Fuzzy fractional functional differential equations under Caputo gH-differentiability

Accepted Manuscript Fuzzy fractional functional differential equations under Caputo gH-differentiability Ngo Van Hoa PII: DOI: Reference:

S1007-5704(14)00371-2 http://dx.doi.org/10.1016/j.cnsns.2014.08.006 CNSNS 3305

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Communications in Nonlinear Science and Numerical Simulation

Please cite this article as: Van Hoa, N., Fuzzy fractional functional differential equations under Caputo gHdifferentiability, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/ 10.1016/j.cnsns.2014.08.006

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Fuzzy fractional functional differential equations under Caputo gH-differentiability Ngo Van Hoa∗ Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.

Abstract In this paper the fuzzy fractional functional differential equations (FFFDEs) under the Caputo generalized Hukuhara differentiability are introduced. We study the existence and uniqueness results of solutions for FFFDEs under some suitable conditions. Also the solution to fuzzy fractional functional initial value problem under Caputotype fuzzy fractional derivatives by a modified Adams-Bashforth-Moulton method (MABMM) is presented. The method is illustrated by solving some examples. Keyword: fractional differential equations; fuzzy functional differential equations; Lipschitz generalized condition; Adams-Bashforth-Moulton method.

1 Introduction The theory of fractional calculus, which deals with the investigation and applications of derivatives and integrals of arbitrary order has a long history. The theory of fractional calculus developed mainly as a pure theoretical field of mathematics, in the last decades it has been used in various fields as rheology, viscoelasticity, electrochemistry, diffusion processes, etc. Fractional calculus and fractional differential equations have undergone expanded study in recent years as a considerable interest both in mathematics and in applications. They were applied in modeling of many physical and chemical processes and in engineering. One of the recently influential works on the subject of fractional calculus is the monograph of Podlubny [25] and the other is the monograph of Kilbas et al. [33]. Consequently, several research papers were done to investigate the theory and solutions of fractional differential equations (see [18, 21, 28] and references therein). In real world systems, delays can be recognised everywhere and there has been widespread interest in the study of delay differential equations for many years. Therefore, delay differential equations (or, as they are called, functional differential equations) play an important role in an increasing number of system models in biology, engineering, ∗

E-mail corresponding author: ngovanhoa [email protected]

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physics and other sciences. There exists an extensive amount of literature dealing with delay differential equations and their applications; the reader is referred to the monographs [11, 12], and the references therein. The study of fuzzy delay differential equations is expanding as a new branch of fuzzy mathematics. Both theory and applications have been actively discussed over the last few years. In the literature, the study of fuzzy delay differential equations has several interpretations. The first one is based on the notion of Hukuhara derivative [29]. Under this interpretation, Lupulescu established the local and global existence and uniqueness results for fuzzy delay differential equations. The second interpretation was suggested by Khastan et al. [34] and Hoa et al. [31]. In this setting, Khastan et al. proved the existence of two fuzzy solutions for fuzzy delay differential equations using the concept of generalized differentiability. Hoa et al. established the global existence and uniqueness results for fuzzy delay differential equations using the concept of generalized differentiability. Moreover, authors have extended and generalized some comparison theorems and stability theorem for fuzzy delays differential equations with definition a new Lyapunov-like function. Besides that, some very important extensions of the fuzzy delay differential equations are the fuzzy delay integro-differential equations and the random fuzzy delay differential equations [32, 38, 39]. Combining the two aspects introduced, fractional calculus and fuzzy delay differential equations, we get fuzzy fractional delay differential equations. Furthermore, fuzzy fractional delay differential equations (FFDDEs) are a very recent topic. We noticed that recently Agarwal et al. [35] proposed the concept of solutions for fractional differential equations with uncertainty. They have considered the RiemannLiouville differentiability concept based on the Hukuhara differentiability to solve fuzzy fractional differential equations. Arshad and Lupulescu in [10] proved some results on the existence and uniqueness of solution to fuzzy fractional differential equation under Hukuhara fractional Riemann-Liouville differentiability. Allahviranloo et al. in [4, 5, 6] have studied the concepts about generalized Hukuhara fractional Riemann-Liouville and Caputo differentiability of fuzzy-valued functions. Later, authors have proved the existence and uniqueness of solution for fuzzy fractional differential equation by using different methods. Alikhani et al. in [3] have proved the existence and uniqueness results for nonlinear fuzzy fractional integral and integrodifferential equations by using the method of upper and lower solutions. Mazandarani et.al. [8] studied the solution to fuzzy fractional initial value problem under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method. Besides, authors studied some results on the existence and uniqueness of solution to fuzzy fractional differential equation under Caputo type-2 fuzzy fractional derivative and the definition of Laplace transform of type-2 fuzzy number-valued functions [9]. Salahshour et al. [6] proposed some new results toward existence and uniqueness of solution of fuzzy fractional differential equation. The solutions of fuzzy fractional differential equations are investigated by using the fuzzy Laplace transforms in [7]. According to the concept of Caputo-type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability, Fard et al. [16] extended and established some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional Euler-Lagrange equation for both constrained and unconstrained fuzzy fractional variational problems. We believed that mathematical models of physical phenomena should have the prop-

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erties that existence and uniqueness of solution and solution’s behavior changes continuously with the initial conditions. The importance of existence and uniqueness theorems in the study of initial value problems is well-known due to their relevance in establishing the well-posedness of the real-world problems arising in physical, engineering systems. Uniqueness results play a significant role in continuation of solutions and in the theory of autonomous systems. While the uniqueness results almost always come with the cost of stringent conditions, they are valuable, for without such uniqueness results it is impossible to make predictions about the behaviour of physical systems. Therefore, in this paper we study the existence, uniqueness and approximations solutions of fuzzy fractional functional differential equations under the Caputo generalized Hukuhara differentiability, which provide mathematical models for real-world problem in which the fractional rate of change depends on the influence of their hereditary effects. This direction of research is motivated by the results of Bede and Stefanini [15], Agarwal et al. [35, 36], Arshad et al. [10], Lupulescu [29, 30], Allahviranloo et al. [4, 5, 6], Salahshour [7], Mazandarani et al. [8]. This paper is organized as follows: In Section 2, we present the basic notations of the Riemann-Liouville fractional integral and Caputo fractional derivative for fuzzy functions. In Section 3, we study the existence, uniqueness solutions of some classes of fuzzy fractional functional differential equation under the Caputo generalized Hukuhara differentiability. Finally, the solution to fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives by a modified MABMM is presented in Section 4.

2 Preliminaries The basic definition of fuzzy numbers is mentioned in [22]. Let E denote the set of fuzzy subsets of the real axis, if ω : R → [0, 1], satisfying the following properties: (i) ω is normal, that is, there exists z0 ∈ R such that ω(z0 ) = 1; (ii) ω is fuzzy convex, that is, for 0 ≤ λ ≤ 1 ω(λz1 + (1 − λ)z2 ) ≥ min{ω(z1 ), ω(z2 )}, for any z1 , z2 ∈ R; (iii) ω is upper semicontinuous on R; (iv) [ω]0 = cl{z ∈ R : ω(z) > 0} is compact, where cl denotes the closure in (R, | · |). Then E is called the space of fuzzy numbers. For r ∈ (0, 1], define [ω]r = {z ∈ R | ω(z) ≥ r} = [ω(r), ω(r)]. Then from (i) to (iv), it follows that the r−level set [ω]r is a closed bounded interval for all r ∈ [0, 1]. Especially, for addition and scalar multiplication in fuzzy set space E, we have [ω1 + ω2 ]r = [ω1 ]r + [ω2 ]r , [λω1 ]r = λ[ω1 ]r . The notation [ω]r = [ω(r), ω(r)], where ω denotes the left-hand endpoint of [ω]r and ω denotes the right-hand endpoint of [ω]r . It should be noted that for a ≤ b ≤ c, a, b, c ∈ R, a triangular fuzzy number ω = (a, b, c) is given such that ω(α) = a + (b − a)α and ω(α) = c − (c − b)α are the endpoints of the α−cut for all α ∈ [0, 1]. For ω ∈ E, we define the length of ω as len(ω) = ω(r) − ω(r). The Hausdorff distance between fuzzy numbers is given by D0 [ω1 , ω2 ] = sup {|ω1 (r) − ω2 (r)|, |ω1 (r) − ω2 (r)|}. 0≤r≤1

3

The metric space (E, D0 ) is complete, separable and locally compact and the following properties of the metric D0 are valid (see [22]). D0 [ω1 + ω3 , ω2 + ω3 ] = D0 [ω1 , ω2 ], D0 [λω1 , λω2 ] = |λ|D0 [ω1 , ω2 ], D0 [ω1 , ω2 ] ≤ D0 [ω1 , ω3 ] + D0 [ω3 , ω2 ], for all ω1 , ω2 , ω3 ∈ E and λ ∈ R. Let ω1 , ω2 ∈ E. If there exists ω3 ∈ E such that ω1 = ω2 + ω3 , then ω3 is called the H-difference of ω1 , ω2 and it is denoted by ω1 ω2 . Let us remark that ˆ ω1 ω2 , ω1 + (−1)ω2 . Let us denote 0ˆ ∈ E the zero element of E as follows: 0(z) = 1 if ˆ = 0 if z , 0, where 0 is the zero element of R. z = 0 and 0(z) Definition 2.1. [15] The generalized Hukuhara difference of two fuzzy numbers ω1 , ω2 ∈ E (gH-difference for short) is defined as follows ( (i) ω1 = ω2 + ω3 , (2.1) ω1 gH ω2 = ω3 ⇔ or (ii) ω2 = ω1 + (−1)ω3 . The generalized Hukuhara differentiability was introduced in [15]. Definition 2.2. Let t ∈ (a, b) and h be such that t + h ∈ (a, b), then the generalized Hukuhara derivative of a fuzzy-valued function x : (a, b) → E at t is defined as x(t + h) gH x(t) . (2.2) h→0 h If D gH x(t) ∈ E satisfying (2.2) exists, we say that x is generalized Hukuhara differentiable (gH-differentiable for short) at t. Also, we say that x is [(i)-gH]-differentiable at t if 0 (i) D gH x(t) = [x0 (t, r), x (t, r)], and that x is [(ii)-gH]-differentiable at t if (ii) D gH x(t) = 0 [x (t, r), x0 (t, r)], r ∈ [0, 1]. D gH x(t) = lim

