We will consider variational iteration method (VIM) and Padé approximant, for finding analytical solutions of the Glauert-jet (self-similar wall jet over an impermeable, resting plane surface) problem. The solutions are compared with the exact solution. The results illustrate that VIM is an attractive method in solving the systems of nonlinear equations. It is predicted that VIM can have a found wide application in engineering problems

Hamed Shahmohamadi

Mohammad Mehdi Rashidi

Mathematical Problems in Engineering

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Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 501476, 7 pages
doi:10.1155/2010/501476
Research Article
A Novel Solution for the Glauert-Jet Problem by
Variational Iteration Method-Pade´ Approximant
Hamed Shahmohamadi and Mohammad Mehdi Rashidi
Engineering Faculty of Bu-Ali Sina University, P. O. Box 65175-4161, 3146984738 Hamedan, Iran
Correspondence should be addressed to Hamed Shahmohamadi, hamed shah2006@yahoo.com
Received 19 December 2009; Accepted 11 February 2010
Academic Editor: Jihuan Huan He
Copyright q 2010 H. Shahmohamadi and M. M. Rashidi. 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 will consider variational iteration method VIM and Pade´ approximant, for finding analytical
solutions of the Glauert-jet self-similar wall jet over an impermeable, resting plane surface
problem. The solutions are compared with the exact solution. The results illustrate that VIM is
an attractive method in solving the systems of nonlinear equations. It is predicted that VIM can
have a found wide application in engineering problems.
1. Introduction
Nonlinear phenomena play a crucial role in applied mathematics and physics. We know that
most of engineering problems are nonlinear, and it is diﬃcult to solve them analytically.
Various powerful mathematical methods have been proposed for obtaining exact and
approximate analytic solutions.
The VIM was first proposed by He 1, 2 systematically illustrated in 1999 3, and
used to give approximate solutions of the problem of seepage flow in porous media with
fractional derivatives. The VIM is useful to obtain exact and approximate solutions of
linear and nonlinear diﬀerential equations. In this method, general Lagrange multipliers
are introduced to construct correction functionals for the problems. The multipliers can
be identified optimally via the variational theory. There is no need of linearization or
discretization, and large computational work and round-oﬀ errors are avoided. It has been
used to solve eﬀectively, easily, and accurately a large class of nonlinear problems with
approximation 4, 5. It was shown by many authors 6–16 that this method is more powerful
than existing techniques such as the Adomian method 17, 18. He et al. 19 proposed
three standard variational iteration algorithms for solving diﬀerential equations, integro-
diﬀerential equations, fractional diﬀerential equations, diﬀerential-diﬀerence equations, and
fractional/fractal diﬀerential-diﬀerence equations. The algorithm used in this paper belongs
2 Mathematical Problems in Engineering
to the variational iteration algorithm-I according to 19. Recently Herisanu and Marinca
20 suggested an optimal variational iteration algorithm using the variational iteration
method for nonlinear oscillator. Ismail and Abde Rabboh 21 presented a restrictive
Pade´ approximation for the generalized Fisher and Burger-Fisher equations. Rashidi and
Shahmohamadi 22 provided a new technique for solution of three-dimensional Navier-
Stokes equations for the flow near an infinite rotating disk by combination of variational
iteration method and Pade´ approximant.
The motivation of this letter is to extend variational iteration method and Pade´
approximant to solve Glauert-jet problem 23, also known in the Russian literature as the
Akatnov results 24. The Glauert-jet named also the plane wall jet is a thin jet of fluid that
flows tangentially to an impermeable, resting wall surrounded by fluid of the same type
in a quiescent ambient flow. It consists of an inner region wherein the flow resembles the
boundary layer and an outer region wherein the flow is like a free shear layer. The wall jets
are of great engineering importance with many applications. Main applications are turbine
blade cooling, paint spray, and air-foils in high-lift configurations 25.
We consider the self-similar plane wall jet formed over an impermeable resting wall
governed by the equation of a steady boundary layer over a flat plate see 26
∂ψ
(
x;y
)
∂y
∂2ψ
(
x;y
)
∂x∂y
− ∂ψ
(
x;y
)
∂x
∂2ψ
(
x;y
)
∂y2

