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Applied Mathematics, 2011, 2, 987-992
doi:10.4236/am.2011.28136 Published Online August 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
A New Analytical Approach for Solving Nonlinear
Boundary Value Problems in Finite Domains
Jafar Biazar1, Behzad Ghanbari2*
1Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2Department of Mat hem at i cs, Kermanshah University of Technology, Kermanshah, Iran
E-mail: [email protected]
Received November 4, 2010; revised May 25, 2011; accepted July 4, 2011
Abstract
Based on the homotopy analysis method (HAM), we propose an analytical approach for solving the follow-
ing type of nonlinear boundary value problems in finite domain. In framework of HAM a convenient way to
adjust and control the convergence region and rate of convergence of the obtained series solutions, by defin-
ing the so-called control parameter , is provided. This paper aims to propose an efficient way of finding
the proper values of . Such values of parameter can be determined at the any order of approximations of
HAM series solutions by solving of a nonlinear polynomial equation. Some examples of nonlinear initial
value problems in finite domain are used to illustrate the validity of the proposed approach. Numerical re-
sults confirm that obtained series solutions agree very well with the exact solutions.
Keywords: Homotopy Analysis Method, Boundary Value Problems, Finite Domain
1. Introduction
In this work, we consider the following typ e of nonlinear
boundary value problems in a finite domain (as consid-
ered in [1]):


()( 1)
,, ,,,
nn
uxfxuu uaxb
 
2
2
(1)
subject to th e two-point boundary conditions
 
 
()
01
(2)
01
,,,
,() ,,
rr
nr nr
uau aua
ub ubub
 
 



  (2)
where is an integer.
0rn
Also f is a polynomial in and
(1)
,,,
n,
x
uxu xux
 01 01 2
,,,,,,,
rnr


are real constants.
Since, such type of boundary value problems arise in
the mathematical modeling of the viscoelastic flows and
other branches of mathematical, physical and engineer-
ing sciences, the approximate solutions of these prob-
lems are of great importance. See [2-4] and the refer-
ences therein.
Some numerical and analytical methods such as
shooting method [5], Finite-element method [6], sinc-
Galerkin method [7], finite-difference method [8], Ado-
mian technique [9], variational iteration method (VIM)
[10], homotopy perturbation method (HPM) [11], analy-
sis method (HAM) [12], have been studied for obtaining
approximate solutions to boundary value problems.
The homotopy analysis method [13-17] is a general
analytic approach to get series solutions of various types
of nonlinear equations, including ordinary differential
equations, partial differential equations, differential-in-
tegral equations, differential-difference equation, and
coupled equations of them. Unlike perturbation methods,
the HAM is independent of small/large physical parame-
ters, and thus is valid no matter whether a nonlinear
problem contains small/large physical parameters or not.
More importantly, different from all perturbation and
traditional non-perturbation methods, the HAM provides
us a simple way to ensure the convergence of solution
series, and therefore, the HAM is valid even for strongly
nonlinear problems. Besides, different from all perturba-
tion and previous non-perturbation methods, the HAM
provides us with great freedom to choose proper base
functions to approximate a nonlinear problem.
These advantages make the method to be a powerful
and flexible tool in mathematics and engineering, which
can be readily distinguished from existing numerically
and analytically methods.
Up to now, this method has been successfully applying
this method to various nonlinear problems in science and
J. BIAZAR ET AL.
988
;
engineering. A systematic description of the method and
its applications are found in [15].
This paper is arranged in the following manner. In
Section 2, the HAM is applied to solve the problem of
nonlinear boundary value problems. In Section 3, the
basic idea of the present approach is described. Further-
more, some numerical examples are presented in Section
4. Finally, conclusions are drawn in Section 5.
2. The Implement of HAM to BVPs
In order to obtain a convergent series solution to the
nonlinear problem (1, 2), we first construct the zeroth
order deformation equation
  
0
1;pLxpu xpNxp



(3)
where is an embedding parameter,
[0,1]p0
is
a convergence-control parameter, and
;
x
p
is an
unknown function, respectively. According to (1), the
auxiliary linear operator is given by
L
 
;
;n
n
x
p
Lxp x


(4)
and the nonlinear operator is given by
N

1
1
;,,,
n
nn
Nxp fxx
xx
,
n


  



(5)
From (3), when and ,
0p1p
 
0
;0
x
ux
and
 
;1
x
ux
both hold. Therefore, as increases from 0 to 1, the
p
solution
;
x
p
varies from the initial guess
0
uxto
the solution
ux. Expanding
;
x
p
in Taylor series
with respect to, one has
p
 
1
;;0 i
i
i
x
px ux


p (6)
where
 
0
;
1
!
m
mm
p
xp
ux mp
(7)
Assuming that the series (6) is convergent at 1p
,
The solution series
 
