Application of a spectral method to Fractional Differential Equations under uncertainty 1

Abstract

The main result obtained in this paper is constructed the fractional Chebyshev operational matrix based on generalized shifted fractional-order Chebyshev functions of the first and second kind, is applied this operational matrix to the problem for numerically solving fuzzy fractional differential equations of order 0 < ν < 1 with fuzzy initial condition. We shown through numerical result that a new tau method is effective to the good approximate solution of Kelvin-Voiget equation, the model of viscosity behavior for non-Newtonian fluid and fuzzy fractional differential equation with variable coefficient. The numerical accuracy are compared with the results obtained by generalized fractional-order Legendre functions, Chebyshev polynomials and Jacobi polynomials.

Keywords

Fuzzy fractional differential equation fractional Chebyshev operational matrix Caputo-type fuzzy fractional derivative

1 Introduction

The applied field of fractional calculus has collected the attention of many researchers in several areas of behaviors of physical phenomena [1, 2], control engineering [3, 4], signal processing [5], electromagnetism [6]. Most of fractional differential equations(FDEs) modeling the real world do not have the exact solutions, the finding approximate and numerical solution of FDE is the important task. Also scientists have interest for the theory of the existence and uniqueness of FDE [7 –9]. The representative methods are predictor-corrector method [10], Adomian’s decomposition method [11], variational iteration method [12] and homotopy analysis methods [13]. By using operational matrix based on orthogonal functions can obtain the numerical solution of high accuracy of FDEs [14 –17]. The main advantage of this technique is that it reduces the problem for solving FDEs to those for solving a system of algebraic equations. Recently for obtaining the highly accurate solutions of FDEs with initial conditions, S. Kazem et al. [18] introduced fractional-order Legendre functions and A.H. Bhrawy et al. [19] proposed shifted fractional-order Jacobi orthogonal functions. Also E.H. Doha [20] advanced shifted Chebyshev operational matrix. Furthermore the real world has the problems with uncertainty. Those problems can be expressed by fuzzy fractional differential equations(FFDEs) with modeling of the real world phenomenon. Recently many scientists had advance the research of FFDEs. Agarwal et al. [21] presented the new concept of solution for FFDE with initial condition under the Riemann-Liouville’s differentiability. To solving FFDEs, the researchers introduced analytical method [22, 23], fuzzy variation iteration method [24], Banach fixed point theorem [25], the approximate method using linearization formula [26]. As shown in FDEs, finding exact solutions for FFDEs with initial conditions is the difficult questions. So these are the several attempts for obtained approximate or numerical solutions of FFDEs with initial conditions. These typical methods are fuzzy Laplace transforms [27], modified fractional Euler method [28], shifted Jacobi operation matrix [29], shifted Legendre operational matrix [30].

In this paper, from the viewpoint of accuracy, when we apply the generalized shifted fractional-order Chebyshev functions of first and second kind to FFDEs with initial value under Caputo’s differentiable, we presented the error bound of the approximate solution using fractional Chebyshev operational matrix and improved the accuracy of numerical solution of FFDEs.

This paper is constructed as follows. In Section 2, we are going to describe definitions and properties of fuzzy number space and fractional calculus. Some definitions and theorems of the fuzzy fractional calculus is illustrated in Section 3. In Section 4, we introduced the properties of generalized shifted fractional-order Chebyshev functions of first and second kind. Next Section 5 is discussed the application of the fractional Chebyshev operational matrix for solving fuzzy linear fractional differential equations. The numerical results and the graphs of Kelvin-Voiget equation, the model of viscosity behavior for non-Newtonian fluid and fuzzy fractional differential equation with variable coefficient by using presented method are shown in Section 6. The conclusion is given in last Section.

2 Preliminaries

We shall introduce some definitions and useful notation for fuzzy calculus and fractional calculus which used throughout this paper.

2.1 Space of fuzzy numbers and Fuzzy calculus

Let $ℝ$ be a set of real number. We denote the class of fuzzy subsets u of $ℝ$ by $𝔼$ . Now let $u : ℝ \to [0, 1]$ satisfy the following properties: (i) u is upper semi-continuous, (ii) u is fuzzy convex, (iii) u is normal, i . e ., $\exists x_{0} \in ℝ$ for which u (x₀) =1, (iv) supp $u = {x \in ℝ$ | u (x) >0} is the support of the u, and its closure cl(suppu) is compact. Then, we say that $𝔼$ is the space of fuzzy numbers. The r-level set of a fuzzy number $u \in 𝔼$ is defined as follows $[u]^{r} = {\begin{matrix} {x \in ℝ | u (x) \geq r} & if 0< r≤ 1, \\ cl (supp u) & if r=0 . \end{matrix}$ (1)

We can understand easily that the r-level set of a fuzzy number is a closed and bounded interval [ $u_{-}^{r}$ , $u_{+}^{r}$ ] on $ℝ$ , where $u_{-}^{r}$ denotes the left-hand endpoint of [u] ^r and $u_{+}^{r}$ denotes the right-hand endpoint of [u] ^r. The addition and scaler multiplication of fuzzy number in $𝔼$ are defined as $[u \oplus v]^{r} = [u_{-}^{r} + v_{-}^{r}, u_{+}^{r} + v_{+}^{r}],$ $[λ ⊙ u]^{r} = [λ u_{-}^{r}, λ u_{+}^{r}], λ \geq 0$ . The Hausdorff distance D between fuzzy numbers on $𝔼$ ( $D : 𝔼 \times 𝔼 \to ℝ_{+} \cup 0)$ are defined as follows: $D (u, v) = sup_{r \in [0, 1]} Hausdorff distance ([u]^{r}, [v]^{r}) .$ (2)

Definition 1. [31] Let f and g be the two fuzzy number valued functions on the interval [a, b], i . e ., $f, g : [a, b] \to 𝔼$ . The uniform distance between fuzzy number valued functions is defined by $D^{*} (f, g) = sup_{x \in [a, b]} D (f (x), g (x)) .$ (3)

Definition 2. [32] Let $x, y \in 𝔼$ . If there exists $z \in 𝔼$ such that x = y ⊕ z, then z is called the H-difference of x and y, and it is denoted by x ⊖ y. In this paper, the sign "⊖" means Hukuhara-difference (H-difference). Also we assume that the H-difference and generalized Hukuhara difference (gH-difference) [33] exist in throughout of paper.

Definition 3. [34, 35] Let $f : (a, b) \to 𝔼$ and x₀ ∈ (a, b). We say that f is strongly generalized differential at x₀, if there exists an element $f^{'} (x_{0}) \in 𝔼$ , such that (i) ∀h > 0 sufficiently small, ∃f (x₀ + h) ⊖ f (x₀), ∃f (x₀) ⊖ f (x₀ - h) and the limits (in the metric D) $\begin{matrix} f^{'} (x_{0}) & = lim_{h \to 0^{+}} \frac{f (x_{0} + h) ⊖ f (x_{0})}{h} \\ = lim_{h \to 0^{+}} \frac{f (x_{0}) ⊖ f (x_{0} - h)}{h}, \end{matrix}$ (4) (ii) ∀h > 0 sufficiently small, ∃f (x₀) ⊖ f (x₀ + h), ∃f (x₀ - h) ⊖ f (x₀) and the limits (in the metric D) $\begin{matrix} f^{'} (x_{0}) & = lim_{h \to 0^{+}} \frac{f (x_{0}) ⊖ f (x_{0} + h)}{- h} \\ = lim_{h \to 0^{+}} \frac{f (x_{0} - h) ⊖ f (x_{0})}{- h} . \end{matrix}$ (5)

Remark 1: f is so-called (1)-differentiable on (a, b), if f is differentiable in the sense (i) of Definition 3 and also f is (2)-differentiable on (a, b), if f is differentiable in the sense (ii) of Definition 3.

2.2 Fractional calculus

Definition 4. [36] The Riemann-Liouville fractional integral operator of order ν (ν ≥ 0) is defined as $(I^{ν} f) (x) = \frac{1}{Γ (ν)} \int_{0}^{x} \frac{f (t)}{(x - t)^{1 - ν}} dt, ν > 0, x > 0,$ (6)

Definition 5. [36] The Caputo fractional derivatives of order ν is defined as $c^{D^{ν}} f (x) = \frac{1}{Γ (m - ν)} \int_{0}^{x} (x - t)^{m - ν - 1} f^{m} (t) dt,$ (7) where D^m is the classical differential operator of order m and m - 1 < ν ≤ m, x > 0 .

For the Caputo derivative, we have: $\begin{matrix} c^{D^{ν}} C = 0 (C is a constant), \\ c^{D^{ν}} x^{β} = {\begin{matrix} 0, & for β \in ℕ_{0}, β < ⌈ ν ⌉, \\ \frac{Γ (β + 1) x^{β - ν}}{Γ (β + 1 - ν)}, & \begin{matrix} for β \in ℕ_{0}, β \geq ⌈ ν ⌉ \\ or β \notin ℕ, β > ⌊ ν ⌋ . \end{matrix} \end{matrix} \end{matrix}$ (8) Here $ℕ = {1, 2, \dots}$ and $ℕ_{0} = {0, 1, 2, \dots}$ .

Theorem 1. [36] Similar to the differential equation of integer order, the Caputo’s fractional differentiation is a linear operator, i . e ., $c^{D^{ν}} (λ f (x) + μ g (x)) = λ c^{D^{ν}} f (x) + μ c^{D^{ν}} g (x),$ (9)where λ and μ are constants.

