Journal of Fuzzy Set Valued Analysis 2014 (2014) 1-11
Available online at www.ispacs.com/jfsva Volume 2014, Year 2014 Article ID jfsva-00196, 11 Pages doi:10.5899/2014/jfsva-00196 Research Article
Variational Iteration Method for Solving a Fuzzy Generalized Pantograph Equation A. Amiri∗ Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran c A. Amiri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted Copyright 2014 ⃝ use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract A numerical method for solving the fuzzy generalized pantograph equation under fuzzy initial value conditions is presented. This technique provides a sequence of functions which converges to the exact solution to the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. To display the validity and applicability of the numerical method two illustrative examples are presented. Keywords: Variational iteration method; Generalized Pantograph equation; Fuzzy differential equation; Generalized Hukuhara dif-
ferentiability
1 Introduction Fuzzy differential equations are utilized for the purpose of the modelling problems in science and engineering. Most of the problems in science and engineering require the solutions of a fuzzy differential equation which are satisfied with fuzzy initial conditions, therefore a fuzzy initial value problem is occurring, and should be solved. The topic of fuzzy differential equations has been rapidly growing in recent years. The concept of the fuzzy derivative was first introduced by Chang and Zadeh [15], it was followed up by Dubois and Prade [19]. Kaleva in [26] and [27] proposed fuzzy differential equations using H-derivative and it was developed by some other authors (see [[31], [11], [14], [13], [16], [18], [25], [7], [5], [6], [9], [21]]). The numerical methods for solving fuzzy differential equations are introduced in [1], [2], [4], [3]. The name pantograph originated from the study [30] by Ockendon and Tayler. These equations arise in industrial applications and in studies based on biology, economy, control theory and electrodynamics, among others. Properties of the analytic solution of these equations as well as numerical methods have been studied by several authors [17], [29], [20]. In [28], the authors introduced a numerical method based on the Taylor polynomials for the approximate solution of the fuzzy pantograph equation with linear functional argument. The purpose of the present paper is to develop and apply variation iteration methods to the generalized pantograph equation with fuzzy initial conditions. This paper is organized as follows: In Section 2, we describe the basic notations and preliminaries. In Section 3, the variational iteration method is briefly described. We define the fuzzy generalized pantograph equation under generalized Hukuhara differentiability in Section 4 and according to the type of differentiability, the solutions of the fuzzy generalized pantograph equation are investigated. Some numerical examples are given to clarify the details and efficiency of the method in Section 5. We end up with some conclusions. ∗ Correspondence
author. Email address:
[email protected], Tel:+989102477750
