# Application of He's Variational Iteration Method to Solve Semidifferential Equations ofnth Order

Authors: Asghar Ghorbani; Abdolsaeed Alavi
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

He's variational iteration method is applied to solventh order semidifferential equations. Comparison is made between collocation spline method based on Lagrange interpolation and the present method. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions, or perturbation theory. Some examples are given to illustrate the effectiveness of the method; the results show that He's method provides a straightforward and powerful mathematical tool for solving various semidifferential equations of the nth order.


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## Body

## 1. Introduction

He’s variational iteration method (HVIM) [1–14] proposed by He has been recently intensively studied by scientists and engineers and
favorably applied to various kinds of nonlinear problems [15–25]. The method has been shown to solve effectively, easily, and accurately a large class of nonlinear problems, generally one or two iterations lead to high accurate solutions. This method is, in fact, a modifying of the general Lagrange multiplier method into an iteration method, which is called correction functional. Generally speaking, the solution procedure of He’s method is very effective, straightforward, and convenient. For a relatively comprehensive survey on the method, new interpretation, and new development, the reader is referred to the review
articles [14, 26].A fractional differential equation of the
form [27]:
(1.1)[Dn/2+c1D(n−1)/2+⋯+cnD0]y(t)=f(t),
is called a semidifferential equation of nth order, where c1,…,cn∈R and f(t) is a given function from I into R, I is the interval [0, T].The analytic results on existence and
uniqueness of solutions to fractional differential equations have been
investigated by many authors (see, e.g., [28–30]).In (1.1)Dq denotes the
fractional differential operator of order q∉N in the sense of Caputo, and
is given by [29–31]
(1.2)Dqy(t)=Jm−qDmy(t)=1Γ(m−q)∫0t(t−s)m−q−1Dmy(s)ds,
where m∈NDm is the
usual integer differential operator of order m, and Jq is the
Riemann-Liouville fractional integral operator of order q>0, defined by
(1.3)Jqy(t)=1Γ(q)∫0t(t−s)q−1y(s)ds,t>0.Forq, p>0, m−1<q≤m, and γ>−1, we have the following
properties:
(1.4)(JqJpy)(t)=(JpJqy)(t)=(Jq+py)(t),Dqtγ=Γ(γ+1)Γ(γ+1+q)tγ+q,(JqDqy)(t)=y(t)−∑k=0m−1y(k)(0+)tkk!.The Caputo fractional derivative is considered
here because it allows traditional, initial, and boundary conditions to be
included in the formulation of the problem. For more details on the geometric
and physical interpretation for fractional derivatives of both the Riemann-Liouville and
Caputo types see [29–31].The theory for the derivatives of fractional
order was developed in the 19th century. In earlier work, the main application
of the fractional calculus has been as a technique for solving integral
equations, for more details (see, [31]). Recently, fractional derivatives have
proved to be tools in the modeling of many physical phenomena (see, [32–36]). We mention
the important example: the Bagley-Torvik equation,
(1.5)[D2+c1D3/2+c4D0]y(t)=f(t),y(0)=a1,y′(0)=a2,
which
arises, for example, in the modeling of the motion of a rigid plate immersed in
a Newtonian fluid. Studying the numerical solution of (1.1) has been increased in the last two decades.
A survey of some numerical methods is given by Podlubny [37]. Blank [38] proposed the collocation spline
method and also Rawashdeh [27] applied the collocation spline method to solve
semidifferential equations. It is well-known that the main disadvantage of the presented
methods in [27, 37, 38] is the complex and difficult procedure. In
order to overcome the demerit, in this paper we will apply He’s variational
iteration method for solving semidifferential equations of nth order.
This will make the solution procedure easier, more effective, and more
straightforward.