3 Fuzzy fractional Caputo’s derivative

At first, we introduce some notation for understand this paper $• L_{p} (I, 𝔼)$ , 1≤ p < ∞ is the set of all fuzzy-valued measurable functions f on I where the norm is defined by $∥ f ∥_{p} = (\int_{0}^{L} (d (f (t), 0))^{p} dt)^{\frac{1}{p}}$ $• C (I, 𝔼)$ is a space of fuzzy-valued functions, which are continuous on I, where I = [0, L] and L is the constant.

Definition 6. [27] Let $f \in C (I, 𝔼) ⋂ L_{1} (I, 𝔼)$ . The fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as follow $(I_{0 +}^{ν} f) (x) = \frac{1}{Γ (ν)} \int_{0}^{x} (x - t)^{ν - 1} f (t) dt, x > 0,$ (10) where 0 < ν ≤ 1 .

Definition 7. [1] Let $f \in C (I, 𝔼) ⋂ L_{1} (I, 𝔼)$ and x₀ ∈ I and $Φ (x) = \frac{1}{Γ (1 - ν)} \int_{0}^{x} \frac{f (t) ⊖ \sum_{k = 0}^{1} \frac{t^{k}}{k!} f_{0}^{(k)}}{(x - t)^{ν}} dt$ , where $f_{0}^{(k)} = \frac{d^{k} f (x)}{{dx}^{k}} |_{x = 0}$ . We say that f (x) is fuzzy Caputo fractional differentiable of order 0 < ν ≤ 1 at x₀, if there exists an element $(c^{D_{0 +}^{ν}} f) (x_{0}) \in C (I, 𝔼)$ such that for h > 0 sufficiently near zero on either; $\begin{matrix} (1) (c^{D_{a +}^{ν}} f) (x_{0}) & = lim_{h \to 0^{+}} \frac{Φ (x_{0} + h) ⊖ Φ (x_{0})}{h} \\ = lim_{h \to 0^{+}} \frac{Φ (x_{0}) ⊖ Φ (x_{0} - h)}{h}, \end{matrix}$ (11) or $\begin{matrix} (2) (c^{D_{a +}^{ν}} f) (x_{0}) & = lim_{h \to 0^{+}} \frac{Φ (x_{0}) ⊖ Φ (x_{0} + h)}{- h} \\ = lim_{h \to 0^{+}} \frac{Φ (x_{0} - h) ⊖ Φ (x_{0})}{- h} . \end{matrix}$ (12)Definition 8. [37] Let $f \in C (I, 𝔼) ⋂ L_{1} (I, 𝔼)$ is a fuzzy-valued function, then f is said to be Caputo’s fuzzy differentiable at x when $(c^{D_{0 +}^{ν}} f) (x) = \frac{1}{Γ (1 - ν)} \int_{0}^{x} (x - t)^{- ν} f^{'} (t) dt,$ (13) where 0 < ν ≤ 1 .

Remark 2: We say that a fuzzy-valued function f is ^c [(1) - ν]-differentiable if it is differentiable as in the Definition 7 (1) and f is ^c [(2) - ν]-differentiable if it is differentiable as in the Definition 7 (2).

Theorem 2. [28] Let $f (x) \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ be a fuzzy valued function and $[f (x)]^{r} = [f_{-}^{r} (x), f_{+}^{r} (x)]$ for r ∈ [0, 1] and x₀ ∈ (0, L], then 1) If f (x) is fuzzy Caputo-type fractional differentiable function in the first form, then for 0 < ν ≤ 1 $[c^{D^{ν}} f (x_{0})]^{r} = [c^{D^{ν}} f_{-}^{r} (x_{0}), c^{D^{ν}} f_{+}^{r} (x_{0})],$ (14) 2) If f (x) is fuzzy Caputo-type fractional differentiable function in the second form, then for 0 < ν ≤ 1 $[c^{D^{ν}} f (x_{0})]^{r} = [c^{D^{ν}} f_{+}^{r} (x_{0}), c^{D^{ν}} f_{-}^{r} (x_{0})],$ (15)where $c^{D^{ν}} f_{-}^{r} (x_{0}) = [\frac{1}{Γ (1 - ν)} \int_{0}^{x} \frac{{Df}_{-}^{r} (t)}{(x - t)^{ν}} dt]_{x = x_{0}},$ (16) $c^{D^{ν}} f_{+}^{r} (x_{0}) = [\frac{1}{Γ (1 - ν)} \int_{0}^{x} \frac{{Df}_{+}^{r} (t)}{(x - t)^{ν}} dt]_{x = x_{0}} .$ (17)

4 Generalized shifted fractional-order Chebyshev functions of first and second kind

Shifted fractional-order Chebyshev functions of first and second kind is a special case of shifted fractional-order Jacobi functions. They have been used widely in mathematical analysis and practical applications. When y (x) is differential, the advantages of using fractional-order Chebyshev functions is the good representation of smooth function by finite Chebyshev expansion. In this Section, we will introduce generalized shifted fractional-order Chebyshev functions of first and second kind in order to complete our aim. In this paper we consider 0 < λ < 1. If λ = 1, shifted fractional-order Chebyshev functions is equivalent to shifted Chebyshev polynomials.

4.1 Generalized shifted fractional-order Chebyshev functions of first and second kind

4.1.1 Generalized shifted fractional-order Chebyshev functions of first kind $T_{n}^{(λ)} (x)$

Shifted fractional-order Chebyshev functions of first kind $T_{n}^{(λ)} (x)$ (0 < λ < 1) are the eigenfunctions of the Sturm-Liouville differential equation $\begin{matrix} (1 - t^{λ}) t^{- λ + 2} v^{″} (t) & + t (- \frac{λ}{2} + (1 - \frac{λ}{2}) (1 - t^{λ}) t^{- λ}) \cdot \\ \cdot v^{'} (t) + λ^{2} n^{2} v (t) = 0, t \in [0, 1] . \end{matrix}$ (18) They satisfy the following recurrence formula on [0, 1] $T_{n + 1}^{(λ)} (t) = 2 (2 t^{λ} - 1) T_{n}^{(λ)} (t) - T_{n - 1}^{(λ)} (t), n = 1, 2, \dots,$ (19) Then the analytic expression of shifted fractional-order Chebyshev functions of first kind $T_{n}^{(λ)} (x)$ of degree nλ is as follow $\begin{matrix} T_{n}^{(λ)} (t) = & \sum_{i = 0}^{n} E_{1, i}^{(n)} t^{i λ}, t \in [0, 1], \\ E_{1, i}^{(n)} = & \frac{n (- 1)^{n - i} (n + i - 1)! 2^{2 i}}{(n - i)! (2 i)!}, n = 0, 1, 2, \dots . \end{matrix}$ (20) Clearly, we have $T_{n}^{(λ)} (0) = (- 1)^{n}, T_{n}^{(λ)} (1) = 1 .$ (21)

In order to expand the shifted fractional-order Chebyshev functions $T_{n}^{(λ)} (t)$ of first kind defined on [0, 1] to interval [0, L], we define generalized shifted fractional-order Chebyshev functions of first kind (GFCF1) $T_{L, n}^{(λ)} (x)$ by using change of variable x = Lt. Then generalized shifted fractional-order Chebyshev functions of first kind $T_{L, n}^{(λ)} (x)$ are satisfied following recurrence formula. $\begin{matrix} T_{L, n + 1}^{(λ)} (x) = 2 (2 (\frac{x}{L})^{λ} - 1) & T_{L, n}^{(λ)} (x) - T_{L, n - 1}^{(λ)} (x), \\ n = 1, 2, \dots, \end{matrix}$ (22) where $T_{L, 0}^{(λ)} (x) = 1$ , $T_{L, 1}^{(λ)} (x) = 2 (\frac{x}{L})^{λ} - 1$ . Also the analytic expression of generalized shifted fractional-order Chebyshev functions of first kind $T_{L, n}^{(λ)} (x)$ of degree nλ is presented as follow $T_{L, n}^{(λ)} (x) = \sum_{i = 0}^{n} E_{1, i}^{(n)} \frac{x^{i λ}}{L^{i λ}}, n = 0, 1, 2, \dots .$ (23) The orthogonality property of generalized shifted fractional-order Chebyshev functions of first kind with respect to the weight function $w_{L, 1}^{(λ)} (x) = λ x^{\frac{λ}{2} - 1} \cdot$ $\cdot (1 - (\frac{x}{L})^{λ})^{- \frac{1}{2}}$ on interval [0, L] can be taken as follows $\int_{0}^{L} T_{L, i}^{(λ)} (x) T_{L, j}^{(λ)} (x) w_{L, 1}^{(λ)} (x) dx = L^{\frac{λ}{2}} h_{1, j} δ_{ij},$ (24) where δ_ij is the Kronecker function and $h_{1, j} = {\begin{matrix} π, & j = 0, \\ \frac{π}{2}, & j = 1, 2, \dots . \end{matrix}$ (25) Let u (x) ∈ L₂ [0, L]. Then the expansion of u (x) with respect to the generalized shifted fractional-order Chebyshev functions of first kind are presented as follows $u (x) = \sum_{i = 0}^{\infty} u_{i} T_{L, i}^{(λ)} (x),$ (26) where u_i is $u_{i} = \frac{1}{L^{\frac{λ}{2}} h_{1, i}} \int_{0}^{L} u (x) T_{L, i}^{(λ)} (x) w_{L, 1}^{(λ)} (x) dx .$ (27) In fact, only the first N terms of generalized shifted fractional-order Chebyshev functions of first kind in Eq. (eq6) are considered. Then we take $u (x) \approx u_{N} = \sum_{i = 0}^{N - 1} u_{i} T_{L, i}^{(λ)} (x) .$ (28)

