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A novel method for nonlinear singular fourth order four-point boundary
value problems
X.Y. Li
∗
, B.Y. Wu
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, PR China
a r t i c l e
i n f o
Article history: Received 27 December 2009 Accepted 13 April 2011 Keywords: NonlinearSingular fourth order four-point boundary value problems
Homotopy perturbation method Reproducing kernel method
a b s t r a c t
In this paper, a novel method is proposed for solving nonlinear singular fourth order
four-point boundary value problems (BVPs) by combining advantages of the homotopy
perturbed method (HPM) and the reproducing kernel method (RKM). Some numerical
examples are presented to illustrate the strength of the method.
©
2011 Elsevier Ltd. All rights reserved.
1. Introduction
In this paper, we consider the following nonlinear singular fourth order four-point boundary value problems:
a
0(
x
)
u
′′′′(
x
)
+
a
1(
x
)
u
′′′(
x
)
+
a
2(
x
)
u
′′(
x
)
+
a
3(
x
)
u
′(
x
)
+
a
4(
x
)
u
(
x
)
+
g
(
x
,
u
)
=
f
(
x
),
0
<
x
<
1
,
u
(
0
)
=
0
,
u
(α)
=
0
,
u
(β)
=
0
,
u
(
1
)
=
0
,
(1.1)
where
α, β
∈
(
0
,
1
),
a
i(
x
),
i
=
0
,
1
,
2
,
3
,
4
,
f
(
x
)
∈
C[
0
,
1
]
,
g
(
x
,
u
)
is a nonlinear function of
u
(
x
)
, and maybe
a
0(
0
)
=
0 or
a
0(
1
)
=
0. That is, the equation may be singular at
x
=
0
,
1. We consider
u
(
0
)
=
0
,
u
(α)
=
0
,
u
(β)
=
0
,
u
(
1
)
=
0 since
the boundary conditions
u
(
0
)
=
γ
0,
u
(α)
=
γ
1,
u
(β)
=
γ
2,
u
(
1
)
=
γ
3can be reduced to
u
(
0
)
=
0
,
u
(α)
=
0
,
u
(β)
=
0
,
u
(
1
)
=
0.
Multi-point boundary value problems arise in various areas of applied mathematics and physics. Two-point boundary
value problems have been extensively studied in the literature. Also, the existence and multiplicity of solutions of fourth
order four-point boundary value problems have been studied by many authors; see [
1–8
] and the references therein.
However, the research for methods for solving multi-point boundary value problems has proceeded very slowly. Geng [
9
]
proposed a method for a class of second order three-point BVPs by converting the original problem into an equivalent
integro-differential equation. Geng and Cui [
10
] presented a method for solving nonlinear multi-point boundary value
problems by combining homotopy perturbation and variational iteration methods. Dehghan and Tatari [
11
] used the
Adomian decomposition method for solving multi-point boundary value problems. However, for numerical methods for
solving singular multi-point BVPs, few works are available. Li and Wu [
12
] introduced the reproducing kernel method for
solving linear singular fourth order four-point BVPs.
In this paper, we will give the analytic approximation of nonlinear singular fourth order four-point boundary value
problems
(1.1)
by combining HPM and RKM for solving linear singular fourth order four-point BVPs presented in [
12
].
∗
Corresponding author.E-mail address:[email protected](X.Y. Li).
0898-1221/$ – see front matter©2011 Elsevier Ltd. All rights reserved.
Reproducing kernel theory has important applications in numerical analysis, differential equation, probability and
statistics and so on [
9
,
10
,
12–25
]. Recently, Cui, Geng, Lin and Chen have presented RKM for singular linear two-point
boundary value problem, singular nonlinear two-point periodic boundary value problem, nonlinear system of boundary
value problems and nonlinear partial differential equations.
The HPM was proposed originally by He [
26–36
] and was applied to many nonlinear problems [
37–45
]. This method is
based on the use of the traditional perturbation method and homotopy technique. Using this method, a rapid convergent
series solution can be obtained in most cases.
In terms of HPM,
(1.1)
can be reduced to a series of linear singular fourth order four-point BVPs. In our previous work,
it is shown that RKM is an effective method for solving linear singular fourth order four-point BVPs. Hence, we introduce a
new reliable method for nonlinear singular fourth order four-point BVPs by the conjunction of HPM and RKM.