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2 Basic concepts In this section, we represent some definitions and introduce the necessary notation which will be used throughout the paper. Definition 2.1. A fuzzy number is a function such as u : Rn → [0, 1] satisfying the following properties: (i) u is normal, i. e, ∃ t0 ∈ R with u(t0 ) = 1; (ii) u is convex fuzzy set i. e, u(λ x + (1 − λ )y) ≥ min(u(x), u(y)) ∀x, y ∈ R, λ ∈ [0, 1]; (iii) u is upper semi-continuous on R; (iv) closure of {t ∈ Rn |u(t) > 0} is compact. Then RF is called the space of fuzzy numbers. } { For 0 < α ≤ 1 denote u(α ) = t ∈ Rn u(t) ≥ α = [u(α ), u(α )]. Then from (i) to (iv), it follows that the α -level set u(α ) is a closed interval for all α ∈ [0, 1]. For arbitrary(u, v ∈) RF and k ∈ R, the addition and scalar multiplication are defined by (u + v)(α ) = u(α ) + v(α ) , (ku)(α ) = k u(α ) respectively. [8] If u ⊕ v = w, then w ⊖ v = u; here, ⊖ is the Hukuhara difference. Definition 2.2. [12] Given two fuzzy numbers u, v ∈ RF , the generalized Hukuhara difference (gH-difference for short) is the fuzzy number w, if it exists, such that { (i) u = v + w, u ⊖gH v = w ⇐⇒ (2.1) or (ii) v = u + (−1)w. It is easy to show that(i) and (ii) are both valid if and only if w is a crisp number. In terms of α -cuts we have [u ⊖gH ν ]α = [min{u(α ) − v(α ), u(α ) − v(α )}, max{u(α ) − v(α ), u(α ) − v(α )}] and if the H-difference exists, then u ⊖ v = u ⊖gH v; the conditions for the existence of w = u ⊖gH v ∈ RF are { case(i) { case(ii)
w(α ) = u(α ) − v(α ) and w(α ) = u(α ) − v(α ) with w(α ) increasing, w(α ) decreasing, w(α ) ≤ w(α ),
∀α ∈ [0, 1],
(2.2)
w(α ) = u(α ) − v(α ) and w(α ) = u(α ) − v(α ) with w(α ) increasing, w(α ) decreasing, w(α ) ≤ w(α ),
∀α ∈ [0, 1],
(2.3)
Let α -level representation of fuzzy-valued function f : [a, b] → RF is expressed by f (t; α ) = [ f (t; α ), f (t; α )], t ∈ [a, b], for each α ∈ [0, 1]. Definition 2.3. [12]. Let t0 ∈ ]a, b[ and h be such that t0 + h ∈ ]a, b[, then the gH-derivative of a function f : ]a, b[→ RF at t0 is defined as ′ fgH (t0 ) = lim
h→0
f (t0 + h) ⊖gH f (t0 ) . h
(2.4)
′ (t ) ∈ R If fgH 0 F satisfying (2.4) exists, we say that f is generalized Hukuhara differentiable (gH-differentiable for short) at t0 .
Theorem 2.1. [12] Let f :]a, b[→ RF be such that f (t; α ) = [ f (t; α ), f (t; α )]. Suppose that the functions f (t; α ) and f (t; α ) are real-valued functions, differentiable w.r.t. t , uniformly w.r.t. α ∈ [0, 1]. Then the function f (t) is gH-differentiable at a fixed t ∈]a, b[ if and only if one of the following two cases holds: ′
′
(a) f ′ (t; α ) is increasing, f (t; α ) is decreasing as functions of α , and f ′ (1; α ) ≤ f (1; α ), or
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′
′
(b) f ′ (t; α ) is decreasing, f (t; α ) is increasing as functions of α , and f (1; α ) ≤ f ′ (1; α ). also, ∀α ∈ [0, 1] we have ′
′
′ fgH (t; α ) = [min{ f ′ (t; α ), f (t; α )}, max{ f ′ (t; α ), f (t; α )}].
(2.5)
Definition 2.4. [12] Let f : [a, b] → RF and t0 ∈ ]a, b[ with f α (t; α ) and f α (t; α ) both differentiable at t0 . We say that : • f is (i)-gH-differentiable at t0 if ′
′ fi.gH (t0 ; α ) = [ f ′ (t0 ; α ), f (t0 ; α )],
∀α ∈ [0, 1],
(2.6)
∀α ∈ [0, 1].
(2.7)
• f is (ii)-gH-differentiable at t0 if ′
′ fii.gH (t0 ; α ) = [ f (t0 ; α ), f ′ (t0 ; α )],
Definition 2.5. [10] The second generalized Hukuhara derivative of a fuzzy-valued function f : (a, b) −→ RF at t0 is defined as f ′ (t0 + h) ⊖gH f ′ (t0 ) ′′ fgH (t0 ) = lim , h→0 h ′′ (t ) ∈ R , we say that f ′ (t) is generalized Hukuhara differentiable at t . if fgH 0 0 F gH ′ (t) is (i)-gH-differentiable at t if Also we say that fgH 0
{ ′′ fi.gH (t0 ; α ) =
′′
[ f ′′ (t0 ; α ), f (t0 ; α )], if f be (i)-gH-differentiable on (a, b); ′′ [ f (t0 ; α ), f ′′ (t0 ; α )], if f be (ii)-gH-differentiable on (a, b).
