The Variational Iteration Method for Solving Fuzzy Duffing's Equation

* Corresponding Author. Email address: [email protected], Tel: +60175551703.

The Variational Iteration Method for Solving

Fuzzy Duffing's Equation

A.F. Jameel*

School of Mathematical Sciences, 11800 USM, University Science Malaysia

Copyright 2014 © A.F. Jameel. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

Abstract

In this paper, we use the Variational Iteration Method (VIM) to find the approximate analytical solution

for an initial value problem involving the fuzzy Duffing ordinary differential equation. VIM allows for the

solution of the differential equation to be calculated in the form of an infinite series in which the

components can be easily computed. Numerical examples were solved to illustrate the capability of VIM

in this regard.

Keywords: Fuzzy numbers, Fuzzy differential equations, Variational Iteration Method and Duffing Equation.

1 Introduction

Nonlinear ordinary differential equations are used to model a wide class of problems. The dynamics of

these equations can often be understood by reference to a simple nonlinear ordinary differential equation.

One such equation is the second order Duffing equation given by [9]

( ) ( ) ( ) ( ) ( ) (1.1)

The initial conditions associated with it are

( ) ( )

where , , , , and are real constants. Equation (1.1) has been discussed in many works that arise in

various scientific fields such as physics, engineering, biology, and communication theory [9, 24, 28, 35].

Various problems involving nonlinear control systems in physics, engineering and communication theory

have increased interest in the periodical forced Duffing's equation [23, 25, 28]. In some cases, uncertainty

leads to the fuzzy Duffing’s equation (FDE). These models are used in various applications including

population models [36], quantum optics gravity [13], and medicine [6, 8]. Fuzzy Initial value problem

Journal of Interpolation and Approximation in Scientific Computing 2014 (2014) 1-14

Available online at www.ispacs.com/jiasc

Volume: 2014, Year 2014 Article ID: jiasc-00053, 14 Pages

doi:10.5899/2014/jiasc-00053

Research Article

of 142Journal of Interpolation and Approximation in Scientific Computing

http://www.ispacs.com/journals/jiasc/2014/jiasc-00053/

International Scientific Publications and Consulting Services

(FIVP) involving ordinary differential equations are suitable mathematical models to model certain

dynamical systems in which there exist uncertainties or vagueness.

In recent years approximate-analytical methods such as Adomian Decomposition Method (ADM),

Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) have been used to solve

fuzzy initial value problems which involve ordinary differential equations. In [16], the HPM was used to

solve first order linear fuzzy initial value problems involving ordinary differential equations. The ADM

was employed in [2, 7, 17] to solve first order linear and nonlinear fuzzy initial value problems. VIM is an

approximate - analytical method that was first proposed by He [18-22] and has been applied to many

physics and engineering problems [5, 27, 31, 34]. VIM was used in [17] to obtain the approximate solution

for first order linear FIVP. Furthermore, it was found that VIM is more effective than ADM and the

convergence of VIM is much faster than ADM. In [1] VIM also was used to find the approximate-

analytical solution for fuzzy differential equations including nonlinear first order FIVP. Abbasbandy et al.

in [3] used VIM to find the approximate solution for high order linear FIVP by converting it into a first

order system of fuzzy differential equations. The convergence of VIM for this system was also proved.

In this paper, we use VIM to obtain an approximate-analytical solution of nonlinear FIVP involving fuzzy

Duffing’s equation directly without converting into a first order system. The convergence of VIM is also

investigated. The outline of this paper is as follows. In section 2, some basic definitions and notations are

given as they will be used in other sections. In section 3, the structure of VIM is formulated for solving

second order FIVP. In section 4, the convergence analysis of VIM is presented and proved in detail. In

section 5, we consider some examples and finally, in section 6, we give the conclusions of this study.