4.1.2 Generalized shifted fractional-order Chebyshev functions of second kind

U_{n}^{(λ)} (x)

Shifted fractional-order Chebyshev functions of second kind $U_{n}^{(λ)} (t)$ (0 < λ < 1) are similar to shifted fractional-order Chebyshev functions of first kind $T_{n}^{(λ)} (t)$ and are the eigenfunctions of the Sturm-Liouville differential equation as following $\begin{matrix} (1 - & t^{λ}) t^{- λ + 2} v^{″} (t) + t (- \frac{3}{2} λ + (1 + \frac{λ}{2}) (1 - t^{λ}) \cdot \\ \cdot t^{- λ}) v^{'} (t) + λ^{2} n (n + 2) v (t) = 0, t \in [0, 1] . \end{matrix}$ (29) Also the recurrence relations that $U_{n}^{(λ)} (t)$ satisfy on [0, 1] is given by $\begin{matrix} U_{n + 1}^{(λ)} (t) = 2 (2 t^{λ} - 1) U_{n}^{(λ)} (t) - U_{n - 1}^{(λ)} (t), \\ n = 1, 2, \dots, \end{matrix}$ (30) where $U_{0}^{(λ)} (t) = 1$ , $U_{1}^{(λ)} (t) = 2 (2 t^{λ} - 1)$ . Also the analytic expression of shifted fractional-order Chebyshev functions of second kind $U_{n}^{(λ)} (t)$ of degree nλ can be expressed $\begin{matrix} U_{n}^{(λ)} (t) = & \sum_{i = 0}^{n} E_{2, i}^{(n)} t^{i λ}, t \in [0, 1], \\ E_{2, i}^{(n)} = & \frac{(- 1)^{n - i} (n + i + 1)! 2^{2 i}}{(n - i)! (2 i + 1)!}, n = 0, 1, 2, \dots . \end{matrix}$ (31) Then we can know $U_{n}^{(λ)} (0) = (- 1)^{n} (n + 1), U_{n}^{(λ)} (1) = (n + 1) .$ (32) Similar to above first kind method, we define generalized shifted fractional-order Chebyshev functions of second kind (GFCF2) $U_{L, n}^{(λ)} (x)$ on interval [0, L] by using change of variable x = Lt. Also the recurrence formula of generalized shifted fractional-order Chebyshev functions of second kind $U_{L, n}^{(λ)} (x)$ are as follows. $\begin{matrix} U_{L, n + 1}^{(λ)} (x) = 2 (2 (\frac{x}{L})^{λ} - 1) & U_{L, n}^{(λ)} (x) - U_{L, n - 1}^{(λ)} (x), \\ n = 1, 2, \dots, \end{matrix}$ (33) where $U_{L, 0}^{(λ)} (x) = 1$ , $U_{L, 1}^{(λ)} (x) = 2 (2 (\frac{x}{L})^{λ} - 1)$ . Also the analytic expression of generalized shifted fractional-order Chebyshev functions of second kind $U_{L, n}^{(λ)} (x)$ of degree nλ is as follow $U_{L, n}^{(λ)} (x) = \sum_{i = 0}^{n} E_{2, i}^{(n)} \frac{x^{i λ}}{L^{i λ}}, n = 0, 1, 2, \dots .$ (34) The orthogonality property of generalized shifted fractional-order Chebyshev functions of second kind with respect to the weight function $w_{L, 2}^{(λ)} (x) = λ x^{\frac{3}{2} λ - 1} \cdot$ $\cdot (1 - (\frac{x}{L})^{λ})^{\frac{1}{2}}$ on interval [0, L] can be taken as follows $\int_{0}^{L} U_{L, i}^{(λ)} (x) U_{L, j}^{(λ)} (x) w_{L, 2}^{(λ)} (x) dx = L^{\frac{3}{2} λ} h_{2, j} δ_{ij},$ (35) where δ_ij is the Kronecker function and $h_{2, j} = \frac{π}{8}, j = 0, 1, 2, \dots .$ (36) The expansion of u (x) ∈ L₂ [0, L] with respect to the generalized shifted fractional-order Chebyshev functions of second kind are as follows $u (x) = \sum_{i = 0}^{\infty} u_{i} U_{L, i}^{(λ)} (x),$ (37) where u_i is $u_{i} = \frac{1}{L^{\frac{3}{2} λ} h_{2, i}} \int_{0}^{L} u (x) U_{L, i}^{(λ)} (x) w_{L, 2}^{(λ)} (x) dx .$ (38) Actually, we consider the first N terms generalized shifted fractional-order Chebyshev functions of second kind only in Eq. (eq9). $u (x) \approx u_{N} = \sum_{i = 0}^{N - 1} u_{i} U_{L, i}^{(λ)} (x) .$ (39)

4.2 Fractional Chebyshev Operational matrix of Caputo’s derivative of order ν

Here we introduce fractional Chebyshev operational matrix of first and second kind of Caputo’s derivative of order ν. The following theorem is Generalized Taylors formula.

Theorem 3. (Generalized Taylors formula) Suppose that D^iλu (x) ∈ C [0, L] for i = 0, 1, 2, ⋯ , N. Then we have $u (x) = \sum_{i = 0}^{N - 1} \frac{x^{i λ} \cdot D^{i λ} u (0^{+})}{Γ (i λ + 1)} + \frac{x^{N λ}}{Γ (N λ + 1)} D^{N λ} u (ξ),$ (40)where 0 < ξ ≤ x, ∀x ∈ [0, L]. Also one has $| u (x) - \sum_{i = 0}^{N - 1} \frac{x^{i λ}}{Γ (i λ + 1)} D^{i λ} u (0^{+}) | \leq M_{λ} \frac{x^{N λ}}{Γ (N λ + 1)},$ (41)where M_λ ≥ |D^Nλu (ξ) |. In case of λ = 1, the generalized Taylor’s formula reduces to the classical Taylors formula.

Theorem 4. Suppose that D^kλu (x) ∈ C [0, 1] for k = 0, 1, ⋯ , N. 1) If u_N (x) is the best approximation to u (x) from $F_{1, N} = Span {T_{L, n}^{(λ)} : 0 \leq n \leq N}$ , then the error bound is presented as follows $\begin{matrix} {∥ u (x) - u_{N} (x) ∥}_{w_{L, 1}^{(λ)}} \\ \leq \frac{M_{1, λ}}{Γ (N λ + 1)} \sqrt{\frac{Γ (\frac{1}{2}) Γ (2 N + \frac{1}{2}) L^{2 N λ + \frac{1}{2} λ}}{Γ (2 N + 1)}}, \end{matrix}$ (42)where M_1,λ ≥ |D^Nλu (x) |, x ∈ [0, L]. 2) If u_N (x) is the best approximation to u (x) from $F_{2, N} = Span {U_{L, n}^{(λ)} : 0 \leq n \leq N}$ , then the error bound is presented as follows $\begin{matrix} {u (x) - u_{N} (x)}_{w_{L, 2}^{(λ)}} \\ \leq \frac{M_{2, λ}}{Γ (N λ + 1)} \sqrt{\frac{Γ (\frac{3}{2}) Γ (2 N + \frac{3}{2}) L^{2 N λ + \frac{3}{2} λ}}{Γ (2 N + 3)}}, \end{matrix}$ (43)where M_2,λ ≥ |D^Nλu (x) |, x ∈ [0, L].

Proof. Now, we consider generalized shifted fractional-order Chebyshev functions of first kind. Let u_N (x) is the best approximation to u (x) from $F_{1, N}$ . Then by using the definition of the best approximation, we obtain for $\forall v_{N} (x) \in F_{1, N}$ ${∥ u (x) - u_{N} (x) ∥}_{w_{L, 1}^{(λ)}} \leq {∥ u (x) - v_{N} (x) ∥}_{w_{L, 1}^{(λ)}} .$ (44) Considering the Generalized Taylors formula, then we have $| u (x) - \sum_{i = 0}^{N - 1} \frac{x^{i λ}}{Γ (i λ + 1)} D^{i λ} u (0^{+}) | \leq M_{1, λ} \frac{x^{N λ}}{Γ (N λ + 1)} .$ (45) Since $\sum_{i = 0}^{N - 1} \frac{x^{i λ}}{Γ (i λ + 1)} D^{i λ} u (0^{+}) \in F_{1, N}$ , we can obtain as following $\begin{matrix} {∥ u (x) - u_{N} (x) ∥}_{w_{L, 1}^{(λ)}}^{2} \\ \leq ∥ u (x) - \sum_{i = 0}^{N - 1} \frac{x^{i λ}}{Γ (i λ + 1)} D^{i λ} u (0^{+}) ∥_{w_{L, 1}^{(λ)}}^{2} \\ \leq \frac{M_{1, λ}^{2}}{Γ (N λ + 1)^{2}} \int_{0}^{L} x^{2 N λ} w_{L, 1}^{(λ)} dx \\ \leq \frac{M_{1, λ}^{2}}{Γ (N λ + 1)^{2}} \frac{Γ (\frac{1}{2}) Γ (2 N + \frac{1}{2}) L^{2 N λ + \frac{1}{2} λ}}{Γ (2 N + 1)} . \end{matrix}$ (46) For generalized shifted fractional-order Chebyshev functions of second kind, we can apply similar method. By taking the square roots for above equation, the proof of the Theorem is completed. □

Now we discuss the fractional Chebyshev operational matrix based on generalized shifted fractional-order Chebyshev functions of first and second kind. If $Φ^{T} (x) = [T_{L, 0}^{(λ)} (x), T_{L, 1}^{(λ)} (x), \dots, T_{L, N - 1}^{(λ)} (x)]$ or $Φ^{T} (x) = [U_{L, 0}^{(λ)} (x), U_{L, 1}^{(λ)} (x), \dots, U_{L, N - 1}^{(λ)} (x)],$ then we can present D^νΦ (x) ≈ D^(ν)Φ (x).