Remark 2.1. If x(t) = (z1 (t), z2 (t), z3 (t)) is triangular number valued function, then we say that x is [(i)-gH]-differentiable at t if (i) D gH x(t) = (z01 (t), z02 (t), z03 (t)), and that x is [(ii)-gH]differentiable at t if (ii) D gH x(t) = (z03 (t), z02 (t), z01 (t)). Lemma 2.1. (see [14]) If x(t) = (z1 (t), z2 (t), z3 (t)) is triangular number valued function, then (i) If x is [(i)-gH]-differentiable at t ∈ [a, b] then D gH x(t) = (z01 (t), z02 (t), z03 (t)). (ii) If x is [(ii)-gH]-differentiable at t ∈ [a, b] then D gH x(t) = (z03 (t), z02 (t), z01 (t)). Next, we recall some notations and concepts presented in detail in recent works of Allahviranloo et al. (see [4]). Let us denote C([a, b], E) as the space of all continuous fuzzy-valued functions on [a, b]. Also, we denote the space of all Lebesque integrable fuzzy-valued functions on [a, b] by L([a, b], E). The fuzzy gH-fractional Caputo differentiability of fuzzyvalued functions was introduced in [4]. For a detail discussion on fractional derivatives and fuzzy fractional derivatives, we refer to [4, 5, 25, 35]. Definition 2.3. The Riemann-Liouville fractional integral operator of order α > 0, of a real-valued function ϕ ∈ L1 [a, b], is defined as Iaα+ ϕ(t)

1 = Γ(α)

Zt a

where Γ(·) is the Eluer gamma function. 4

(t − s)α−1 ϕ(s)ds,

(2.3)

Definition 2.4. Let ϕ : [a, b] → R, the Caputo fractional derivative of order α > 0, Rt 1 m − 1 < α < m, m ∈ N, is defined as C Dαa+ ϕ(t) = (t − s)m−α−1 ϕm (s)ds, where the Γ(m − α) a function ϕ(t) has absolutely continuous derivatives up to order (m − 1). If α ∈ (0, 1), then, Rt 1 d C α (t − s)−α ϕ(s)ds. Da+ ϕ(t) = Γ(1 − α) a ds Definition 2.5. [5] Let x : [a, b] → E, the fuzzy Riemann-Liouville integral of fuzzy-valued function x is defined as follows: (=αa+ x)(t)

1 = Γ(α)

Zt

(t − s)α−1 x(s)ds

(2.4)

a

for a ≤ t, and 0 < α ≤ 1. For α = 1, we set =1a = I, the identity operator. The fuzzy gH-fractional Caputo differentiability was introduced in [4]. Definition 2.6. Let D(m) ∈ C([a, b], E) ∩ L([a, b], E). The fuzzy gH-fractional Caputo difgH ferentiability of fuzzy-valued function x ( [gH]Cα − differentiable for short) is defined as following: C α gH Da+ x(t)

=

=am−α (D(m) x)(t) + gH

1 = Γ(m − α)

Zt

(t − s)m−α−1 (D(m) x)(s)ds gH

(2.5)

a

where m − 1 < α ≤ m, t > a. If α ∈ (0, 1), then, α C gH Da+ x(t)

=

=1−α a+ (D gH x)(t)

1 = Γ(1 − α)

Zt

(t − s)α (D gH x)(s)ds.

(2.6)

a

Lemma 2.2. [4] Suppose that x : [a, b] → E be a fuzzy function and D gH x(t) ∈ C([a, b], E) ∩ L([a, b], E). Then =αa+ (CgH Dαa+ x)(t) = x(t) gH x(a). Theorem 2.1. [4] Let D gH x(t) ∈ C([a, b], E) ∩ L([a, b], E) where [x(t)]r = [x(t, r), x(t, r)] for 0 ≤ r ≤ 1, t ∈ [a, b]. The function x(t) is [gH]Cα − differentiable if and only if x(t, r) and x(t, r) are Caputo fractional differentiable functions. Furthermore h

ir

α C gH Da+ x(t)

o n oi h n = min C Dαa+ x(t, r), C Dαa+ x(t, r) , max C Dαa+ x(t, r), C Dαa+ x(t, r) ,

(2.7)

where C Dαa+ x(t, r) and C Dαa+ x(t, r) defined in Definition 2.4. Lemma 2.3. If x(t) = (z1 (t), z2 (t), z3 (t)) is triangular number valued function, then   (i) If x is [(i)−gH]−differentiable at t ∈ [a, b] then (CgH Dαa+ x)(t) = C Dαa+ z1 (t), C Dαa+ z2 (t), C Dαa+ z3 (t) . (ii) If x is [(ii)−gH]−differentiable at t ∈ [a, b] then (CgH Dαa+ x)(t) =

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C

 Dαa+ z3 (t), C Dαa+ z2 (t), C Dαa+ z1 (t) .

Proof. The proof of (i) is as follows. Suppose that x is [(i) − gH]−differentiable at t ∈ [a, b], then according Lemma 2.1, we have 1 (CgH Dαa+ x)(t) = Γ(1 − α) 1 = Γ(1 − α)

Zt

(t − s)α (z01 , z02 , z03 )(s)ds

a

Zt

(t −

s)α z01 (s)ds,

a

Zt a

(t −

s)α z02 (s)ds,

Zt

(t −

s)α z03 (s)ds

a

!

.

In view of Definition 2.3, we have (CgH Dαa+ x)(t) =



C

 Dαa+ z1 (t), C Dαa+ z2 (t), C Dαa+ z3 (t) .

The proof of (ii) follows the same steps as those of (i), so we omit it.



Definition 2.7. Let x : [a, b] → E be [gH]Cα − differentiable at t ∈ (a, b). We say that x is [(i) − gH]Cα −differentiable at t ∈ [a, b] if (i)

h

ir h i (CgH Dαa+ x)(t) = C Dαa+ x(t, r), C Dαa+ x(t, r) , 0 ≤ r ≤ 1

(2.8)

and that x is [(ii) − gH]Cα −differentiable at t if (ii)

h

ir h i (CgH Dαa+ x)(t) = C Dαa+ x(t, r), C Dαa+ x(t, r) , 0 ≤ r ≤ 1

(2.9)

where 1 C α Da+ x(t, r) = Γ(1 − α)

Zt

d (t − s)−α x(s, r)ds, ds

1 C α Da+ x(t, r) = Γ(1 − α)

a

Zt

(t − s)−α

d x(s, r)ds. ds

a

Definition 2.8. [4] Let x : [a, b] → E be a fuzzy function. A point t ∈ (a, b) is said to be a switching point for the [gH]Cα −differentiability of x, if in any neighborhood V of t there exist points t1 < t < t2 such that type(I) at t1 (2.8) holds while (2.9) does not hold and at t2 (2.9) holds and (2.8) does not holds, or type(II) at t1 (2.9) holds while (2.8) does not hold and at t2 (2.8) holds and (2.9) does not holds. Theorem 2.2. [4] Let x : [a, b] → E be a fuzzy-valued function on [a, b]. h ir h i (i) If x is [(i) − gH]−differentiable at t ∈ [a, b] then (CgH Dαa+ x)(t) = C Dαa+ x(t, r), C Dαa+ x(t, r) . i ir h h (ii) If x is [(i) − gH]−differentiable at t ∈ [a, b] then (CgH Dαa+ x)(t) = C Dαa+ x(t, r), C Dαa+ x(t, r) .

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3 Main results For σ > 0 let Cσ = C([−σ, 0], E) denote the space of continuous mappings from [−σ, 0]     to E. Define a metric Dσ in Cσ by Dσ x, y = sup D0 x(t), y(t) . Let p > 0. Denote t∈[−σ,0]

I = [a, a + p], J = [a − σ, a] ∪ I = [a − σ, a + p]. For any t ∈ I denote by the element of Cσ defined by xt (s) = x(t + s) for s ∈ [−σ, 0]. We consider the following fuzzy fractional functional differential equation (FFFDE) of order α ∈ (0, 1) with the initial condition: C α  D + x(t) = f (t, xt ), t > a, 0 < α < 1,   gH a (3.1)    x(t) = ϕ(t − a), a > t > a − σ

where CgH Dαa+ is the Caputo’s generalized Hukuhara derivative from Definition 2.6, f : I × Cσ → E. In this paper, we consider only [(i) − gH]Cα −differentiable type and [(ii) − gH]Cα −differentiable type solutions, i.e. such that there are no switching points in I. By a solution to equation (3.1) we mean a fuzzy mapping x ∈ C(J, E), that satisfies: x(t) = ϕ(t−a) for t ∈ [a − σ, a], x is differentiable on [a, a + p] and CgH Dαa+ x(t) = f (t, xt ) for t ∈ I. Lemma 3.1. (see [29]) Assume that f ∈ C(I × Cσ , E) and x ∈ C(J, E). Then the fuzzy mapping t → f (t, xt ) belongs to C(I, E). Remark 3.1. Under assumptions of the Lemma 3.1 we have the mapping t → f (t, xt ) is integrable over the interval I. By Lemma 2.2 and the technique used in [26], we get the equivalent between solutions of (3.1) and functional integral equations. The following lemma is similar to the result proved in [26]. Lemma 3.2. Let α ∈ (0, 1), the fuzzy fractional functional differential equation (3.1) is equivalent to the following integral equation   x(t) = ϕ(t − a) for t ∈ [a − σ, a]      Zt  (3.2)  1   x(t) gH ϕ(0) = (t − s)α−1 f (s, xs )ds, t ∈ I.    Γ(α)  a

Two cases of existence of the generalized H-difference imply that the integral equation in the Lemma 3.2 is actually a unified formulation for one the integral equations     x(t) = ϕ(t − a) for t ∈ [a − σ, a] x(t) = ϕ(t − a) for t ∈ [a − σ, a]           Zt Zt   and   1 1   α−1   ϕ(0) x(t) = (−1) (t − s) f (s, xs )ds, (t − s)α−1 f (s, xs )ds. x(t) ϕ(0) =       Γ(α) Γ(α)   a

a

with being the classical Hukuhara difference. Now, we consider b x and e x are solutions Eq. (3.1) in [(i) − gH]Cα -differentiability type and [(ii) − gH]Cα -differentiability type, respectively, then by using Lemma 3.2, we have   b x(t) = ϕ(t − a) for t ∈ [a − σ, a]      Zt  (3.3)  1   b x(t) = ϕ(0) + (t − s)α−1 f (s, b xs )ds, t ∈ I,    Γ(α)  a

7

and   e x(t) = ϕ(t − a) for t ∈ [a − σ, a]      Zt   1   e x(t) = ϕ(0) (−1) (t − s)α−1 f (s, e xs )ds,    Γ(α) 

t ∈ I.

(3.4)

a

Definition 3.1. Let x : J → E be a fuzzy function which is [(i) − gH]Cα -differentiable ([(ii) − gH]Cα -differentiable). If x and its derivative satisfy problem (3.1), we say x is a (i)-solution ((ii)-solution) of problem (3.1). The following comparison principle is fundamental in investigation of the local existence of solutions of fuzzy fractional functional differential equations. Theorem 3.1. [28] Let m ∈ C([a−σ, ∞), R) and satisfy the inequality C Dαa+ m(t) ≤ g(t, |mt |σ ), t > a, where g ∈ C([a, ∞) × R+ , R+ ). Assume that r(t) = r(t, a, u0 ) is the maximal solution of the IVP C

Dαa+ u = g(t, u),

u(a) = u0 ≥ 0,

existing on [a, ∞). Then, if |ma |σ ≤ u0 , we have m(t) ≤ r(t), t ∈ [a, ∞). Let ρ > 0 be a given constant, and let Ω(x0 , ρ) = {x ∈ E : D0 [x, x0 ] ≤ ρ} and S(x0 , ρ) = {ξ ∈ Cσ : Dσ [ξ, x0 ] ≤ ρ}. Let us consider the mappings f : I ×S(x0 , ρ) → E and g : I ×[0, ρ] → R+ . Where     ϕ(t − a), t ∈ [a − σ, a] , x0 (t) =    ϕ(0), t ∈ I.