The rest of the paper is organized as follows. In the next section, the HPM is introduced. The conjunction of HPM and
RKM are applied to
(1.1)
in Section
3
. The numerical examples are presented in Section
4
. Section
5
ends this paper with a
brief conclusion.
2. Analysis of HPM
To illustrate the basic ideas of this method, we consider the following nonlinear differential equation:
A
(
u
)
−
f
(
r
)
=
0
,
r
∈
Ω
,
(2.1)
with the boundary conditions of
B
(
u
, ∂
u
/∂
n
)
=
0
,
r
∈
Γ
,
(2.2)
where
A
is a general differential operator,
B
a boundary operator,
f
(
r
)
a known analytical function and
Γ
is the boundary of
the domain
Ω
.
Generally speaking, the operator
A
can be divided into parts which are
L
and
N
, where
L
is linear, but
N
is nonlinear.
(2.1)
can therefore be rewritten as
L
(
u
)
+
N
(
u
)
−
f
(
r
)
=
0
,
r
∈
Ω
.
(2.3)
By the homotopy technique, we construct a homotopy
V
(
r
,
p
)
:
Ω
× [
0
,
1
] →
R
which satisfies:
H
(
V
,
p
)
=
(
1
−
p
)
[L
(
V
)
−
L
(
u
0)
] +
p[A
(
V
)
−
f
(
r
)
] =
0
,
p
∈ [
0
,
1
]
,
r
∈
Ω
,
(2.4)
or
H
(
V
,
p
)
=
L
(
V
)
−
L
(
u
0)
+
pL
(
u
0)
+
p[N
(
V
)
−
f
(
r
)
] =
0
,
p
∈ [
0
,
1
]
,
r
∈
Ω
,
(2.5)
where
p
∈ [
0
,
1
]
is an embedding parameter,
u
0is an initial approximation of
(2.1)
, which satisfies the boundary conditions.
Obviously, from
(2.4)
or
(2.5)
, one obtains
H
(
V
,
0
)
=
L
(
V
)
−
L
(
u
0)
=
0
,
(2.6)
H
(
V
,
1
)
=
A
(
V
)
−
f
(
r
)
=
0
,
(2.7)
the changing process of
p
from zero to unity is just that of
V
(
r
,
p
)
from
u
0(
r
)
to
u
(
r
)
. In topology, this is called deformation,
and
L
(
V
)
−
L
(
u
0)
and
A
(
V
)
−
f
(
r
)
are called homotopy.
According to the HMP, we can first use the embedding parameter
p
as a ‘‘small parameter’’, and assume that the solution
of
(2.4)
or
(2.5)
can be written as a power series in
p
:
V
=
V
0+
pV
1+
p
2V
2+ · · ·
.
(2.8)
Setting
p
=
1 results in the approximate solution of Eq.
(2.1)
:
u
=
lim
p→1V
=
V
0+
V
1+
V
2+ · · ·
.
(2.9)
The combination of the perturbation method and the homotopy method is called the HPM, which has eliminated the
limitations of traditional perturbation methods. On the other hand, this technique is of full advantage of traditional
perturbation techniques. The series
(2.9)
is convergent in most cases. However, the convergent rate depends on the nonlinear
operator
A
(
V
)
(the following opinions are suggested by He [
34
])
(1) The second derivative of
N
(
V
)
with respect to
V
must be small because the parameter may be relatively large, i.e.,
p
→
1.
(2) The norm of
L
−1(∂
N
/∂
V
)
must be smaller than one so that the series converges.
3. Combined HP–RKM for
(1.1)
For
(1.1)
, according to the HPM, we construct a homotopy as follows:
a
0(
x
)
u
′′′′(
x
)
+
a
1(
x
)
u
′′′(
x
)
+
a
2(
x
)
u
′′(
x
)
+
a
3(
x
)
u
′(
x
)
+
a
4(
x
)
u
(
x
)
−
f
(
x
)
+
pg
(
x
,
u
)
=
0
(3.1)
where
p
∈ [
0
,
1
]
is an embedding parameter. In case
p
=
0,
(3.1)
becomes a linear equation, which is easy to be solved, and
when
p
=
1,
(3.1)
turns out to be the original one,
(1.1)
.