Theorem 5. If D^(ν) is N × N the fractional Chebyshev operational matrix based on generalized shifted fractional-order Chebyshev functions of first or second kind, then elements ${d_{ij}}_{i, j = 0}^{N - 1}$ of D^(ν) are expressed as 1) in case of generalized shifted fractional-order Chebyshev functions of first kind $\begin{matrix} d_{ij} = \frac{L^{- ν}}{h_{1, j}} \cdot \sum_{k = 0}^{i} \cdot \sum_{s = 0}^{j} \\ E_{1, k}^{(i)^{'}} E_{1, s}^{(j)} \frac{Γ (\frac{1}{2}) Γ (k λ + 1) Γ (k + s + \frac{1}{2} - \frac{ν}{λ})}{Γ (k λ + 1 - ν) Γ (k + s + 1 - \frac{ν}{λ})}, \end{matrix}$ (47) 2) in case of generalized shifted fractional-order Chebyshev functions of second kind $\begin{matrix} d_{ij} = \frac{L^{- ν}}{h_{2, j}} \cdot \sum_{k = 0}^{i} \cdot \sum_{s = 0}^{j} \\ E_{2, k}^{(i)^{'}} E_{2, s}^{(j)} \frac{Γ (\frac{3}{2}) Γ (k λ + 1) Γ (k + s + \frac{3}{2} - \frac{ν}{λ})}{Γ (k λ + 1 - ν) Γ (k + s + 3 - \frac{ν}{λ})}, \end{matrix}$ (48)where $E_{j, k}^{(i)^{'}} = {\begin{matrix} 0, for k λ \in ℕ_{0} and k λ < ⌈ ν ⌉, \\ E_{j, k}^{(i)}, \begin{matrix} for k λ \in ℕ_{0} and k λ \geq ⌈ ν ⌉ \\ or k λ \notin ℕ and k λ > ⌊ ν ⌋, \end{matrix} \end{matrix}$ (49)j = 1, 2 .

Proof. First, we consider the generalized shifted fractional-order Chebyshev functions of first kind. By the orthogonality property of generalized shifted fractional-order Chebyshev functions of first kind and the property of Caputo’s derivative Eq. eq2, $\begin{matrix} D^{ν} T_{L, i}^{(λ)} & (x) = \sum_{k = 0}^{i} E_{1, k}^{(i)} D^{ν} \frac{x^{k λ}}{L^{k λ}} \\ = \sum_{k = 0}^{i} \frac{E_{1, k}^{(i)^{'}}}{L^{k λ}} \frac{Γ (k λ + 1)}{Γ (k λ + 1 - ν)} \cdot x^{k λ - ν}, \end{matrix}$ (50) where $E_{1, k}^{(i)^{'}}$ is obtained by Eq. eq14. Now, let $x^{k λ - ν} = \sum_{j = 0}^{N - 1} b_{j} T_{L, j}^{(λ)} (x) .$ (51) Then the coefficients b_j are as following $\begin{matrix} b_{j} = \frac{1}{L^{\frac{λ}{2}} h_{1, j}} \int_{0}^{L} x^{k λ - ν} \cdot T_{L, j}^{(λ)} (x) w_{L, 1}^{(λ)} (x) dx \\ = \frac{L^{k λ - ν}}{h_{1, j}} \sum_{s = 0}^{j} E_{1, s}^{(j)} \frac{Γ (\frac{1}{2}) Γ (k + s + \frac{1}{2} - \frac{ν}{λ})}{Γ (k + s + 1 - \frac{ν}{λ})} . \end{matrix}$ (52) By Eqs. eq15, eq16, eq17, $\begin{matrix} D^{ν} T_{L, i}^{(λ)} & (x) = \sum_{j = 0}^{N - 1} \frac{L^{- ν}}{h_{1, j}} \sum_{k = 0}^{i} \sum_{s = 0}^{j} E_{1, k}^{(i)^{'}} E_{1, s}^{(j)} \cdot \\ \cdot \frac{Γ (\frac{1}{2}) Γ (k λ + 1) Γ (k + s + \frac{1}{2} - \frac{ν}{λ})}{Γ (k λ + 1 - ν) Γ (k + s + 1 - \frac{ν}{λ})} T_{L, j}^{(λ)} (x) . \end{matrix}$ (53)

Similarly, we can verify the result for the generalized shifted fractional-order Chebyshev functions of second kind $\begin{matrix} D^{ν} U_{L, i}^{(λ)} & (x) = \sum_{j = 0}^{N - 1} \frac{L^{- ν}}{h_{2, j}} \sum_{k = 0}^{i} \sum_{s = 0}^{j} E_{2, k}^{(i)^{'}} E_{2, s}^{(j)} \cdot \\ \cdot \frac{Γ (\frac{3}{2}) Γ (k λ + 1) Γ (k + s + \frac{3}{2} - \frac{ν}{λ})}{Γ (k λ + 1 - ν) Γ (k + s + 3 - \frac{ν}{λ})} U_{L, j}^{(λ)} (x), \end{matrix}$ (54) in which $E_{2, k}^{(i)^{'}}$ is presented Eq. eq14. □

5 Application of the fractional Chebyshev operational matrix for the fuzzy-value function

5.1 Approximation of the fuzzy-valued function

The fuzzy valued function $y \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ are presented as following form by generalized shifted fractional-order Chebyshev functions of first kind based on the interval I = [0, L]. (see [41]) $y (x) = \sum_{i = 0}^{+ \infty} *^{θ_{i}} ⊙ T_{L, i}^{(λ)} (x), x \in I, 0 < λ < 1,$ (55) where fuzzy coefficient θ_i (i = 0, 1, ⋯) is given $θ_{i} = \frac{1}{L^{\frac{λ}{2}} h_{1, i}} \int_{0}^{L} y (x) ⊙ T_{L, i}^{(λ)} (x) ⊙ w_{L, 1}^{(λ)} (x) dx,$ (56) in which ∑^* means the addition ⊕ in $𝔼$ and $w_{L, 1}^{(λ)} (x) = λ ⊙ x^{\frac{λ}{2} - 1} ⊙ (1 - (\frac{x}{L})^{λ})^{- \frac{1}{2}}$ , $T_{L, i}^{(λ)} (x)$ is the generalized shifted fractional-order Chebyshev functions of first kind. In fact, only the first N-terms of generalized shifted fractional-order Chebyshev functions of first kind are considered in above Eq. eq18. Then we obtain $y (x) ≃ y_{N} (x) = \sum_{i = 0}^{N - 1} *^{θ_{i}} ⊙ T_{L, i}^{(λ)} (x) = Θ^{T} ⊙ Φ_{1} (x),$ (57) where Θ ^T = [θ₀, θ₁, ⋯ , θ_N-1] and $Φ_{1}^{T} (x) = [T_{L, 0}^{(λ)} (x),$ $T_{L, 1}^{(λ)} (x), \dots, T_{L, N - 1}^{(λ)} (x)]$ .

Definition 9. Let $y \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ . The parametric form of approximation of fuzzy valued function y (x) , 0 ≤ r ≤ 1 is $\begin{matrix} [y (x)]^{r} ≃ [y_{N} (x)]^{r} \\ = [\sum_{i = 0}^{N - 1} θ_{i, -}^{r} T_{L, i}^{(λ)} (x), \sum_{i = 0}^{N - 1} θ_{i, +}^{r} T_{L, i}^{(λ)} (x)], \end{matrix}$ (58) The following Theorem 6 supplies an upper error bound of the fuzzy approximation function based on the generalized shifted fractional-order Chebyshev functions of first and second kind. By this theorem, we can show that the fuzzy approximate function converges to the original function.