Under generalized Lipschitz condition we obtain the existence and uniqueness of two solutions to FFFDE. To prove this assertion we use an idea of successive approximations. Theorem 3.2. Suppose that the following conditions hold: (i) f ∈ C(I × S(x0 , ρ), E) and ˆ ≤ M1 , ∀(t, ξ) ∈ I × S(x0 , ρ); (ii) g ∈ C(I × [0, ρ], R+ ), g(t, 0) ≡ 0 and 0 ≤ D0 [ f (t, ξ), 0] g(t, u) ≤ M2 , ∀t ∈ I, 0 ≤ u ≤ ρ, such that g(t, u) is nondecreasing in u and g is such that the IVP (C Dαa+ u) = g(t, u),

u(a) = 0

(3.5)

has only the solution u(t) ≡ 0 on I; (iii) D0 [(t1 − s)α−1 f (t, ξ), (t2 − s)α−1 f (t, ψ)] ≤ |(t1 − s)α−1 − (t2 − s)α−1 |g(t, Dσ [ξ, ψ]), ∀(t, ξ), (t, ψ) ∈ I × S(x0 , ρ), and Dσ [ξ, ψ] ≤ M3 . Then, the following successive approximations given by     ϕ(t − a), t ∈ [a − σ, a] , b x0 (t) =  (3.6)   ϕ(0), t ∈ I,

and for n = 0, 1, 2, ...

  ϕ(t − a), t ∈ [a − σ, a] ,      Zt  b xn+1 (t) =   " n
1  ϕ(0) + (t − s)α−1 f s, b xs ds, t ∈ I,    Γ(α)  a

8

(3.7)

for case [(i) − gH]Cα -differentiability, and     ϕ(t − a), t ∈ [a − σ, a] , e x0 (t) =    ϕ(0), t ∈ [a, a + γ],

(3.8)

and for n = 0, 1, 2, ...

  ϕ(t − a), t ∈ [a − σ, a] ,      Zt  e xn+1 (t) =  " n
1   ϕ(0) (−1) (t − s)α−1 f s, e xs ds, t ∈ [a, a + γ],    Γ(α) 

(3.9)

a

for case [(ii) − gH]Cα -differentiability (where 0 < γ < d such that equation (3.9) is well defined, i.e. the foregoing Hukuhara differences do exists), converge uniformly to two unique solutions b x(t) ) ( h ρΓ(α + 1) i α1 h ρΓ(α + 1) i α1 . , and e x(t) of (3.1), respectively, on [a, a + d] where d = min p, M1 M2 Proof. Without loss of generality, we prove for case [(ii) − gH]Cα -differentiability. The proof of the second case is completely similar to previous one and so it is omitted. For a ≤ t1 ≤ t2 ≤ a + d and M = max{M1 , M2 }, we have #

1 D0 [e x (t1 ), e x (t2 )] = D0 (−1) Γ(α) n

n

≤

M Γ(α)

Zt1

Zt1

α−1

(t1 − s)

a

|(t1 − s)α−1 − (t2 − s)α−1 |ds +

a

provided |t1 − t2 | < δ, where δ = Similarly, 0

Zt2 t1

h εΓ(α + 1) i1/α 2M

1 D0 [e x (t), e x (t)] ≤ Γ(α) 1

f (s, e xn−1 s )ds, (−1)

Zt a

1 Γ(α)

Zt2

α−1

(t2 − s)

a

! 2M α−1 (t2 − s) ds ≤ |t1 − t2 |α < ε Γ(α)

, proving that e xn (t) is continuous on [a, a + d].

" 0
ˆ D0 [ f s, e xs , 0] Mrα ds ≤ ≤ ρ. Γ(α + 1) (t − s)1−α

Thus, it is easily obtain that the successive approximations are continuous and satisfy the following relation: D0 [e xn+1 (t), e x0 (t)] ≤ ρ

, ∀t ∈ [a, a + d], n = 0, 1, 2, 3, ....

Hence, e xn+1 ∈ C([a, a + d], Ω(x0 , ρ)). Now, we define the following successive approximah ρΓ(α + 1) i1/α }: tions of (3.5) for d = min{p, M Zt M(t − a)α 1 0 n+1 u (t) = , u (t) = (t − s)α−1 g(s, un (s))ds, t ∈ [a, a + d], n = 0, 1, 2, .... Γ(α + 1) Γ(α) a

Then, we get immediately 1 u (t) = Γ(α) 1

Zt

(t − s)α−1 g(s, u0 (s))ds ≤

M2 (t − a)α ≤ u0 (t) ≤ ρ, Γ(α + 1)

a

9

%

f (s, e xn−1 s )ds

∀t ∈ [a, a + d].

Hence, by the inductive method and in view that g(t, u) is nondecreasing on u, we get 0 ≤ un+1 (t) ≤ un (t) ≤ ρ,

∀t ∈ [a, a + d], n = 0, 1, 2, 3....

As (C Dαa+ u)(t) = |g(t, un (t))| ≤ M2 the sequence {un } is equicontinuous. Hence, we can conclude by Ascoli-Arzela theorem and the monotonicity of the sequence {un (t)} that 1 Rt n limn→∞ u (t) = u(t) uniformly on [a, a + d] and u(t) = (t − s)α−1 g(s, u(s))ds. Thus, Γ(α) a u ∈ C([a, a + d], [0, ρ]) and u(t) is the solution of the initial value problem (3.5). From assumption (ii), we get u(t) ≡ 0. In addition, we have

and

1 D0 [e x1 (t), e x0 (t)] ≤ Γ(α)

Zt a

M1 (t − a)α ˆ ≤ u0 (t), ≤ (t − s)α−1 D0 [ f (s, e x0s ), 0]ds Γ(α + 1)

1 sup sup D0 [e x (t + ς), e x (t + ς)] ≤ Γ(α) ς∈[−σ,0] ς∈[−σ,0] 1

0

≤

Zt+ς

1 sup Γ(α) θ∈[t−σ,t]

a

Zθ a

ˆ (t − s)α−1 D0 [ f (s, e x0s ), 0]ds ˆ (t − s)α−1 D0 [ f (s, e x0s ), 0]ds

M1 ≤ sup (θ − a)α ≤ u0 (t). Γ(α + 1) θ∈[t−σ,t]

Suppose D0 [xi (t), xi−1 (t)] ≤ ui−1 (t) and sup D0 [xi (t + ς), xi−1 (t + ς)] ≤ ui−1 (t) then by the ς∈[−σ,0]

assumption (iii), we get 1 D0 [e xi+1 (t), e xi (t)] ≤ Γ(α) 1 ≤ Γ(α)

Zt a

Zt

(t − s)α−1 g(s, sup D0 [e xi (s + ς), e xi−1 (s + ς)])ds ς∈[−σ,0]

(t − s)α−1 g(s, ui−1 (s))ds = ui (t).

a

Thus, by mathematical induction, we obtain : D0 [e xn+1 (t), e xn (t)] ≤ un (t), ∀t ∈ [a, a + d], n = 0, 1, 2, .... Applying this property we have, for t ∈ [a, a + d] and for n = 0, 1, 2, ..., n−1 D0 [CgH Dαa+ e xn+1 (t), CgH Dαa+ e xn (t)] ≤ D0 [ f (t, e xnt ), f (t, e xn−1 xnt , e xn−1 (t)). t )] ≤ g(t, Dσ [e t ]) ≤ g(t, u

Assume m ≥ n, then one can easily obtain

D0 [CgH Dαa+ e xn (t), CgH Dαa+ e xm (t)] ≤ D0 [CgH Dαa+ e xn (t), CgH Dαa+ e xn+1 (t)]

xm (t)] xm+1 (t), CgH Dαa+ e xm+1 (t)] + D0 [CgH Dαa+ e xn+1 (t), CgH Dαa+ e + D0 [CgH Dαa+ e ≤ 2g(t, un−1 (t)) + g(t, Dσ [e xn (t), e xm (t)]).

Therefore, we obtain the Dini derivative C D+α of the function D0 [e xn (t), e xm (t)] as follows: a+ C

D+α xn (t), e xm (t)] ≤ 2g(t, un−1 (t)) + g(t, Dσ [e xn (t), e xm (t)]). a+ D0 [e 10

Since g(t, un−1 (t)) uniformly converges to 0, then for arbitrary ε > 0, there exists a natural number n0 such that C

D+α xn (t), e xm (t)] ≤ g(t, Dσ [e xn (t), e xm (t)]) + ε for m ≥ n ≥ n0 . a D0 [e

From the fact that D0 [e xn (a), e xm (a)] = 0 < ε and by using Theorem 3.1, we have D0 [e xn (t), e xm (t)] ≤ uε (t),

t ∈ [a, a + d], m ≥ n ≥ n0 ,

(3.10)

where uε (t) is the maximal solution to the following IVP for each n : (C Dαa+ u)(t) = g(t, uε (t)) + ε. Due to Proposition 2.2 in [37] one can infer that {uε (·)} converges uniformly to the maximal solution u(t) ≡ 0 of (3.5) on [a, a + d] as ε → 0. Hence, by virtue of (3.8), we infer that {e xn } converges uniformly to a continuous function e x : [a, a + d] → Ω(x0 , ρ). Note that e x is the desired solution to (3.1). Indeed, for every t ∈ [a, a + d], we have h

1 D0 ϕ(0), e x(t) + (−1) Γ(α)

Zt a

i (t − s)α−1 f (s, e xs )ds ≤ D0 [e xn (t), e x(t)] 1 + Γ(α)

Zt a

(t − s)α−1 D0 [ f (s, e xn−1 xs )]ds. s ), f (s, e

The summands in last expression converge to 0. Due to Lemma 3.2 the function e x is the (ii)-solution to (3.1). Now, we show the uniqueness of (ii)-solution of Eq. (3.1). Suppose e y(t) be another local (ii)-solution to (3.1) on the interval [a, a + d]. Define m(t) = D0 [e x(t), e y(t)], then m(a) = 0 and C α e C αe xt , e yt ]) ≤ g(t, |mt |σ ). (C D+α a+ m)(t) = D0 [ gH Da+ x(t), gH Da+ y(t)] ≤ g(t, Dσ [e Hence Theorem 3.1, we have m(t) ≤ u(t) for all t ∈ [a, a + d], where u(t) ≡ 0 is the maximal solution of IVP (3.5). Therefore e x(t) ≡ e y(t), which completes the proof.  Corollary 3.1. Let ϕ(t − a) ∈ Cσ and suppose that f ∈ C(I × S(x0 , ρ), E) satisfies the condition: there exists a constant L > 0 such that for every ξ, ψ ∈ S(x0 , ρ) it holds h "
i   D0 (t1 − s)α−1 f (t, ξ) , (t2 − s)α−1 f t, ψ 6 |(t1 − s)α−1 − (t2 − s)α−1 |LDσ ξ, ψ .