2 Preliminaries

The definitions and theorems given in this section are required in our work. We cite articles from where

they were substantively taken

Definition 2.1. A fuzzy number is a function such as , - which satisfies the following

properties[17]

1. ( ) is normal, i.e with ( ) ,

2. ( ) is convex fuzzy set ,i.e. ( ( ) ) * ( ) ( )+ , -,

3. upper semi-continuous on ,

4. * ( ) + is compact.

A fuzzy number can also be represented in the parametric form

Definition 2.2. In the parametric form, a fuzzy number is represented by an ordered pair of

functions ( ( ) ( )) , , - which satisfies [24]:

i. ( ) is a bounded left continuous non-decreasing function over , -.

ii. ( ) is a bounded left continuous non-increasing function over , -.

iii. ( ) ( ), , -

A crisp number r is simply represented by ( ) ( ) , , -

Definition 2.3. If be the set of all fuzzy numbers, we say that ( ) is a fuzzy function [17] if




Definition 2.4. A mapping for some interval is called a fuzzy function process [14] and

we denote the r-level set by:

, ( )- 0 ( ) ( )1 , -

The r-level sets of a fuzzy number are much more effective as representation forms of fuzzy set. Fuzzy sets

can be defined by the families of their r-level sets based on the resolution, identity theorem [12, 33].

Definition 2.5. The fuzzy integral of fuzzy process [31], ( ) ∫ ( )

for and , - is defined by:

∫ ( )

6∫ ( )

∫ ( )

7

Definition 2.6. Each function induces another function ( ) ( ) defined for each fuzzy

interval in by[37]:

( )( ) 8 ( ) ( ) ( )

( )

This is called the Zadeh extension principle.

Definition 2.7. Consider . If there exists such that , then is called the H-

difference[30] (Hukuhara difference) of x and y and is denoted by .

Definition 2.8. [31] Let ( ) be the Hausdorff distance between two fuzzy sets ,

( ) * (, - , - )| , -+

and ( ) is a complete metric space.

Definition 2.9. Let and , where , -. We say that Hukuhara differentiable at if

there exists an element [ ]

such that for all h > 0 sufficiently near to 0, ( )

( ) ( ) ( ) exists where the limits are taken in the metric space ( )[31]

( ) ( )

( ) ( )

The fuzzy set ( ) is called Hukuhara derivative of at .

Theorem 2.1. [31] Let , - be Hukuhara differentiable and denote

[ ( )]

0 ( ) ( )1 .

Then the boundary functions 0 ( ) ( )1 are differentiable functions and

[ ( )]

[( ( ))

. ( )/ ] , , -




Theorem 2.2. [31] Let , - be Hukuhara differentiable n times and

[ ( )]

0 ( ) ( )1

0 ( ) ( )1 , -

that ( ) ( ) are differentiable. We can write for nth order fuzzy derivative

[ ( )( )]

6( ( )( ))

4 ( )

( )5

7 , - .

3 Fuzzification and Defuzzification of VIM for Second Order FIVP

The general structures of VIM for solving crisp nonlinear problems have been presented in [18-22]

amongst others. The general defuzzification of nth order FIVP was presented in [31]. The basic principles

of VIM for FIVP have been described in [1, 17, 31]. In this section the structure of VIM for the

approximate solution of second order FIVP is described followings the work of others. Consider the

following second order FIVP:

( ) ( ( ) ( )) ( ) , - (3.2)

( ) ( )

(3.3)

Here ( ) is the fuzzy function of the crisp variable with being a fuzzy function of the crisp variable,

fuzzy variable and the first order fuzzy Hukuhara derivative ( ). According to theorem 2.2 section 2

( ) is the second order fuzzy Hukuhara derivative of the fuzzy function with are fuzzy

numbers as in definitions (2.1-2.2) section 2. We denote the fuzzy function by , - , - , for

, - and , -. It means that the r-level set of ( ) can be defined as:

{, ( )- 0 ( ) ( )1

, ( )- 0 ( ) ( )1

(3.4)

and

{, ( )- 0 ( ) ( )1

, ( )- 0 ( ) ( )1

(3.5)

