Research Journal of Mathematics and Statistics 5(1-2): 01-04, 2013
ISSN: 2042-2024, e-ISSN: 2040-7505
© Maxwell Scientific Organization, 2013
Submitted: September 25, 2012
Accepted: November 23, 2012
Published: May 30, 2013
The Application of Homotopy Perturbation Method to Blasius Equations
1
Hossein Towsyfyan, 1Adnan Adnani Salehi and 2Gholamreza Davoudi
Department of Manufacturing Engineering, Arvandan Nonprofit Higher
Education Institute, Khorramsahr, Iran
2
Department of Manufacturing, Technical and Vocational University, Ahvaz, Iran
P
P
P
P
P
P
1
3TP
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P
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Abstract: In this study, the Homotopy Perturbation Method (HPM), which is a powerful and easy to use analytic
tool for nonlinear problems, is compared with the Variation Iteration Method (VIM) in the fluid mechanics field.
Thus, the basic idea of the (HPM) is introduced and two forms of Blasius equations are solved using HPM and
compare is made against VIM.
Keywords: Blasius equation, HPM, VIM
INTRODUCTION
In this study, the basic idea of the HPM is
introduced and the Homotopy Perturbation Method
(HPM) has been applied to achieve the solution of
Blasius equation which is one of the basic equations of
fluid dynamics. Thus, two forms of Blasius equation are
solved through HPM, then compare is made with exact
solution of Variation Iteration Method (VIM) achieved
by Abdul-Majid (2007).
Blasius equation is one of the fundamental and
basic equations of fluid dynamics, which described the
velocity profile of the fluid in the boundary layer theory
on a half infinite interval (Belhachmi et al., 2000;
Datta, 2003). Kuiken (1981) and He (1998) have
investigated analytical and numerical solution methods
to handle this problem. Two forms of Blasius equation
appear in the fluid mechanic theory, where each is
subjected to specific physical conditions. The equation
has the forms as follows:
1
u ( x)u '' ( x) = 0
2
u (0) = 0, u '' (0) = 1, u ' (∞) = 0
u ''' ( x ) +
BASIC IDEA OF HOMOTOPY PERTURBATION
METHOD (HPM)
To illustrate the basic ideas of this method, we
consider the following equation:
(1)
A(u ) − f (r ) = 0, r ∈ Ω
(3)
And:
With the boundary condition of:
1
u ( x) + u ( x)u '' ( x) = 0
2
u (0) = 0, u ' (0) = 0, u ' (∞) = 1
'''
(2)
 ∂Ω 
B u ,
 = 0, r ∈ Γ
 ∂n 
It is well known that the Blasius equation is the
mother of all boundary layer equations in fluid
mechanics. Obviously, it is difficult to solve these
differential equations analytically. Perturbation method
is one of the well-known methods to solve nonlinear
problems; it is based on the existence of small/large
parameters, the so called perturbation quantity (Nayfeh,
2000). Many nonlinear problems do not contain such
kind of the perturbation quantity and we can use non
perturbation methods such as the Adomain’s
decomposition method (Adomian, 1994). In this study,
the Homotopy Perturbation Method (HPM) has been
applied to achieve the solution of Blasius equation
which is one of the basic equations of fluid dynamics.
(4)
where,
A : A general differential operator
B : A boundary operator
f (r) : A known analytical function
Γ : The boundary of the domain Ω
A can be divided into two parts which are L and N,
where L is linear and N is nonlinear.
Equation (3) can therefore be rewritten as follows:
L(u ) + N (u ) − f (r ) = 0, r ∈ Ω
(5)
Homotopy perturbation structure is shown as follows:
Corresponding Author: Gholamreza Davoudi, Department of Manufacturing, Technical and Vocational University, Ahvaz, Iran
1
Res. J. Math. Stat., 5(1-2): 01-04, 2013
H (u , P) = (1 − p)[L(u ) − L(u0 )] + p[ A(u ) − f (r )] = 0 (6)
P0 :
where,
u (r , p ) : Ω × [0,1] → R
In Eq. (6), P ∈ [0, 1] is an embedding parameter
and u 0 is the first approximation that satisfies the
boundary condition. We can assume that the solution of
Eq. (6) can be written as a power series in p, as
following:
u = u 0 + pu1 + p 2 u 2 + p 3 u 3 + ....
P2 :
(11)
(12)
d3
d2
d2
u ( x) + 0 .5u0 ( x)( 2 u1 ( x) + 0.5u1 ( x)( 2 u0 ( x)) = 0 (13)
3 2
dx
dx
dx
Solving Eq. (11) to (13) subject to initial condition
given by Eq. (1), we have:
(8)
And the best approximation for solution is:
u = lim p →1 u = u0 + u1 + u2 + u3 + ....
d2
d3
(
)
+
0
.
5
(
)(
u0 ( x ) = 0
u
x
u
x
1
0
dx 2
dx 3
P1 :
(7)
d3
u0 ( x ) = 0
dx 3
u0 =
1
Ax 2 + x
2
u1 = −
1 2 5 1
A x −
Ax 4
240
48
(9)
u2 =
Interested readers may refer to works of Ganji and
Sadighi (2007) for a detailed discussion on the above
convergence and the principle of the HPM.