Theorem 6. Let $y \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ . Also $D^{k λ} y (x) \in C (I, 𝔼)$ , k = 0, 1, ⋯ , N - 1. If y_N (x) is the best approximation of fuzzy function y (x) using generalized shifted fractional-order Chebyshev functions of first or second kind, then the error bound is presented as follows 1) In case of generalized shifted fractional-order Chebyshev functions of first kind $\begin{matrix} D^{*} ( & y (x), y_{N} (x)) \\ \leq \frac{M_{1, λ}^{r}}{Γ (N λ + 1)} \sqrt{\frac{Γ (\frac{1}{2}) Γ (2 N + \frac{1}{2}) L^{2 N λ + \frac{1}{2} λ}}{Γ (2 N + 1)}} . \end{matrix}$ (59) 2) In case of generalized shifted fractional-order Chebyshev functions of second kind $\begin{matrix} D^{*} ( & y (x), y_{N} (x)) \\ \leq \frac{M_{2, λ}^{r}}{Γ (N λ + 1)} \sqrt{\frac{Γ (\frac{3}{2}) Γ (2 N + \frac{3}{2}) L^{2 N λ + \frac{3}{2} λ}}{Γ (2 N + 3)}} . \end{matrix}$ (60)where $M_{i, λ}^{r} = \max {M_{i, λ}^{r_{-}}, M_{i, λ}^{r_{+}}}$ , i = 1, 2. Proof. Now we consider generalized shifted fractional-order Chebyshev functions of first kind. By the definition of D^*, $D^{*} (y (x), y_{N} (x)) = sup_{x \in [0, L]} D (y (x), y_{N} (x))$ (61) $\begin{matrix} = sup_{x \in [0, L]} sup_{r \in [0, 1]} \max {| y_{-}^{r} (x) - & y_{N, -}^{r} (x) | \\ , | y_{+}^{r} (x) - y_{N, +}^{r} (x) |} \end{matrix}$ (62) $\begin{matrix} = sup_{r \in [0, 1]} \max {∥ y_{-}^{r} (x) - & y_{N, -}^{r} (x) ∥_{\infty} \\ , ∥ y_{+}^{r} (x) - y_{N, +}^{r} (x) ∥_{\infty}} \end{matrix}$ (63) $\begin{matrix} \leq sup_{r \in [0, 1]} & \max {\frac{D^{N λ} y_{-}^{r} (x)}{Γ (N λ + 1)}, \frac{D^{N λ} y_{+}^{r} (x)}{Γ (N λ + 1)}} \cdot \\ \cdot \sqrt{\frac{Γ (\frac{1}{2}) Γ (2 N + \frac{1}{2}) L^{2 N λ + \frac{1}{2} λ}}{Γ (2 N + 1)}} \end{matrix}$ (64)

$\leq \frac{M_{1, λ}^{r}}{Γ (N λ + 1)} \sqrt{\frac{Γ (\frac{1}{2}) Γ (2 N + \frac{1}{2}) L^{2 N λ + \frac{1}{2} λ}}{Γ (2 N + 1)}} .$ (65) Similarly, we can proof the result for the generalized shifted fractional-order Chebyshev functions of second kind. □

5.2 Application for the fuzzy linear fractional differential equation with the variable coefficients

Consider the following fuzzy linear fractional differential equation with the variable coefficients ${\begin{matrix} (c^{D_{0^{+}}^{ν}} y) (x) = a (x) ⊙ y (x) \oplus f (x), \\ y (0) = y_{0} \in 𝔼, 0 < ν \leq 1, \end{matrix}$ (66) where $y \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ , $c^{D_{0^{+}}^{ν}}$ denotes the fuzzy Caputo fractional derivative of order ν and $f (x) : [0, L] \to 𝔼$ , $a (x) : [0, L] \to ℝ$ . The following theorem have proved the fact that the above FFDEs can be equivalent with an parameter form equations. Indeed the this results have been presented and proved in many papers ([1, 28] and so on).

Theorem 7. Let $y \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ and 0 < ν ≤ 1. Also we assume that $a \in C (I, ℝ)$ . 1) In case of a (x) >0, x ∈ [0, L], let y (x) is ^c [(1) - ν]-differentiability. Then Eq. eq21 is equivalent to following parameter form equations ${\begin{matrix} c^{D_{0^{+}}^{ν}} y_{-}^{r} (x) = a (x) \cdot y_{-}^{r} (x) + f (x), \\ c^{D_{0^{+}}^{ν}} y_{+}^{r} (x) = a (x) \cdot y_{+}^{r} (x) + f (x), \\ y^{r} (0) = [y_{0, -}^{r}, y_{0, +}^{r}], 0 < ν \leq 1 . \end{matrix}$ (67) 2) In case of a (x) <0, x ∈ [0, L], let y (x) is ^c [(2) - ν]-differentiability. Then Eq. eq21 is equivalent to following parameter form equations ${\begin{matrix} c^{D_{0^{+}}^{ν}} y_{+}^{r} (x) = a (x) \cdot y_{+}^{r} (x) + f (x), \\ c^{D_{0^{+}}^{ν}} y_{-}^{r} (x) = a (x) \cdot y_{-}^{r} (x) + f (x), \\ y^{r} (0) = [y_{0, -}^{r}, y_{0, +}^{r}], 0 < ν \leq 1 . \end{matrix}$ (68)Proof. We first consider that y (x) is ^c [(1) - ν]-differentiability in case of a (x) >0, x ∈ [0, L]. Then by using Theorem 2 and Theorem 4.2 in [1], we have $(c^{D_{0^{+}}^{ν}} y)^{r} (x) = [c^{D_{0^{+}}^{ν}} y_{-}^{r} (x), c^{D_{0^{+}}^{ν}} y_{+}^{r} (x)],$ (69) Also by assumption of a (x), The interval family $[a (x) \cdot y_{-}^{r} (x), a (x) \cdot y_{-}^{r} (x)]$ generates a fuzzy number a (x) ⊙ y (x) such that $[a (x) ⊙ y (x)]^{r} = [a (x) \cdot y_{-}^{r} (x), a (x) \cdot y_{-}^{r} (x)] .$ (70) Therefore we can show that FFDEs (ref eq21) is equivalent to parameter form equations (ref eq:22). For the Eq. eq:23, the proof method is almost similar. □ Now $\begin{matrix} [y (x)]^{r} ≃ [y_{N} (x)]^{r} = & [\sum_{i = 0}^{N - 1} *^{θ_{i}} ⊙ T_{L, i}^{(λ)} (x)]^{r} \\ = Θ^{T} (r) ⊙ Φ_{1} (x), \end{matrix}$ (71) where Θ ^T (r) = [ Θ _- (r) , Θ ₊ (r)] ^T is the vector of the fuzzy unknown coefficients and $Φ_{1}^{T} (x) = [T_{L, 0}^{(λ)} (x),$ $T_{L, 1}^{(λ)} (x), \dots, T_{L, N - 1}^{(λ)} (x)]$ . $f (x) ≃ f_{N} (x) = \sum_{i = 0}^{N - 1} *^{f_{i}} ⊙ T_{L, i}^{(λ)} (x) = F^{T} (r) ⊙ Φ_{1} (x),$ (72) $F^{T} (r) = [f_{0}, f_{1}, \dots, f_{N - 1}],$ (73) $a (x) ≃ a_{N} (x) = \sum_{i = 0}^{N - 1} *^{a_{i}} ⊙ T_{L, i}^{(λ)} (x) = A^{T} ⊙ Φ_{1} (x),$ (74) $A^{T} = [a_{0}, a_{1}, \dots, a_{N - 1}],$ (75) $f_{i} = \frac{1}{L^{\frac{λ}{2}} h_{1, i}} ⊙ \int_{0}^{L} f (x) ⊙ T_{L, i}^{(λ)} (x) ⊙ w_{L, 1}^{(λ)} (x) dx,$ (76) $a_{i} = \frac{1}{L^{\frac{λ}{2}} h_{1, i}} ⊙ \int_{0}^{L} a (x) ⊙ T_{L, i}^{(λ)} (x) ⊙ w_{L, 1}^{(λ)} (x) dx .$ (77) Also we discuss the production of two fuzzy approximation vectors Φ ₁ (x) and Φ ₁ (x) $Φ_{1} (x) Φ_{1}^{T} (x) ⊙ A ≃ \tilde{A} ⊙ Φ_{1} (x)$ (78) where $\tilde{A}$ is the N × N product operational matrix of the vector A . By the above Eq. eq24 and the orthogonal property of generalized shifted fractional-order Chebyshev functions of first kind, the elements ${a_{ij}}_{i, j = 0}^{N - 1}$ of the $\tilde{A}$ are presented as follow $a_{ij} = \frac{1}{L^{\frac{λ}{2}} h_{1, i}} ⊙ \sum_{k = 0}^{N - 1} *^{a_{k}} ⊙ g_{ijk},$ (79) in which, $g_{ijk} = \int_{0}^{L} T_{L, i}^{(λ)} (x) T_{L, j}^{(λ)} (x) T_{L, k}^{(λ)} (x) w_{L, 1}^{(λ)} (x) dx .$ (80)