ˆ ≤ M1 . Then, the following successive Moreover, there exists a M1 > 0 such that D0 [ f (t, ξ), 0] approximations given by Eq. (3.6) and Eq. (3.8) converge uniformly to two unique solutions b x(t) ) ( h ρΓ(α + 1) i α1 h ρΓ(α + 1) i α1 , . and e x(t) of (3.1), respectively, on [a, a + d] where d = min p, M1 M2

Proof. The proof is obtained immediately by taking in Theorem 3.4, g(t, u) = LDσ [ξ, ψ].  Example 3.3. Consider the fuzzy fractional functional differential equation  C α    ( gH D0+ x)(t) = λx(t − 1), 1 ≥ t > 0    x(t) = (1 − t, 2 − t, 3 − t), t ∈ [−1, 0]

where λ ∈ [−1, 1]\{0}.

11

(3.11)

It can be checked that f : [0, 1] × S(x0 , 3) → E in problem (3.11) is a continuous mapping and satisfies conditions of Theorem 3.4. In particular - for (t, ξ) ∈ [0, 1] × S(x0 , 3) ˆ = |λ|Dσ [ξ, 0] ˆ ≤ Dσ [ξ, 0] ˆ ≤ 6, D0 [ f (t, ξ), 0] - for t ∈ [0, 1], ξ, ψ ∈ S(x0 , 3) D0 [ f (t, ξ), f (t, ψ)] = |λ|Dσ [ξ, ψ]. Hence as the function g which appears in Theorem 3.4 we can put g(t, u) = |λ|u for (t, u) ∈ [0, 1] × [0, 6]. Then we can easy show that g satisfies assumptions of Theorem 3.4 and g(t, u) ≤ 6 for (t, u) ∈ [0, 1] × [0, 6]. Based on the types of differentiability, we have to solve problem (3.11) in two cases. We observer that the sequence of successive approximations for case [(i) − gH]Cα −differentiable is well defined all t ∈ [0, 1], while the sequence of successive approximations for case [(ii) − gH]Cα −differentiable is well defined on [0, T] by choice of T. Case 1. Consider λ ∈ (0, 1]. For t ∈ [0, 1] the the sequence of successive approxima1 Rt tions x0 (t) = ϕ(0) and xn (t) = x0 (t) + (t − s)α−1 f (s, xn−1 s )ds is well defined. Also, by Γ(α) 0 recursion we obtain that xn (t) ∈ S(x0 , 3) for t ∈ [0, 1]. Therefore, by Theorem 3.4, there exists a unique (i)-solution to the problem (3.11). Using Definition 2.7 we obtain systems of fractional functional differential equations  C α  D0+ x(t, r) = λx(t − 1, r), 1 ≥ t > 0,      C α    D0+ x(t, r) = λx(t − 1, r), 1 ≥ t > 0, (3.12)    x(t, r) = ϕ(t, r),       x(t, r) = ϕ(t, r).

By solving (3.12), we obtain exact solution as follows: % # λ(4 − r)tα λ(2 + r)tα λtα+1 λtα+1 r − , (3 − r) + − , [x(t)] = (1 + r) + Γ(α + 1) Γ(α + 2) Γ(α + 1) Γ(α + 2) where t ∈ [0, 1].This solution is shown in Figure 1.

Case 2. Consider λ ∈ [−1, 0). For t ∈ [0, T] the the sequence of successive approxima1 Rt 0 n 0 tions x (t) = ϕ(0) and x (t) = x (t) (−1) (t − s)α−1 f (s, xn−1 s )ds is well defined. Indeed, Γ(α) 0 firstly observer that for (t, ξ) ∈ [0, 1] × S(x0 , 3) we have len( f (t, ξ)) = |λ|len(ξ) = 8|λ|. It is been that 1 len(ϕ(0)) = 2(1 − r) ≥ Γ(α)

Zt

|λ|(t − s)α−1 len(ξs )ds =

0

12

|λ| 8t, Γ(α + 1)

1 Rt which implies that the Hukuhara differences x0 (t) (−1) (t − s)α−1 f (s, xn−1 s )ds exist in Γ(α) 0 ) ( 2(1 − r)Γ(α + 1) . Also, by recursion we obtain the case t ∈ [0, T], where T = min 1, 8|λ| that xn (t) ∈ S(x0 , 3) for t ∈ [0, T]. Thus, by Theorem 3.4, there exists a unique (ii)-solution ( defined on interval [0, T]) to the problem (3.11). Using Definition 2.7 we obtain systems of fractional functional differential equations  C α  D0+ x(t, r) = λx(t − 1, r), T ≥ t > 0,      C α    D0+ x(t, r) = λx(t − 1, r), T ≥ t > 0, (3.13)    x(t, r) = ϕ(t, r),       x(t, r) = ϕ(t, r). By solving (3.13), we obtain exact solution as follows: % # λ(4 − r)tα λ(2 + r)tα λtα+1 λtα+1 r , − , (3 − r) + − [x(t)] = (1 + r) + Γ(α + 1) Γ(α + 2) Γ(α + 1) Γ(α + 2)

5

4

4.5

3.5

4

3

3.5

2.5 x(t)

x(t)

where t ∈ [0, T], λ ∈ [−1, 0). The graph of solution is drawn in Figure 2.

3

2

2.5

1.5

2

1

1.5

0.5

1 −1

−0.8

−0.6

−0.4

−0.2

0 t

0.2

0.4

0.6

0.8

0 −1

1

Figure 1. Solution of Example 3.3, Case 1. (λ = 0.5, α = 0.5)

−0.8

−0.6

−0.4

−0.2

0 t

0.2

0.4

0.6

0.8

Figure 2. Solution of Example 3.3, Case 2. (λ = −0.5, α = 0.5)

Next, we show the existence of the solutions of the fuzzy fractional functional differential equation of the form C α    gH Da+ x(t) = f (t, x(t), xt ), t > a, 0 < α < 1, (3.14)    x(t) = ϕ(t − a), a > t > a − σ Now, we shall prove existence and uniqueness results for (3.14) by using the contraction principle, which studied in [29]. In the following, for a given k > 0, we consider the set Sk of all continuous fuzzy functions x ∈ C([a − σ, ∞), E) such that x(t) = ϕ(t − a) on [a − σ, a] ˆ exp(−kt) < ∞ . On Sk we can define the following metric and sup {D0 [x(t, ω), 0] t≥a−σ

Dk [x, y] = sup {D0 [x(t), y(t)] exp(−kt)}. t≥a−σ

13

(3.15)

1

  where k > 0 is chosen suitably later. We easily prove that the space Sk , Dk of continuous fuzzy functions x : [a, ∞) → E is a complete metric space with distance (3.15). Theorem 3.4. Suppose that the following conditions hold: (i) f ∈ C([a, ∞) × E × Cσ , E) and there exist constants L1 , L2 > 0 such that D0 [ f (t, y, ξ), f (t, z, ψ)] ≤ L1 D0 [y, z] + L2 Dσ [ξ, ψ] for all y, z, ∈ E, ξ, ψ ∈ Cσ and t, s ≥ a; (ii) there exist constants M > 0 and b > 0 such that ˆ 0), ˆ 0] ˆ ≤ M exp(bt) for all t ≥ a, where b < k. Then the FFFDE (3.14) has a unique D0 [ f (t, 0, solution for each case on [a, ∞). Proof. For t ≥ a, equation (3.14) can be written as =1−α a+ (D gH x)(t)

1 = Γ(1 − α)

Zt

(t − s)α (D gH x)(s)ds = f (t, x(t), xt ).

a

Applying the operator by =αa on both sides of the above equation and using Lemma 2.2, we obtain 1−α x(t) gH x(a) = =1−α a+ f (t, x(t), xt ), or x(t) gH ϕ(0) = =a+ f (t, x(t), xt ).

Since the way of the proof is similar for all two cases, we only consider case [(i) − gH]Cα −differentiable for x, i.e. 1 x(t) = ϕ(0) + Γ(α)

Zt

(t − s)α−1 f (s, x(s), xs )ds.

(3.16)

a

Now, we consider the complete metric space (Sk , Dk ), and define an operator P : Sk → Sk   ϕ (t − a) if t ∈ [a − σ, a]      Zt  (Px) (t) =  1   (t − s)α−1 f (s, x(s), xs ) if t > a. ϕ(0) +    Γ(α)  a

We can choose a big enough value for k such that P is a contraction, so the Banach fixed point theorem provides the existence of a unique fixed point for P, that is, a unique solution for (3.14). The first, we shall prove that P(Sk ) ∈ Sk with assumption k > b. Indeed, let x ∈ Sk . For each t ≥ a, we get % ) ( # o n h i 1 Rt α−1 ˆ ˆ (t − s) f (s, x(s), xs )ds, 0 exp(−kt) sup D0 (Px) (t) , 0 exp(−kt) = sup D0 ϕ(0) + Γ(α) a t>a t>a ( ) n h h i  i h   io 1 Rt α−1 −kt ˆ ˆ ˆ ˆ ˆ ˆ 6 sup D0 ϕ (0) , 0 + (t − s) D0 f (s, x(s), xs ) , f s, 0, 0 + D0 f s, 0, 0 , 0 dse Γ(α) a t>a   h i i  h i Rt   h  L −ks k(s−t)  α−1 ˆ ˆ ˆ ds D0 x(s), 0 + sup D0 x(s + θ), 0 e e ≤ sup   D0 ϕ(0), 0 + Γ(α) (t − s) t≥a θ∈[−σ,0] a ( ) M Rt + sup (t − s)α−1 ebs−kt ds Γ(α) t≥a a ˆ < where L = max{L1 , L2 }. Further, since x ∈ Sk , there exists ρ > 0 such that D0 [x(t), 0]

14

h i ρ exp(kt) < ∞. It follows that sup D0 x(s + θ), 0ˆ ≤ ρ exp(−kt) for all t ≥ a. Therefore, for θ∈[−σ,0]

all t ≥ a, we obtain

  k(t−a) Z     o i h i h   2ρL   α−1 −u ˆ u e du ϕ(0), 0 + D sup D0 (Px) (t) , 0ˆ exp(−kt) ≤ sup   0   α   k Γ(α) t≥a  t>a  0   b(t−a) Z       M   α−1 −u + sup  . u e du    α   k Γ(α) t≥a   n