′ (t) is (ii)-gH-differentiable at t if for all α ∈ [0, 1], and that fgH 0
{ ′′ fii.gH (t0 ; α ) =
′′
[ f (t0 ; α ), f ′′ (t0 ; α )], if f be (i)-gH-differentiable on (a, b); ′′ [ f ′′ (t0 ; α ), f (t0 ; α )], if f be (ii)-gH-differentiable on (a, b).
for all α ∈ [0, 1]. Definition 2.6. [12] We say that a point t0 ∈ ]a, b[ is a switching point for the gH-differentiability of f , if in any neighborhood V of t0 there exist points t1 < t0 < t2 such that (type-I) at t1 (2.6) holds while (2.7) does not hold and at t2 (2.7) holds and (2.6) does not hold, or (type-II) at t1 (2.7) holds while (2.6) does not hold and at t2 (2.6)holds and (2.7) does not hold. 3 Variational Iteration Method (VIM) Consider the following general nonlinear initial value problem. L[y(t)] + N[y(t)] = g(t)
(3.8)
Where L is a linear operator, N a nonlinear operator and g(t) is a known analytical function. According to VIM, we can construct a correction functional as follows. ∫ t
un+1 (t) = un (t) +
0
λ (s)[Lun (s) + N(u˜n )(s) − g(s)]ds
n ≥ 0.
(3.9)
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Where λ is a general Lagrangian multiplier [24] which can be identified optimally via the variational theory, the subscript n denotes the nth-order approximation, uen is considered as a restricted variation i.e. δ uen = 0, [23], [22]. Therefore, we first determine the Lagrange multiplier λ that will be identified optimally via integration by parts. The successive approximations un+1 , n ≥ 0 of the solution u will be readily obtained upon using the determined Lagrangian multiplier and any selective function u0 . Consequently, the solution is given by u = limn→∞ un . Now, consider the following system of generalized pantograph equations { ′ U (t) = β U(t) + f (t, U(t), U(ε1 (t)), U(ε2 (t)), ..., U(εl (t))) (3.10) U(0) = U0 where U(t) = (u1 , u2 , ...., un )t in which ui , i = 1, 2, ..., n are unknown real functions of variable t, U(ε j (t)) = (u1 (ε j (t)), u2 (ε j (t)), ..., un (ε j (t))), j = 1, 2, ..., n and moreover f and εi (t), i = 1, 2, ..., l are analytical function and β ∈ R+ , we consider
εi (t) = qit
i = 1, 2, ..., l.
where 0 < q1 < ql−1 < ... < ql < 1. Now consider Eq. (3.10), according to VIM, we consider the correction functional in the following form, ∫ t [ ] Un+1 (t) = Un (t) + λ (s) Un′ (s) − β Un (s) − fe(s, Un (s), Un (ε1 (s)), ..., Un (εl (s))) ds 0
to find the optimal value of λ we have ∫ t [ ] δ Un+1 (t) = δ Un (t) + δ λ (s) Un′ (s) − β Un (s) ds 0
then
δ Un+1 (t) = δ Un (t) + λ δ Un (s)|s=t −
∫ t[ 0
] λ ′ (s) + β λ (s) δ Un (s)ds = 0.
Thus we have the following stationary conditions: { 1 + λ (s)|s=t = 0, λ ′ (s) + β λ (s)|s=t = 0, The Lagrange multiplier, therefore, can be identified :
λ = −e−β (s−t) As a result, we obtain the following iteration formula: ∫ t [ ] Un+1 (t) = Un (t) − e−β (s−t) Un′ (s) − β Un (s) − f (s, Un (s), Un (ε1 (s)), ..., Un (εl (s)) ds.
(3.11)
0
Consider following system of generalized pantograph equation { ′′ U (t) = f (t, U(t), U(ε1 (t)), U(ε2 (t)), ..., U(εl (t))) U(0) = λ0 , U ′ (0) = λ1
(3.12)
In accordance with the process described for Eq. (3.10), the correct functional for Eq. (3.12) can be written as ∫ t [ ] Un+1 (t) = Un (t) + λ (s) Un′′ (s) − fe(s, Un (s), Un (ε1 (s)), ..., Un (εl (s))) ds. 0
and, we obtain the following Lagrange multiplier
λ (s) = (s − t) Therefore, we have the following iteration formula: ∫ t ] [ Un+1 (t) = Un (t) + (s − t) Un′′ (s) − f (s, Un (s), Un (ε1 (s)), Un (ε2 (s)), ..., Un (εl (s))) ds.