## 2. He’s Variational Iteration Method (HVIM)

To illustrate its basic idea of the method, He [14, 26]
considered the following general nonlinear equation:
(2.1)Lu(t)+Nu(t)=f(t),
where L is a linear operator, N is a
nonlinear operator, and f(t) is a given continuous function. The basic
character of the method is to construct a correction functional for the system,
which reads
(2.2)un+1(t)=un(t)+∫0tλ(s){Lun(s)+Nu˜n−f(s)}ds,
where λ is a Lagrange multiplier which can be
identified optimally via variational theory [7],
un is the nth approximate solution,
and u˜n denotes a restricted variation, that is, δu˜n=0. In most cases, the integration in (2.2) is not
easily evaluated or needs a huge computational work. Therefore, in the
following, we introduce an application of the HVIM to reduce the size of
calculations and to overcome the difficulty arising in calculating complicated
integrals.An application of He’s variational
iteration method can be proposed based on this assumption that the functionf(s) can be divided into two parts, namely, f0(s) and f1(s),
(2.3)f(s)=f0(s)+f1(s).According
to this assumption,f(s)=f0(s)+f1(s), we construct the
following iteration formula:
(2.4)u1(t)=u0(t)+∫0tλ(s){Lu0(s)+Nu0−f0(s)}ds,un+1(t)=un(t)+∫0tλ(s){Lun(s)+Nun−(f0(s)+f1(s))}ds,n≥1.It is shown that this method is very
effective and easy for linear problem, its exact solution can be obtained by
only one iteration becauseλ can be exactly
identified. But for nonlinear problems, there are secular terms, which should
be considered
[1]. Therefore, we first determine the
Lagrange multiplier λ that will be identified optimally via
integration by parts. The successive approximation un(t), n≥0 of the solution u(t) will be readily obtained upon using the
obtained Lagrange multiplier and by using any selective function u0. The zeroth
approximation u0 may be selected any function that just
satisfies at least the initial and boundary conditions. With λ determined, then several approximations un(t), n≥0 follow immediately. Consequently, the exact
solution may be obtained by using
(2.5)u(t)=limn→∞un(t).The HVIM has been shown to solve
effectively, easily, and accurately a large class of nonlinear problems with
approximations converges rapidly to accurate solutions.


## 3. Implementation of the Method

In this
section, an application of He’s variational iteration method is discussed, for
solving the
Bagley-Torvik equation
and the initial value problem, which are semidifferential equations of order 4,
[27].Example 3.1.
As an
example that arises in application, we solve the Bagley-Torvik equation
[27]
(3.1)[D2+D3/2+D0]y(t)=2+4t/π+t2,y(0)=y′(0)=0.
According to He’s variational iteration
method, we consider the correction functional in the following form (see, [14, 26]):
(3.2)yn+1(t)=yn(t)+∫0tλ{yn(2)(s)+y˜n(3/2)(s)+y˜n(s)−2−4s/π−s2}ⅆs, where λ is the general Lagrange
multiplier, y0 is an initial approximation
which must be chosen suitably, and y˜n is the restricted
variation, that is, δy˜n=0. Under these conditions, its
stationary conditions of the above correction functional can be expressed as
follows:
(3.3)∂2λ(t,s)∂s2=0,1−∂λ(t,s)∂s|t=s=0,∂λ(t,s)|t=s=0.
The
Lagrange multiplier, therefore, can be easily identified as(3.4)λ=s−t, leading to
the following iteration formula:
(3.5)yn+1(t)=yn(t)+∫0t(s−t){yn(2)(s)+yn(3/2)(s)+yn(s)−2−4s/π−s2}ⅆs.
We start with the initial approximationy0(t)=0. In Table 1, we have listed the
approximation solution y3(t) of (3.5), the exact solution, and the absolute
error.
Here,
an application of the HVIM can be introduced based on the assumption that the
functionf(t) can be divided into two parts, namely, f0(t) and f1(t),
(3.6)f(t)=f0(t)+f1(t).
According
to this assumption,f(t)=f0(t)+f1(t), we construct the
following iteration formula:
(3.7)y1(t)=y0(t)+∫0t(s−t){y0(2)(s)+y0(3/2)(s)+y0(s)−f0(s)}ⅆs,y2(t)=y1(t)+∫0t(s−t){y1(2)(s)+y1(3/2)(s)+y1(s)−(f0(s)+f1(s))}ⅆs,⋮yn+1(t)=yn(t)+∫0t(s−t){yn(2)(s)+yn(3/2)(s)+yn(s)−(f0(s)+f1(s))}ⅆs,n≥1.
Consequently, begin with the initial approximationy0(t)=0, in
view of the iteration formulas (3.7) and taking  f0(t)=2 and f1(t)=4t/π+t2, we can obtain the
following approximations:
(3.8)y0(t)=0,y1(t)=t2,yn(t)=t2,n≥1.
We, therefore,
obtain(3.9)y(t)=t2.
This
is the exact solution. We note that the success of obtaining the
exact solution is a result of the proper selection off0(t) and f1(t).Table 1
Numerical solutions and absolute errors of Example3.1 using the original HVIM.
ty(t)=t2y3(t)|t2−y3(t)|0.10.010.1005487432e−10.5487432e−40.20.040.4063125562e−10.63125562e−30.30.090.9266557142e−10.266557142e−20.40.160.16748012190.74801219e−20.50.250.26679592080.167959208e−10.60.360.39277307220.327730722e−10.70.490.54806535200.580653520e−10.80.640.73588508230.958850823e−10.90.810.96007684480.15007684481.01.001.22519950000.2251995000Example 3.2.
Consider the following initial
value problem [27]
(3.10)[D2−2D+D1/2+D0]y(t)=6t−6t2+165πt5/2+t3,y(0)=y′(0)=0.
In
view of the correction functional (2.2), the Lagrange multiplier can be
identified as
(3.11)λ=s−t.
As a result, we obtain the following iteration
formula:(3.12)yn+1(t)=yn(t)+∫0t(s−t){yn(2)(s)−yn(1)+yn(1/2)(s)+yn(s)−6s+6s2−165πs5/2−s3}ⅆs.
We start with the initial approximationy0(t)=0. The comparison between
the exact solution and the approximation solutiony3(x) of (3.12) can
be seen in Table 2.
Now, we choose(3.13)f0(s)=6s,f1(s)=−6s2+165πS5/2−s3.
Under
the above assumption, we construct the following iteration formulas:(3.14)y1(t)=y0(t)+∫0t(s−t){y0(2)(s)−y0(1)+y0(1/2)(s)+y0(s)−6s}ⅆs,y2(t)=y1(t)+∫0t(s−t){y1(2)(s)−y1(1)+y1(1/2)(s)+y1(s)+6s2−165πs5/2−s3}ⅆs,⋮yn+1(t)=yn(t)+∫0t(s−t){yn(2)(s)−yn(1)+yn(1/2)(s)+yn(s)+6s2−165πs5/2−s3}ⅆs.
With
starting the initial approximationu0(t)=0 and in view of the iteration formulas (3.14), therefore, obtain the
following approximations:
(3.15)y0(t)=0,y1(t)=t3,yn(t)=t3,n≥1.
Thus we obtain(3.16)y(t)=t3, which is the exact solution.Table 2
Numerical solutions and absolute errors of Example3.2using the original HVIM.
ty(t)=t3y3(t)|t3−y3(t)|0.10.0010.9745626669e−30.254373331e−40.20.0080.7586906052e−20.413093948e−30.30.0270.2488146380e−10.211853620e−20.40.0640.5722919465e−10.677080535e−20.50.1250.10831259790.166874021e−10.60.2160.18112600170.348739983e−10.70.3430.27799106350.650089365e−10.80.5120.40058707630.11141292370.90.7290.54999536290.17900463711.01.0000.72675677450.2732432255


