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Computers and Mathematics with Applications
journal homepage:www.elsevier.com/locate/camwa
Modified variational iteration method (nonlinear homogeneous initial
value problem)
Tamer A. Abassy
∗Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Benha 13518, Egypt Department of Mathematics, Faculty of Arts and Science in Wadi Addwasir, King Saud University, Saudi Arabia
a r t i c l e i n f o Article history:
Received 26 February 2009
Received in revised form 5 September 2009 Accepted 7 October 2009
Keywords:
Nonlinear initial value problem Variational iteration method Modified variational iteration method
a b s t r a c t
The modified variational iteration method is applied for analytical treatment of nonlinear homogeneous initial value problem. The modified variational iteration method accelerates the convergence of the power series solution and reduces the size of work. A comparison between modified variational iteration method (MVIM) and variational iteration method (VIM) was made. The comparison enhances the use of the modified variational iteration method if we wish to obtain an approximate power series solution that converges faster to the closed form solution. The method is very simple and easy.
© 2009 Elsevier Ltd. All rights reserved. 1. Introduction
Nonlinear phenomena that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid dynamics, mathematical biology and chemical kinetics, can be modeled by nonlinear differential equations [1–3]. A broad class of analytical solutions methods and numerical solutions methods were used in to handle these problems such as Backlund transformation [4,5], Hirota’s bilinear method [6,7], Darboux transformation [8], symmetry method [9], the inverse scattering transformation [10], the tanh method [11,12], the Adomian decomposition method [13] and other asymptotic methods for strongly nonlinear equations [14].
The variational iteration method (VIM) was proposed by ‘‘He’’ in [15–18] and had been proved by many authors [18–22] to be a powerful mathematical tool for solving various types of nonlinear problems, which represent a plenty of modern science branches.
In some applications when series solution is searched for, variational iteration method has some drawbacks which reduce the efficiency of the method due to repeated calculations and calculations of massive unneeded terms. Abassy et al. proposed the modified variational iteration method (MVIM) and used it to give an approximate power series solutions for some well-known nonlinear problems [23,24]. The modified variational iteration method (MVIM) facilitates the computational work and minimizes it. This method can effectively improve the speed of convergence [23,24]. Abassy et al. also proposed further treatments on MVIM results by using Padé approximants and Laplace transform [25,26]. The treatment improves the convergence and gives the closed form solution in some cases; for more details see [23–26].
In this work we aim to introduce analytical treatment for nonlinear initial value problems by using the modified variational iteration method (MVIM). The modified form introduces a change in the formulation of variational iteration recursive relation and provides a qualitative improvement over standard variational iteration method. While this change is rather simple, it does demonstrate the reliability, efficiency, and power of the method. This paper is an extension of the work done in [23–26], which a new application of MVIM for nonlinear homogeneous initial value problems is described.
∗Corresponding author at: Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Benha 13518, Egypt. E-mail address:[email protected].
URL:http://faculty.ksu.edu.sa/abassy.
0898-1221/$ – see front matter©2009 Elsevier Ltd. All rights reserved.
MVIM is used for solving the general nonlinear initial value problem Lu
(
x,
t) +
Ru(
x,
t) +
Nu(
x,
t) =
0,
u(
x,
0) =
f0(
x),
∂
u(
x,
t)
∂
tt=0
=
f1(
x),
...
∂
s−1u(
x,
t)
∂
ts−1t=0
=
fs−1(
x),
(1) where L=
∂∂sts
,
s=
1,
2,
3, . . .
is the highest partial derivative with respect to t, R is a linear operator and Nu(
x,
t)
is the nonlinear term. Ru(
x,
t)
and Nu(
x,
t)
are free of partial derivative with respect to t.MVIM gives an approximate series solution that converges to Eq.(1)closed form solution. A comparison was made between variational iteration method, modified variational iteration method.
2. The modified variational iteration method 2.1. Previous work (s
=
1 & s=
2)In Paper [23], Eq.(1)(with s
=
1) is solved using the MVIM formula:Un+1
=
Un+
Z
t0
λ {
R
(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ,
(2)where U−1
=
0, U0=
f0(
x)
and Gn(
x,
t)
is a polynomial of degree n in t and is obtained from NUn(
x,
t) =
Gn(
x,
t) +
O(
tn+1)
,and
λ = −
1 which is the Lagrange multiplier and is identified via variational theory. See [23] for more details and illustrative examples.In Paper [24], Eq.(1)(with s
=
2) is solved using the MVIM formula:Un+1
(
x,
t) =
Un(
x,
t) +
Z
t0
λ(τ) {
R
(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ
(3)where U−1
=
0, U0=
f0(
x) +
f1(
x)
t and Gn(
x,
t)
is a polynomials of degree(
2n+
1)
in t and is obtained fromNUn
(
x,
t) =
Gn(
x,
t) +
O(
t2(n+1))
, andλ = −(
t−
τ)
which is the Lagrange multiplier and is identified optimally viavariational theory; for more details and illustrative example, see [24].