∂3ψ
(
x;y
)
∂y3
, 1.1
with the dimensionless stream function
ψ
(
x, y
)
 4x1/4ft, t  x−3/4y, 1.2
where ψx, y is the stream function. The system deals with the following impermeability,
no-slip, and asymptotic conditions:
ψx, 0  0,
∂ψx; 0
∂y
 0,
∂ψ
(
x;y
)
∂y
−→ 0 as y −→ ∞. 1.3
The self-similar part ft satisfies the ordinary diﬀerential equation
f ′′′ 	 ff ′′ 	 2f
′2  0, 1.4
along with the boundary conditions
f0  0, f ′0  0, f ′t −→ 0 as t −→ ∞. 1.5
The Glauert-jet solution corresponds to the normalisation f∞  1 of the stream function. It
leads to the well-known implicit form of the analytical solution of the problem 1.4 and 1.5
found by Glauert 23
t  31/2 arctan
[(
3f
)1/2
2 	 f1/2
]
	 ln
⎡
⎣
(
1 	 f 	 f1/2
)1/2
1 − f1/2
⎤
⎦. 1.6
Mathematical Problems in Engineering 3
The downstream velocity profile f ′t of the Glauert-jet solution is
f ′t 
2
3
√
f
(
1 −
(√
f
)3)
, 1.7
with the skin friction
f ′′0 
2
9
. 1.8
2. Basic Concepts of VIM
To illustrate the basic concepts of VIM, we consider the following diﬀerential equation:
Lu 	Nu  gt, 2.1
where L, N, and gt are the linear operator, the nonlinear operator, and a heterogeneous
term, respectively. The variational iteration method was proposed by He where a correction
functional for 2.1 can be written as
un	1t  unt 	
∫ t
0
λ
[
Lunτ 	Nu˜nτ − gτ
]
dτ, n ≥ 0. 2.2
It is obvious that the successive approximations, uj, j ≥ 0 can be established by determining
λ, a general Lagrangian multiplier, which can be identified optimally via the variational
theory. The function u˜n is a restricted variation which means δu˜n  0. Therefore, we first
determine the Lagrange multiplier λ that will be identified optimally via integration by parts.
The successive approximations un	1t, n ≥ 0 of the solution ut will be readily obtained
upon using the obtained Lagrange multiplier and by using any selective function u0. With λ
determined, then several approximations ujt, j ≥ 0, follow immediately. Consequently, the
exact solution may be obtained by using
u  lim
n→∞
un. 2.3
3. Analytical Solution
In order to obtain VIM solution of 1.4, we construct a correction functional which reads
fn	1t  fnt 	
∫ t
0
λ
⎛
⎝∂
3fnτ
∂τ3
	 f˜nτ
∂2f˜nτ
∂τ2
	 2
(
∂f˜nτ
∂τ
)2⎞
⎠dτ, 3.1
where λ is the general Lagrangian multiplier which is to be determined later and f˜nτ is
considered as a restricted variation, that is, δf˜nτ  0.
4 Mathematical Problems in Engineering
Its stationary conditions can be obtained as follows:
1 	 λ′′τ
∣
∣
τt  0 , λτ|τt  0 , λ′τ
∣
∣
τt  0, λ
′′′τ  0. 3.2
The Lagrange multiplier can be identified as
λ  −1
2
τ − t2. 3.3
As a result, the following variational iteration formula can be obtained
fn	1t  fnt − 12
∫ t
0
τ − t2
⎛
⎝∂
3fnτ
∂τ3
	 f˜nτ
∂2f˜nτ
∂τ2
	 2
(
∂f˜nτ
∂τ
)2⎞
⎠dτ. 3.4
Now we must start with an arbitrary initial approximation. Therefore according to 1.5 and
1.8, it is straight-forward to choose an initial guess
fot 
1
9
t2. 3.5
Using the above variational formula 3.4, we have
f1t  f0t − 12
∫ t
0
τ − t2
⎛
⎝∂
3f0τ
∂τ3
	 f˜0τ
∂2f˜0τ
∂τ2
	 2
(
∂f˜0τ
∂τ
)2⎞
⎠dτ. 3.6
Substituting 3.5 into 3.6, we have
f1t 
1
9
t2 − 1
486
t5. 3.7
By the same way, we can obtain f2t, f3t, . . . .
After obtaining the result of 7th iteration, we will apply the pade´ approximation using
symbolic software such as Mathematica; we have the following:
ft12,12  2184t
2
(
787679232592245840 	 17256587913623520t3 	 99910515826755t6
	135721670638t9
)/(
15482622995833184231040 	 625910732693761394880t3
	7676359653888591120t6 	 29381362861683924t9 	 19389517508143t12
)
.
3.8
Therefore, we are able to give an approximate solution of the considered problem.
Mathematical Problems in Engineering 5
0