0

h i Let K = sup D0 ϕ(θ − a), 0ˆ . Then θ∈[a−σ,a]

h i 1 Dk Px, 0ˆ ≤ K + α (2ρL + M) < ∞. k and thus Px ∈ Sk . The following steps, we shall prove that P is a contraction by metric Dk . Let x, y ∈ Sk . Then for −σ ≤ s ≤ 0, D0 [(Px)(a + s), (Py)(a + s)] = 0. For each t ≥ a, we have h


i D0 (Px) (t) , Py (t) ≤ "

≤

1 Γ(α)

Zt

  (t − s)α−1 L1 D0 [x(s), y(s)] + L2 Dσ [xs , ys ] ds

L Γ(α)

Zt

  (t − s)α−1 D0 [x(s), y(s)] + sup D0 [x(θ), y(θ)] ds.

a

a

θ∈[s−σ,s]

From (3.15) it follows that D0 [x(s), y(s)] ≤ Dk [x, y] exp(ks) for all s ≥ a − σ. Hence sup D0 [x(θ), y(θ)] ≤ Dk [x, y] exp(ks) for all s ≥ a. Further, for each t ≥ a, we obtain θ∈[s−σ,s]

h

"
i D0 (Px) (t) , Py (t) ≤

L Γ(α)

Zt

  (t − s)α−1 D0 [x(s), y(s)] + sup D0 [x(θ), y(θ)] ds

Zt

  (t − s)α−1 Dk [x, y]eks + Dk [x, y]eks ds

a

L ≤ Γ(α)

a

θ∈[s−σ,s]

and so Z n h i o LDk [x, y] "
i Dk Px, Py = sup D0 (Px) (t) , Py (t) exp(−kt) ≤ sup (t − s)α−1 ek(s−t) ds Γ(α) t≥a−σ t≥a t

h

a

≤

LDk [x, y] sup kα Γ(α) t≥a

k(t−a) Z

uα−1 e−u ds ≤

LDk [x, y] ≤ Dk [x, y]. kα

0

Now, choose k large enough such that L/kα < 1, we have the operator P on Sk is a contraction. By using Banach fixed point theorem provides the existence of a unique fixed point for P and the unique fixed of P is in the space Sk , that is a unique solution for (3.14) in case [(i) − gH]Cα −differentiable.  15

4 Formulation of the Numerical Method for FFFDEs In this section, the modified Adams-Bashforth-Moulton method for solving fuzzy functional differential equation of fractional order under the Caputo-type fuzzy fractional derivative will be investigated. For this purpose, well consider, for simplicity and without lose of generality, we consider the following FFFDE described by C α    gH D0+ x(t) = f (t, x(t), x(t − τ)), t > 0, 0 < α < 1, (4.1)    x(t) = ϕ(t), t ∈ [−τ, 0],

where τ > 0, b > 0 are such that b = l.τ for given l ∈ N∗ , f : I = [0, b] × E × Cσ → E and α ∈ (0, 1) is the order of the differential equation, ϕ(t) is the initial value. Let [x(t)]r = [x(t, r), x(t, r)]. By using Zadeh’s extension principle, we obtain [ f (t, x(t), x(t − τ))]r = [ f (t, r, u1 , v1 , u2 , v2 ), f (t, r, u1 , v1 , u2 , v2 )], where f (t, r, u1 , v1 , u2 , v2 ) = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), f (t, r, u1 , v1 , u2 , v2 ) = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), for r ∈ [0, 1]. In this equation (4.1) we shall solve it by two types of Caputo fractional generalized Hukuhara derivative. Consequently, based on the types of differentiability, we have the following two cases. i ir h h Case 1. If x(t) is [(i) − gH]Cα −differentiable then (CgH Dα0+ x)(t) = C Dα0+ x(t, r), C Dα0+ x(t, r) and (4.1) is translated into the following fractional functional differential system: C α  D0+ x(t, r) = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), t ≥ 0      C α    D0+ x(t, r) = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), t ≥ 0    x(t, r) = ϕ(t, r), t ∈ [−τ, 0]       x(t, r) = ϕ(t, r), t ∈ [−τ, 0].

(4.2)

ir h h i Case 2. If x(t) is [(ii) − gH]Cα −differentiable, then (CgH Dα0+ x)(t) = C Dα0+ x(t, r), C Dα0+ x(t, r) and (4.1) is translated into the following fractional functional differential system: C α  D0+ x(t, r) = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), t ≥ 0      C α    D0+ x(t, r) = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), t ≥ 0    x(t, r) = ϕ(t, r), t ∈ [−τ, 0]       x(t, r) = ϕ(t, r), t ∈ [−τ, 0].

(4.3)

Remark 4.1. If we ensure that the solutions (x(t, r), x(t, r)) of the systems (4.2) and (4.3) respectively are valid level sets of fuzzy number valued functions and if the derivatives (C Dα0+ x(t, r), C Dα0+ x(t, r)) are valid level sets of fuzzy numbers valued functions with two kinds differentiability respectively, then we can contruct the solution of the FFFDE (4.1). In the following theorem we show that the converse result holds as well and the FFFDE (4.1) will be equivalent to the systems (4.2) and (4.3) for each kind of differentiability. 16

Theorem 4.1. Let us consider the FFFDE (4.1) where f : I = [0, b] × E × Cσ → E is such that h (i) [ f (t, x(t), x(t − τ))]r = f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)), f (t, r, x(t, r), x(t, r), x(t − i τ, r), x(t − τ, r)) ; (ii) f and f are equicontinuous and uniformly bounded on any bounded set ( i.e. there exists o n a M > 0 such that max | f (t, r, y1 , z1 , y2 , z2 )|, | f (t, r, y1 , z1 , y2 , z2 )| ≤ M, where yi , zi , ( i = 1, 2) are bounded) ; (iii) there exists L > 0 such that n o max | f (t, r, y1 , z1 , y2 , z2 ) − f (t, r, u1 , v1 , u2 , v2 )|, | f (t, r, y1 , z1 , y2 , z2 ) − f (t, r, u1 , v1 , u2 , v2 )|  ≤ L max |y1 − u1 | + |y2 − u2 |σ , |z1 − v1 | + |z2 − v2 |σ ∀r ∈ [0, 1].

Then the FFFDE (4.1) and the systems of (4.2) and (4.3) are equivalent for each case. Proof. The first the requirement (i) is fulfilled by any fuzzy-valued function obtained from a continuous function by Zadeh’s extension principle. So this condition is not too restrictive. Next, the equicontinuity property in (ii) ensures that f satisfies a continuous property. Further, the Lipschitz property in (iii) of f and f implies the Lipschitz property of the fuzzy-valued function f . Indeed, o n sup max | f (t, r, y1 , z1 , y2 , z2 ) − f (t, r, u1 , v1 , u2 , v2 )|, | f (t, r, y1 , z1 , y2 , z2 ) − f (t, r, u1 , v1 , u2 , v2 )| r∈[0,1]

 ≤ L sup max |y1 − u1 | + |y2 − u2 |σ , |z1 − v1 | + |z2 − v2 |σ , r∈[0,1]

i.e., h i D0 [ f (t, x(t), x(t − τ)), f (t, y(t), y(t − τ))] ≤ L D0 [x(t), y(t)] + Dσ [x(t − τ), y(t − τ)] .

(4.4)

By the continuity of f , from Lipschitz condition and the boundedness condition in (ii) it follows that FFFDE (4.1) has a unique solution for each case on [0, T]. In Case 1, the solution of the problem FFFDE (4.1) is [(i)− gH]Cα −differentiable and so, the functions x(t, r) and x(t, r) are Caputo differentiable, and so (x(t, r), x(t, r)) is a solution of the system (4.2). Similarly for Case 2. Conversely. We consider system (4.2). Let us suppose that we have a solution (x(t, r), x(t, r)) of the system (4.2). Also, the Lipschitz condition (4.4) and the boundedness condition on bounded set imply the existence and uniqueness of the fuzzy solution x for case [(i) − gH]Cα −differentiable on [0, b]. Now, since x is [(i) − gH]Cα −differentiable, x(t, r), x(t, r) the endpoint of [x(t)]r is a solution of the system (4.2). Further, the solution of (4.2) is unique, we have [x(t)]r = [x(t, r), x(t, r)], that is the problem FFFDE (4.1) and the system (4.2) are equivalent. Finally, the equivalence of problem (4.1) and the system (4.3) is proved as in Case 1.  The Theorem 4.1 show us the hint on how to deal with numerical solutions of FFFDEs. We can translate the original FFFDE equivalently into a system of fractional functional 17

differential equations (FFDEs). The numerical solutions of the FFDEs are extremely well studied in the literature, so any numerical method for the system of FFDEs we can use them for solving solution of the FFFDE. Mazandarani and Kamyad [8] proposed the modified fractional Euler method (MFEM) for solving fuzzy fractional initial value problem under Caputo type fuzzy fractional derivatives. The MFEM based on a generalized Taylor’s formula [23] and a modified trapezoidal rule [24] is used for solving fuzzy fractional differential equation of order α ∈ (0, 1). Following the modified Adams-Bashforth-Moulton method (MABMM) is proposed for solving fuzzy functional differential equation of fractional order under the Caputo-type fuzzy fractional derivative. Before introducing the MABMM for FFFDEs, we first briefly recall the idea of the classical one-step Adams-Bashforth-Moulton algorithm for the following ODE: du = g(t, u(t)), u(0) = u0 . dt

(4.5)

Let t ∈ [0, b], t j = jh, j = 0, 1, ..., N, and h = b/N be the time step. The basic idea is, assuming that we have already calculated approximations u j ≈ u(t j ), j = 1, 2, ..., k that we try to obtain the approximation uk+1 by means of the equation u(tk+1 ) = u(tk ) +

Ztk+1

g(ξ, u(ξ))dξ.

(4.6)

tk

This equation follows upon integration of (4.5) on the interval [tk , tk+1 ]. Since the integral in the right-hand side of (4.6) can be approximated by trapezoidal quadrature formula, Ztk+1

g(ξ, u(ξ))dξ ≈

tk

 h g(tk , u(tk )) + g(tk+1 , u(tk+1 )) . 2

(4.7)

Therefore, we can get the approximation to uk+1 as follows: uk+1 = uk +

 h g(tk , uk ) + g(tk+1 , uk+1 ) , 2

(4.8)

which is an implicit equation. The problem with this equation is that the unknown quantity uk+1 appears on both sides, and due to the nonlinear nature of the function g, we cannot solve for uk+1 directly, in general. However, one can adopt another numerical method to approximate uk+1 is the right-hand side preliminarily. The preliminarily approximation p uk+1 is called predictor and can be obtained in a very similar way, only replacing the trapezoidal quadrature formula by the rectangle rule giving the explicit ( forward Euler method or any other explicit method ) method; here we take forward Euler method as an example; that is, p

uk+1 = uk + hg(tk , uk ).

