Research Article
Constructing Uniform Approximate Analytical Solutions for the
Blasius Problem
Beong In Yun
Department of Statistics and Computer Science, Kunsan National University, Gunsan 573-701, Republic of Korea
Correspondence should be addressed to Beong In Yun; paulllyun@gmail.com
Received 17 November 2013; Revised 5 February 2014; Accepted 6 February 2014; Published 12 March 2014
Academic Editor: Lucas Jodar
Copyright © 2014 Beong In Yun. his 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-ininite interval [0,∞). he method is based on a weight function having an S-shape to relect a series solution
near the origin =0 and a reference solution far from the origin. Numerical results show the eiciency of the proposed method.
1. Introduction
For the Blasius problem
():=
()+()
()=0, 0≤<∞, (1)
subject to the boundary conditions
(0)=
(0)=0,
(∞)=, (2)
we recall the well-known properties [1–3] of the so-called
Blasius function () as follows:
(i)
(0)==√
3
0
with
0
=0.4695999883⋅⋅⋅
(ii) lim
→∞
{() − } = √ (/)
0
with
0
=
−1.2167806216⋅⋅⋅ .
hough the Blasius problem looks simple, search for an
approximate analytical solution is known to be quite diicult.
Until now, in the literature [4–22], lots of analytical methods
have been proposed. Recently, in approximation of the
solutions of nonlinear diferential equations in unbounded
domain, several eicient spectral methods [23–27] have
been proposed. hese 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 0 for < /2 and to 1
for > /2 when is large enough. hen, 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
(;). he presented
analytical solution
,
(;), a smooth function on interval
[0,], highly relects the near origin solution
() for
< /2 and the faraway solution () for > /2.
Furthermore, the solution
,
(;) can be continuously
extended to the semi-ininite 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 relects
the behavior of the deviation
,
(;) − (). Results of
numerical experiments, compared with the aforementioned
existing method [27], illustrate availability of the proposed
method.
2. Series Solutions and a Reference Solution
For simplicity we consider the case of =1/2 and =1. he
power series of the Blasius stream function () for this case
is known as
()=
∞
∑
=0
(−
1
2
)
+1
(3+2)!
3+2
, (3)
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 495734, 6 pages
http://dx.doi.org/10.1155/2014/495734