Constructing Uniform Approximate Analytical Solutions for the Blasius Problem
We propose a simple constructive method which assures uniform accuracy of the approximate analytical solutions for the Blasius problem on the semi-infinite interval . The method is based on a weight function having an S-shape to reflect a series solution near the origin and a reference solution far from the origin. Numerical results show the efficiency of the proposed method.
Though the Blasius problem looks simple, search for an approximate analytical solution is known to be quite difficult. Until now, in the literature [4–22], lots of analytical methods have been proposed. Recently, in approximation of the solutions of nonlinear differential equations in unbounded domain, several efficient spectral methods [23–27] 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 in (8) whose values cluster to for and to for when is large enough. Then, employing a series approximate solutions for the Blasius function near the origin 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 , highly reflects the near origin solution for and the faraway solution for . Furthermore, the solution can be continuously extended to the semi-infinite interval . 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 , illustrate availability of the proposed method.
2. Series Solutions and a Reference Solution
For simplicity we consider the case of and . The power series of the Blasius stream function for this case is known as where and the coefficients are computed from the recurrence  In fact, the series becomes This series, however, converges for . In this paper, we will use a partial sum with an integer , 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 on the interval . 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 and are included in Figure 2, where is replaced by the numerical solution.
|(a) Series solutions|
|(b) Reference solution|
3. Uniform Approximate Analytical Solutions
For some and we introduce a weight function defined as It should be noted that and it is strictly increasing on the interval with for any . In addition, for a large it follows that This implies that the value of goes close to 0 for and to 1 for as increases. Figure 3 shows the graphs of with and , 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 -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 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 in the series solution which overshoots the Blasius function (see Figure 1).(S2)Choose a length of the interval for some satisfying or .(S3)Find the optimal exponent of which minimizes 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 .
For example, when we take , from Figure 2, we can find and thus we may set . The optimal exponent is 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, , where indicates the nearest integer to the optimal exponent .
Figure 4 illustrates the availability of the presented approximate solution with given in Table 1. Additionally, numerical results for the -norm errors of the approximate solution and its derivatives are given in Table 2.
4. Further Improvement of the Approximate Solution
In a particular case of , observing the behavior of the difference error , we propose a correction formula by adding an auxiliary term to the formula as follows: where is the maximum of the absolute error at the point . Values of and are numerically evaluated as Numerical implementation for results in the errors Comparing the results with those in Table 2, one can find that the corrected approximation and its first derivative 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  such as based on the generalized Laguerre polynomials for and a scaling parameter . 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.
Table 3 includes numerical results of the relative errors , and , with the parameters , for the Blasius function . Additionally, numerical results of and , and for the first derivative are given in Table 4. In the tables, the relative errors are defined as for an approximation to the Blasius function . Therein, and are replaced by the numerical solutions for a set of nodes . From Tables 3 and 4 we can see that the presented approximations and are less accurate than and on the region , 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 and .
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.
To improve the accuracy, we introduced a correction method 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.
Comparing the presented solutions and with the existing solution , a solution from the generalized Laguerre spectral approach  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 + , 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 highly improves accuracy of with a small number of terms , and numerical results show that it is comparable to the spectral solution with .(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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF-2013R1A1A4A03005079).
S. Finch, “Prandtl-Blasius flow,” 2008, http://www.people.fas.harvard.edu/~sfinch/csolve/bla.pdf.View at: Google Scholar
H. Schlichting, Boundary-Layer Theory, McGraw-Hill, New York, NY, USA, 7th edition, 1979.View at: MathSciNet
J. Biazar, M. Gholami Porshokuhi, and B. Ghanbari, “Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 622–628, 2010.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Bao and J. Shen, “A generalized-Laguerre-Hermite pseudospectral method for computing symmetric and central vortex states in Bose-Einstein condensates,” Journal of Computational Physics, vol. 227, no. 23, pp. 9778–9793, 2008.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, “An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method,” Computer Physics Communications, vol. 181, no. 6, pp. 1096–1108, 2010.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Parand, M. Dehghan, and A. Taghavi, “Modified generalized Laguerre function tau method for solving laminar viscous flow: the Blasius equation,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20, no. 6-7, pp. 728–743, 2010.View at: Publisher Site | Google Scholar | MathSciNet
A. Saadatmandi and M. Dehghan, “Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method,” Communications in Numerical Methods in Engineering with Biomedical Applications, vol. 24, no. 11, pp. 1467–1474, 2008.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet