Solving fuzzy fractional differential equations using fuzzy Sumudu transform

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1 Available online at J. Nonlinear Sci. Appl. Vol. (201X), Research Article Solving fuzzy fractional differential equations using fuzzy Sumudu transform Norazrizal Aswad Abdul Rahman, Muhammad Zaini Ahmad Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, Arau, Perlis, Malaysia. Abstract In this paper, we apply fuzzy Sumudu transform (FST) for solving fuzzy fractional differential equations (FFDEs) involving Caputo fuzzy fractional derivative. It followed by suggesting a new result on the property of FST for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of FFDEs and finally we demonstrate a numerical example. Keywords: Caputo fuzzy fractional derivative, fuzzy Sumudu transform, fuzzy fractional differential equation. MSC: 26A33, 44A05, 34A08, 34A07 1. Introduction Fractional calculus is the generalization of ordinary calculus. This includes the functions derivative of arbitrary order. The topic has been explored and studied by various researchers in many fields such as engineering, mathematics and so forth [29, 13, 31, 3, 19. One of the major contributions in this field was the work studied by [33, which discussed the topic intensively. Later, it was studied by [27, where the authors proposed some applications. When dealing with fractional differential equations, the terms such as Riemann-Liouville, Grünwald-Letnikov and Caputo fractional derivative are considered by many authors [21, 18, 26. Of the three definitions of derivative stated, Riemann-Liouville and Caputo fractional derivatives appeared to be more popular. As times moving on, the fractional differential equations seems to have some drawbacks. One of them is the initial value assigned to the model. In general, the determination of initial values is very difficult. It always involves uncertainty quantities. This is true when dealing with real physical phenomena. To Corresponding author addresses: norazrizalaswad@gmail.com (Norazrizal Aswad Abdul Rahman), mzaini@unimap.edu.my (Muhammad Zaini Ahmad) Received 201X-X-X

2 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), handle uncertainty quantities, many researchers proposed several new concepts. The one that stands out among the concepts is fuzzy set theory [43. This theory is able to deal with differential equations possessing uncertainties at initial values. The first contribution on handling fractional differential equations with uncertainties was studied by [2. This has influenced many researchers to further explore the subject [9, 8, 6, 5, 37, 25. Integral transforms have long been used in solving ordinary differential equations, as well as fractional differential equations. The integral transforms were preceded by Fourier transform. Later, several new integral transforms have been proposed, namely, Laplace, Mellin, and Hankel transforms [35, 28, 38. One of the recent integral transforms introduced in the literature is the Sumudu transform [39, 40. The virtue of this transform is that it holds a scale preserving property which resulting in the original function to be similar with the transformed function. It can also be seen in the literature, there exist several discussions on solving few types of fractional differential equations, as we stated previously, using Sumudu transform [23, 14, 15. Recently, fuzzy Laplace transform [7 has been used to solve FFDEs involving Riemann-Liouville fractional derivative [34. However, this type of fractional derivative has a drawback. It requires a quantity of fractional H-derivative of an unknown solution at the fuzzy initial point, which is not practical in real life situation. In this paper, we propose a new solution of FFDEs involving Caputo fuzzy fractional derivative using FST. The FST is first proposed by [4 followed by [1. The arrangement of this paper is as the following. In Section 2, we revise some fundamental theories on fuzzy numbers and fuzzy functions. Plus, some definitions and theorems on Caputo fuzzy fractional derivative will also be provided. It is followed by the definition of the FST in Section 3. In this section, we also propose a new property of FST for Caputo fuzzy fractional derivative. Next, in Section 4, we provide a procedure on solving FFDEs possessing Caputo fuzzy fractional derivative using the FST in detail. A numerical example is demonstrated in Section 5 and finally in Section 6, the conclusions is drawn. 2. Basic concepts and theories Here, we revisit several definitions and theorems for a better understanding of this paper Fuzzy numbers and fuzzy functions Throughout this paper, R denotes the set of real numbers. Fuzzy number is defined as follows. Definition 2.1. [42 A fuzzy number is a mapping ũ : R [0, 1 with the following criteria: 1. ũ is normal, i.e. there exists x 0 R such that ũ(x 0 ) = 1, 2. ũ is convex, i.e. for all and λ [0, 1, x, y R, holds, 3. ũ is upper semi continuous, i.e. for any x 0 R, ũ(λx + (1 λ)y) minũ(x), ũ(y)}, ũ(x 0 ) lim ũ(x) x x ± 0 4. supp ũ = x R ũ(x) > 0} is the support of ũ, and its closure cl(supp ũ) is compact,. Definition 2.2. [22 Let ũ be a fuzzy number defined in F(R). The α-level set of ũ, for any α [0, 1, denoted by ũ α, is a crisp set that contains all elements in R, such that the membership value of ũ is greater or equal to α, that is: ũ α = x R ũ(x) α}.

3 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), Whenever we represent the fuzzy number with α-level set, we can see that it is closed and bounded. It is denoted by [u α, u α, where they represent the lower and upper bound α-level set of the fuzzy number, respectively. As the fuzzy number is resolved by the interval ũ α, researchers [17, 30 defined another representation, parametrically, of fuzzy numbers as in the following definition. Definition 2.3. A fuzzy number ũ in parametric form is a pair [u α, u α of functions u α and u α, for any α [0, 1 which satisfy the following requirements: 1. u α is a bounded non-decreasing left continuous function in (0, 1, 2. u α is a bounded non-increasing left continuous function in (0, 1, 3. u α u α. Some researchers classified the fuzzy numbers into several types of fuzzy membership function. To the deepest of our study, triangular fuzzy membership function or also often referred to as triangular fuzzy number is the most widely used membership function. Definition 2.4. [24 A triangular fuzzy number ũ can be defined by a triplet (a 1, a 2, a 3 ), the membership function is defined as follows: 0, if x < a 1, x a 1, if a 1 x < a 2, ũ(x) = a 2 a 1 a 3 x, if a 2 x a 3, a 3 a 2 0, if x > a 3, The α-level of the fuzzy number ũ is ũ α = [a 1 + (a 2 a 3 )α, a 3 (a 3 a 2 )α, for any α [0, 1. The definition of the operations on fuzzy numbers can be referred in the paper by [36. Theorem 2.5. [41 Let f : R F(R) and it is represented by [f α (x), f α (x). For any fixed α [0, 1, assume f α (x) and f α (x) are Riemann-integrable on [a, b for every b a, and assume there are two positive M α and M α such that b a f α (x) dx M α and b a f α(x) dx M α for every b a. Then, f(x) is improper fuzzy Riemann-integrable on [a, ) and the improper fuzzy Riemann-integrable is a fuzzy number. Furthermore, we have [ f(x)dx = f α (x)dx, f α (x)dx. H-difference of fuzzy numbers is defined as follows. a a Definition 2.6. If ũ, ṽ F(R) and if there exists a fuzzy subset ξ F(R) such that ξ + ũ = ṽ, then xi is unique. In this case, ξ is called the Hukuhara difference, or simply H-difference, of u and v and is denoted by ṽ H ũ. In the next definition, the strongly generalized differentiability concept is provided. Definition 2.7. [11, 12 Let f : (a, b) F(R) and x 0 (a, b). We say that f is strongly generalized differentiable at x 0, if there exists an element f (x 0 ) F(R), such that 1. for all h > 0 sufficiently small, there exist f(x 0 + h) H f(x0 ), f(x 0 ) H f(x0 h) and the limits (in the metric D) a or: lim h 0 f(x 0 + h) H f(x0 ) h f(x0 ) H f(x0 h) = lim = h 0 h f (x 0 ),

4 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), for all h > 0 sufficiently small, there exist f(x 0 ) H f(x0 + h), f(x 0 h) H f(x0 ) and the limits (in the metric D) lim h 0 f(x 0 ) H f(x0 + h) h f(x0 h) H f(x0 ) = lim = h 0 h f (x 0 ). In this paper, we denote the space of all continuous fuzzy functions on [a, b R and the space of all Lebesgue integrable fuzzy functions on the bounded interval [a, b by C F [a, b and L F [a, b, respectively. Definition 2.8. [34 Let f C F [a, b L F [a, b be a fuzzy function. The fuzzy Riemann-Liouville integral of the fuzzy function f is defined as follows ( ) I β f (x) = 1 x f(t) Γ(β) 0 (x t) 1 β dt, x, β R +. Theorem 2.9. [10 Let f C F [a, b L F [a, b be a fuzzy function. The fuzzy Riemann-Liouville integral of the fuzzy function f is as as follows: [( ) [ I β f (x) α = I β f α (x), I β f α (x), 0 α 1, where ) (I β f α (x) = 1 x Γ(β) 0 ) (I β f α (x) = 1 x Γ(β) 0 f α (t) (x t) 1 β dt, x, β R +, f α (t) (x t) 1 β dt, x, β R Caputo Fuzzy Fractional Derivative In this subsection, we provide some definitions and theorems on Caputo fuzzy fractional derivative. [32 extended the Caputo fractional derivative of crisp case into fuzzy setting. Here, we provide some of the concept proposed. Lemma [32 Let f(x) be a crisp continuous function and ( β )-times differentiable in the independent variable x over the interval of differentiation (integration) [0, x. Then the relation β C D β f(x) = RL D β x f(x) k k! f (k) 0, β (n 1, n, n N, k=0 is hold, where f (k) 0 = dk f(x) dx k and C D β denotes Caputo derivative operator. While β and β are the value x=0 β rounded up and down to the closest integer number, respectively. RL D β is the common Riemann-Liouville fractional derivative operator which is defined as follows RL D β 1 f(x) = Γ( β β) d β x dx β 0 Definition [32 Let f(x) C F [0, b L F [0, b, G(x) = 1 lim h 0 + G(x 0 +h) G(x 0 ) G(x 0 ) G(x 0 h) f(t) dt (x t) 1 β +β f(t) t k β k=0 x Γ( β β) 0 G(x 0 ) G(x 0 +h) f (k) 0 k! dt, and H(x (x t) 1 β +β 0 ) = G(x 0 h) G(x 0 ) h. h = lim h 0 + h and L(x 0 ) = lim h 0 + h = lim h 0 + f(x) is Caputo fuzzy fractional differentiable function of order 0 < β 1, if there exists an element C D β f(x0 ) C F such that for all 0 α 1 and for h > 0 sufficiently near zero, either:

5 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), C D β f(x0 ) = lim h 0 + G(x 0 + h) G(x 0 ) h = lim h 0 + G(x 0 ) G(x 0 h), or h 2. C D β f(x0 ) = lim h 0 + for 0 < β 1. G(x 0 ) G(x 0 + h) h = lim h 0 + G(x 0 h) G(x 0 ), h If the fuzzy function f(x) is differentiable as in Definition 2.11 (1), it is called Caputo fuzzy differentiable in the first form. If f(x) is differentiable as in Definition 2.11 (2), it is called Caputo fuzzy differentiable in the second form. Theorem [32 Let f(x) C F [0, b L F [0, b be a fuzzy function and [ f(x) α = [f α (x), f α (x), for α [0, 1 and x 0 (0, b). Then 1. If f(x) is Caputo fuzzy fractional differentiable in the first form, then for every 0 < β 1, [ C D β f(x0 ) α = [ C D β f α (x 0 ), C D β f α (x 0 ). 2. If f(x) is Caputo fuzzy fractional differentiable in the second form, then for every 0 < β 1, where [ C D β f(x0 ) α = [ C D β f α (x 0 ), C D β f α (x 0 ), [ C D β 1 f α (x 0 ) = Γ( β β) [ C D β 1 f α (x 0 ) = Γ( β β) x 0 x D k f(t) = dk f(t) dt k. 0 D β f α (t) (x t) 1 β +β D β f α (t) (x t) 1 β +β x=x 0. x=x 0, Next, we give the definition for the classical Sumudu transform when dealing with Caputo s fractional derivative of crisp type. Definition [16, 20 The classical Sumudu transform of the Caputo s fractional derivative of the function f is given by n 1 s[ C D β f(x)(u) = u β G(u) k=0 f k (0) u β k, β (n 1, n. Since in this paper, we only consider 0 < β < 1, Definition 2.13 can be simplified as follows. s[ C D β f(x)(u) = G(u) f 0 (0) β (0, 1. Note that when β = 1, the definition is similar to the definition of Sumudu transform for first order derivative.

6 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), Fuzzy Sumudu transform for Caputo fuzzy fractional derivative In this part, we recall the definition of FST and later we propose a new result on the property of FST for Caputo fuzzy fractional derivative. Definition 3.1. [4, 1 Let f : R F(R) be a continuous fuzzy function. Suppose f(ux) e x is improper fuzzy Riemann-integrable on [0, ), then 0 f(ux) e x dx is called fuzzy Sumudu transform and is denoted by G(u) = S[ f(x)(u) = f(ux) e x dx, u [ τ 1, τ 2, 0 where the variable u is used to factor the variable x in the argument of the fuzzy function and τ 1, τ 2 > 0. The FST can also be written into the following parametric form. S[ f(x)(u) = [s[f α (x)(u), s[f α (x)(u). In the following theorem, we introduce a new property of FST for Caputo fuzzy fractional derivative. This is done by directly extending the definition for classical Sumudu transform of Caputo fractional derivative into fuzzy setting. Theorem 3.2. Let f(x) C F [0, b L F [0, b be a continuous fuzzy function, and C D β f is the Caputo fuzzy fractional derivative of f on [0, ). Then, for 0 < β 1, we have S[ C D β f(x)(u) = G(u) H f(x0 ) where f is Caputo fuzzy fractional differentiable in the first form, or S[ C D β f(x)(u) = f(x 0 ) H ( G(u)) where f is Caputo fuzzy fractional differentiable in the second form. Proof. First, we assume f is Caputo fuzzy fractional differentiable in the first form (Theorem 2.12 (1)). Therefore: G(u) H [ f(0) s[f(x)(u) f(0) u β = s[f(x)(u) f(0) u β. From the classical Sumudu transform for Caputo fractional derivative, we know that and s[ C D β f(x)(u) = s[f(x)(u) f(0) Then, s[ C D β f(x)(u) = s[f(x)(u) f(0) u β. G(u) H f(0) [ u β = s[ C D β f(x)(u), s[ C D β f(x)(u). Since f is Caputo fuzzy fractional differentiable in the first form G(u) H f(x0 ) u β = S[ C D β f(x)(u).

7 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), Now, we assume that f is Caputo fuzzy fractional differentiable in the second form (Theorem 2.12 (2)). Therefore, This is analogous to f(x 0 ) H [ ( G(u)) f(x0 ) ( s[f(x)(u)) u β = f(x 0) ( s[f(x)(u)) u β. f(x 0 ) H [ ( G(u)) s[f(x)(u) f(0) u β = s[f(x)(u) f(0) From the classical Sumudu transform for Caputo fractional derivative, finally we have f(x 0 ) H ( G(u)) [ u β = s[ C D β f(x)(u), s[ C D β f(x)(u). Since f is Caputo fuzzy fractional differentiable in the second form, then we finally have u β. The proof is complete. f(x 0 ) H ( G(u)) u β = S[ C D β f(x)(u). 4. Procedures for Solving FFDEs using FST Consider the following FFDE C D β ỹ(x) = f[x, ỹ(x), ỹ α (x 0 ) = [y α (0), y α (0). (4.1) where f C F (a, b) L F (a, b) and x 0 (a, b). By using FST on both side of Eq. (4.1), we have [ S C D ỹ(x) β (u) = S [f(x, ỹ(x)) (u). Case 1 : If we consider f to be Caputo fuzzy fractional differentiable in the first form, then from Theorem 2.12 (1), we get [ C D β ỹ(x 0 ) α = [ C D β y α (x 0 ), C D β y α (x 0 ). Now, we obtain the following system where β (0, 1. From Theorem 3.2, we have C D β y α (x) = f[x, ỹ(x) = f α [x, ỹ(x), y α (x 0 ) = y α (0), C D β y α (x) = f[x, ỹ(x) = f α [x, ỹ(x), y α (x 0 ) = y α (0), Therefore, S[ C D β ỹ(x)(u) = G(u) H ỹ(t 0 ) u β.

8 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), where, and, s[f α (x, ỹ(x))(u) = s[y α (x)(u) y α (0) s[f α (x, ỹ(x))(u) = s[y α(x)(u) y α (0) f α (x, ỹ(x)) = min f(x, u) u [y α (x), y α (x)}, f α (x, ỹ(x)) = max f(x, u) u [y α (x), y α (x)}. (4.2) To solve Eq. (4.2), first we assume that s[y α (x)(u) = L 1 α(u), s[y α (x)(u) = U 1 α(u). L 1 α(u) and U 1 α(u) are the solutions of Eq. (4.2) under this case. We obtain y α (x) and y α (x) using the inverse FST as the following. y α (x) = s 1 [L 1 α(u), y α (x) = s 1 [U 1 α(u). Case 2 : If we consider f to be Caputo fuzzy fractional differentiable in the second form, then from Theorem 2.12 (2), we get [ C D β f(x0 ) α = [ C D β y α (x 0 ), C D β y α (x 0 ). Now, we obtain the following system where β (0, 1. From Theorem 3.2, C D β y α (x) = f[x, ỹ(x) = f α [x, ỹ(x), y α (x 0 ) = y α (0), C D β y α (x) = f[x, ỹ(x) = f α [x, ỹ(x), y α (x 0 ) = y α (0), Therefore, where, S[ C D β ỹ(x)(u) = ỹ(t 0) H ( G(u)) u β. s[f α (x, ỹ(x))(u) = s[y α (x)(u) y α (0) s[f α (x, ỹ(x))(u) = s[y α(x)(u) y α (0) f α (x, ỹ(x)) = min f(x, u) u [y α (x), y α (x)}, (4.3)

9 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), and, To solve Eq. (4.3), first we assume that f α (x, ỹ(x)) = max f(x, u) u [y α (x), y α (x)}. s[y α (x)(u) = L 2 α(u), s[y α (x)(u) = U 2 α(u). L 2 α(u) and U 2 α(u) are the solutions of Eq. (4.3) or this case. We have y α (x) and y α (x) by the inverse of FST as the following. y α (x) = s 1 [L 2 α(u), y α (x) = s 1 [U 2 α(u). 5. A numerical example In this part, the method proposed will be demonstrated on a FFDE. This is to show that the method is practicable. Example 5.1. The following FFDE is considered. C D β ỹ(x) = ỹ(x), ỹ(x 0 ) = [y α (0), y α (0). (5.1) Case 1: By taking fuzzy Sumudu transform on both side of (5.1), we have [ S C D ỹ(x) β (u) = S[ỹ(x)(u). From Theorem 2.12 (1) for Caputo fuzzy fractional differentiability in the first form, we have and from Theorem 3.2, we have [ C D β ỹ(x 0 ) α = [ C D β y α (x 0 ), C D β y α (x 0 ), S[ C D β ỹ(x)(u) = G(u) H ỹ(x 0 ) s[y α (x)(u) = s[y α (x)(u) y α (0) Then, we obtain s[y α (x)(u) = s[y α(x)(u) y α (0) u β. (1 u β )s[y α (x)(u) = y α (0), (1 u β )s[y α (x)(u) = y α (0). By applying inverse Sumudu transform, we obtain