Now for defuzzification of Eq. (3.2), we let ( ) ( ) ( ) be the fuzzy function of ( ) ( ), such

that

( ) 0 ( ) ( )1 0 ( ) ( ) ( ) ( )1

Also, we can write

[ ( )]

0 ( ) ( )1 (3.6)

with

{ ( ) 0 1

( ) 0 1

(3.7)




Since ( ) ( ( )) and by using the fuzzy extension principle as in section 2, we can define the

following membership function

{ ( ( ) ) { ( ( ))| ( ) ( )}

( ( ) ) { ( ( ))| ( ) ( )} (3.8)

where

{ ( ( ) ) ( ( ) ( ))

( ( ) ) ( ( ) ( )) (3.9)

Now we can rewrite Eq. (1.1) as follows

( ) ( ( ) ( )) ( ) (3.10)

( ) ( ( ) ( )) ( ) (3.11)

According to the VIM procedure, we can construct the following correction functional as follows

( ) ( ) ∫ ( ) 2 ( ) . ( )

⏞ / ( )3

(3.12)

( ) ( ) ∫ ( ) 2

( ) . ( )

⏞ / ( )3

(3.13)

where , , -, ( ) is a general Lagrange multiplier. Now we let

⏞ . ( )

⏞ / ( ) (3.14)

⏞ . ( )

⏞ / ( ) (3.15)

where and are nonlinear operators , then we can rewrite Esq. (3.14) and (3.15) in the following forms

( ) ( ) ∫ ( ) 2 ( )

⏞3

(3.16)

( ) ( ) ∫ ( ) 2

( )

⏞3

(3.17)

⏞ is considered as restricted variation i.e

⏞ . The general Lagrangian multiplier ( ) related

with Eq. (3.2) can be determined as:

( ) ( ) ∫ ( ) 2 ( )

⏞3

( ) ( ) ∫ ( ) 2 ( )3

Integrating by parts we obtain the following:

( ) , ( )- ( ) ( ) ( ) ∫ ( ) ( )

Therefore, have the following stationary conditions:

{

( )

( ) |

( )|

(3.18)




Similarly, we have the same Lagrangian multiplier for the upper bound of Eq. (3.2). From these conditions

and according to the order of the Eq. (3.16) we can determine the Lagrangian multiplier

( ) ( ) (3.19)

Here the initial guesses that satisfies the initial conditions in Eq. (3.3) are given by

{ ( ) ( ) ( )

( ) ( ) ( )

(3.20)

where ( ) ( ) are fuzzy constants numbers as in definition (2.1) of section 2 and can be determined

from the initial conditions in Eq. (3.3) for all , -.

The successive approximations in Esq. (3.12)-(3.13) of VIM will be readily obtained by choosing all the

above-mentioned parameters ( ) and ( ). Consequently, for , the exact solution may

be obtained by

( ) ( ) *, - + (3.21)

where ( ) [ ( ) ( )] is the exact solution of Eq. (3.2) and the VIM approximate solution is

( ) 0 ( ) ( )1. We reiterate that basic principles of VIM for FIVP have been described in

the work of previous researchers as cited in this section.

4 Convergence of VIM

In [3] the convergence of the approximate solution of system of linear FIVP obtained by VIM have been

proved. Also the proof of convergence of VIM for second order crisp nonlinear differential equation was

presented in [15]. Using these previous works as a basis, we investigate the convergence of VIM for a

nonlinear FIVP such as fuzzy Duffing equation. The variational iteration formula generates a recurring

sequence *, - +. Clearly, the limit of the sequence will be the solution of Eq. (3.2) if the sequence is

convergent. In this section, we give a proof of convergence of VIM by introducing a new iterative

formulation of this procedure. The proof follows the procedure used by and ideas of earlier researchers.