(3.13)
0
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4 Fuzzy Generalized Pantogragh Equation Consider function u(t) : R → RF satisfying the fuzzy generalized pantograph equations Problem 1 {
u′gH (t) = β ⊙ u(t) ⊕ f (t, u(t), u(α1 (t)), u(α2 (t)), ..., u(αl (t))) u(0) = λ0
(4.14)
u′′gH (t) = f (t, u(t), u(ε1 (t)), u(ε2 (t)), ..., u(εl (t))) u(0) = λ0 , u′ (0) = λ1
(4.15)
Problem 2 {
where f and εi (t), i = 1, 2, ..., l are analytical function and β ∈ R+ and λi ∈ RF , i = 0, 1. We consider
εi (t) = qit
i = 1, 2, ..., l.
where 0 < ql < ql−1 < ... < q1 < 1. We choose the derivative type of solution for Problem 4.14 or 4.15 and translate functional equation to the corresponding system and VIM is applied for solving this system, for example if u(t) is (i)-gH-differentiable for Problem 4.14, then we obtain ′ u (t; α ) = β u(t; α ) + f (t, u(t), u(ε1 (t)), u(ε2 (t)), ..., u(εl (t)); α ), for all α ∈ [0, 1]; u(0; α ) = u0 (α ) u′ (t; α ) = β u(t; α ) + f (t, u(t), u(ε1 (t)), u(ε2 (t)), ..., u(εl (t)); α ), for all α ∈ [0, 1]; u(0; α ) = u0 (α ) 5 Example Example 5.1. Consider the following pantograph equation { ′′ ugH = 34 u(t) + u( 2t ) ⊖gH [−3α + 4 12 α + 12 ]t 2 + [α + 1 u(0; α ) = 0 u′ (0; α ) = 0 0 ≤ t ≤ 1.
− 6α + 8]
where 0 denotes the crisp set {0}. The exact solution of this equation is equal u(t; α ) = [ 21 α + 12 In this example u and u′ are (i) − gH differentiable 2.8, so we have the following system: ′′ 3 t 1 1 2 u (t; α ) = 4 u(t; α ) +′ u( 2 ; α ) − ( 2 α + 2 )t + (α + 1) u (0; α ) = 0 u(0, α ) = 0 , ′′ (t; α ) = 3 u(t; α ) + u( t ; α ) − (−3α + 4)t 2 + (−6α + 8) u 4 2 u(0, α ) = 0 , u′ (0; α ) = 0
− 3α + 4]t 2 .
Now we have the following iteration formula ∫ t [ ] 3 s 1 1 un+1 (t; α ) = un (t; α ) + (s − t) u′′ n (s; α ) − un (s; α ) − un ( ; α ) + ( α + )s2 − (α + 1) ds 4 2 2 2 0 and un+1 (t; α ) = un (t; α ) +
∫ t 0
[ ] s 3 (s − t) u′′ n (s; α ) − un (s; α ) − un ( ; α ) + (−3α + 4)s2 − (−6α + 8) ds 4 2
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Let us start with an initial approximation u0 (t; α ) = 0, u0 (t; α ) = 0: u1 (t; α ) = (α + 1)t 2 − (α + 1)
t4 12
u1 (t; α ) = (−6α + 8)t 2 + (−6α + 8)
t4 12
for n = 2 we have t2 t4 13 6 + (α + 1) − (α + 1) t 2 24 5760 t2 t4 13 6 u2 (t; α ) = (−6α + 8) + (−6α + 8) − (−6α + 8) t 2 24 5760 u2 (t; α ) = (α + 1)
for n = 3 we have: t2 13 6 + (α + 1) t − (α + 1)0.000031 t 8 2 11520 t2 13 6 t − (−6α + 8)0.000031 t 8 u3 (t; α ) = (−6α + 8) + (−6α + 8) 2 11520 u3 (t; α ) = (α + 1)
for n = 4 we have: t2 + (α + 1)0.000015 t 8 2 t2 u4 (t; α ) = (−6α + 8) + (−6α + 8)0.000015 t 8 2 u4 (t; α ) = (α + 1)
t 1 2 3 4 5 6 7 8 9 10
1 2)
α=
u4 (t, 2.30000 × 10−13 5.88800 × 10−11 1.50902 × 10−9 1.50732 × 10−8 8.98437 × 10−8 3.86311 × 10−7 1.32590 × 10−6 3.85875 × 10−6 9.90074 × 10−6 2.29999 × 10−5
1 2
u4 (t, 21 ) 7.50000 × 10−13 1.91999 × 10−10 4.92075 × 10−9 4.91520 × 10−8 4.91520 × 10−7 1.25971 × 10−6 4.32360 × 10−6 1.25829 × 10−5 3.22850 × 10−5 7.49999 × 10−5
Table 1: Numerical results for Example.5.1
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1 Exact solution Approximate solution
0.9 0.8 0.7
α
0.6 0.5 0.4 0.3 0.2 0.1 0 0.1
0.2
0.3
0.4
0.5
0.6 u(0.5,α)
0.7
0.8
0.9
1
1.1
Figure 1: Graph of the VIM approximation error of Numerical illustration for example 5.1.