11
11
1
A3 x 8 +
A2 x 7 +
Ax 6
161280
20160
960
And when P →1, then we can obtain:
u = u 0 + u1 + u 2 =
APPLICATION OF HPM TO
BLASIUS EQUATION
11
11
1
A3 x 8 +
A2 x 7 +
Ax 6
161280
20160
960
1
1
1
−
A2 x5 −
Ax 4 + Ax 2 + x
240
48
2
The first form of Blasius equation: We first consider
the first form of Blasius equation:
1
u ( x)u '' ( x) = 0
2
u (0) = 0, u '' (0) = 1, u ' (∞) = 0
u ''' ( x) +
d2
d3
d3
u ( x)) + P ( 3 u ( x) + 0.5u ( x)( 2 u ( x)) = 0 (10)
3
dx
dx
dx
This is exactly the same as that obtained by AbdulMajid (2007) through VIM. Figure 1 presents a
comparison between VIM and HPM solution.
It can be concluded that the curves are
satisfactorily feet. Since Blasius equation cannot be
easily solved by analytical methods, therefore, the
perturbation method is valid for first form of Blasius
equation.
Substituting U from Eq. (8) into (10) and
rearranging based on powers of p-terms, we have:
The second form of Blasius equation: We first
consider the second form of Blasius equation:
Now we construct the following homotopy for Eq. (1):
(1 − p )(
Fig. 1: The obtained solution by HPM in comparison with VIM
Continuous line: HPM; Dashed line: VIM
2
Res. J. Math. Stat., 5(1-2): 01-04, 2013
Fig. 2: The obtained solution by HPM in comparison with VIM
Continuous line: HPM; Dashed line: VIM
1
u ( x)u '' ( x) = 0
2
u (0) = 0, u ' (0) = 0, u ' (∞) = 1
u ''' ( x) +
We reformulate (2) by introducing a new
dependent variable:
y ( x) = Bu ( Bx)
P2 :
(14)
1 1
y ( x)
B B
1
y ( x) y " ( x) = 0
2
(21)
d3
d2
d2
y ( x) + 0 .5 y 0 ( x)( 2 y1 ( x)) + 0.5 y1 ( x)( 2 y 0 ( x)) = 0 (22)
3 2
dx
dx
dx
1
2
x2
1 5
x
240
1145833333 8
y 2 ( x) =
x
168 × 1011
(16)
And when P →1, then we can obtain:
y = y 0 + y1 + y 2 =
This also means that we can impose the condition
y '' (0) = 1 , which means that u '' (0) = 1 . Moreover, it is
1145833333 8
1 5 1 2
x −
x + x
240
2
168 × 1011
The exact solution of VIM obtained by AbdulMajid (2007) is compared with HPM in Fig. 2.
According to Fig. 2, it can be seen that the curves
are approximately agreed.
B3
clear from Eq. (17) that:
(18)
CONCLUSION
Now we construct the following homotopy for Eq. (2):
(1 − p)(
d3
d2
y1 ( x) + 0 .5 y0 ( x)( 2 y0 ( x) = 0
3
dx
dx
y 1 ( x) = −
y (0) = 0, y ' (0) = 0, y ' ( x) = B 2u ' ( Bx) → B 2 as x → ∞ (17)
y ' (∞ )
P1 :
y 0 ( x) =
Consequently, we have the conditions:
B=
(20)
(15)
B is a parameter that should be determined. As a
result, Eq. (2) will be the same as the equation for u:
y "' ( x ) +
d3
y0 ( x) = 0
dx 3
Solving Eq. (20) to (22) subjected to initial
condition given by Eq. (17), we have:
That is equivalent:
u ( x) =
P0 :
In this study, the Homotopy Perturbation Method
(HPM) has been applied to achieve the solution of
Blasius equation which is one of the basic equations of
fluid dynamics. It has been shown that HPM provides
very accurate numerical solutions for non-liner
problems in comparison with Variation Iteration
Method (VIM). The numerical results we achieved here
d3
d3
d2
y ( x)) + P( 3 y ( x) + 0.5U ( X )( 2 y ( x)) = 0 (19)
3
dx
dx
dx
Substituting y from Eq. (8) into (19) and
rearranging based on powers of p-terms, we have:
3
Res. J. Math. Stat., 5(1-2): 01-04, 2013
justify the advantage of this method and it can be
concluded that HPM can successfully solve the two
forms of Blasius equations; however, the results are
more accurate for first form of Blasius equation.
Belhachmi, Z., B. Bright and K. Taous, 2000. On the
concave solutions of the Blasius equations. Acta
Mat. Univ., Comenianae, LXIX(2): 199-214.
Datta, B.K., 2003. Analytic solution for the Blasius
equation. Indian J. Pure Appl. Math., 34(2):
237-240.
Ganji, D.D. and A. Sadighi, 2007. Application of
Homotopy perturbation and variation iteration
methods to nonlinear heat transfer and porous
media equations. J. Comput. Appl. Math., 207(1,
1): 24-34.
He, J.H., 1998. Approximate analytical solution of
Blasius equation. Commun. Nonlinear Sci. Numer.
Simul., 3(4): 260-263.
Kuiken, H.K., 1981. A backward free-convective
boundary layer. Quart. J. Mech. Appl. Math., 34:
397-413.
Nayfeh, A.H., 2000. Perturbation Methods. Wiley, New
York.
ACKNOWLEDGMENT
The authors would like to thank, Prof. Abdul-Majid
Wazwaz, who sent us some of his valuable papers.
Also, the first author would like to appreciate reviewers
and journal’s editor-in-chief.
REFERENCES
Abdul-Majid, W., 2007. The variation iteration method
for solving two forms of Blasius equation on a
half-infinite domain. Appl. Math. Comput., 188:
485-491.
Adomian, G., 1994. Solving Frontier Problems of
Physics: The Decomposition Method. Kluwer
Academic Press, Dordrecht.
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