Here, , - denotes the class of all real valued functions defined on , - which have continuous

second order derivatives. Before we continue the discussion of convergence details we note than one can

rewrite equations (14-15) in the following form

, ( )- , ( )- (4.22)

where the linear operator

can be defined as

, ( )- 8 ( )

( )

(4.23)

and the nonlinear operator can be defined as

, ( )- { ( . ( )/ ( ))

. . ( )/ ( ) / (4.24)

Lemma 4.1. If for any i, , - , -, then the variational iteration formula (12-13) is equivalent to

the following iterative relation for all , -:

( ( ) ( )) ( ( ) ( )

( )) (4.25)

.

( ) ( )/ .

( )

( )

( )/ (4.26)




where

{ ( ( )

( ) ( )) ( ) . ( )/ ( )

. ( )

( )

( )/

( ) . ( )/ ( )

(4.27)

and as in Eq.(4.22) .

Proof. First we take the lower bound solution of Eq. (4.22). Suppose ( ) ( ) satisfies the

variational iteration formula (3.12). Applying to both sides of (3.12) we have

( ( ) ( )) ∫

( )

( )

|

[ ( )| ]

Now, by using the conditions (3.18) and ( )

|

, we obtain

( ( ) ( )) ( ( )

( ) ( ))

Conversely, suppose ( ) ( ) satisfies (4.25). From the definition of in (4.22) and

( ) , multiplying (4.24) by ( ) and then integrating from both sides of the resulting term from 0

to t we obtain

∫ ( ) ( ( )

( ))

∫ ( )

(4.28)

Integration Eq. (4.28) by parts, then Eq. (4.25) becomes

( )| ( ( )

( )) ( )

|

( ( ) ( )) ∫ ( )

( ( )

( )) ∫ ( )

which exact results in Eq. (3.12) upon imposing the conditions (3.18). The upper bound of Eq. (3.2) can be

proved in a similar manner.

Theorem 4.1. If the sequence (3.21) converges, where *, - + is produced by the VIM formulation of (12-

13), then it must be the exact solution of the Eq. (3.2) for all , -.

Proof. If the sequence *, - + converges, we can write

8 ( ) ( )

( ) ( )

(4.29)

and it holds for

8 ( ) ( )

( )

( ) (4.30)

Now for the lower bound of Eq. (3.2), we use Eq. (4.29), and the definition of in (4.23), we can obtain

( ( ) ( )) ( ( ) ( )) (4.31)

From Eq. (4.31) and according to the Lemma 4.1, we can obtain




( ( ) ( )) ( ( ) ( )

( )) (4.32)

which gives us

( ( ) ( )

( )) (4.33)

Since . ( )/ ( ( ) ( )) and Eq. (3.2) is a second order equation and if ,

may be include both ( ) ( ) or one of them such that for the lower bound

( ( )) ( ) , ( ( )) (

( ))

( )

Now from (4.32) and (4.26), one can derive the following

( ( ) ( )

( )) ( ( ) . ( )/ ( )) (4.34)

(

( ))

( . ( )/) . ( )/

. ( ) ( ( ) ( )) ( )/

From the equations (4.33) and (4.34),

( ) ( ( ) ( )) ( ) for (4.35)

On the other hand, using the specified initial conditions in (3.3) and the initial guess in (3.20), we have

{ ( )

( ) ( )

( )

( ) ( )

(4.36)

Therefore, according to expressions (4.33)-(4.35), ( ) must be the exact solution of the Eq. (3.2) , and

similarly we work on the upper bound of Eq. (3.2). This ends the proof.

It is clear that the convergence of the sequence (3.21) depends upon the initial guess in (3.20) and the

linear operator with the number of terms of sequence (3.21). Fortunately, as long as ( ) and are

chosen that the sequence (3.21) converges in a region , it must converge to the exact solution in

this region.

5 Numerical Examples

In order to assess the advantages and the accuracy of VIM for solving fuzzy Duffing differential

equations, we have applied the method to two different examples. All the results are calculated by using

the Mathematica codes.