(4.9)

Then, the so-called one-step Adams-Bashforth-Moulton method is uk+1 = uk +

 h p g(tk , uk ) + g(tk+1 , uk+1 ) , 2 18

(4.10)

in which (4.9) is called the predictor, and (4.10) is called the corrector. It is well known that this method is convergent of order 2, i.e., max |u(t j ) − u j | = O(h2 ), if u is sufficiently j=0,1,...,N

smooth. We now give a generalization of trapezoidal rule to approximation the fractional integral I0α+ g(t) of order α > 0. Theorem 4.2. [24] Suppose that the interval [0, b] is subdivided into N subintervals [t j , t j+1 ] of b equal width h = by using the nodes t j = jh, for j = 0, 1, ..., N. The modified trapezoidal rule N  hα g(0)  hα g(b) + T(g, h, α) = (N − 1)α+1 − (N − α − 1)Nα Γ(α + 2) Γ(α + 2)   α N−1  X  h g tj "
α+1 "
α+1 "
α+1 + N− j+1 −2 N− j + N− j−1 Γ (α + 2) j=1 is an approximation to fractional integral I0α+ g(b) = T(g, h, α) − O(h2 ). After some manipulations, the initial value problems (4.2) and (4.3) can be equivalent to the following fractional integral equations   x(t, r) = I0α+ f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)) + ϕ(0, r), t ≥ 0        x(t, r) = I0α+ f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)) + ϕ(0, r), t ≥ 0     (4.11) x(t, r) = ϕ(t, r), t ∈ [−τ, 0]        x(t, r) = ϕ(t, r), t ∈ [−τ, 0]      t ∈ [0, b], α ∈ (0, 1) for Case 1, and   x(t, r) = I0α+ f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)) + ϕ(0, r), t ≥ 0         x(t, r) = I0α+ f (t, r, x(t, r), x(t, r), x(t − τ, r), x(t − τ, r)) + ϕ(0, r), t ≥ 0      x(t, r) = ϕ(t, r), t ∈ [−τ, 0]        x(t, r) = ϕ(t, r), t ∈ [−τ, 0]      t ∈ [0, b], α ∈ (0, 1)

(4.12)

for Case 2. Consider a uniform grid {tn = nh : n = −k, −k + 1, ..., −1, 0, 1, ..., N} where k and N are integers such that h = T/N and h = τ/k. Having introduced the Adams-BashforthMoulton algorithm, now the key problem is to establish approximation to the delayed terms x(t − τ, r) and x(t − τ, r), respectively. Since the way of the establishment is similar for two terms, we only establish for case of the delayed term x(t − τ, r). Let x(t j , r) = ϕ(t j , r), j = −k, −k + 1, ..., −1, 0. Therefore, x(t − τ, r) can be approximated by    x (r), if j = 0, 1, ..., N,   j−k x(t j − τ, r) ≈  (4.13)    ϕ j (r) if j = −k, −k + 1, ..., −1, 0. 19

Thus, we have the following relations: for j = 1, ...., N x(t j , r) → x j (r) for j = 1, ...., N x(t j − τ, r) → x j−k (r) for j = −k, −k + 1, ..., 0 x(t j , r) → ϕ (r). j

Using the modified trapezoidal rule in Theorem 4.2, the numerical scheme for (4.11), (4.12) can be depicted as:   hα p   f (tn , r, xpn (r), xn (r), xn−k (r), xn−k (r)) (r) = ϕ(0, r) + x  n   Γ(α + 2)     n−1 X   hα   a j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r)) +    Γ(α + 2) j=0        hα p  x (r) = ϕ(0, r) +  f (tn , r, xpn (r), xn (r), xn−k (r), xn−k (r))  n   Γ(α + 2) (4.14)   n−1  X α  h    + a j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r))   Γ(α + 2) j=0          (n − 1)α+1 − (n − α − 1)nα ,   j=0     a j,n =    α+1 α+1 α+1   (n − j + 1) + (n − j − 1) − 2(n − j) , j ∈ [1, n − 1]        t ∈ [0, b], α ∈ (0, 1), for Case 1, and   hα p   x (r) = ϕ(0, r) + f (tn , r, xpn (r), xn (r), xn−k (r), xn−k (r))  n   Γ(α + 2)     n−1 X   hα   + a j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r))    Γ(α + 2) j=0        hα p   f (tn , r, xpn (r), xn (r), xn−k (r), xn−k (r)) x (r) = ϕ(0, r) +  n   Γ(α + 2)   n−1  X  hα    + a j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r))   Γ(α + 2) j=0          (n − 1)α+1 − (n − α − 1)nα ,   j=0     a j,n =    α+1 α+1 α+1   (n − j + 1) + (n − j − 1) − 2(n − j) , j ∈ [1, n − 1]        t ∈ [0, b], α ∈ (0, 1), p

p

(4.15)

for Case 2. The approximations xn (r) and xn (r) are used in (4.14), (4.15) to evaluate predictor terms X 1 b j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r)), Γ(α + 1) j=0 n−1

xpn (r) = ϕ(0, r) +

X 1 = ϕ(0, r) + b j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r)), Γ(α + 1) j=0 n−1

p xn (r)

20

for Case 1, and xpn (r)

X 1 = ϕ(0, r) + b j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r)), Γ(α + 1) j=0

p xn (r)

X 1 = ϕ(0, r) + b j,n f (t j , r, x j (r), x j (r), x j−k (r), x j−k (r)), Γ(α + 1) j=0

n−1

n−1

  for Case 2, where b j,n = hα (n − j)α − (n − j − 1)α . Next, we present some simple examples to solve the fuzzy fractional functional initial value problem under Caputo type fuzzy fractional derivatives. t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0.082 0.064 0.056 0.052 0.049 0.047 0.045 0.043 0.042 0.041

0.1 0 0.094 0.075 0.067 0.063 0.060 0.058 0.056 0.054 0.053 0.052

0.2 0 0.105 0.086 0.079 0.074 0.071 0.069 0.067 0.065 0.064 0.063

0.3 0 0.116 0.097 0.090 0.085 0.082 0.080 0.078 0.076 0.075 0.074

0.4 0 0.127 0.108 0.101 0.096 0.093 0.091 0.089 0.087 0.086 0.085

0.5 0 0.138 0.119 0.112 0.107 0.104 0.102 0.100 0.098 0.097 0.096

0.6 0 0.149 0.130 0.123 0.118 0.115 0.113 0.111 0.110 0.108 0.107

0.7 0 0.160 0.141 0.134 0.129 0.126 0.124 0.122 0.121 0.119 0.118

0.8 0 0.171 0.152 0.145 0.140 0.137 0.135 0.133 0.132 0.130 0.129

0.9 0 0.182 0.163 0.156 0.152 0.148 0.146 0.144 0.143 0.141 0.140

1 0 0.193 0.174 0.167 0.163 0.160 0.157 0.155 0.154 0.152 0.151

Table 1. Error of numerical solution in Case 1 (λ = 1) - x(t, r) t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0.082 0.064 0.056 0.052 0.049 0.047 0.045 0.043 0.042 0.041

0.1 0 0.072 0.053 0.046 0.041 0.038 0.036 0.034 0.033 0.031 0.030

0.2 0 0.061 0.042 0.035 0.030 0.027 0.025 0.023 0.021 0.020 0.019

0.3 0 0.050 0.031 0.024 0.019 0.016 0.014 0.012 0.010 0.009 0.008

0.4 0 0.039 0.020 0.013 0.008 0.005 0.003 0.001 0.001 0.002 0.003

0.5 0 0.028 0.009 0.002 0.003 0.006 0.008 0.010 0.012 0.013 0.014

0.6 0 0.017 0.002 0.009 0.014 0.017 0.019 0.021 0.023 0.024 0.025

0.7 0 0.006 0.013 0.021 0.025 0.028 0.030 0.032 0.034 0.035 0.036

0.8 0 0.006 0.024 0.032 0.036 0.039 0.041 0.043 0.045 0.046 0.047

0.9 0 0.017 0.035 0.043 0.047 0.050 0.052 0.054 0.056 0.057 0.058

1 0 0.028 0.046 0.054 0.058 0.061 0.064 0.065 0.067 0.068 0.069

Table 2. Error of numerical solution in Case 1 (λ = 1) - x(t, r) Example 4.3. Consider again the fuzzy fractional functional differential equation  C 0.75    ( gH D0+ x)(t) = λx(t − 1), t ∈ [0, 1]    x(t) = (1 − t, 2 − t, 3 − t), t ∈ [−1, 0]

where λ ∈ [−1, 1]\{0}.

21

(4.16)

1 0.8

0.6

0.6 r

r

1 0.8

0.4

0.4

0.2

0.2 0 7

0 8 6

6

1

0 0

0 0 0.041 0.032 0.028 0.026 0.025 0.024 0.023 0.022 0.021 0.021

[x(t)]

t

0.1 0 0.047 0.038 0.034 0.032 0.030 0.029 0.028 0.027 0.027 0.026

0.2 0 0.053 0.043 0.039 0.037 0.036 0.035 0.034 0.033 0.032 0.032

0.4

2

0.2

0.3 0 0.058 0.049 0.045 0.043 0.041 0.040 0.039 0.038 0.038 0.037

r

1 0

0.2 t

Figure 4. The approximations (i)solution of Example 4.3 with λ = 1.

Figure 3. The analytical (i)-solution of Example 4.3 with λ = 1.

t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.6

3

0.4

2 [x(t)]

0.8

4

0.6

r

1

5

0.8

4

0.4 0 0.064 0.054 0.051 0.048 0.047 0.046 0.045 0.044 0.043 0.043

0.5 0 0.069 0.060 0.056 0.054 0.052 0.051 0.050 0.049 0.049 0.048

0.6 0 0.075 0.065 0.062 0.059 0.058 0.057 0.056 0.055 0.054 0.054

0.7 0 0.080 0.071 0.067 0.065 0.063 0.062 0.061 0.061 0.060 0.059

0.8 0 0.086 0.076 0.073 0.070 0.069 0.068 0.067 0.066 0.065 0.065

0.9 0 0.091 0.082 0.078 0.076 0.074 0.073 0.072 0.072 0.071 0.070

1 0 0.097 0.087 0.084 0.082 0.080 0.079 0.078 0.077 0.076 0.076

Table 3. Error of numerical solution in Case 2 (λ = −1/2) - x(t, r) Case 1. Consider λ ∈ (0, 1]. If we consider (CgH D0.75 x)(t) in the sense of [(i)−gH]C0.75 −differentiable, 0+ the analytical solution of (4.16) is given by % # λ(2 + r)t0.75 λ(4 − r)t0.75 λt1.75 λt1.75 , t ∈ [0, 1]. [x(t)] = (1 + r) + − , (3 − r) + − Γ(1.75) Γ(2.75) Γ(1.75) Γ(2.75) r

Case 2. Consider λ ∈ [−1, 0). If we consider (CgH D0.75 x)(t) in the sense of [(ii)−gH]C0.75 −differentiable, 0+ the analytical solution of (4.16) is given by % # λ(4 − r)t0.75 λ(2 + r)t0.75 λt1.75 λt1.75 − , (3 − r) + − , t ∈ [0, 1]. [x(t)] = (1 + r) + Γ(1.75) Γ(2.75) Γ(1.75) Γ(2.75) r

The exact and approximate solutions are both depicted in Figs. 3,4,5 and 6. As can be seen, there is a high agreement between the approximate solutions obtained by the modified Adams-Bashforth-Moulton method and the exact solutions. For more comparison, Tables 1,2,3 and 4 show the errors between the analytical and approximations solutions. The results show that the proposed method is quite effective.