10 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), By relation, [ y α (x) = y α (0)s [ y α (x) = y α (0)s u β. finally, we have: s[x γ 1 E β,γ ( x β )(u) = y α (x) = y α (0)E β,1 [x β, y α (x) = y α (0)E β,1 [x β. uγ 1 1 ± E β,1 [x β is the Mittag-Leffler function defined by E β,1 [x β = k=0 (x β ) k Γ(βk + 1). Case 2: Using fuzzy Sumudu transform on Eq. (5.1), we have [ S C D ỹ(x) β (u) = S[ỹ(x)(u). From Theorem 2.12 (2) for Caputo fuzzy fractional differentiability in the second form: and from Theorem 3.2 [ C D β ỹ(x 0 ) α = [ C D β y α (x 0 ), C D β y α (x 0 ), we have S[ C D β f(x)(u) = f(x 0 ) H ( G(u)) s[y α (x)(u) = y α(0) [ s[y α (x)(u) u β. equivalent to s[y α (x)(u) = y α (0) [ s[y α (x)(u) s[y α (x)(u) = s[y α(x)(u) y α (0) u β. Then, we obtain s[y α (x)(u) = s[y α (x)(u) y α (0)

11 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), By applying inverse Sumudu transform, we have By relation, (1 + u 2β )s[y α (x)(u) = y α (0) u β y α (0), (1 + u 2β )s[y α (x)(u) = y α (0) u β y α (0). [ [ y α (x) = y α (0)s u 2β y α (0)s 1 u β 1 + u2 β, [ [ y α (x) = y α (0)s u 2β y α (0)s 1 u β 1 + u2 β. finally, we have s[x γ 1 E β,γ ( x β )(u) = uγ 1 1 ± y α (x) = y α (0)E 2β,1 [ x 2β y α (0)x β E 2β,β+1 [ x 2β, y α (x) = y α (0)E 2β,1 [ x 2β y α (0)x β E 2β,β+1 [ x 2β, where, and are the Mittag-Leffler functions. E 2β,1 [ x 2β = E 2β,β+1 [ x 2β = k=0 k=0 ( x 2β ) k Γ(2βk + 1), ( x 2β ) k Γ(2βk + β + 1). Assume that ỹ(x 0 ) = [1 + α, 3 α. The numerical solutions of (5.1) for Cases 1 and 2 at x = 2 are listed in Tables 1 and 2, respectively. For graphical results, please see in Figs. 1 and 2. Numerical solutions for both cases are obtained by expending the Mittag-Leffler functions up to 11 terms. It can be concluded that the solutions of Eq. (5.1) is in agreement with the solutions of fuzzy differential equation as β approaches to Conclusions In this paper, we have proposed a new analytical method for dealing with fuzzy fractional differential equations involving Caputo fuzzy fractional derivatives. A new property of fuzzy Sumudu transform for Caputo fuzzy fractional derivative has been introduced. The new property has been used to construct a procedure for solving FFDEs. A numerical example has been solved to show that FST is functional. Acknowledgement This research received funding by Ministry of Science, Technology and Innovation, Malaysia under the Fundamental Research Grant Scheme, project code

12 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), Table 1: Numerical solutions of Eq. (5.1) for Case 1 using different values of β. β = 0.4 β = 0.6 β = 0.8 β = 1.0 α y β (x) y β (x) y β (x) y β (x) y β (x) y β (x) y β (x) y β (x) Figure 1: Numerical solutions of (5.1) for (a) β = 0.4 (b) β = 0.6, (c) β = 0.8 and (d) β = 1 (Case 1). References [1 N A Abdul Rahman and M Z Ahmad, Applications of the fuzzy Sumudu transform for the solution of first order fuzzy differential equations, Entropy 17 (2015), no. 7, , 3.1

13 N.A. Abdul Rahman, M.Z. Ahmad, J. Nonlinear Sci. Appl. Vol. (201X), Table 2: Numerical solutions of Eq. (5.1) for Case 2 using different values of β. β = 0.4 β = 0.6 β = 0.8 β = 1.0 α y β (x) y β (x) y β (x) y β (x) y β (x) y β (x) y β (x) y β (x) Figure 2: Numerical solutions of (5.1) for (a) β = 0.4 (b) β = 0.6, (c) β = 0.8 and (d) β = 1 (Case 2). [2 R P Agarwal, V Lakshmikantham, and J J Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods & Applications 72 (2010), no. 6,

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