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Computers and Mathematics with Applications
journal homepage:www.elsevier.com/locate/camwa
The variational iteration method for Cauchy problems
Xin-Wei Zhou
∗, Li Yao
Department of Mathematics, Kunming University, 2 Kunshi Road Kunming, Yunnan 650031, PR China
a r t i c l e i n f o
Article history:
Received 21 January 2010
Received in revised form 13 May 2010 Accepted 14 May 2010
Keywords:
Variational iteration method Cauchy problem
Transport equation Inviscid Burgers’ equation
a b s t r a c t
In this paper, the variational iteration method is used to solve the Cauchy problem. Some examples are given to elucidate the solution procedure and reliability of the obtained results. The variational iteration algorithm leads to exact solutions in the present study.
© 2010 Elsevier Ltd. All rights reserved.
1. Introduction
In Ref. [1], Ji-Huan He gave a very lucid as well as elementary discussion of the variational iteration method. The variational iteration algorithm [1–5] is of unheard simplicity, and the obtained results are of surprising accuracy (sometimes exact solutions can be obtained), the method can be easily understood by non-mathematical students and applied to various nonlinear problems. In addition, many authors gave great effort to give sophisticated theoretical verification of the variational iteration method, for example, Hesameddini and Latifizadeh [6] re-constructed the variational iteration algorithms using the well-known Laplace transform; Odibat [7], Salkuyeh [8], and Tatari & Dehghan [9] proved the variational iteration algorithm leads to convergent results. The method has been widely used to handle nonlinear models. The main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently, for example, Wazwaz applied the method to various nonlinear systems including nonlinear wave equations and nonlinear diffusion equations, and concluded that the variational iteration method is a reliable analytical tool for solving nonlinear wave equations [10–13]; Goh, et al. obtained a similar conclusion that the variational iteration method is a reliable treatment for hyperchaotic systems [14]; Saberi-Nadjafi and Tamamgar pointed out that the method is a highly promising method for integro-differential equations [15]; Uremen and Yildirim obtained exact solutions of the Poisson equation [16]; Sadighi and Ganji also obtained exact solutions of nonlinear diffusion equations [17]; Altintan & Ugur [18], Hemeda [19] and Yusufoglu & Bekir [20] applied the method to, respectively, Sturm–Liouville equation, wave equation, and the regularized long wave equation with great success. There are also many modifications of the variational iteration method [21–23], among which Herişanu and Marinca’s modification (an optimal variational iteration algorithm) [23] is much more attractive, where the variational iteration method is coupled with the least squares technology, and only one iteration leads to ideal results. Recently, He et al. [5] summarized various variational iteration algorithms for various nonlinear equations including fractional–differential equation, fractal–differential equation, differential–difference equations. Though much achievement has been achieved, application of the variational iteration method to Cauchy problems has not yet been dealt with.
In this paper, we use the method to discuss the first-order partial differential equation in the form [24]:
ut
(
x,
t) +
a(
x,
t)
ux(
x,
t) = φ(
x),
x∈
R,
t>
0 (1)u
(
x,
0) = ψ(
x),
x∈
R.
(2)∗Corresponding author.
E-mail address:[email protected](X.-W. Zhou).
0898-1221/$ – see front matter©2010 Elsevier Ltd. All rights reserved.
When a
(
x,
t) =
a is a constant andφ(
x) =
0. Eq.(1)is a linear equation called the transport equation which can describe many interesting phenomena such as the spread of AIDS, the moving of wind. When a(
x,
t) =
u(
x,
t)
, the equation is called the inviscid Burgers’ equation arising in one-dimensional stream of particles or fluid having zero viscosity.2. He’s variational iteration method
To illustrate the basic concepts of the variational iteration method, we consider the following general nonlinear system:
Lu
+
Nu=
φ(
x)
(3)where L is a linear operator part while N is the nonlinear operator part, and
φ(
x)
is a known analytic function. According to the variational iteration method, a correction functional can be constructed as follows:un+1
(
x) =
un(
x) +
Z
x0
λ
Lun(ξ) +
Nu˜
n(ξ) − φ(ξ)
dξ
(4) whereλ
is a general Lagrange multiplier [1–5], which can be identified optimally via the variational theory, the subscript n denotes the nth approximation, andu˜
n(ξ)
is considered as a restricted variation [5], i.e.δ˜
un=
0.The initial guess, u0
(
x)
, can be freely chosen with possible unknown constants; it can also be solved from its corresponding linear homogeneous equationLu
=
0.
(5)The variational iteration method can solve effectively, easily, and accurately a large class of nonlinear problems with approximations converging rapidly to the accurate solution.
3. Applications
Since our focus is on the ideas and basic principles, we shall consider only the simplest possible equations to clearly illustrated the solution procedure. In particular, we will focus on pure Cauchy problems. These problems are initial value problems.
According to Eq.(4), we can construct a correction functional to the Eq.(1)which reads un+1
(
x,
t) =
un(
x,
t) +
Z
t 0λ
∂
un(
x, ξ)
∂ξ
+
a(
x, ξ)
∂
un(
x, ξ)
∂
x−
φ(
x)
dξ.
(6)The stationary conditions are given by
1
+
λ =
0,
λ
0|
ξ=t=
0.
(7)So that
λ = −
1.As a result, we obtain the following iteration formula: un+1
(
x,
t) =
un(
x,
t) −
Z
t 0∂
un(
x, ξ)
∂ξ
+
a(
x, ξ)
∂
un(
x, ξ)
∂
x−
φ(
x)
dξ.