22

t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 0.041 0.032 0.028 0.026 0.025 0.024 0.023 0.022 0.021 0.021

0.1 0 0.036 0.027 0.023 0.021 0.019 0.018 0.017 0.016 0.016 0.015

0.2 0 0.030 0.021 0.017 0.015 0.014 0.012 0.012 0.011 0.010 0.010

0.3 0 0.025 0.015 0.012 0.010 0.008 0.007 0.006 0.005 0.005 0.004

0.4 0 0.019 0.010 0.006 0.004 0.003 0.001 0.000 0.000 0.001 0.002

0.5 0 0.014 0.004 0.001 0.001 0.003 0.004 0.005 0.006 0.006 0.007

0.6 0 0.008 0.001 0.005 0.007 0.008 0.010 0.011 0.011 0.012 0.013

0.7 0 0.003 0.007 0.010 0.012 0.014 0.015 0.016 0.017 0.018 0.018

0.8 0 0.003 0.012 0.016 0.018 0.020 0.021 0.022 0.022 0.023 0.024

0.9 0 0.008 0.018 0.021 0.024 0.025 0.026 0.027 0.028 0.029 0.029

1 0 0.014 0.023 0.027 0.029 0.031 0.032 0.033 0.033 0.034 0.035

Table 4. Error of numerical solution in Case 2 (λ = −1/2) - x(t, r)

1 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 3

r

1

0 3

1

2 1

2

0.8 0.6

1 [x(t)]r

0.4 0

0.4 0 0

0.8 0.6

1 0.2 0

0.2 t

Figure 6. The approximations (ii)solution of Example 4.3 with λ = −0.5.

Figure 5. The analytical (ii)-solution of Example 4.3 with λ = −0.5.

Example 4.4. Consider the following fuzzy fractional functional initial value problem  C 0.5 2 2 2    ( gH D0+ x)(t) = λtx(t − 1) + (t , 2t , 3t ), t ∈ [0, 1] (4.17)    x(t) = ϕ(t) = (t + 1, 2t + 2, 3t + 3), t ∈ [−1, 0] where λ ∈ [−1, 1]\{0}.

Case 1. Consider λ ∈ (0, 1]. If we consider (CgH D0.5 x)(t) in the sense of [(i)−gH]C0.5 −differentiable, 0+ one has to solve the corresponding delay differential system as indicated in (4.2), i.e.  C 0.5  D x(t, r) = λtx(t − 1, r) + (1 + r)t2 , t ≥ 0    0+   C 0.5 2    D0+ x(t, r) = λtx(t − 1, r) + (3 − r)t , t ≥ 0 (4.18)    x(t, r) = ϕ(t, r) = (1 + r)(t + 1), t ∈ [−τ, 0]       x(t, r) = ϕ(t, r) = (3 − r)(t + 1), t ∈ [−τ, 0]. 23

t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 .0010 .0014 .0018 .0021 .0024 .0027 .0029 .0031 .0033 .0035

0.1 0 .0008 .0006 .0001 .0015 .0033 .0056 .0085 .0118 .0156 .0199

0.2 0 .0007 .0002 .0021 .0050 .0090 .0139 .0198 .0267 .0345 .0434

0.3 0 .0005 .0010 .0041 .0086 .0147 .0222 .0312 .0416 .0535 .0668

0.4 0 .0004 .0018 .0060 .0122 .0204 .0305 .0426 .0565 .0724 .0902

0.5 0 .0002 .0026 .0080 .0158 .0261 .0388 .0539 .0714 .0914 .1137

0.6 0 .0001 .0034 .0099 .0194 .0318 .0471 .0653 .0864 .1103 .1371

0.7 0 .0001 .0042 .0119 .0230 .0375 .0554 .0767 .1013 .1292 .1605

0.8 0 .0002 .0050 .0139 .0266 .0432 .0637 .0880 .1162 .1482 .1840

0.9 0 .0003 .0058 .0158 .0302 .0489 .0720 .0994 .1311 .1671 .2074

1 0 .0005 .0066 .0178 .0338 .0546 .0803 .1108 .1460 .1860 .2308

Table 5. Error of numerical solution in Case 1 (λ = 1) - x(t, r) t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 .0029 .0043 .0055 .0064 .0073 .0080 .0087 .0094 .0100 .0106

0.1 0 .0030 .0051 .0074 .0100 .0130 .0163 .0201 .0243 .0289 .0340

0.2 0 .0031 .0059 .0094 .0136 .0187 .0246 .0314 .0392 .0478 .0574

0.3 0 .0033 .0067 .0113 .0172 .0244 .0329 .0428 .0541 .0668 .0809

0.4 0 .0034 .0076 .0133 .0208 .0301 .0412 .0542 .0690 .0857 .1043

0.5 0 .0036 .0084 .0153 .0244 .0358 .0495 .0655 .0839 .1047 .1277

0.6 0 .0037 .0092 .0172 .0280 .0415 .0578 .0769 .0988 .1236 .1512

0.7 0 .0039 .0100 .0192 .0316 .0472 .0661 .0883 .1138 .1425 .1746

0.8 0 .0040 .0108 .0211 .0352 .0529 .0744 .0996 .1287 .1615 .1980

0.9 0 .0041 .0116 .0231 .0387 .0586 .0827 .1110 .1436 .1804 .2215

1 0 .0043 .0124 .0250 .0423 .0643 .0910 .1224 .1585 .1993 .2449

Table 6. Error of numerical solution in Case 1 (λ = 1) - x(t, r) By solving (4.18), we obtain analytical solution as follows (see Figure 7):  h  2(1 + λ)t5/2 i 2(1 + λ)t5/2  , (3 − r) 1 + , t ∈ [0, 1]. [x(t)]r = (1 + r) 1 + Γ(7/2) Γ(7/2) The result obtained using the numerical method proposed in this paper is shown in Figure 8. The errors between the analytical and approximations solutions are listed in Tables 5 and 6. x)(t) in the sense of [(ii)−gH]C0.5 −differentiable, Case 2. Consider λ ∈ [−1, 0). If we consider (CgH D0.5 0+ one has to solve the corresponding delay differential system as indicated in (4.3), i.e.  C 0.5  D x(t, r) = λtx(t − 1, r) + (3 − r)t2 , t ≥ 0    0+   C 0.5 2    D0+ x(t, r) = λtx(t − 1, r) + (1 + r)t , t ≥ 0 (4.19)    x(t, r) = ϕ(t, r) = (1 + r)(t + 1), t ∈ [−τ, 0]       x(t, r) = ϕ(t, r) = (3 − r)(t + 1), t ∈ [−τ, 0]. By solving (4.19), we obtain analytical solution as follows (see Figure 9): 24

1 0.8

0.6

0.6

r

r

1 0.8

0.4

0.4

0.2

0.2 0 10

0 10 8

8

1 6

[x(t)]

0

0 0 .0013 .0020 .0025 .0030 .0034 .0037 .0041 .0044 .0047 .0049

[x(t)]

t

0.1 0 .0013 .0021 .0029 .0036 .0044 .0053 .0062 .0072 .0083 .0095

0.2 0 .0013 .0022 .0032 .0043 .0055 .0068 .0084 .0101 .0120 .0140

0.4

2

0.2 0

Figure 7. The analytical (i)-solution of Example 4.4 with λ = 1. t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.8 0.6

4

0.4

2 r

1 6

0.8 0.6

4

0.3 0 .0013 .0023 .0035 .0049 .0065 .0084 .0105 .0129 .0156 .0186

r

0

0.2 0 t

Figure 8. The approximations (i)solution of Example 4.4 with λ = 1. 0.4 0 .0013 .0024 .0038 .0055 .0076 .0099 .0127 .0158 .0193 .0231

0.5 0 .0013 .0025 .0041 .0062 .0086 .0115 .0149 .0187 .0229 .0277

0.6 0 .0013 .0026 .0045 .0068 .0097 .0131 .0170 .0215 .0266 .0322

0.7 0 .0013 .0027 .0048 .0074 .0107 .0146 .0192 .0244 .0302 .0368

0.8 0 .0013 .0028 .0051 .0081 .0117 .0162 .0213 .0272 .0339 .0413

0.9 0 .0012 .0030 .0054 .0087 .0128 .0177 .0235 .0301 .0375 .0458

1 0 .0012 .0031 .0057 .0093 .0138 .0193 .0256 .0329 .0412 .0504

Table 7. Error of numerical solution in Case 2 (λ = −0.2) - x(t, r). h 2(1 + 3λ)t5/2 r(2λ − 2)t5/2 i 2(3 + λ)t5/2 r(2λ − 2)t5/2 , t ∈ [0, 1]. + , (3 − r) + − [x(t)]r = (1 + r) + Γ(7/2) Γ(7/2) Γ(7/2) Γ(7/2) The result obtained using the numerical method proposed in this paper is shown in Figure 10. The errors between the analytical and approximations solutions are listed in Tables 7 and 8. In the sequel, we would like to realize the stability of our approach when the input initial measured data are contaminated by random noise for different problems. We can evaluate the stability by increasing the different levels of random disturbance in the initial data: ˆ = ϕ(t) + εR(i), ϕ(t) where ϕ(t) ∈ Cσ is the initial exact data and R(i) is the random noise. We employ the function RANDOM NUMBER given in Matlab to generate the noisy data R(i), which are random numbers in [−1, 1], and ε means the level of absolute noise. Then, the initial noisy ˆ are used in the calculations. data ϕ(t)

25

t\r 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0 .0002 .0003 .0004 .0004 .0005 .0005 .0006 .0006 .0007 .0007

0.1 0 .0002 .0002 .0000 .0002 .0006 .0010 .0016 .0022 .0030 .0038

0.2 0 .0002 .0001 .0003 .0008 .0016 .0026 .0037 .0051 .0066 .0084

0.3 0 .0002 .0001 .0006 .0015 .0026 .0041 .0059 .0079 .0103 .0129

0.4 0 .0002 .0001 .0009 .0021 .0037 .0057 .0080 .0108 .0140 .0175

0.5 0 .0002 .0002 .0012 .0027 .0047 .0072 .0102 .0137 .0176 .0220

0.6 0 .0002 .0003 .0016 .0034 .0058 .0088 .0124 .0165 .0213 .0266

0.7 0 .0003 .0004 .0019 .0040 .0068 .0103 .0145 .0194 .0249 .0311

0.8 0 .0003 .0005 .0022 .0046 .0079 .0119 .0167 .0222 .0286 .0357

0.9 0 .0003 .0006 .0025 .0053 .0089 .0134 .0188 .0251 .0322 .0402

1 0 .0003 .0008 .0028 .0059 .0100 .0150 .0210 .0280 .0359 .0448

Table 8. Error of numerical solution in Case 2 (λ = −0.2) - x(t, r).