Now we can discuss fuzzy-like residual R_N (t) for the given Eq. eq21. Let $〈 \cdot, \cdot 〉_{𝔼}$ denotes the fuzzy inner product over $L_{2} (I, 𝔼)$ . Similarity to typical tau method, we obtain following equation $\begin{matrix} [R_{N} (x)]^{r} = [R_{N, -}^{r} (x), R_{N, +}^{r} (x)] \\ = [Θ_{-}^{T} (r) (D^{(ν)} Φ_{1} (x) - a (x) Φ_{1} (x)) - F^{T} (r) Φ_{1} (x), \\ Θ_{+}^{T} (r) (D^{(ν)} Φ_{1} (x) - a (x) Φ_{1} (x)) - F^{T} (r) Φ_{1} (x)], \end{matrix}$ (81) $〈 [R_{N} (x)]^{r}, T_{L, j}^{(λ)} (x) 〉_{𝔼} = \tilde{0}, j = 0, 1, \dots, N - 2,$ (82) or we can write $\begin{matrix} 〈 Θ^{T} (r) ⊙ D^{(ν)} Φ_{1} (x), T_{L, j}^{(λ)} (x) 〉_{𝔼} = \\ = 〈 a (x) ⊙ Θ^{T} (r) ⊙ Φ_{1} (x), T_{L, j}^{(λ)} (x) 〉_{𝔼} \\ + 〈 F^{T} (r) ⊙ Φ_{1} (x), T_{L, j}^{(λ)} (x) 〉_{𝔼}, j = 0, 1, \dots, N - 2, \end{matrix}$ (83) and for the fuzzy initial condition $[y (0)]^{r} ≃ [y_{N} (0)]^{r} = Θ^{T} (r) ⊙ Φ_{1} (0) .$ (84) From Eq.ref eq26 and ref eq27 ${\begin{matrix} \begin{matrix} \sum_{i = 0}^{N - 1} & θ_{i} (r) ⊙ 〈 \sum_{k = 0}^{N - 1} d_{ik} T_{L, k}^{(λ)} (x), T_{L, j}^{(λ)} (x) 〉_{w_{L, 1}^{λ}} \\ = \sum_{i = 0}^{N - 1} θ_{i} (r) ⊙ 〈 \sum_{k = 0}^{N - 1} a_{ik} T_{L, k}^{(λ)} (x), T_{L, j}^{(λ)} (x) 〉_{w_{L, 1}^{λ}} \\ + \sum_{i = 0}^{N - 1} f_{i} (r) ⊙ 〈 T_{L, i}^{(λ)} (x), T_{L, j}^{(λ)} (x) 〉_{w_{L, 1}^{λ}}, \\ j = 0, 1, \dots, N - 2, \end{matrix} \\ \sum_{i = 0}^{N - 1} θ_{i} (r) ⊙ T_{L, i}^{(λ)} (0) = [y (0)]^{r} . \end{matrix}$ (85) Above equation is N fuzzy linear algebraic equations. Also in which $r \in [0, 1], 〈 T_{L, i}^{(λ)} (x), T_{L, j}^{(λ)} (x) 〉_{w_{L, 1}^{λ}}$ $= \int_{0}^{L} T_{L, i}^{(λ)} (x) T_{L, j}^{(λ)} (x) w_{L, 1}^{λ} (x) dx$ and a_ik are determined by Eq. eq24.

We introduced the numerical process for fractional Chebyshev operational matrix based on the generalized shifted fractional-order Chebyshev functions of first kind. Similarly, we can think that the fractional Chebyshev operational matrix based on the generalized shifted fractional-order Chebyshev functions of second kind are applies for solving Eq. eq21.

6 Numerical Examples

In this Section we are going to discuss three examples for the model of viscosity behavior for non-Newtonian fluid, Kelvin-Voiget equation in [40] and the fuzzy fractional differential equation with variable coefficient in [42]. Example 1. In [40], a general model of viscosity behavior of for non-Newtonian fluid under uncertainty was introduced as following $(c^{D_{0^{+}}^{ν}} ω) (t) \oplus B ⊙ ω (t) = \frac{1}{β} ⊙ ρ (t),$ (86) where $ω \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ is a continuous fuzzy-valued function, $c^{D_{0^{+}}^{ν}}$ represents the Caputo-type of the fractional derivative under fuzzy notion with order ν ∈ (0, 1] and L ∈ (0, 1]. Let us consider above Eq. eq29 such that: ${\begin{matrix} (c^{D_{0^{+}}^{ν}} ω) (t) \oplus B ⊙ ω (t) = \frac{1}{β} ⊙ ρ (t), \\ [ω (0)]^{r} = [- 1 + r, 1 - r], 0 \leq r \leq 1, \end{matrix}$ (87) where ω (t) is the fuzzy stress-strain function. We consider that the viscous coefficient E and the modulus of elasticity β is equal to 1 and ρ (t) = te^-t. According to the definition of ^c [(1) - ν]-differentiability and Theorem 2, we have ${\begin{matrix} (c^{D_{0^{+}}^{ν}} ω_{-}^{r}) (t) + ω_{-}^{r} (t) = {te}^{- t}, 0 < ν \leq 1, \\ ω_{-}^{r} (0) = - 1 + r, 0 \leq r \leq 1 . \end{matrix}$ (88) and ${\begin{matrix} (c^{D_{0^{+}}^{ν}} ω_{+}^{r}) (t) + ω_{+}^{r} (t) = {te}^{- t}, 0 < ν \leq 1, \\ ω_{+}^{r} (0) = 1 - r, 0 \leq r \leq 1 . \end{matrix}$ (89)

Then the exact solution of the above problem are presented as following ${\begin{matrix} \begin{matrix} ω_{e, -}^{r} & (t) = (- 1 + r) E_{ν, 1} [- t^{ν}] \\ + \int_{0}^{t} (t - θ)^{ν - 1} E_{ν, ν} [- (t - θ)^{ν}] ρ (θ) d θ, \end{matrix} \\ \begin{matrix} ω_{e, +}^{r} & (t) = (1 - r) E_{ν, 1} [- t^{ν}] \\ + \int_{0}^{t} (t - θ)^{ν - 1} E_{ν, ν} [- (t - θ)^{ν}] ρ (θ) d θ . \end{matrix} \end{matrix}$ (90) where $ρ (θ) = θ e^{a θ}, 0 < ν \leq 1, 0 < r \leq 1, E_{v, u} (t) = \sum_{k = 0}^{\infty} \frac{t^{k}}{Γ (vk + u)}$ is the generalized Mittag-Leffler function. Now According presented method in Section 5, we have computed the several absolute errors to compare the accuracy with other paper. $E_{gc}^{1} (r)$ and $E_{gc}^{2} (r)$ are the absolute errors obtained using generalized shifted fractional Chebyshev functions of first and second kind. The absolute errors of the tau method with the generalized shifted fractional-order Legendre functions in [40], Chebyshev operational matrix method in [1] and Jacobi operational matrix method in [29] are E_gl (r), E_c (r) and E_J (r), respectively. Tables 1 and 2 are obtained in t ∈ [0, 1]. From Tables 1 and 2, we can know that when λ = ν, is gained a high accuracy approximate solution. Figs. 1 and 2 shown the graph of absolute errors obtained in case of the generalized shifted fractional Chebyshev functions of first and second kind.

Table 1

The absolute error of Example 1 with ν = λ = 0.95, N = 7

r	Our method 1	Our method 2	Method in [40]	Method in [1]	Method in [29]
	$\underline{E} {gc}^{1} (r)$	$\underline{E} {gc}^{2} (r)$	$\underline{E} gl (r)$	$\underline{E} c (r)$	$\underline{E} J (r)$
0.0	4.44089e-16	3.33067e-16	2.22045e-16	3.33067e-16	3.33067e-16
0.1	9.56961e-06	1.58763e-07	6.68595e-06	1.37726e-03	2.69512e-04
0.2	9.67130e-06	5.84985e-06	4.69361e-06	7.12204e-04	4.21414e-04
0.3	1.44490e-06	5.75669e-06	2.77385e-06	3.37502e-05	4.75742e-04
0.4	1.27315e-05	6.22648e-07	5.13635e-06	1.35743e-03	2.72262e-04
0.5	9.18429e-06	7.61139e-07	3.85925e-06	1.14016e-03	1.79861e-04
0.6	4.60027e-06	4.47808e-06	3.15308e-06	1.54893e-04	2.63922e-04
0.7	7.72533e-06	3.97548e-06	3.72507e-06	5.75769e-04	2.98495e-04
0.8	3.36645e-06	3.02849e-07	2.86612e-06	3.80208e-04	1.78849e-04
0.9	6.53707e-06	1.76213e-06	2.02665e-06	7.24659e-04	1.39996e-04
1.0	6.20416e-07	1.32664e-06	3.89700e-07	6.26593e-05	1.57806e-04

Table 2

The absolute error of Example 1 with ν = λ = 0.95, N = 7

r	Our method 1	Our method 2	Method in [40]	Method in [1]	Method in [29]
	${\bar{E}}_{gc}^{1} (r)$	${\bar{E}}_{gc}^{2} (r)$	${\bar{E}}_{gl} (r)$	${\bar{E}}_{c} (r)$	${\bar{E}}_{J} (r)$
0.0	2.22045e-16	1.11022e-16	2.22045e-16	1.11022e-16	1.11022e-16
0.1	9.56810e-06	1.58890e-07	6.68480e-06	1.39634e-03	2.75326e-04
0.2	9.67025e-06	5.84929e-06	4.69317e-06	7.30094e-04	4.33373e-04
0.3	1.44355e-06	5.75530e-06	2.77268e-06	4.45166e-05	4.94979e-04
0.4	1.27301e-05	6.22642e-07	5.13560e-06	1.38264e-03	2.80833e-04
0.5	9.18203e-06	7.61679e-07	3.85787e-06	1.16635e-03	1.82249e-04
0.6	4.59993e-06	4.47731e-06	3.15262e-06	1.59726e-04	2.72194e-04
0.7	7.72327e-06	3.97430e-06	3.72370e-06	5.94095e-04	3.10661e-04
0.8	3.36635e-06	3.02393e-07	2.86557e-06	3.85209e-04	1.84162e-04
0.9	6.53554e-06	1.76208e-06	2.02583e-06	7.40434e-04	1.42941e-04
1.0	6.20116e-07	1.32562e-06	3.89602e-07	6.16944e-05	1.62697e-04

Fig.1

The graph of absolute error $E_{gc}^{1} (r)$ of Example 1 with ν = λ = 0.95, N = 7.

Fig.2

The graph of absolute error $E_{gc}^{2} (r)$ of Example 1 with ν = λ = 0.95, N = 7.