Example 5.2. Consider the pantograph equation of first order with variable coefficients { −3t −t u′gH (t) = ⊖gH u(t) ⊖gH e 2 sin( 2t )u( 2t ) ⊖gH 2e 4 cos( 2t )sin( 4t )u( 4t ), 0 ≤ t ≤ 1; u(0; α ) = [α + 1 − 2α + 4], for all α ∈ [0, 1]. Note that the exact solution of this problem is u(t; α ) = [α + 1 − 2α + 4]e−t cos(t) Since u(t) has (ii)-gH-differentiable, the (ii)-gH-differentiable solution is obtained by solving −t −3t u′ (t; α ) = −u(t; α ) − e 2 sin( 2t )u( 2t ; α ) − 2e 4 cos( 2t )sin( 4t )u( 4t ; α ); −t −3t u′ (t; α ) = −u(t; α ) − e 2 sin( 2t )u( 2t ; α ) − 2e 4 cos( 2t )sin( 4t )u( 4t ; α ); u(0; α ) = [α + 1 − 2α + 4] To solve this equation using the VIM, based on the iteration formula (3.11) we get ∫ t [ ] −s −3s s s s s s un+1 (t; α ) = un (t; α ) − e(s−t) u′ (s; α ) + u(s; α ) + e 2 sin( )u( ; α ) + 2e 4 cos( )sin( )u( ; α ) ds (5.16) 2 2 2 4 4 0 ∫ t [ ] −s −3s s s s s s un+1 (t; α ) = un (t; α ) − e(s−t) u′ (s; α ) + u(s; α ) + e 2 sin( )u( ; α ) + 2e 4 cos( )sin( )u( ; α ) ds (5.17) 2 2 2 4 4 0 We start with initial approximation u0 (t; α ) = (α + 1)cos(t) and u0 (t; α ) = (−2α + 4)cos(t), the other terms of the sequence un (t; α ) are computed easily by substituting this equation into Eqs. (5.16) and (5.17), we have n=1 5 (α + 1)t 3 + ... 24 5 u1 (t; α ) = (−2α + 4) − (−2α + 4)t + (−2α + 4)t 3 + ... 24 u1 (t; α ) = (α + 1) − (α + 1)t +
n=2 1 1 539 u2 (t; α ) = (α + 1) − (α + 1)t + (α + 1)t 3 − (α + 1)t 4 + (α + 1)t 5 + ... 3 6 15360 1 1 539 u2 (t; α ) = (−2α + 4) − (−2α + 4)t + (−2α + 4)t 3 − (−2α + 4)t 4 + (−2α + 4)t 5 + ... 3 6 15360
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n=3 1 1 1 1051249 u3 (t; α ) = (α + 1) − (α + 1)t + (α + 1)t 3 − (α + 1)t 4 + (α + 1)t 5 − (α + 1)t 7 + ... 3 6 30 660602880 1 1 1 u3 (t; α ) = (−2α + 4) − (−2α + 4)t + (−2α + 4)t 3 − (−2α + 4)t 4 + (−2α + 4)t 5 3 6 30 1051249 7 − (−2α + 4)t + ... 660602880 .. . In the other hand , we have 1 1 1 1 u(t; α ) = (α + 1)e−t cos(t) = (α + 1) − (α + 1)t + (α + 1)t 3 − (α + 1)t 4 + (α + 1)t 5 − (α + 1)t 7 + ... 3 6 30 630 1 1 u(t; α ) = (−2α + 4)e−t cos(t) = (−2α + 4) − (−2α + 4)t + (−2α + 4)t 3 − (−2α + 4)t 4 3 6 1 1 + (−2α + 4)t 5 − (−2α + 4)t 7 + ... 30 630 We see that the approximation solutions obtained by VIM have good agreement with exact solution of this problem. In Table 2 the absolute errors of the present method for n = 3 and α = 31 .