In view of the HPM, we use the homotopy parameter
p
to expand the solution
u
=
u
0+
pu
1+
p
2u
2+
p
3u
3· · ·
.
(3.2)
The approximate solution of
(1.1)
can be obtained by setting
p
=
1
u
=
u
0+
u
1+
u
2+
u
3· · ·
.
(3.3)
Substituting
(3.2)
into
(3.1)
, and equating coefficients of the identical powers of
p
yields the following equations:
p
i:
Lu
i(
x
)
=
f
i(
x
),
u
i(
0
)
=
0
,
u
i(α)
=
0
,
u
i(β)
=
0
,
u
i(
1
)
=
0
,
i
=
0
,
1
,
2
, . . .
where
Lu
(
x
)
=
a
0(
x
)
u
′′′′(
x
)
+
a
1(
x
)
u
′′′(
x
)
+
a
2(
x
)
u
′′(
x
)
+
a
3(
x
)
u
′(
x
)
+
a
4(
x
)
u
(
x
),
f
0(
x
)
=
f
(
x
)
,
f
i(
x
)
= −
d
i−1g
(
x
,
u
)
(
i
−
1
)
!
d
p
i−1
p=0(
x
),
i
=
1
,
2
, . . . .
By using RKM presented in [
12
] for solving linear singular fourth order four-point BVPs, one can obtain
u
i,
i
=
0
,
1
,
2
. . .
u
i(
x
)
=
∞−
j=1 j−
k=1β
jkf
i(
x
k)ψ
j(
x
).
(3.4)
Therefore, the approximate solution of
(1.1)
and
m
-term approximation to this solution are obtained
U
(
x
)
=
∞−
k=0u
k(
x
),
U
m(
x
)
=
m−1−
k=0u
k(
x
).
(3.5)
Now, the approximate solution
U
m,n(
x
)
can be obtained by the
n
-term intercept of the
u
k(
x
),
k
=
0
,
1
,
2
, . . .
, and
U
m,n(
x
)
=
m−1−
k=0 n−
i=1A
ikψ
i(
x
),
(3.6)
where
A
ik=
∑
i j=1β
ijf
k(
x
j)
.
4. Numerical examples
In this section, we present and discuss the numerical results by employing the present method for two examples. Results
demonstrate the present method is remarkably effective.
Example 4.1.
Consider the following singular fourth order four-point boundary value problem
x
4(
1
−
x
)
u
′′′′(
x
)
+
300e
2xu
′′′(
x
)
+
200 sin
√
xu
′′(
x
)
+
2
u
′(
x
)
+
xu
3(
x
)
=
f
(
x
),
0
≤
x
≤
1
,
u
(
0
)
=
0
,
u
1
2
=
sinh
1
2
,
u
1
4
=
sinh
1
4
,
u
(
1
)
=
sinh 1
,
where
f
(
x
)
=
(
2
+
300e
x/2)
cosh
x
+
sinh
x
(
−x
5+
x
4+
x
sinh
2x+
200 sin
(
√
x
))
. The exact solution is given by
u
(
x
)
=
sinh
x
.
According to
(3.4)–(3.6)
, one can obtain the approximation
U
m,n(
x
)
of
u
(
x
)
. When we take
m
=
3
,
n
=
6
,
11
,
51, and
x
i=
ni,
i
=
1
,
2
, . . . ,
n
, the absolute errors
|u
n(
x
)
−
u
(
x
)
|
between the approximate solution and exact solution are given in
Fig. 1
.
Example 4.2.