0.2
0.4
0.6
0.8
1
f
0 1 2 3 4 5 6 7
t
Exact
VIM Pade`
Figure 1: Comparison of the exact solution 1.6 with the VIM-Pade´ solution 3.8.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
f ′
0 1 2 3 4 5 6 7
t
Exact
VIM Pade`
Figure 2: Comparison of the exact derivative solution 1.7 with the derivative of the VIM-Pade´ solution
3.8.
4. Discussion
In this paper, VIM and Pade´ approximants are used to find approximate solutions of the
famous Glauert-jet problem. The problem of fluid jet along an impermeable, resting wall
surrounded by fluid of the same type at rest has been considered. The closed-form solution
of the corresponding boundary layer equations 1.6 was given by Glauert 23, known as the
plane wall jet, e-jet, or the exponentially decaying wall jet 25.
The VIM-Pade´ solution of ft has been compared with the exact solution in Figure 1.
Table 1 demonstrates the values of absolute error between VIM-Pade´ solution and the exact
solution for diﬀerent values of t. In order to give a comprehensive approach of the problem,
the derivative of the VIM-Pade´ solution 3.8 graph is given as well as the exact downstream
velocity profile 1.7 graph, as shown in Figure 2. Table 2 demonstrates the values of absolute
error between the derivative of the VIM-Pade´ solution 3.8 and the exact derivative solution
6 Mathematical Problems in Engineering
Table 1: Comparison of the exact solution 1.6 with the VIM-Pade´ solution 3.8.
t VIM-Pade´ Exact results Absolute error
0.5 0.0277 0.0277 0
1 0.1091 0.1091 0
1.5 0.2354 0.2354 0
2 0.3879 0.3879 0
2.5 0.5421 0.5421 0
3 0.6774 0.6774 0
3.5 0.7833 0.7833 0
4 0.8595 0.8595 0
4.5 0.9111 0.9111 0
5 0.9446 0.9446 0
5.5 0.9657 0.9658 0.0001
6 0.9786 0.9791 0.0005
6.5 0.9863 0.9872 0.0009
7 0.9901 0.9922 0.0021
Table 2: Comparison of the exact derivative solution 1.7 with the derivative of the VIM-Pade´ solution
3.8.
t VIM-Pade´ Exact results Absolute error
0.5 0.1105 0.1105 0
1 0.2123 0.2123 0
1.5 0.2865 0.2865 0
2 0.3149 0.3149 0
2.5 0.2949 0.2949 0
3 0.2428 0.2428 0
3.5 0.1810 0.1810 0
4 0.1256 0.1256 0
4.5 0.0829 0.0830 0.0001
5 0.0530 0.0531 0.0001
5.5 0.0329 0.0333 0.0004
6 0.0198 0.0206 0.0008
6.5 0.0111 0.0126 0.0015
7 0.0049 0.0077 0.0028
1.7 for diﬀerent values of t.The accuracy of the method is very good, and the obtained
results are near to the exact solution. The result show that the VIM is a powerful methods has
high accuracy, and, very eﬃcient.We sincerely hope this method can be applied in a wider
range.
References
1 J.-H. He, “Approximate solution of nonlinear diﬀerential equations with convolution product
nonlinearities,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 69–73,
1998.
2 J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous
media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 12, pp. 57–68, 1998.
Mathematical Problems in Engineering 7
3 J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,”
International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
4 H. Tari, D. D. Ganji, and M. Rostamian, “Approximate solutions of K2,2, KdV and modified KdV
equations by variational iteration method, homotopy perturbation method and homotopy analysis
method,” International Journal of Nonlinear Science and Numerical Simulation, vol. 8, no. 2, pp. 203–210,
2007.
5 E. Yusufoglu, “Variational iteration method for construction of some compact and noncompact
structures of Klein-Gordon equations,” International Journal of Nonlinear Sciences and Numerical
Simulation, vol. 8, no. 2, pp. 153–158, 2007.
6 M. M. Rashidi, H. Shahmohamadi, and G. Domairry, “Variational iteration method for solving three-