2.2. Third case (s
=
3)Substituting by s
=
3 in Eq.(1), we obtain the following nonlinear initial value problem∂
3∂
t3u(
x,
t) +
Ru(
x,
t) +
Nu(
x,
t) =
0,
u(
x,
0) =
f0(
x),
∂
u(
x,
t)
∂
tt=0
=
f1(
x),
∂
2u(
x,
t)
∂
t2t=0
=
f2(
x),
(4)where R is a linear operator and Nu
(
x,
t)
is the nonlinear term. Ru(
x,
t)
and Nu(
x,
t)
are free of partial derivative with respect to t.Following the VIM procedure [14,15] to calculate the Lagrange multipliers
λ
from the following variational iteration formula: Un+1(
x,
t) =
Un(
x,
t) +
Z
t 0λ
∂
3∂τ
3Un+
∼z}|{
RUn+
∼z}|{
NUn
dτ,
(5)the Lagrange multiplier
λ
can be identified optimally via variational theory,∼
z}|{
RUn, and
∼
z}|{
NUnare considered to have restricted
variations, i.e.
δ
∼z}|{
RUn=
0, δ
∼z}|{
NUn=
0.Calculating variation,
δ
Un+1(
x,
t) = δ
Un(
x,
t) + δ
Z
t 0λ(τ)
∂
3U n∂τ
3+
∼z}|{
RUn+
∼z}|{
NUn
dτ,
(6) i.e.δ
Un+1(
x,
t) = δ
Un(
x,
t) + δ
Z
t 0λ(τ)
∂
3Un(
x, τ)
∂τ
3 dτ,
(7) i.e.δ
Un+1(
x,
t) = δ
Un(
x, τ)(
1+
λ
00(τ))
τ= t
−
δ
∂
Un(
x, τ)
∂τ
(λ
0(τ))
τ= t
+
δ
∂
2U n(
x, τ)
∂τ
2(λ(τ))
τ= t
−
Z
t 0δ
Un(
x, τ)
∂
3λ(τ)
∂τ
3 dτ.
(8)Consequently, the following stationary conditions are obtained:
δ
un:
λ
000(τ) =
0,
δ
u00n:
λ(τ)|τ=
t=
0,
δ
u0n:
λ
0(τ)
τ= t
=
0,
δ
un:
1−
λ
00(τ)
τ= t
=
0.
(9)The Lagrange multiplier, therefore, can be identified as
λ =
−(t−2τ)2. Now, we have the following VIM formula:Un+1
(
x,
t) =
Un(
x,
t) −
Z
t 0(
t−
τ)
2 2∂
3∂τ
3Un+
RUn+
NUn dτ.
(10)2.2.1. Remarks on variational iteration method results
Analyzing the results obtained from iteration formula(10), it can be verified that Untakes the following form:
U0
=
B00+
B 1 0t+
B 2 0t 2,
U1=
B01+
B11t+
B21t2+
B31t3+
B41t4+
B15t5+
B61t6+ · · ·
...
Un=
B0n+
B 1 nt+
B 2 nt 2+ · · · +
B3n+1 n t 3n+1+
B3n+2 n t 3n+2+
B3n+3 n t 3n+3+ · · ·
,
(11) where Bmn is the coefficient of tmwhich is settled and takes the same value for each Unas
(
3n+
2) ≥
m and is not settledand does not take the same value for each Unas
(
3n+
2) <
m.2.2.1.1. Remark 1. Concerning our case of L
=
∂3∂t3, there are repeated calculation in each step
(
n>
0)
. To cancel some ofthe repeated calculation, the iteration formula(10)can be handled as follows:
Un+1
(
x,
t) =
Un(
x,
t) +
Z
t 0−
(
t−
τ)
2 2(
Un(
x, τ))τττ
dτ +
Z
t 0−
(
t−
τ)
2 2{
R(
Un) +
NUn}
dτ
=
Un(
x,
0) +
t(
Un)
t(
x,
0) +
t2 2(
Un)
tt(
x,
0) +
Z
t 0−
(
t−
τ)
2 2{
R(
Un) +
NUn}
dτ,
but Un(
x,
0) +
t(
Un)
t(
x,
0) +
t2 2(
Un)
tt(
x,
0) =
f0(
x) +
f1(
x)
t+
f2(
x)
2 t 2=
U 0(
x,
t).