Example 5.1. Consider the fuzzy Duffing’s equation [4] of the following type

( ) ( ) ( ) (5.37)

( ) , - ( ) , -

, -

The exact solution of Eq. (5.37) was given in [4] in which it was solved by the Laplace transform

decomposition algorithm. The initial approximation guesses of Eq. (5.37) are given by:

( ) and ( ) ( )




According to section 3 the variational formula of this problem is given as follows:

( ) ( ) ∫ ( ){ ( )( ) ( ) , ( )- }

(5.38)

In order to solve this problem and get a convergent solution we expand in Maclaurin

series expansions of order 6 and substitute these series in Eq. (5.38) such that

( )

and ( )

We use formula (5.38) to obtain 5-order VIM approximate solution ( ) results and compare with the

exact ( ) and Maple numerical solutions ( ) of Eq. (5.37) when for all , - as

shown in the following tables and figure:

Table 1: Lower 5- order VIM approximate, exact and Maples numerical solution of Eq. (5.37) at t = 0.1

Table 2: Upper 5- order VIM approximate, exact and Maples numerical solution of Eq. (5.37) at t = 0.1

( ) ( ) ( )

Fig 1: Approximate 5-order VIM compares with the exact and Maple numerical solutions of Eq. (5.37) when t =0.1.

Table 2: Upper 5- order VIM approximate, exact and Maple numerical solutions of

Eq. (37) at t =0.1.

r (0.1; ) (0.1, ) (0.1, )

0 1.1965247538384560 1.19652475400 1.1965247998087429200

0.25 0.9185228031915503 0.91852280370 0.9185229272108376140

0.5 0.6437715602672260 0.64377156090 0.6437716048754136280

0.75 0.3712203400592153 0.37122034070 0.3712203458632439840

1 0.0998334160144086 0.09983341664 0.0998334151895059942

r-level

Table 1: Lower 5- order VIM approximate, exact and Maples numerical solutions of

Eq. (37) at t =0.1

r (0.1; ) (0.1, ) (0.1, )

0 0.99465487450978190 0.9946548739 0.994654868827032580

0.25 0.71762190762759080 0.7176219071 0.7176219140579600520

0.5 0.44355788058277457 0.4435578799 0.4435578163872099240

0.75 0.17141703887020315 0.1714170382 0.1714169861319961300

1 0.099833416014408650 0.09983341664 0.09983341518950599420




From figure (1) and tables (1)-(2) one can see that the exact and the numerical results satisfies the

concepts of the fuzzy numbers in section 2. The following tables compare the accuracy of 5- order VIM

approximate solution and Maple numerical solution with the exact solution of Eq. (5.37) when t = 0.1 for

all 0 ≤ r ≤1 such that

, - [ ]

| ( ) ( )| [ ]

[ ]

| ( ) ( )|

Table 3: Absolute error for the lower and upper bound approximate solution of Eq. (5.37) at t= 0.1 by 5- order VIM

exact and Maple numerical solutions

It is clear from table 3 that the 5- order VIM approximate- analytical solution is more accurate than

results from the Maple numerical solution when compared of with exact the analytical solution of Eq.

(4.36) when for all , -.

Example 5.2. Consider the Duffing’s equation [9] with the fuzzier version of the following type

( ) ( ) ( ) (5.39)

( ) , - ( ) , -

, -

The initial approximation guesses of Eq. (5.39) are given by:

( ) ( ) and ( ) ( )

According to section 3 the variational formula of this problem is given as follows:

( ) ( ) ∫ ( ){ ( )( )

( ) , ( )- }

(5.40)

Since Eq. (5.39) does not have an exact analytical solution and in order to show the accuracy of 5- terms

VIM approximate solution we define the following residual error [23]

( ) | ( ) ( ) , ( )- |

We use formula (5.40) to obtain 5-order VIM approximate solution of Eq. (5.39) when for all

, - as shown in the table below:

Table 3: Absolute error for the lower and upper bound approximate solution of

Eq. (37) at t =0.1 by 5-order VIM exact and Maple numerical solutions

r (0.1; ) (0.1, ) (0.1, ) (0.1, )