Example 2. The fuzzy fractional Kelvin-Voiget(KV) model under the Caputo gH-differentiability was defined as following in [40] $ρ (t) = E ⊙ δ (t) \oplus η D_{0^{+}}^{ν} δ (t)$ (91) where $δ \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ is a continuous fuzzy-valued function, $c^{D_{0^{+}}^{ν}}$ presents the Caputo-type derivative under fuzzy notion with order ν ∈ (0, 1] and L ∈ (0, 1]. And E and ρ (t) can be defined as fuzzy parameter and fuzzy set-valued function, respectively, base on the conditions of the model. Now let η = 1, E = 1 and $ρ (t) = \sqrt{t} + \frac{\sqrt{π}}{2}$ . The above KV equation ref eq31 is presented as follow ${\begin{matrix} \sqrt{t} + \frac{\sqrt{π}}{2} = δ (t) \oplus c^{D_{0^{+}}^{ν}} δ (t), 0 \leq t \leq 1 \\ [δ (0)]^{r} = [- 1 + r, 1 - r], 0 \leq r \leq 1, \end{matrix}$ (92) in which 0 < ν ≤ 1. Also The exact solution of this problem under ^c-differentiability are represented following parametric form ${\begin{matrix} \begin{matrix} Δ_{e, -}^{r} & (t) = (- 1 + r) E_{ν, 1} [- t^{ν}] \\ + \int_{0}^{t} (t - θ)^{ν - 1} E_{ν, ν} [- (t - θ)^{ν}] ρ (θ) d θ, \end{matrix} \\ \begin{matrix} Δ_{e, +}^{r} & (x) = (1 - r) E_{ν, 1} [- t^{ν}] \\ + \int_{0}^{t} (t - θ)^{ν - 1} E_{ν, ν} [- (t - θ)^{ν}] ρ (θ) d θ . \end{matrix} \end{matrix}$ (93) where $ρ (θ) = \sqrt{θ} + \frac{\sqrt{π}}{2}, 0 < ν \leq 1, 0 \leq r \leq 1$ .

Similarly, as shown in Example 1, Tables 3 and 4 presented the absolute errors between the exact solutions and approximate solutions of Example 2 for ν = λ = 0.75 in t ∈ [0, 1]. Also Figs. 3 and 4 shown the graph of absolute errors in case of the generalized shifted fractional Chebyshev functions of first and second kind.

Fig.3

The graph of absolute error $E_{gc}^{1} (r)$ of Example 2 with ν = λ = 0.75, N = 7.

Fig.4

The graph of absolute error $E_{gc}^{1} (r)$ of Example 2 with ν = λ = 0.75, N = 7.

Table 3

The absolute error of Example 2 with ν = λ = 0.75, N = 7

	Our method 1	Our method 2	Method in [40]	Method in [1]	Method in [29]
r	${\bar{E}}_{gc}^{1} (r)$	${\bar{E}}_{gc}^{2} (r)$	${\bar{E}}_{gl} (r)$	${\bar{E}}_{c} (r)$	${\bar{E}}_{J} (r)$
0.0	7.77156e-16	7.77156e-16	5.55112e-16	8.88178e-16	6.66134e-16
0.1	2.84428e-04	3.13397e-05	7.87898e-05	2.03973e-02	2.93148e-04
0.2	3.29905e-05	5.64758e-05	4.82521e-05	2.22489e-02	4.70912e-03
0.3	3.60891e-04	6.11827e-05	1.51864e-04	2.40427e-03	5.41399e-03
0.4	1.67161e-04	3.12479e-05	7.80171e-05	2.62943e-02	9.38535e-04
0.5	2.28074e-04	6.15500e-05	1.00475e-04	1.62966e-02	1.08984e-04
0.6	2.99324e-04	5.74378e-05	1.13973e-04	1.28168e-02	2.57398e-03
0.7	1.10935e-05	3.86547e-05	4.85550e-05	1.59330e-02	3.20814e-03
0.8	2.66784e-04	6.01448e-05	1.32883e-04	1.00823e-02	3.08755e-04
0.9	5.12229e-05	6.87968e-05	4.23509e-05	1.19773e-02	4.70310e-04
1.0	4.76523e-05	1.44945e-04	5.90976e-05	1.74296e-03	3.90054e-04

Table 4

The absolute error of Example 2 with ν = λ = 0.75, N = 7

	Our method 1	Our method 2	Method in [40]	Method in [1]	Method in [29]
0.0	2.22045e-16	5.13478e-16	3.88578e-16	5.55111e-16	7.21645e-16
0.1	2.84436e-04	3.13356e-05	7.87889e-05	9.22651e-03	1.61545e-04
0.2	3.30099e-05	5.64899e-05	4.82682e-05	1.00750e-02	2.11302e-03
0.3	3.60915e-04	6.11904e-05	1.51878e-04	1.03066e-03	2.40280e-03
0.4	1.67191e-04	3.12608e-05	7.80369e-05	1.19342e-02	4.29361e-04
0.5	2.28084e-04	6.15581e-05	1.00484e-04	7.36851e-03	5.33264e-05
0.6	2.99360e-04	5.74544e-05	1.13997e-04	5.80586e-03	1.14923e-03
0.7	1.10828e-05	3.86597e-05	4.85553e-05	7.19300e-03	1.42202e-03
0.8	2.66813e-04	6.01633e-05	1.32908e-04	4.58057e-03	1.46703e-04
0.9	5.12337e-05	6.88121e-05	4.23556e-05	5.42365e-03	2.18376e-04
1.0	4.76585e-05	1.44986e-04	5.91088e-05	7.81793e-04	1.81152e-04

Example 3. The fuzzy fractional differential equation with variable coefficient in [42] discussed as following ${\begin{matrix} (c^{D_{0^{+}}^{ν}} u) (x) = \sqrt{x} u (x) + x e^{- x}, \\ u (0) = u_{0}, x \in [0, 1] . \end{matrix}$ (94) where $u \in C (I, 𝔼) \cap L_{1} (I, 𝔼)$ is a continuous fuzzy-valued function, $c^{D_{0^{+}}^{ν}}$ indicate the fuzzy fractional derivative order ν ∈ [0, 1] of Caputo type, [u₀] ^r = [-1 + r, 1 - r] and 0 < ν ≤ 1.

We consider above equation under ^c-differentia-bility and $\sqrt{x} \geq 0, x \in [0, 1]$ . According to the definition of ^c [(1) - ν]-differentiability and Theorem 2, then the above equation ref eq33 are equal to as following ${\begin{matrix} c^{D_{0^{+}}^{ν}} u_{-}^{r} (x) = \sqrt{x} u_{-}^{r} (x) + {xe}^{x}, \\ u_{-}^{r} (0) = - 1 + r, 0 \leq r \leq 1 . \end{matrix}$ (95) and ${\begin{matrix} c^{D_{0^{+}}^{ν}} u_{+}^{r} (x) = \sqrt{x} u_{+}^{r} (x) + {xe}^{x}, \\ u_{+}^{r} (0) = 1 - r, 0 \leq r \leq 1 . \end{matrix}$ (96) We applied the presented method in Section 5 to Example 3. As in crisp sense, we obtained the approximate solution of above example by using the tau method based on fractional Chebyshev operational matrix. Here we assumed the true that $a (x) = \sqrt{x}$ is the nonnegative on the interval [0, 1] and $c^{D_{0^{+}}^{ν}}$ is ^c [(1) - ν]-differentiability of Caputo-type. Also The better approximate solutions are related to the choice of the parameter λ. Indeed as example 1 and 2, we gained the good approximate solution in case of ν = λ for example 3. The graph of the approximate solutions of Example 3 obtained by using generalized shifted Chebyshev functions of first and second kind was shown in Figs. 5 and 6. Tables 5 and 6 are the numerical values of Figs. 5 and 6 obtained in x ∈ [0, 1].

Fig.5

The graph of approximate solution of Example 3 with α = β = 0.75, N = 7 using GFCF1.

Fig.6

The graph of approximate solution of Example 3 with α = β = 0.75, N = 7 using GFCF1.

Table 5

The approximate value of Example 3 with ν = λ = 0.75, N = 7using GFCF1

x \| r	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
0.0	-1.0000	-0.9000	-0.8000	-0.7000	-0.6000	-0.5000	-0.4000	-0.3000	-0.2000	-0.1000	0.0000
0.1	-1.0349	-0.9304	-0.8259	-0.7213	-0.6168	-0.5122	-0.4077	-0.3031	-0.1986	-0.0940	0.0105
0.2	-1.0776	-0.9664	-0.8552	-0.7441	-0.6329	-0.5217	-0.4106	-0.2994	-0.1882	-0.0771	0.0341
0.3	-1.1261	-1.0068	-0.8874	-0.7680	-0.6487	-0.5293	-0.4099	-0.2906	-0.1713	-0.0519	0.0674
0.4	-1.1829	-1.0537	-0.9245	-0.7953	-0.6661	-0.5369	-0.4078	-0.2786	-0.1494	-0.0202	0.1090
0.5	-1.2495	-1.1087	-0.9679	-0.8272	-0.6864	-0.5456	-0.4049	-0.2641	-0.1233	0.0174	0.1582
0.6	-1.3276	-1.1733	-1.0191	-0.8648	-0.7105	-0.5562	-0.4020	-0.2477	-0.0934	0.0608	0.2151
0.7	-1.4198	-1.2498	-1.0798	-0.9098	-0.7398	-0.5698	-0.3999	-0.2299	-0.0599	0.1101	0.2801
0.8	-1.5296	-1.3412	-1.1528	-0.9645	-0.7761	-0.5878	-0.3994	-0.2111	-0.0227	0.1657	0.3540
0.9	-1.6608	-1.4510	-1.2411	-1.0312	-0.8213	-0.6114	-0.4015	-0.1917	0.0182	0.2281	0.4380
1.0	-1.8171	-1.5820	-1.3470	-1.1119	-0.8769	-0.6418	-0.4067	-0.1717	0.0634	0.2985	0.5335