So Un+1(
x,
t) =
U0(
x,
t) +
Z
t 0−
(
t−
τ)
2 2{
R(
Un) +
NUn}
dτ.
(12)Via the iteration formula(12)some repeated computations are canceled. 2.2.1.2. Remark 2. If we rewrite Eq.(11)in the form
Un
(
x,
t) =
Unst(
x,
t) +
U nsn
(
x,
t),
(13)It is observed that the addition of the term Unns
(
x,
t)
deteriorates the convergence to the exact solution since the coefficients of ts in Unsn
(
x,
t)
are not the exact coefficients of ts. So, Unns(
x,
t)
does not lead to better convergence butdeteriorates the convergence and should be canceled.
To overcome this problem and eliminate Unns
(
x,
t)
, the following modification on the recursive formula(12)is suggested: Un+1(
x,
t) =
U0(
x,
t) +
Z
t 0λ(τ) {
R(
Un) +
Gn(
x, τ)}
dτ,
(14) where U0=
f0(
x) +
f1(
x)
t+
f2(2x)t2,λ(τ) =
−(t−τ)22 and Gn
(
x,
t)
is obtained from NUn(
x,
t) =
Gn(
x,
t) +
O(
t 3n+3)
.2.2.1.3. Remark 3. To cancel all repeated computation, let us rewrite Eq.(14)in the following iteration formula:
Un+1
=
U0+
Z
t 0λ(τ) {
R(
Un−1) +
Gn−1}
dτ +
Z
t 0λ(τ) {
R(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ.
(15)But it is known from(14)that
Un
=
U0+
Z
t0
λ(τ) {
R
(
Un−1) +
Gn−1}
dτ.
(16)Substituting by Eq.(16)in Eq.(15), we get:
Un+1
=
Un+
Z
t 0λ(τ) {
R(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ,
(17) where U−1=
0, U0=
f0(
x) +
f1(
x)
t+
f2(2x)t2,λ(τ) =
−(t−τ)22 and Gn
(
x,
t)
is obtained from NUn(
x,
t) =
Gn(
x,
t) +
O(
t 3n+3)
.2.2.2. MVIM for s
=
3Eq.(17)is the MVIM formula that is used with Eq.(1)with s
=
3. This final modified formula(17)cancels all the repeated calculations and the unsettled terms in VIM.2.2.2.1. Illustrative example. Consider the nonlinear initial value problem d3u
(
x,
t)
dt3+
du(
x,
t)
dx−
2x(
u(
x,
t))
2+
6(
u(
x,
t))
4=
0,
u(
x,
0) =
−
1 x2,
ut(
x,
0) =
−
1 x4,
utt(
x,
0) =
−
2 x6.
(18)Solving Eq.(18)using MVIM we found that:
Ru
(
t,
x) =
du(
t,
x)
dx
,
Nu(
t,
x) = −
2x(
u(
t,
x))
2
+
6(
u(
t,
x))
4,
and the following iteration formula is used: Un+1
=
Un−
Z
t 0(
t−
τ)
2 2{
R(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ,
(19) where U−1=
0, U0= −
x12−
1 x4t−
1 x6t 2, and Gn
(
x,
t)
is a polynomial of degree(
3n+
2)
in t and is obtained from−
(
Un(
t,
x))
2+
6(
Un(
t,
x))
4=
Gn(
t,
x) +
O(
t3(n+1)).
The following results are obtained U1
=
U0−
1 x8t 3−
1 x10t 4−
1 x12t 5,
U2=
U1−
1 x14t 6−
1 x16t 7−
1 x18t 8,
U3=
U2−
1 x20t 9−
1 x22t 10−
1 x24t 11,
....
(20)This is a power series expansion of the closed form solution u
(
t,
x) =
1Fig. 1. Solid gray line represents the absolute error between UVIM
2 and closed form solution(21)and dashed line represents the absolute error between
UMVIM
2 and closed form solution(21)(x=3).
Solving Eq.(18)using VIM we found that:
Ru
(
t,
x) =
du(
t,
x)
dx
,
Nu(
t,
x) = −
2x(
u(
t,
x))
2
+
6(
u(
t,
x))
4,
and the following iteration formula is used: Un+1
=
Un−
Z
t 0(
t−
τ)
2 2{
(
Un(
x, τ))τ
+
R(
Un(
x, τ)) +
N(
Un(
x, τ))}
dτ,
(22) where U0= −
x12−
1 x4t−
1x6t2. The following results are obtained
U1(VIM)
=
U1(MVIM)+
O(
t6),
U2(VIM)=
U2(MVIM)+
O(
t9),
U3(VIM)=
U3(MVIM)+
O(
t12),
...