0 6.09781 × 10 10 5.07296 × 10 9 1.61543 × 10 10 4.58087 × 10 8

0.25 5.27591 × 10 10 6.95796 × 10 9 5.08449 × 10 10 1.23511 × 10 7

0.5 6.82774 × 10 10 6.35127 × 10 8 6.32773 × 10 10 4.39754 × 10 8

0.75 6.70203 × 10 10 5.20680 × 10 8 6.40784 × 10 10 5.16324 × 10 9

1 6.25591 × 10 10 1.450494 × 10 9 6.25591 × 10 10 1.450494 × 10 9




Table 4: Result and accuracy for the lower and upper bound approximate solution of

Eq. (5.39) at t= 0.1 by 5- order VIM

It is clear from table 4 that the 5-terms of VIM approximate solution of Eq. (5.39) satisfy the concepts of

the fuzzy numbers in section 2.

6 Conclusions

In this paper, the Variational Iteration Method (VIM) has been successfully introduced and applied to

solve second order fuzzy Duffing’s equation to obtain an approximate solution. The problem was solved

directly without it first being reduced to a first order system. The convergence analyses of VIM have been

investigated in detail with regard to the fuzzy Duffing’s equation following the procedures and ideas of the

previous researchers. The obtained results in this paper show that the VIM is capable of accurately solving

second order nonlinear fuzzy initial value problems.

Acknowledgement

Many thanks to Professor Ahmad Izani Md Ismail for his valuable remarks and comments which have

substantially improved the presentation of this paper.

References

[1] T. Allahviranloo, A. Panahi, H. Rouhparvar, A Computational Method To Find An Approximate

Analytical Solution For Fuzzy Differential Equations, An. S,t. Univ. Ovidius Constant, 17 (1) (2009)

5-14.

[2] T. Allahviranloo, S. Khezerloo, M. Mohammadzaki, Numerical Solution for Differential Inclusion by

Adomian Decomposition Method, Journal of Applied Mathematics, 5 (17) (2008) 51-62.

[3] S. Abbasbandy, T. Allahviranloo, P. Darabi, O. Sedaghatfar, Variational Iteration Method for Solving

N-th order Fuzzy Differential Equations, Mathematical and Computational Applications, 16 (4)

(2011).

[4] N. I. Ahmad, M. Mamat, J. Kavikumar, N. A. Hamzad, Solving Fuzzy Duffing’s Equation by the

Laplace Transform Decomposition, Applied Mathematical Sciences, 6 (59) (2012) 2935-2944.

[5] L. Assas, Variational Iteration Method for Solving Coupled-Kdv Equations, Chaos Solutions Fractals,

38 (2008) 1225-1228.

http://dx.doi.org/10.1016/j.chaos.2007.02.012

Table 4: Result and accuracy for the lower and upper bound approximate solution of

Eq. (39) at t =0.1 by 5-order VIM

r (0.1; ) (0.1, ) (0.1, ) (0.1, )

0 0.9817642960058663 1.73342 × 10 7 1.1975986372194212 2.88251 × 10 7

0.25 1.0088113021609082 1.84973 × 10 7 1.1706904673788245 2.70845 × 10 7

0.5 1.0358399373112013 1.97301 × 10 7 1.1437610043010307 2.54402 × 10 7

0.75 1.0628497128059113 2.10366 × 10 7 1.1168107333604598 2.38873 × 10 7

1 1.0628497128059113 2.24209 × 10 7 1.0628497128059113 2.24209 × 10 7





[6] M. F. Abbod, D. G. Von Keyserlingk, D. A Linkens, M. Mahfouf, Survey of Utilization of Fuzzy

Technology in Medicine and Healthcare, Fuzzy sets and system, 120 (2001) 331-349.

http://dx.doi.org/10.1016/S0165-0114(99)00148-7

[7] E. Babolian, H. Sadeghi, Sh. Javadi, Numerically Solution of Fuzzy Differential Equations by

Adomian Method, Applied Mathematics and Computation, 149 (2004) 547-557.

http://dx.doi.org/10.1016/S0096-3003(03)00160-7

[8] S. Barro, R. Marin, Fuzzy Logic in Medicine, Heidelberg: Physica-Verlag, (2002).

http://dx.doi.org/10.1007/978-3-7908-1804-8

[9] B. Berna, S. Mehmet, Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix

Method, Journal of Applied Mathematics, 6 (2013).