1 0.8

0.8

0.6 r

1

0.4

0.4

0.2

0.2

0 3.5

r

0.6

3

0 4

1 2.5

0.8

3

0.8

[x(t)]r

0.4 1

0.4

1.5

0.6

2

0.6

2

1

[x(t)]

0.2

r

1

0.2 0 t

0 t

Figure 10. The approximations (ii)solution of Example 4.4 with λ = −0.2.

Figure 9. The analytical (ii)-solution of Example 4.4 with λ = −0.2.

Example 4.5. Consider the following fuzzy fractional functional initial value problem

where λ ∈ [−1, 1]\{0}.

 2−t   (C D0.75 x)(t) = λx(t) + x(t − 1), t ∈ [0, 1]    gH 0+ 4   1 2 3    x(t) = ϕ(t) = ( , , ), t ∈ [−1, 0] 1−t 1−t 1−t

(4.20)

Since the exact solution cannot be found analytically, we use the numerical method proposed in this study. Case 1. Consider λ ∈ (0, 1] and [(i) − gH]Cα −differentiable. Using the modified AdamsBashforth-Moulton method (4.14), the approximation solutions [x(t)]0 = [x(t, 0), x(t, 0)] to ˆ 0)] to (4.20) with ˆ 0 = [ˆx(t, 0), x(t, (4.20) with ε = 0 (without the random noise) and [x(t)] ε = 10−1 ( with the absolute random noise) are shown in Figure 11 and Table 9. Moreover, numerical comparisons between the approximations solutions with the different levels of random disturbance in the initial data (ε = 0, ε = 10−1 and ε = 10−3 ) are given in Tables 9 26

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x(t, 0) 1 1.1765437 1.2806761 1.3834048 1.4853482 1.5878194 1.6916579 1.7974463 1.9056240 2.0165453 2.1305120

xˆ (t, 0) 1 1.1796942 1.2845423 1.3888642 1.4958840 1.6009553 1.7039047 1.8054685 1.9100296 2.0220905 2.1373696

Relative error 0 0.0031505 0.0038662 0.0054595 0.0105359 0.0131359 0.0122468 0.0080222 0.0044056 0.0055452 0.0068576

x(t, 0) 3 3.5296311 3.8420283 4.1502142 4.4560445 4.7634582 5.0749737 5.3923389 5.7168718 6.0496357 6.3915360

ˆ 0) x(t, 3 3.5248971 3.8400694 4.1471581 4.4550768 4.7609250 5.0676735 5.3854138 5.7141256 6.0467259 6.3873480

Relative error 0 0.0047340 0.0019589 0.0030561 0.0009676 0.0025332 0.0073001 0.0069251 0.0027462 0.0029097 0.0041880

Table 9. Numerical results for Example 4.5 in Case 1 (λ = 1) with ε = 0 (x(t, 0), x(t, 0)) and ˆ 0)). ε = 10−1 (ˆx(t, 0), x(t, t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x(t, 0) 1 1.1765437 1.2806761 1.3834048 1.4853482 1.5878194 1.6916579 1.7974463 1.9056240 2.0165453 2.1305120

xˆ (t, 0) 1 1.1765957 1.2807808 1.3835389 1.4855074 1.5879828 1.6918359 1.7976576 1.9058236 2.0167247 2.1306934

Relative error 0 5.21E-05 1.05E-04 1.34E-04 1.59E-04 1.63E-04 1.78E-04 2.11E-04 2.00E-04 1.79E-04 1.81E-04

x(t, 0) 3 3.5296311 3.8420283 4.1502142 4.4560445 4.7634582 5.0749737 5.3923389 5.7168718 6.0496357 6.3915360

ˆ 0) x(t, 3 3.5296836 3.8420915 4.1502928 4.4560996 4.7634792 5.0749745 5.3923011 5.7168267 6.0496131 6.3915218

Relative error 0 5.25E-05 6.31E-05 7.85E-05 5.51E-05 2.09E-05 7.95E-07 3.78E-05 4.50E-05 2.26E-05 1.42E-05

Table 10. Numerical results for Example 4.5 in Case 1 (λ = 1) with ε = 0 (x(t, 0), x(t, 0)) ˆ 0)). and ε = 10−3 (ˆx(t, 0), x(t, and 10. Case 2. Consider λ ∈ [−1, 0) and [(ii) − gH]Cα −differentiable. Using the modified AdamsBashforth-Moulton method (4.15), the approximation solutions [x(t)]0 = [x(t, 0), x(t, 0)] to ˆ 0)] to (4.20) with ε = 10−1 are shown in Figure 12 ˆ 0 = [ˆx(t, 0), x(t, (4.20) with ε = 0 and [x(t)] and Tables 11. Moreover, numerical comparisons between the approximations solutions with the different levels of random disturbance in the initial data (ε = 0, ε = 10−1 and ε = 10−3 ) are given in Tables 11 and 12. Remark 4.2. The numerical errors of our approach are in the order O(10−2 ) − O(10−6 ). Therefore, it can be concluded that the present MABMM is accurate, stable, effective, and insensitive to disturbance on initial data.

27

1 0.8

0.8

0.6 r

1

0.4

0.4

0.2

0.2

0 3.5

r

0.6

3

0 7

0.8

0.8

4 0.4

2

[x(t)]r

1

[x(t)]

0.2

r

1

0.2 0 t

0 t

Figure 12. The approximations (ii)solution of Example 4.5 with λ = −0.2.

Figure 11. The approximations (i)solution of Example 4.5 with λ = 1. x(t, 0) 1 1.1630491 1.2183239 1.2713067 1.3201438 1.3655048 1.4079825 1.4480239 1.4859674 1.5220751 1.5565549

0.4

1.5

0.6

3

0.6

2

1

5

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 2.5

6

xˆ (t, 0) 1 1.1625201 1.2186846 1.2718588 1.3184260 1.3653639 1.4069858 1.4492460 1.4920895 1.5265370 1.5590845

Relative error 0 5.29E-04 3.61E-04 5.52E-04 1.72E-03 1.41E-04 9.97E-04 1.22E-03 6.12E-03 4.46E-03 2.53E-03

x(t, 0) 3 2.9534378 2.9038226 2.8641428 2.8292984 2.7976989 2.7685325 2.7413060 2.7156879 2.6914403 2.6683842

ˆ 0) x(t, 3 2.9522989 2.9039291 2.8612752 2.8291788 2.8036438 2.7750752 2.7443403 2.7188386 2.6967142 2.6736413

Relative error 0 1.14E-03 1.07E-04 2.87E-03 1.20E-04 5.94E-03 6.54E-03 3.03E-03 3.15E-03 5.27E-03 5.26E-03

Table 11. Numerical results for Example 4.5 in Case 2 (λ = −0.2) with ε = 0 (x(t, 0), x(t, 0)) ˆ 0)). and ε = 10−1 (ˆx(t, 0), x(t,

5 Conclusions In this study, the existence and uniqueness theorems of fuzzy functional fractional differential equations have been investigated by the methods of successive approximations and the contraction principle. Results here might be used in further research on fuzzy functional fractional integro-differential equations. Other possible directions of research could be an approach for fuzzy functional fractional differential equations using concept of Riemann-Liouville type fuzzy fractional derivatives (for concept of Riemann-Liouville type fractional differentiability on fuzzy setting see [5] ). Moreover, the method of successive approximations under generalized Lipschitz conditions presented for proving fuzzy problems can be used to proving crisp differential equations, where the problem have been investigated under stronger conditions, on involving functions such as boundedness and Lipschitz continuity. In general, it is not easy to derive the analytical solutions to most of the fuzzy functional

28

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x(t, 0) 1 1.1630491 1.2183239 1.2713067 1.3201438 1.3655048 1.4079825 1.4480239 1.4859674 1.5220751 1.5565549

xˆ (t, 0) 1 1.1630747 1.2183250 1.2713097 1.3201551 1.3654837 1.4079135 1.4479214 1.4858438 1.5219629 1.5564604

Relative error 0 2.56E-05 1.16E-06 2.96E-06 1.13E-05 2.11E-05 6.90E-05 1.03E-04 1.24E-04 1.12E-04 9.45E-05

x(t, 0) 3 2.9534378 2.9038226 2.8641428 2.8292984 2.7976989 2.7685325 2.7413060 2.7156879 2.6914403 2.6683842

ˆ 0) x(t, 3 2.9534272 2.9038065 2.8640965 2.8292831 2.7976721 2.7684836 2.7412433 2.7156037 2.6913415 2.6682904

Relative error 0 1.05E-05 1.61E-05 4.63E-05 1.53E-05 2.67E-05 4.89E-05 6.28E-05 8.43E-05 9.88E-05 9.38E-05

Table 12. Numerical results for Example 4.5 in Case 2 (λ = −0.2) with ε = 0 (x(t, 0), x(t, 0)) ˆ 0)). and ε = 10−3 (ˆx(t, 0), x(t, fractional differential equations. Therefore, it is vital to develop some reliable and efficient techniques to solve fuzzy functional fractional differential equations. In this study, the modified Adams-Bashforth-Moulton method (MABMM) as a known and simpler method is preferred for solving fuzzy functional fractional differential equations. In order to compare with the exact analytical solution, some numerical examples are provided to illustrate the effectiveness of the proposed method. Moreover, we are interested in the stability of our approach when the input initial measured data are polluted by random noise for different problems. We can evaluate the stability by increasing the different levels of random noise in the initial data. We noticed that recently the semi analytical method, Adomian Decomposition Method (ADM) that is widely used for solving nonlinear delay differential equations [17] and fuzzy differential equations [13]. The topic of the ADM has been rapidly growing in recent years. The decomposition method was introduced by Adomian [1, 2] in the 1980s in order to solve linear and nonlinear functional equations (algebraic, differential, partial differential, integral, ... (see, e.g., [17, 19, 20])). Therefore, in the future studies effort is made to use the Adomian decomposition method for examining fuzzy functional fractional differential equations under the Caputo-type fuzzy fractional derivatives of order α ∈ (0, 1). Acknowledgements: The author would like to express his gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper.

References [1] Adomian G. Nonlinear stochastic systems and application to physics. Kluwer, Dordecht, 1989. [2] Adomian G. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Dordecht, 1994.

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Highlights • • • •

The existence and uniqueness theorems of FFDEs are investigated by the different methods. The modified Adams-Bashforth-Moulton method is proposed for solving fuzzy functional fractional differential equations. Some numerical examples are provided to illustrate the effectiveness of the proposed method. We are interested in the stability of our approach when the input initial measured data are polluted by random noise.