(23)
andFig. 1shows the absolute error.
2.3. MVIM for general case
Now, consider the general nonlinear initial value problem(1). Following the procedures used in [23,24] and the previous Section2.2. The following stationary conditions are obtained:
δ
un:
λ
(s)(τ) =
0,
δ
u(ns−1):
λ(τ)|τ=
t=
0,
δ
u(ns−2):
λ
0(τ)
τ=t
=
0,
...
δ
un:
1−
λ
(s−1)(τ)
τ= t
=
0.
(24)The Lagrange multiplier, therefore, can be identified as
λ =
−(t(−s−τ)1()!s−1). So, the modified variational iteration formula takes the formUn+1
(
x,
t) =
Un(
x,
t) +
Z
t0
λ(τ) {
R(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ,
(25)where
λ(τ)
is called a general Lagrange multiplier and equals−(t(−s−τ)1()!s−1), which is identified optimally via variational theory and Gn(
x,
t)
is a polynomial of degree(
s(
n+
1) −
1)
in t and is obtained fromThe modified variational iteration formula(25)can be solved iteratively using U−1
=
0,
U0=
f0(
x) +
f1(
x)
t+ · · · +
fs−1(
x)
(
s−
1)!
t s−1,
to obtain an approximate power series solution for Eq.(1).
2.3.1. Illustrative example
Consider the nonlinear initial value problem d4u
(
t)
dt4
−
24(
u(
t))
5
−
40(
u(
t))
3−
16u(
t) =
0u
(
0) =
0,
ut(
0) =
1,
utt(
0) =
0,
uttt(
0) =
2.
(26)
Solving Eq.(26)using MVIM we found that:
Ru
(
t) = −
16u(
t),
Nu(
t) = −
24(
u(
t))
5−
40(
u(
t))
3,
and the following iteration formula is used:Un+1
=
Un−
Z
t0
(
t−
τ)
36
{
R(
Un−
Un−1) + (
Gn−
Gn−1)}
dτ,
(27)where U−1
=
0, U0=
t+
13t3, and Gn(
x,
t)
is a polynomial of degree(
4n+
3)
in t and is obtained from−
24(
Un(
t))
5−
40(
Un(
t))
3=
Gn(
t) +
O(
t4(n+1)).
The following results are obtained U1
=
U0+
2 15t 5+
17 315t 7,
U2=
U1+
62 2835t 9+
1382 155 925t 11,
U3=
U2+
21 844 6 081 075t 13+
929 569 638 512 875t 15,
....
(28)This is a power series expansion of the closed form solution
u
(
t) =
tan(
t).
(29)Solving Eq.(26)using VIM we found that:
Ru
(
t) = −
16u(
t),
Nu(
t) = −
24(
u(
t))
5−
40(
u(
t))
3,
and the following iteration formula is used:Un+1
=
Un−
Z
t0
(
t−
τ)
36
{
(
Un(
x, τ))τ
+
R(
Un(
x, τ)) +
N(
Un(
x, τ))}
dτ,
(30) where U0=
t+
13t3, The following results are obtainedU1(VIM)
=
U1(MVIM)+
O(
t8),
U2(VIM)=
U2(MVIM)+
O(
t12),
U3(VIM)=
U3(MVIM)+
O(
t16),
...
(31)
andFig. 2shows the absolute error.
By observing the results obtained by MVIM and VIM in the illustrative examples, we found that MVIM is faster in calculations and more convergent than VIM. MVIM eliminates all the non-settled terms in VIM.
Fig. 2. Solid gray line represents the absolute error between UVIM
2 and closed form solution(29)and dashed line represents the absolute error between
UMVIM
2 and closed form solution(29).
3. Summary and conclusion
The general nonlinear initial value problem(1)is solved through using the modified variational iteration formula(25). The solution obtained for Eq.(1)is a power series solution, which converges to Eq.(1)closed form solution in the neighborhood of t
=
0.By analyzing and comparing the results obtained and procedures used in modified variational iteration method and variational iteration method we observe that [23,24]:
1. Modified variational iteration method cancels all the unsettled terms in variational iteration method. 2. Modified variational iteration method is faster than variational iteration method and save time. 3. Modified variational iteration method convergence is faster than variational iteration convergence. 4. Modified variational iteration method facilitates the computational work and minimizes it. References
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