[10] G. F. Corliss, Guarented Error Bounds for Ordinary Differential Equations in Theory and Numeric of

Ordinary and Partial Equations, Oxford University press, (1995).

[11] D. Dubois, H. Prade, Towards Fuzzy Differential Calculus, Part 3: Differentiation, Fuzzy Sets and

Systems, 8 (1982) 225-233.

http://dx.doi.org/10.1016/S0165-0114(82)80001-8

[12] D. Dubois, H. Prade, Fuzzy Numbers: An Overview, Analysis of Fuzzy Information, Mathematical

Logic, CRC Press, Boca Raton, FL, 1 (1987) 3-39.

[13] M. S. El Naschie, From Experimental Quantum Optics to Quantum Gravity via a Fuzzy Kahler

Manifold, Chaos Solution and Fractals, 25 (2005) 969-977.


[14] O. S. Fard, An Iterative Scheme for the Solution of Generalized System of Linear Fuzzy Differential

Equations, World Applied Sciences Journal, 7 (2009) 1597-11604.

[15] A. Ghorbanifar, J. Saberi-Nadjafi, Convergence of He’s Variational Iteration Method for Nonlinear

Oscillators, Nonlinear Sci. Lett. A, 1 (4) (2010) 379-384.

[16] M. Ghanbari, Numerical Solution of Fuzzy Initial Value Problems under Generalization

Differentiability by HPM, Int. J. Industrial Mathematics, 1 (1) (2009) 19-39.

[17] M. Ghanbari, Solution of the First Order Linear Fuzzy Differential Equations by Some Reliable

Methods, Journal of fuzzy set and analysis, 2012 (2012) 1-20.

http://dx.doi.org/10.5899/2012/jfsva-00126

[18] J. H. He, Some Asymptotic Methods for Strongly Nonlinear Equations, Internat. J. Modern Phys. B,

20 (2006) 1141- 1199.

http://dx.doi.org/10.1142/S0217979206033796

http://dx.doi.org/10.1016/S0165-0114%2899%2900148-7

http://dx.doi.org/10.1016/S0096-3003%2803%2900160-7

http://dx.doi.org/10.1007/978-3-7908-1804-8

http://dx.doi.org/10.1016/S0165-0114%2882%2980001-8



http://dx.doi.org/10.1142/S0217979206033796




[19] J. H. He, Variational Iteration Method for Delay Differential Equations, Comm. Non-Linear. Sci.

Numer. Simulation, 2 (1997) 235- 236.

http://dx.doi.org/10.1016/S1007-5704(97)90008-3

[20] J. H. He, Approximate Solution of Nonlinear Differential Equations with Convolution Product Non-

Linearity’s, Comput. Methods. Appl. Mech. Engng, 167 (1998) 69-73.

http://dx.doi.org/10.1016/S0045-7825(98)00109-1

[21] J. H. He, Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous

Media, Comput. Methods. Appl. Mech. Engng, 167 (1998) 57-68.

http://dx.doi.org/10.1016/S0045-7825(98)00108-X

[22] J. H. He, Variational Iteration Method- A Kind of Non-Linear Analytical Technique: Some Examples,

Internat. J. Non-Linear Mech, 34 (1999) 699-708.

http://dx.doi.org/10.1016/S0020-7462(98)00048-1

[23] J. C. Jiao, Y. Yamamoto, C. Dang, Y. Ha, An after Treatment Technique for Improving the Accuracy

of Adomian’s Decomposition Method, Comput. Math. Appl, 43 (2002) 783-798.

http://dx.doi.org/10.1016/S0898-1221(01)00321-2

[24] O. Kaleva, A Note on Fuzzy Differential Equations, Nonlinear Analysis, 64 (1) (2006) 895-900.

http://dx.doi.org/10.1016/j.na.2005.01.003

[25] S. A. Khuri, A Laplace Decomposition Algorithm Applied To A Class Of Nonlinear Differential

Equations, J. Appl. Math, 1 (4) (2001) 141-155.

http://dx.doi.org/10.1155/S1110757X01000183

[26] S. Liang, J. J. David, Comparison of Homotopy Analysis Method and Technique, Applied

Mathematics and Computation, 35 (1) (2003) 73-79.