Consider the following singular fourth order four-point boundary value problem
sin
x
(
e
x−
1
)
2u
′′′′(
x
)
+
e
x2u
′′′(
x
)
+
2 sin
√
xu
′′(
x
)
+
sinh
xu
′(
x
)
+
x
sin
u
=
f
(
x
),
0
≤
x
≤
1
,
u
(
0
)
=
0
,
u
1
3
=
sin
1
3
,
u
2
3
=
sin
2
3
,
u
(
1
)
=
sin 1
,
where
f
(
x
)
=
(
−
1
+
e
x)
2sin
2x
−
2 sin
(
√
x
)
sin
x
−
e
x/2cos
x
+
x
sin
(
sin
x
)
+
cos
x
sinh
x
. The exact solution is given by
u
(
x
)
=
sin
x
. According to
(3.4)–(3.6)
, one can obtain the approximation
U
m,n(
x
)
of
u
(
x
)
. When we take
m
=
3
,
n
=
6
,
11
,
51,
and
x
i=
ni,
i
=
1
,
2
, . . . ,
n
, the absolute errors
|
u
n(
x
)
−
u
(
x
)
|
between the approximate solution and exact solution are
given in
Fig. 2
.
Fig. 1. Absolute errors
|
u(
x)
−
U3,6(
x)
|
,
|
u(
x)
−
U3,11(
x)
|
,
|
u(
x)
−
U3,51(
x)
|
forExample 4.1.Fig. 2. Absolute errors
|
u(
x)
−
U3,6(
x)
|
,
|
u(
x)
−
U3,11(
x)
|
,
|
u(
x)
−
U3,51(
x)
|
forExample 4.2.Remark.
For
Examples 4.1
and
4.2
, it is difficult to solve them directly using the existing method such as the decomposition
method and the variational iteration method. However, the present method can solve them successfully.
5. Conclusion
In this paper, the combination of homotopy perturbation and reproducing kernel methods was employed successfully for
solving nonlinear singular fourth order four-point boundary value problems. The numerical results show that the present
method is an accurate and reliable analytical technique for nonlinear singular fourth order four-point boundary value
problems.
Acknowledgments
The authors would like to thank the unknown referees for their careful reading and helpful comments. The work was
supported by the National Natural Science Foundation of China (Grant No. 11026200).
References
[1] S.H. Chen, W. Ni, C.P. Wang, Positive solution of fourth order ordinary differential equation with four-point boundary conditions, Applied Mathematics Letters 19 (2006) 161–168.
[2] I.Y. Karaca, Fourth-order four-point boundary value problem on time scales, Applied Mathematics Letters 21 (2008) 1057–1063.
[3] X.G. Zhang, L.S. Liu, Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator, Journal of Mathematical Analysis and Applications 336 (2007) 1414–1423.
[4] C.Z. Bai, D.D. Yang, H.B. Zhu, Existence of solutions for fourth order differential equation with four-point boundary conditions, Applied Mathematics Letters 20 (2007) 1131–1136.
[5] D.X. Ma, X.Z. Yang, Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 223 (2009) 543–551.
[6] Q. Zhang, S.H. Chen, J.H. Lv, Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 196 (2006) 387–393.
[7] Y.L. Zhong, S.H. Chen, C.P. Wang, Existence results for a fourth-order ordinary differential equation with a four-point boundary condition, Applied Mathematics Letters 21 (2008) 465–470.
[8] R.P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinary differential equations with singularities, Journal of Mathematical Analysis and Applications 297 (2004) 131–151.
[9] F.Z. Geng, Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Applied Mathematics and Computation 215 (2009) 2095–2102.
[10] F.Z. Geng, M.G. Cui, Solving nonlinear multi-point boundary value problems by combining homotopy perturbation and variational iteration methods, International Journal of Nonlinear Sciences and Numerical Simulation 10 (5) (2009) 597–600.
[11] M. Tatari, M. Dehghan, The use of the Adomian decomposition method for solving multipoint boundary value problems, Physica Scripta 73 (2006) 672–676.
[12] X.Y. Li, B.Y. Wu, Reproducing kernel method for singular fourth order four-point boundary value problems, Bulletin of the Malaysian Mathematical Sciences Society 34 (2011) 147–151.
[13] F.Z. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Applied Mathematics and Computation 213 (2009) 163–169.
[14] F.Z. Geng, M.G. Cui, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied Mathematics and Computation 192 (2007) 389–398.
[15] F.Z. Geng, M.G. Cui, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean Mathematical Society 45 (3) (2008) 77–87.
[16] F.Z. Geng, M.G. Cui, Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications 327 (2007) 1167–1181.
