κ=κ0/2 and the coefficients aj are computed from the recurrence  (4)aj={1,j=0,1r=0j-1(3j-13r)araj-r-1,j2. In fact, the series becomes (5)Sf(x)=κ2x2-κ2240x5+11161280κ3x8-7363866880κ4x11+. This series, however, converges for |x|<ρ  =  5.6900380545. In this paper, we will use a partial sum (6)Snf(x)=j=0n(-12)jajκj+1(3j+2)!x3j+2, with an integer n0, for an approximate solution to the Blasius function f(x) near the origin.

On the other hand, for a reference solution approximating f(x) far from the origin, we consider the following linear function: (7)Rf(x)=βx+βαB0=x+2B0, based on the property (ii) in the previous section.

Figure 1(a) illustrates graphs of the series approximate solutions Snf(x) with n=0,1,2,3,4 on the interval [0,8]. Therein, the dotted line indicates the numerical solution for the Blasius function f(x). It is observed that Snf(x) overshoots f(x) when n is even and undershoots when n is odd. In addition, Figure 1(b) shows the graph of the reference solution Rf(x) which undershoots f(x). To illustrate motivation of the main idea proposed in the next section, graphs of the differences Snf(x)-f(x) with n=0,2,4 and Rf(x)-f(x) are included in Figure 2, where f(x) is replaced by the numerical solution.

Approximations of the series solutions Snf(x) for each n=0,1,2,3,4 in (a) and the reference solution Rf(x) in (b).

Series solutions Snf(x)

Reference solution Rf(x)

Differences Snf(x)-f(x) with n=0,2,4 and Rf(x)-f(x) indicated by the thin lines and the thick line, respectively.

3. Uniform Approximate Analytical Solutions

For some k>1 and L>0 we introduce a weight function wL(k;x) defined as (8)wL(k;x)=xkxk+(L-x)k,0xL. It should be noted that 0wL(k;x)1 and it is strictly increasing on the interval [0,L] with wL(k;L/2)=1/2 for any k. In addition, for a large k it follows that (9)wL(k;x)  ={O((xL)k),for  0x<L21+O((Lx-1)k),for  L2<xL. This implies that the value of wL(k;x) goes close to 0 for x<L/2 and to 1 for x>L/2 as k increases. Figure 3 shows the graphs of wL(k;x) with L=10 and k=2,4,8, for example.

Behavior of the weight function wL(k;x) with L=10 for each k=2,4,8.

Moreover, we can find that the inverse function of wL(k;x)=y takes a form of (10)wL-1(k;y)=L·y1/ky1/k+(1-y)1/k=L·w1(1k;y),hhhhhhhhhhhhhhhhhhhhhh0y1.

In order to improve the accuracy of the approximate solutions for the Blasius function, we propose a weighted average of the series solution Snf(x) and the reference solution Rf(x) as (11)fn,L(k;x)={1-wL(k;x)}Snf(x)+wL(k;x)Rf(x),hhhhhhhhhhhhhhhhhhhhhhhhhx[0,L]. Therein, for given n and L, we may take the optimal value of k, denoted by k*, which minimizes the L2-norm of the residual function Nfn,L(k;x) defined as (12)Nfn,L(k;x)22=0L{12fn,L′′′  (k;x)1111111+12fn,L(k;x)fn,L′′(k;x)}2dx.

From the property (9) of the weight function wL(k;x), it follows that for k large enough (13)fn,L(k;x)~{Snf(x),for  0x<L2Rf(x),for  L2<xL with (14)fn,L(k;L2)={Snf(L/2)+Rf(L/2)}2. This implies that the point x=L/2 is a threshold between the near origin series solution Snf(x) and the faraway reference solution Rf(x).

We now summarize the procedure to choose the parameters n, L, and k in the proposed solution fn,L(k;x) in (11) as follows.

Considering the undershoot of the reference solution Rf(x), take an even integer n0 in the series solution Snf which overshoots the Blasius function f(x) (see Figure 1).

Choose a length L=2d of the interval [0,L] for some d satisfying (15)(Snf(d)-f(d))+(Rf(d)-f(d))0

or Snf(d)+Rf(d)2f(d).