[27] M. A. Noor, K. I. Noor, S. T. Mohyud-Din, Variational Iteration Method For Solving Sixth-Order

Boundary Value Problems, Commun. Nonlinear Sci. Numer. Simulat, 14 (2009) 2571-2580.

[28] H. Nijmeijer, H. Berghuis, On Lyapunov Control of the Duffing Equation, IEEE Trans Circuits Syst.

I, 42 (1995) 473-477.

http://dx.doi.org/10.1109/81.404059

[29] W. O. Rebecca, Lower and Upper Solutions and Existence of Solutions of Fuzzy Differential

Equations, South African Journal of Pure and Applied Mathematics, 4 (2009) 29-43.

[30] S. Salahshour, Nth-order Fuzzy Differential Equations under Generalized Differentiability, Journal of

fuzzy set value analysis, 2011 (2011) 1-14.


http://dx.doi.org/10.1016/S1007-5704%2897%2990008-3

http://dx.doi.org/10.1016/S0045-7825%2898%2900109-1

http://dx.doi.org/10.1016/S0045-7825%2898%2900108-X

http://dx.doi.org/10.1016/S0020-7462%2898%2900048-1

http://dx.doi.org/10.1016/S0898-1221%2801%2900321-2

http://dx.doi.org/10.1016/j.na.2005.01.003

http://dx.doi.org/10.1155/S1110757X01000183

http://dx.doi.org/10.1109/81.404059





[31] S. Siah Mansouri, N. Ahmady, Numerical Method For solving Nth order Fuzzy Differential Equation

by using Characterization Theorem, Communication in Numerical Analysis, 2012 (2012) 1-12.

http://dx.doi.org/10.5899/2012/cna-00054

[32] N. H. Sweilam, Variational Iteration Method for Solving Cubic Nonlinear Schro Dinger Equation, J.

Comput.Appl. Math, 207 (2007) 155-163.

http://dx.doi.org/10.1016/j.cam.2006.07.023

[33] S. Seikkala, On the Fuzzy Initial Value Problem, Fuzzy Sets Syst, 24 (1987) 319-330.

http://dx.doi.org/10.1016/0165-0114(87)90030-3

[34] R. Yulita Molliq, M. S. M. Noorani, I. Hashim, Variational Iteration Method for Fractional Heat- And

Wave-Like Equations, Nonlinear Analysis: Real World Applications, 10 (3) (2009) 1854-1869.

http://dx.doi.org/10.1016/j.nonrwa.2008.02.026

[35] E. Yusufoglu, Numerical Solution of Duffing Equation by the Laplace Decomposition Algorithm,

Applied Mathematics and Computation, 177 (2) (2006) 572-580.

http://dx.doi.org/10.1016/j.amc.2005.07.072

[36] M. Zaini, Ahmad Bernard De Baets, A Predator-Prey Model with Fuzzy Initial Populations, IFSA-

EUSFLAT (2009).

[37] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965) 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

http://dx.doi.org/10.5899/2012/cna-00054

http://dx.doi.org/10.1016/j.cam.2006.07.023

http://dx.doi.org/10.1016/0165-0114%2887%2990030-3

http://dx.doi.org/10.1016/j.nonrwa.2008.02.026

http://dx.doi.org/10.1016/j.amc.2005.07.072

http://dx.doi.org/10.1016/S0019-9958%2865%2990241-X

The Variational Iteration Method for Solving Fuzzy Duffing's Equation

Documents