dimensional Navier-Stokes equations of flow between two stretchable disks,” Numerical Methods for
Solving Partial Diﬀerential Equations. In press.
7 S. Momani and S. Abuasad, “Application of He’s variational iteration method to helmholtz equation,”
Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1119–1123, 2005.
8 M. A. Abdou and A. A. Soliman, “Variational iteration method for solving Burger’s and coupled
Burger’s equations,” Journal of Computational and Applied Mathematics, vol. 181, no. 2, pp. 245–251,
2005.
9 A. M. Wazwaz, “A comparison between the variational iteration method and Adomian decomposi-
tion method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 129–136, 2007.
10 A. M. Wazwaz, “The variational iteration method for rational solutions for KdV, K2,2, Burgers, and
cubic Boussinesq equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 18–
23, 2007.
11 A. M. Wazwaz, Partial Diﬀerential Equations: Methods and Applications, Balkema, Rotterdam, The
Netherlands, 2002.
12 A.-M. Wazwaz, “Necessary conditions for the appearance of noise terms in decomposition solution
series,” Applied Mathematics and Computation, vol. 81, no. 2-3, pp. 265–274, 1997.
13 A.-M. Wazwaz, “A new technique for calculating Adomian polynomials for nonlinear polynomials,”
Applied Mathematics and Computation, vol. 111, no. 1, pp. 33–51, 2000.
14 A.-M. Wazwaz, “A new method for solving singular initial value problems in the second-order
ordinary diﬀerential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45–57, 2002.
15 A.-M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, 1997.
16 A.-M. Wazwaz, “Analytical approximations and Pade´ approximants for volterra’s population
model,” Applied Mathematics and Computation, vol. 100, no. 1, pp. 13–25, 1999.
17 G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic
Publishers, Boston, Mass, USA, 1994.
18 G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of
Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.
19 J. H. He, G. C. Wu, and F. Austin, “The variational iteration method which should be followed,”
Nonlinear Science Letters A, vol. 1, no. 1, pp. 1–30, 2010.
20 N. Herisanu and V. Marinca, “A modified variational iteration method for strongly nonlinear
problems,” Nonlinear Science Letters A, vol. 1, pp. 183–192, 2010.
21 H. N. A. Ismail and A. A. Abde Rabboh, “A restrictive Pade´ approximation for the solution of the
generalized Fisher and Burger-Fisher equations,” Applied Mathematics and Computation, vol. 154, no.
1, pp. 203–210, 2004.
22 M. M. Rashidi and H. Shahmohamadi, “Analytical solution of three-dimensional Navier-Stokes
equations for the flow near an infinite rotating disk,” Communications in Nonlinear Science and
Numerical Simulation, vol. 14, no. 7, pp. 2999–3006, 2009.
23 M. B Glauert, “The wall jet,” Journal of Fluid Mechanics, vol. 1, pp. 625–643, 1956.
24 N. I. Akatnov, “Development of 2D laminar fluid jet along a solid surface,” Proceedings of LIP Technical
Hydromechanics, vol. 5, pp. 24–31, 1953.
25 E. Magyari and B. Keller, “The algebraically decaying wall jet,” European Journal of Mechanics B, vol.
23, no. 4, pp. 601–605, 2004.
26 Y. Bouremel, “Explicit series solution for the Glauert-jet problem by means of the homotopy analysis
method,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 5, pp. 714–724,
2007.
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A Novel Solution for the Glauert-Jet Problem by Variational Iteration Method-Padé Approximant

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A Novel Solution for the Glauert-Jet Problem by Variational Iteration Method-Padé Approximant

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