t 1 2 3 4 5 6 7 8 9 10
1 3)
α=
u3 (t, 3.70000 × 10−13 4.88800 × 10−11 6.50702 × 10−9 2.50932 × 10−8 6.98447 × 10−8 1.96801 × 10−7 2.65590 × 10−7 4.87889 × 10−7 6.50173 × 10−6 7.49896 × 10−5
1 3
u3 (t, 31 ) 6.70000 × 10−13 3.94567 × 10−10 2.52618 × 10−9 5.24619 × 10−8 6.47814 × 10−7 5.73890 × 10−7 6.32457 × 10−6 3.00921 × 10−7 2.24567 × 10−6 9.32456 × 10−5
Table 2: Numerical results for Example.5.2
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1 Exact solution Approximate solution
0.9 0.8 0.7
α
0.6 0.5 0.4 0.3 0.2 0.1 0 0.4
0.6
0.8
1
1.2
1.4 u(0.5,α)
1.6
1.8
2
2.2
2.4
Figure 2: Graph of the VIM approximation error of Numerical illustration for example 5.2
6 Conclusion In this paper, we proposed the variation iteration method for solve the fuzzy generalized pantograph equation under generalized Hukuhara differentiability concept. First we choose type of differentiability of solution and convert it to a system of differential equations. Then we use VIM and find the approximate solution of this system. This technique produces the terms of a sequence using the iteration of the correction functional which converges to the exact solution rapidly. References [1] S. Abbasbandy, T. Allahviranloo, Numerical solutions of fuzzy differential equations by taylor method, Computational Methods in Applied Mathematics, 2 (2002) 113-124. http://dx.doi.org/10.2478/cmam-2002-0006 [2] S. Abbasbandy, T. Allahviranloo, O. Lopez-Pouso, J. J. Nieto, Numerical methods for fuzzy differential inclusions, Computer and Mathematics With Applications, 48 (10-11) (2004) 1633-1641. http://dx.doi.org/10.1016/j.camwa.2004.03.009 [3] T. Allahviranloo, N. Ahmady, E. Ahmady, Erratum to Numerical solution of fuzzy differential equations by predictor-corrector method, Information Sciences, 178 (2008) 1780-1782. http://dx.doi.org/10.1016/j.ins.2007.11.009 [4] T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictorcorrector method, Information Sciences, 177 (7) (2007) 1633-1647. http://dx.doi.org/10.1016/j.ins.2006.09.015 [5] T. Allahviranloo, A. Armand, Z. Gouyandeh, H Ghadiri, Existence and uniqueness of solutions for fuzzy fractional Volterra-Fredholm integro-differential equations, Journal of Fuzzy Set Valued Analysis, 2013 (2013) 1-9. http://dx.doi.org/10.5899/2013/jfsva-00163 [6] T. Allahviranloo, A. Armand, S. Pirmohammadi, Variational Homotopy Perturbation Method: An efficient scheme for solving partial differential equations in fluid mechanics, The Journal of Mathematics and Computer Science, 9 (4) 362-369.