01
;1 i
i
uxxuxu x

(8)
must be one of the solutions of the original problem (1,
2), as proved by Liao in [15].
Our next goal is to determine the higher order terms
(1
m
uxm). Define the vector
 
01
,,,uss
x
uxux ux (9)
Differentiating the zeroth order deformation Equation
(3) times with respect to , then setting mp0p
,
finally dividing them by , we obtain the th
order
deformation equation !mm

1mmmm
Lu xuxhRx





1
(10)
and its boundary cond itions
 
() 0
r
mm m
ua uaua
 (11)
 
(2) 0
nr
mm m
ub ubub


where
 

1
11
0
;
1
1!
m
mm
p
Nxp
Rx mp


(12)
and
1, 1
0, 1
m
m
m
(13)
From (4), th order deformation Equation (10) be-
comes m

() ()
1
nn
mmm m
uxu xhRx



1
1
(14)
In this way, the component solutions of , are
not only dependent upon ,
m
um
x
but also the auxiliary pa-
rameter . Thus, the convergence of the series (8) de-
pends on the parameter .
Finally, theth order approximation to the problem
(1, 2) can be generally expres sed by
m
 
01
m
m
i
ux uxux

i
(15)
As we know, to find a proper convergence-control pa-
rameter , to get a convergent series solution or to get a
faster convergent one, there is a classic way of plotting
the so-called ‘‘-curves” or ‘‘curves for convergence-
control parameter”. For example, one can consider the
convergence of
ux
and of a nonlinear diffe-

ux

rential equation
0Nux


to find a region say R
so that, each gives a convergent series solution
of such kind of quantities.
R
Such a region can be found, although approximately,
by plotting the curves of these unknown quantities versus
.
However, it is a pity that curves for convergence- con-
trol parameter (i.e. -curves) give us only a graphically
region and cannot tell us which value of
R
gives
the fastest convergent series. Furthermore, recently in
Copyright © 2011 SciRes. AM
J. BIAZAR ET AL.
989
[18] a misinterpreted usage of-curves has reported.
Although the solution series (15) given by different
values in the valid region of converge to the exact
solution, the convergence rates of these solution series
are usually different.
i
x

3. Proposed Approach
It should be noted that based on the zeroth-order defor-
mation Equation (3), th-order HAM ap proximation
of the solution , giving by
m

ux
 
01
m
mi
ux uux
is also dependent upon the convergence-control parame-
ter .
For
1
2ba

, let
 
 

()( 1)
,,
0
mm
nn
m
PNu
uf uu

,,
u
 
 

 
(16)
donate the residual error of the governing Equation (12)
at the th-order of HAM approximation. m
Since is indeed a polynomial equation of
order , so at each order of approximation , we can
gain the value of by solving only one algebraic equa-
tion.

0
m
P
mm
4. Numerical Examples
To demonstrate the efficiency of the proposed approach,
we consider several examples. For comparison purpose
they were taken from [1,12].
Example 1. Let’s consider the following second order
boundary value problem [1].
 
33
262,1u xuxuxxx
 2
(17)
with the boundary conditions
 
5
12,2 2
uu
(18)
which has the exact solution in the form of

1
uxx
x
 (19)
For the zeroth order deformation Equation (3), the
auxiliary linear operator is given by
L
 
2
2
;
;
x
p
Lxp x


and the nonlinear operator is given by
N

 
223
2
;
;2;6;
xp
Nxpxp xpx
x



2
In view of the boundary conditions (18), the initial
guess is determined as

2
057
22
ux xx

To obtain higher order terms , the th order
deformation Equation (10) and its boundary conditions
(11) are calculated:

m
ux m
11mmm m
uxux hRx

 


10, 20
mm
uu
where


 
3
11
1
1
00
621
2()
u
mm mmm
j
m
mjiji
ji
R xuxuxx
uxuxux

 


 




In this way, we can calculate
(1
m
uxm) recur-
sively.
In this example, from Equation (16) with 10m
, the
proper value for .obtained as .
0.364
As it can be seen in Table 1 and Figure 1, obtained
approximate series solution agrees very well with the
exact solution (19).
Example 2. As the example, let’s have the following
third order boundary value problem [1].

(3)2 10,0 1utututut x
 
 (20)
with the boundary conditions

001uu u
0
 (21)
This problem was considered in [9] via the finite dif-
ference method.
For the zeroth order deformation Equation (3), the
auxiliary linear operator is given by
L
 
3
3
;
;
x
p
Lxp x


and the nonlinear operator is given by
N
Table 1. Absolute errors of different methods in Example 1.
x
Exact
solution Shooting
method [5]10th HAM
for[1]
0.167
10th HAM
for 0.364
1.1x2.00909090.00067230.0001009 4.32e–6
1.3x
2.06923080.00216820.0001164 2.59e–7
1.5x
2.16666670.00330090.0004174 7.44e–6
1.7x2.28823530.00362130.0000210 8.15e–6
1.9x
2.42631580.00218490.0007363 1.80e–5
Copyright © 2011 SciRes. AM
990 J. BIAZAR ET AL.
Figure 1. Symbols: 5th-order HAM approximation for
0.364; solid line: exact solution.
 