[17] F.Z. Geng, M.G. Cui, Analytical approximation to solutions of singularly perturbed boundary value problems, Bulletin of the Malaysian Mathematical Sciences Society 33 (2010) 221–232.
[18] M.G. Cui, F.Z. Geng, Solving singular two-point boundary value problem in reproducing kernel space, Journal of Computational and Applied Mathematics 205 (2007) 6–15.
[19] M.G. Cui, F.Z. Geng, A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied Mathematics and Computation 188 (2007) 1389–1401.
[20] M.G. Cui, Z. Chen, The exact solution of nonlinear age-structured population model, Nonlinear Analysis. Real World Applications 8 (2007) 1096–1112. [21] C.L. Li, M.G. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied Mathematics and
Computation 143 (2–3) (2003) 393–399.
[22] M.G. Cui, Y.Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Space, Nova Science Pub. Inc., 2009.
[23] Alain Berlinet, Christine Thomas-Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, 2004. [24] J. Du, M.G. Cui, Constructive approximation of solution for fourth-order nonlinear boundary value problems, Mathematical Methods in the Applied
Sciences 32 (2009) 723–737.
[25] H.M. Yao, Y.Z. Lin, Solving singular boundary-value problems of higher even-order, Journal of Computational and Applied Mathematics 223 (2009) 703–713.
[26] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262.
[27] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics 35 (1) (2000) 37–43.
[28] J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (1) (2003) 73–79. [29] J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (1) (2004)
287–292.
[30] J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation 156 (2) (2004) 527–539.
[31] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation 6 (2) (2005) 207–208.
[32] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Soliton and Fractal 26 (3) (2005) 695–700. [33] J.H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (1–2) (2006) 87–88.
[34] J.H. He, Non-perturbative methods for strongly nonlinear problems, Disertation, de-Verlag in GmbH, Berlin, 2006.
[35] J.H. He, Some asymptotic methods for strongly nonlinear equation, International Journal of Modern Physics B 20 (10) (2006) 1141–1149. [36] J.H. He, Addendum new interpretation of homotopy pertirbation method, International Journal of Modern Physics B 20 (18) (2006) 2561–2568. [37] M.A. Rana, A.M. Siddiqui, Q.K. Ghori, R. Qamar, Application of He’s homotopy perturbation method to Sumudu transform, International Journal of
Nonlinear Sciences and Numerical Simulation 8 (2) (2007) 185–190.
[38] E. Yusufoˇglu, Homotopy perturbation method for solving a nonlinear system of second order boundary value problems, International Journal of Nonlinear Sciences and Numerical Simulation 8 (3) (2007) 353–358.
[39] A. Ghorbani, J. Saberi-Nadjafi, He’s homotopy perturbation method for calculating adomian polynomials, International Journal of Nonlinear Sciences and Numerical Simulation 8 (2) (2007) 229–232.
[40] A. Beléndez, C. Pascual, A. Márquez, D.I. Méndez, Application of He’s homotopy perturbation method to the relativistic (An)harmonic oscillator. I: comparison between approximate and exact frequencies, International Journal of Nonlinear Sciences and Numerical Simulation 8 (4) (2007) 483–492. [41] A. Beléndez, C. Pascual, D.I. Méndez, M.L. álvarez, C. Neipp, Application of He’s homotopy perturbation method to the relativistic (An)harmonic
oscillator. II: a more accurate approximate solution, International Journal of Nonlinear Sciences and Numerical Simulation 8 (4) (2007) 493–504. [42] A. Yıldırım, An algorithm for solving the fractional nonlinear Schröinger equation by means of the homotopy perturbation method, International
Journal of Nonlinear Sciences and Numerical Simulation 10 (4) (2009) 445–450.
[43] A. Beléndez, C. Pascual, T. Belndez, A. Hernndez, Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities, Nonlinear Analysis. Real World Applications 10 (1) (2009) 416–427.
[44] M. Inc, E. Cavlak, Hes homotopy perturbation method for solving coupled- KdV equations, International Journal of Nonlinear Sciences and Numerical Simulation 10 (3) (2009) 333–340.
[45] D.D. Ganji, A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, Journal of Computational and Applied Mathematics 207 (1) (2007) 24–34.