Find the optimal exponent k=k* of wL(k;x) which minimizes Nfn,L(k;x)2 defined in (12), that is, satisfies (16)Nfn,L(k*;x)2=mink>1Nfn,L(k;x)2.

As a result, we may expect that the presented approximate solution fn,L(k;x) with the parameters (n,L,k) determined by the procedure (S1)–(S3) will become a corrected approximate solution which improves accuracy of both the series solution Snf(x) and the reference solution Rf(x) over the interval [0,L].

In addition, we may extend fn,L(k;x) to the semi-infinite interval [0,) continuously by setting fn,L(k;x)=Rf(x) for all xL, which assures sufficient accuracy over the interval [L,) for L>6 as can be observed in Figures 1(b) and 2.

For example, when we take n=0, from Figure 2, we can find d3 and thus we may set L=2d=6. The optimal exponent is k*4.31 which is obtained by the software, Mathematica V.9. By the similar way, we can choose the values of L and k* for other cases of n. Table 1 includes the results for the some small values, n=0,2,4, where k indicates the nearest integer to the optimal exponent k*.

Values of n, L, and k* obtained by (S1)–(S3).

n    Length (L) Optimal exponent (k*) k
0 6 4.31 4
2 8 7.87 8
4 8.5 11.49 11

Figure 4 illustrates the availability of the presented approximate solution fn,L(k;x) with (n,L,k)=(4,8.5,11) given in Table 1. Additionally, numerical results for the L2-norm errors of the approximate solution fn,L(k;x) and its derivatives are given in Table 2.

L 2 -norm errors of fn,L(k;x) and its derivatives fn,L(k;x) and fn,L(k;x).

( n , L , k ) f - f n , L 2 f - f n , L 2 f - f n , L 2
( 0,6 , 4 ) 1.4 × 10 - 2 1.2 × 10 - 2 3.1 × 10 - 2
( 2,8 , 8 ) 8.5 × 10 - 3 1.3 × 10 - 2 3.9 × 10 - 2
( 4,8.5,11 ) 4.6 × 10 - 4 1.2 × 10 - 3 2.0 × 10 - 2
( 4,8.5,11 . 49 ) 7.1 × 10 - 4 7.1 × 10 - 4 2.0 × 10 - 2

Approximation of the weighted average fn,L(k;x) with the parameters (n,L,k)=(4,8.5,11) and its error in (a) and those of the related velocity profile fn,L(k;x) in (b).

f n , L ( k ; x )

f n , L ( k ; x )

4. Further Improvement of the Approximate Solution

In a particular case of (n,L,k)=(4,8.5,11.49), observing the behavior of the difference error fn,L(k;x)-f(x), we propose a correction formula by adding an auxiliary term to the formula fn,L(k;x) as follows: (17)f~n,L(k;x)=fn,L(k;x)+Ae-(x-c)2, where A is the maximum of the absolute error |fn,L(k;x)-f(x)| at the point x=c. Values of A and c are numerically evaluated as (18)A=0.00064585,c=5.0402. Numerical implementation for f~n,L(k;x) results in the errors (19)f-f~n,L2=3.9×10-5,f-f~n,L2=1.3×10-4,f-f~n,L′′2=2.0×10-2. Comparing the results with those in Table 2, one can find that the corrected approximation f~n,L(k;x) and its first derivative f~n,L(k;x) reasonably improve the accuracy of fn,L(k;x) and fn,L(k;x).

For comparison with the existing approximation method, we consider the modified generalized Laguerre function Tau method introduced in the literature  such as (20)fNpar(x)=exp(-x2l)j=0N-1ajLjα(xl), based on the generalized Laguerre polynomials Ljα(x) for α=0.5,0.8,1,1.3,1.5 and a scaling parameter l>0. For the unknown coefficients aj’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 aj’s as a result.

Table 3 includes numerical results of the relative errors Efn,L(k;xj), Ef~n,L(k;xj) and EfNPar(xj), with the parameters (N,α,l)=(21,1,1), for the Blasius function f(x). Additionally, numerical results of E1fn,L(k;xj) and E1f~n,L(k;xj), and E1fNPar(xj) for the first derivative f(x) are given in Table 4. In the tables, the relative errors are defined as (21)Eg(xj)=|f(xj)-g(xj)f(xj)|,E1g(xj)=|f(xj)-g(xj)f(xj)| for an approximation g(x) to the Blasius function f(x). Therein, f(xj) and f(xj) are replaced by the numerical solutions for a set of nodes {xj}j=19={1,2,,9}. From Tables 3 and 4 we can see that the presented approximations fn,L(k;xj) and fn,L(k;xj) are less accurate than fNPar(xj) and fNPar(xj) on the region 4x7, and vice versa outside the region. However, it is also noticed that the inferiority of the presented approximations is quite overcome by the corrected approximation f~n,L(k;xj) and f~n,L(k;xj).