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[7] T. Allahviranloo, Z. Gouyandeh, A. Armand, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems, (In press). [8] T. Allahviranloo, N. A. Kiani, M. Barkhordari, Toward the existence and uniqueness of solutions of second-order fuzzy differential equations, 179 (8) (2009) 1207-1215. http://dx.doi.org/10.1016/j.ins.2008.11.004 [9] A. Armand, Z. Gouyandeh, Numerical solution of the system of Volterra integral equations of the first kind, International Journal of Industrial Mathematics, 6 (1) 27-35. [10] A. Armand, Z. Gouyandeh, Solving two-point fuzzy boundary value problem using variational iteration method, Communications on Advanced Computational Science with Applications, 2013 (2013) 1-10. http://dx.doi.org/10.5899/2013/cacsa-00006 [11] B. Bede, I. J. Rudas, A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences, 177 (2007) 1648-1662. http://dx.doi.org/10.1016/j.ins.2006.08.021 [12] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013) 119-141. http://dx.doi.org/10.1016/j.fss.2012.10.003 [13] J. J. Buckley, Simulating Continuous Fuzzy Systems, Springer, (2006). [14] J. J. Buckley, T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999) 241-248. http://dx.doi.org/10.1016/S0165-0114(98)00323-6 [15] S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Transactions on Systems Man Cybernetics, 2 (1972) 330-340. [16] W. Congxin, S. Shiji, Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions, Information Sciences, 108 (1998) 123-134. http://dx.doi.org/10.1016/S0020-0255(97)10064-0 [17] G. A. Derfel, F. Vogl, On the asymptotics of solutions of a class of linear functional-differential equations, European J. Appl. Math, 7 (1996) 511-518. http://dx.doi.org/10.1017/S0956792500002527 [18] Z. Ding, M. Ma, A. Kandel, Existence of the solutions of fuzzy differential equations with parameters, Information Sciences, 99 (1997) 205-217. http://dx.doi.org/10.1016/S0020-0255(96)00279-4 [19] D. Dubois, Towards fuzzy differential calculus: part 3, differentiation, Fuzzy Sets and Systems, 8 (1982) 225233. http://dx.doi.org/10.1016/S0165-0114(82)80001-8 [20] A. Feldstein, Y. Liu, On neutral functional-differential equations with variable time delays, Math. Proc. Cambridge Philos. Soc, 124 (1998) 371-384. http://dx.doi.org/10.1017/S0305004198002497 [21] Z. Gouyandeh, A. Armand, Numerical solutions of fuzzy linear system differential equations and application of a Radioactivity decay model, Communications on Advanced Computational Science with Applications, 2013 (2013) 1-11. http://dx.doi.org/10.5899/2013/cacsa-00005
International Scientific Publications and Consulting Services
Journal of Fuzzy Set Valued Analysis http://www.ispacs.com/journals/jfsva/2014/jfsva-00196/
Page 11 of 11
[22] J. H. He, Approximate solution of nonlinear differential equations with convolution product non-linearities, Comput. Methods. Appl. Mech. Engrg, 167 (1998) 69-73. http://dx.doi.org/10.1016/S0045-7825(98)00109-1 [23] J. H. He, Variational iteration method for delay differential equations, Commun. Nonlinear. Sci. Numer. Simul, 2 (1997) 235-236. http://dx.doi.org/10.1016/S1007-5704(97)90008-3 [24] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in non-linear mathematical physics, in: Variational Methods in the Mechanics of Solids, Pergamon Press, Oxford, (1978) 156-162. [25] L. J. Jowers, J. J. Buckley, K. D. Reilly, Simulating continuous fuzzy systems, Information Sciences, 177 (2007) 436-448. http://dx.doi.org/10.1016/j.ins.2006.03.005 [26] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987) 301-317. http://dx.doi.org/10.1016/0165-0114(87)90029-7 [27] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35 (1990) 389-396. http://dx.doi.org/10.1016/0165-0114(90)90010-4 [28] N. Mikaeilvand, L. Hossieni, The Taylor Method for Numerical Solution of Fuzzy Generalized Pantograph Equations with Linear Functional Argument, Int. J. Industrial Mathematics, 2 (2) (2010) 115-127. [29] G. R. Morris, A. Feldstein, E. W. Bowen, The Phragmen-Lindel of principle and a class of functional-differential equations, in Proceedings of NRL-MRC Conference on Ordinary Differential Equations, (1972) 513-540. [30] J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. London Ser. A, 322 (1971) 447-468. http://dx.doi.org/10.1098/rspa.1971.0078 [31] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987) 319-330. http://dx.doi.org/10.1016/0165-0114(87)90030-3
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