## 4. Comparison with Collocation Spline Method

Rawashdeh
in [27] applied the collocation spline method based on Lagrange interpolation
for solving semidifferential equations of order 4. It is clear that the main
disadvantage of the collocation spline method is its complex and difficult
procedure. In order to make solution procedure easier and more effective, in
this paper, we applied He’s VIM to overcome the demerit. Also Rawashdeh in [27]
reported the computed absolute error (error between exact and approximate
value) with N=50,100 (N is the division number of the given interval)
for Examples 3.1 and 3.2, see Table 3. In the studies by Rawashdeh, much time was
spent and boring operations were done by collocation spline method based on
Lagrange interpolation to get approximate solutions. In our study, however, the
exact solutions are computed easily using this technique. Generally speaking,
He’s VIM is reliable and more efficient as compared with collocation spline
method based on Lagrange interpolation.Table 3
Numerical solution ofy(t) in Examples 3.1 and 3.2.
tn (T=5)N(h=T/N)Absolute
error of Example3.1Absolute
error of Example3.20.1500.3×10−100.3×10−111000.2×10−100.18×10−101500.37×10−701000.44×10−70.534×10−92.5500.66×10−60.44×10−61000.25×10−50.25×10−64500.5×10−70.54×10−51000.15×10−60.88×10−55500.18×10−50.15×10−41000.22×10−40.39×10−4


## 5. Conclusion

In this paper, we have applied
He’s variational iteration method to various semidifferential equations ofnth order.
The obtained solution shows that He’s method is a very convenient and effective
for various semidifferential equations of nth order, only one iteration
leads to exact solutions.



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