(8)Example 1. Consider the transport equation [24–26]: ut
(
x,
t) +
aux(
x,
t) =
0,
x∈
R,
t>
0u
(
x,
0) =
x2,
x∈
R.
(9)According to the iteration formula(8), we have un+1
(
x,
t) =
un(
x,
t) −
Z
t 0∂
un(
x, ξ)
∂ξ
+
a∂
un(
x, ξ)
∂
x dξ.
(10)We can select u0
(
x,
t) =
x2 from the given initial condition. Putting this selection into(10)we obtain the following successive approximations u0(
x,
t) =
x2,
u1(
x,
t) =
x2−
2atx,
u2(
x,
t) =
x2−
2atx+
a2t2,
u3(
x,
t) =
x2−
2atx+
a2t2,
...
un(
x,
t) =
x2−
2atx+
a2t2.
(11)Example 2. Consider the nonlinear Cauchy problem [25] ut
(
x,
t) +
xux(
x,
t) =
0,
x∈
R,
t>
0u
(
x,
0) =
x2,
x∈
R.
(12)According to the iteration formula(8), we have un+1
(
x,
t) =
un(
x,
t) −
Z
t 0∂
un(
x, ξ)
∂ξ
+
x∂
un(
x, ξ)
∂
x dξ.
(13)As stated before, we can select u0
(
x,
t) =
x2from the given initial condition. Using this selection, we obtain the following successive approximations u0(
x,
t) =
x2,
u1(
x,
t) =
x2−
2tx2=
x2(
1−
2t) ,
u2(
x,
t) =
x2−
2tx2+
2t2x2=
x2 1−
2t+
(
2t)
2 2,
u3(
x,
t) =
x2−
2tx2+
2t2x2−
4 3t 3x2=
x2 1−
2t+
(
2t)
2 2!
−
(
2t)
3 3!
,
...
un(
x,
t) =
x2 1−
2t+
(
2t)
2 2!
−
(
2t)
3 3!
+
(
2t)
4 4!
−
(
2t)
5 5!
+ · · ·
.
(14)The VIM admits the use of u
=
limn→∞un, which gives the exact solution un(
x,
t) =
x2e−2t. Example 3. Consider the following non-homogeneous Cauchy problem [25]ut
(
x,
t) +
ux(
x,
t) =
x,
x∈
R,
t>
0u
(
x,
0) =
ex,
x∈
R.
(15)According to the iteration formula(8), we have un+1
(
x,
t) =
un(
x,
t) −
Z
t 0∂
un(
x, ξ)
∂ξ
+
∂
un(
x, ξ)
∂
x−
x dξ.
(16)Similarly, the zeroth approximation is chosen as u0
(
x,
t) =
ex. Accordingly, we obtain the following successive approximations: u0(
x,
t) =
ex,
u1(
x,
t) =
ex−
ext+
tx,
u2(
x,
t) =
ex−
ext−
t2 2+
ext2 2+
tx=
t x−
t 2+
ex 1−
t+
t 2 2!
,
u3(
x,
t) =
ex−
ext−
t2 2+
ext2 2−
ext3 6+
tx=
t x−
t 2+
ex 1−
t+
t 2 2!
−
t3 3!
,
...
un(
x,
t) =
t x−
t 2+
ex 1−
t+
t 2 2!
−
t3 3!
+
t4 4!
−
t5 5!
+ · · ·
.
(17)The VIM admits the use of u
=
limn→∞un, which gives the exact solutionu
(
x,
t) =
t x−
t 2+
ex·
e−t=
t x−
t 2+
ex−t.
Example 4. Consider the inviscid Burgers’ equation [25,26] ut
(
x,
t) +
u(
x,
t)
ux(
x,
t) =
0,
x∈
R,
t>
0u
(
x,
0) =
x,
x∈
R.
(18)According to the iteration formula(8), we have un+1
(
x,
t) =
un(
x,
t) −
Z
t 0∂
un(
x, ξ)
∂ξ
+
un(
x, ξ)
∂
un(
x, ξ)
∂
x dξ.
Starting with initial approximation u0
(
x,
t) =
u(
x,
0) =
x, and proceeding in a similar way illustrated above, we obtain the following successive approximations:u0
(
x,
t) =
x,
u1(
x,
t) =
x−
tx,
u2(
x,
t) =
x−
tx+
t2x−
t3x 3,
u3(
x,
t) =
x−
tx+
t2x−
t3x+
2t4x 3−
2t5x 3+
t6x 9−
t7x 63,
u4(
x,
t) =
x−
tx+
t2x−
t3x+
t4x−
13t5x 15+
2t6x 3−
t7x 29+
71t8x 252−
86t 9x 567+
22t10x 315−
5t11x 189+
t12x 126−
t13x 567+
t14x 3969−
t15x 59 535....
(19)Using u
=
limn→∞un, we get the exact solution of the equationu
(
x,
t) =
limn→∞un
(
x,
t) =
nlim→∞ x−
tx+
t2x−
t3x+
t4x−
t5x+ · · · +
(−
1)
ntnx+ · · · =
x 1−
t.
4. Conclusion
As we know, many science and engineering problems may lead to partial differential equations. A clear conclusion can be drawn from the results that the variational iteration method provides us an efficient tool to obtaining exact solutions of partial differential equations. According to a recent review article [5], the present variational iteration algorithm is labeled as variational iteration algorithm-I, there are variational iteration algorithm-II and variational iteration algorithm-III, which are also suitable for the present study.
Acknowledgements
The authors would like to thank the reviewers for their careful reading and helpful comments. The work is supported by Kunming University Research Fund (2009L006) and the office of Education-funded projects in Yunnan Province (08z0079).
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