Relative errors for the Blasius function f(x).

x j Existing method (in ) Presented methods
E f N Par ( x j ) E f n , L ( k ; x j ) E f ~ n , L ( k ; x j )
1 8.1 × 1 0 - 6 1.8 × 1 0 - 7 1.8 × 1 0 - 7
2 1.6 × 1 0 - 5 3.9 × 1 0 - 7 4.9 × 1 0 - 7
3 1.2 × 1 0 - 5 4.3 × 1 0 - 6 2.9 × 1 0 - 6
4 6.7 × 1 0 - 6 9.3 × 1 0 - 5 2.3 × 1 0 - 6
5 5.4 × 1 0 - 6 2.0 × 1 0 - 4 7.3 × 1 0 - 8
6 6.2 × 1 0 - 6 6.2 × 1 0 - 5 1.5 × 1 0 - 6
7 4.4 × 1 0 - 6 4.9 × 1 0 - 6 2.2 × 1 0 - 6
8 3.5 × 1 0 - 6 1.6 × 1 0 - 7 1.5 × 1 0 - 7
9 3.6 × 1 0 - 6 1.2 × 1 0 - 8 1.2 × 1 0 - 8

Relative errors for the first derivative f(x).

x j Existing method (in ) Presented methods
E 1 f N Par ( x j ) E 1 f n , L ( k ; x j ) E 1 f ~ n , L ( k ; x j )
1 4.7 × 1 0 - 5 1.2 × 1 0 - 7 1.2 × 1 0 - 7
2 1.3 × 1 0 - 5 2.0 × 1 0 - 6 2.6 × 1 0 - 6
3 1.7 × 1 0 - 5 5.3 × 1 0 - 5 4.3 × 1 0 - 6
4 7.4 × 1 0 - 6 3.2 × 1 0 - 4 1.5 × 1 0 - 4
5 1.7 × 1 0 - 5 6.5 × 1 0 - 5 1.2 × 1 0 - 5
6 1.5 × 1 0 - 5 4.5 × 1 0 - 4 4.3 × 1 0 - 5
7 5.6 × 1 0 - 6 7.3 × 1 0 - 5 1.9 × 1 0 - 5
8 3.8 × 1 0 - 6 3.8 × 1 0 - 6 3.2 × 1 0 - 6
9 1.5 × 1 0 - 7 1.5 × 1 0 - 7 1.5 × 1 0 - 7
5. Conclusions

For the Blasius problem on the semi-infinite interval we proposed a uniformly accurate approximation formula fn,L(k;x) in (11). The proposed method employs the weight function wL(k;x) in (8) to combine a near origin series solution and a faraway reference solution.

To improve the accuracy, we introduced a correction method f~n,L(k;x) in (17) which includes an additional term reflecting the deviation of fn,L(k;x) from the Blasius function f(x). As a result we can observe that the presented method is available for approximation to f(x) and f(x) while the approximation to the second derivative f′′(x) is not so effective.

Comparing the presented solutions fn,L(k;x) and f~n,L(k;x) with the existing solution fNPar(xj), a solution from the generalized Laguerre spectral approach  based on Tau method, we summarize advantages of the presented method with discussions as follows.

The presented solution fn,L(k;x) is composed of simple forms of known solutions, that is, a series solution Snf(x) and a reference solution Rf(x)=x + 2B0, 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.

The corrected solution f~n,L(k;x) highly improves accuracy of fn,L(k;x) with a small number of terms n=4, and numerical results show that it is comparable to the spectral solution fNPar(xj) with N=21.

There is a room for further improvement of the present method, for example, by replacing the weight function wL(k,x) by some more appropriate one or employing other partial solutions instead of Snf(x) or Rf(x).

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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2013R1A1A4A03005079).

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