32
3
2
;
;;
;1
2
;
x
px
Nxp xp
x
xp
x






 


p
x
In view of the boundary conditions (21), the initial
guess is determined as

32
0
uxx x

ux
.
To obtain higher order terms m, the th order
deformation Equation (10) and its boundary conditions
(11) are calculated:
m

  
(3) (3)1
00,00, 1
mmm m
mmm
uxuxhRx
uuu

 0
where


 
 
1
(3)11
0
1
1
01
um
mmmi mi
i
m
j
mj m
j
Rxuxuxu
uxu x
x






)
In this way , can be do ne re cursively .

(1
m
uxm
0.922
In this case, is obtained by our approach.
As shown in Table 2 and Figure 2, approximate series
solution using such value of parameter, is in excellent
agreement with the numerical solution given by the
Runge-Kutta-Fehlberg 4-5 technique.
Table 2. Absolute errors of different methods in Example 2.
x
Exact
solution Shooting
method [5] 10th HAM for
[1]
1.23
10th HAM for
0.922
0.1x 2.0090909 0.0006723 1.46e–6 3.96e–7
0.3x 2.0692308 0.0021682 1.41e–6 3.00e–11
0.5x 2.1666667 0.0033009 1.74e–6 1.19e–8
0.7x 2.2882353 0.0036213 2.00e–6 6.89e–8
0.9x 2.4263158 0.0021849 2.31e–6 1.02e–7
Figure 2. Solid line: 5th-order HAM approximation for
0.32683722
; symbols: numerical solution.
Example 3. As the example, let’s have the following
nonlinear fourth-order boundary value problem involving
a parameter [12].
c
 
2
(4)1, 02uxcux x
 (22)
with the boundary conditions
 
0022uu uu

0
  (23)
For the zeroth order deformation Equation (3), the
auxiliary linear operator is given by
L
 
4
4
;
;
x
p
Lxp x


and the nonlinear operato r is given by
N


42
4
;
;;
xp
Nxp cxp
x

1


In view of the boundary conditions (23), the initial
guess is determined as
43
044ux xxx2
To obtain higher order terms , the th order
deformation Equation (10) and its boundary conditions
(11) are calculated:

m
ux m

   
(4) (4)1
00, 00,20, 20
mmmm
mmmm
ux uxhRx
uuuu



where


 
1
(4)11
0
1um
mmmmi im
i
Rxuxcuxux


In this way
(1
m
uxm)
, can be done recursi vely.
Copyright © 2011 SciRes. AM
J. BIAZAR ET AL.
991
Table 3. Comparisons of of 10th-order HAM solu-
tions for different values of in Example 3.

t
0.2x 0.6x
1.0x 1.4x1.8x
1 [12] 1.5e–2 8.4e–2 6. 4e–3 7.49e–36.8e–3
1c 0.924 7.1e–7 4.3e–7 3.7e–7 2.0e–69.6e–6
=0.57 [12] 1.1e–2 1.2e–2 1.2e–2 1.2e–21.1e–2
5c =0.954 1.3e–4 1.3e–4 1.2e–4 1.2e–41.4e–4
0.634 [12] 1.3e–1 1.5e–1 1.5e–1 1.5e–11.3e–1
12c 0.655 1.0e–2 9.9e–3 9.4e–3 9.9e–31.1e–2
Here we will discuss following three cases of .
c
1) In the first case, we take as an example. For
and in (16), we get into a nonlinear
equation, which root is .
1c
0.924
1c10m

2) In second case, we take as an example.
Equation (16) with and has the real solu-
tion .
5c
105cm
=0.954
3) Finally, we take as an example. Equation
(16) with and , introduces
as a proper value for .
12c 10m
12c 0.655
Since the closed-form solution to the problem (22),
(23) is not available, the numerical solution numer is
calculated via the Runge-Kutta-Fehlberg 4-5 technique,
then, we compare the relative errors of the 10th order
u
HAM approximations at different points in the
appr
u
interval , using the formula
(0,1)

appr numer
numer
uu
tu
for different cases of , are reported in Table 3.
c
5. Conclusions
In this paper the solutions of a new way of finding the
control parameter in the homotopy analysis method is
proposed. It is shown for obtained values of such pa-
rameter, HAM approximation series leads to exact solu-
tion of problems or produces an approximate results
which are in a highly agreement with exact solution of
problems. All computations were done using Maple 13
with 15 digit floating point arithmetics (Digits: = 15).
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