Table 6

The approximate value of Example 3 with ν = λ = 0.75, N = 7 using GFCF2

x \| r	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
0.0	1.0000	0.9000	0.8000	0.7000	0.6000	0.5000	0.4000	0.3000	0.2000	0.1000	0.0000
0.1	1.0560	0.9515	0.8469	0.7424	0.6378	0.5333	0.4287	0.3242	0.2196	0.1151	0.0105
0.2	1.1458	1.0346	0.9235	0.8123	0.7011	0.5899	0.4788	0.3676	0.2564	0.1453	0.0341
0.3	1.2610	1.1416	1.0223	0.9029	0.7835	0.6642	0.5448	0.4255	0.3061	0.1868	0.0674
0.4	1.4008	1.2717	1.1425	1.0133	0.8841	0.7549	0.6257	0.4965	0.3674	0.2382	0.1090
0.5	1.5659	1.4251	1.2843	1.1436	1.0028	0.8620	0.7213	0.5805	0.4397	0.2990	0.1582
0.6	1.7578	1.6035	1.4492	1.2950	1.1407	0.9864	0.8322	0.6779	0.5236	0.3694	0.2151
0.7	1.9799	1.8099	1.6400	1.4700	1.3000	1.1300	0.9600	0.7901	0.6201	0.4501	0.2801
0.8	2.2376	2.0492	1.8609	1.6725	1.4842	1.2958	1.1074	0.9191	0.7307	0.5424	0.3540
0.9	2.5368	2.3269	2.1170	1.9072	1.6973	1.4874	1.2775	1.0676	0.8578	0.6479	0.4380
1.0	2.8841	2.6491	2.4140	2.1789	1.9439	1.7088	1.4738	1.2387	1.0036	0.7686	0.5335

7 Conclusion

In this paper, we have proposed the effective method for solving the fuzzy fractional differential equations with initial value. We have constructed the fractional Chebyshev operational matrix based on generalized shifted fractional-order Chebyshev functions of first and second kind (GFCF1 and GFCF2) and obtained the approximate solution with good accuracy by applying them to fuzzy fractional differential equation with the model of viscosity behavior for non-Newtonian fluid and Kelvin-Voiget equation with real world. We can know that the choice of λ is important in the accuracy sense through example 1 and 2. Also in example 3, we discussed the approximate solution of fuzzy fractional differential equation with variable coefficient by using presented method. Furthermore, we can understand that the approximate solution converge to exact solution when the number of fractional Chebyshev term increase. The problem we want to study in the future is the choice problem of the parameter λ effecting the accuracy of solution and the approximate method in case of a (x) ≥0 under ^c [(2) - ν]-differentiability and a (x) <0 under ^c [(1) - ν]-differentiability.

References

Ahmadian

, Salahshour

and Chan

, Fractional differential systems: A fuzzy solution based on operational matrix of shifted chebyshev polynomials and its applications, IEEE Trans Fuzzy Syst 25(1) (2016), 218–236.

Machado

J.T.

, Kiryakova

and Mainardi

, Recent history of fractional calculus, Commun Nonlinear Sci Numer Simul 16(3) (2011), 1140–1153.

Agrawal

O.P.

, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn 38 (2004), 323–337.

Dehghan

, Hamedi

E.A.

and Khosravian-Arab

, A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials, J Vib Contr 22(6) (2016), 1547–1559.

Ortigueira

M.D.

and Machado

J.A.T.

, Fractional signal processing and applications, Signal Process 83 (2003), 2285–2286.

Engheta

, On fractional calculus and fractional multipoles in electromagnetism, IEEE T Antenn Propag 44 (1996), 554–566.

Bai

, Sun

and Chen

, The existence and uniqueness of a class of fractional differential equations, Abstr Appl Anal 2014 (2014), 1–6.

, Wang

and Wei

, Existence and uniqueness results for fractional differential equations with boundary value conditions, Opuscula Mathematica 31 (2011), 629–643.

Mahto

and Abbas

, Existence and uniqueness of solution of caputo fractional differential equations, AIP Conference Proceedings 1479 (2012), 896–899.

10.

Diethelm

, Ford

N.J.

and Freed

A.D.

, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn 29 (2002), 3–22.

11.

Daftardar-Gejji

and Jafari

, Adomian decomposition: A tool for solving a system of fractional differential equations, J Math Anal Appl 301 (2005), 508–518.

12.

Das

, Analytical solution of a fractional diffusion equation by variational iteration method, Comput Math Appl 57 (2009), 483–487.

13.

Jafari

and Momani

, Solving fractional diffusion and wave equations by modified homotopy perturbation method, Phys Lett A 370 (2007), 388–396.

14.

Doha

E.H.

, Bhrawy

A.H.

and Ezz-Eldien

S.S.

, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput Math Appl 62 (2011), 2364–2373.

15.

Doha

E.H.

, Bhrawy

A.H.

and Ezz-Eldien

S.S.

, A new Jacobi operational matrix: An application for solving fractional differential equations, Appl Math Model 36 (2012), 4931–4943.

16.

Kazem

, An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations, Appl Math Model 37 (2013), 1126–1136.

17.

Doha

E.H.

, Bhrawy

A.H.

, Baleanu

and Ezz-Eldien

S.S.

, On shifted Jacobi spectral approximations for solving fractional differential equations, Appl Math Comput 219 (2013), 8042–8056.

18.

Kazem

, Abbasbandy

and Kumar

, Fractional-order Legendre functions for solving fractional-order differential equations, Appl Math Model 37 (2013), 5498–5510.

19.

Bhrawy

A.H.

and Zaky

M.A.

, Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations, Appl Math Model 40 (2016), 832–845.

20.

Doha

E.H.

, Bhrawy

A.H.

and Ezz-Eldien

S.S.

, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl Math Model 35 (2011), 5662–5672.

21.

Agarwal

R.P.

, Lakshmikantham

and Nieto

J.J.

, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal 72 (2010), 2859–2862.

22.

Allahviranloo

, Armand

and Gouyandeh

, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J Intell Fuzzy Syst 26 (2014), 1481–1490.

23.

Chehlabi

and Allahviranloo

, Concreted solutions to fuzzy linear fractional differential equations, Appl Soft Comput 44 (2016), 108–116.

24.

Armand

and Allahviranloo

, Fractional relaxation-oscillation differential equations via fuzzy variational iteration method, J Intell Fuzzy Syst 32 (2017), 363–371.

25.

Allahviranloo

, Armand

and Gouyandeh

, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J Intell Fuzzy Syst 26 (2014), 1481–1490.

26.

Ahmadian

, Salahshour

and Ali-Akbari

, et al., A novel approach to approximate fractional derivative with uncertain conditions, Chaos Soliton Fract 104 (2017), 68–76.

27.

Salahshour

, Allahviranloo

and Abbasbandy

, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun Nonlinear Sci Numer Simul 17 (2012), 1372–1381.

28.

Mazandarani

and Vahidian

, Kamyad, Modified fractional Euler method for solving fuzzy fractional initial value problem, Commun. Nonlinear Sci Numer Simul 18 (2013), 12–21.

29.

Ahmadian

, Suleiman

, Salahshour

and Baleanu

D.A.

, Jacobi operational matrix for solving a fuzzy linear fractional differential equation, Adv Differ Equ 104 (2013), 1–29.

30.

Ahmadian

, Suleiman

and Salahshour

, An operational matrix based on Legendre polynomials for solving fuzzy fractionalorder differential equations, Abstr Appl Anal 2013 (2013), 1–29.

31.

Anastassiou

G.A.

, Fuzzy Mathematics: Approximation Theory, Springer, Heidelberg, 2010.

32.

Kaleva

, Fuzzy differential equations, Fuzzy Sets Syst 24 (1987), 301–317.

33.

Bede

and Stefanini

, Generalized differentiability of fuzzyvalued functions, Fuzzy Sets Syst 230 (2013), 119–141.

34.

Bede

and Gal

S.G.

, Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations, Fuzzy Sets Syst 151 (2005), 581–599.

35.

Chalco-Cano

and Román-Flores

, On new solutions of fuzzy differential equations, Chaos Soliton Fract 38 (2008), 112–119.

36.

Diethelm

, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics. Springer, Berlin, 2010.

37.

Salahshour

, Allahviranloo

, Abbasbandy

and Baleanu

, Existence and uniqueness results for fractional differential equations with uncertainty, Adv Differ Equ 112 (2012), 1–12.

38.

Salahshour

, Ahmadian

, Senu

, Baleanu

and Agarwal

, On analytical solutions of the fractional differential equation with uncertainty: Application to the basset problem, Entropy 17 (2015), 885–902.

39.

Allahviranloo

, Salahshour

and Abbasbandy

, Explicit solutions of fractional differential equations with uncertainty, Soft Comput 16 (2012), 297–302.

40.

Ahmadian

, Ismail

, Salahshour

, Baleanu

and Ghaemi

, Uncertain viscoelastic models with fractional order: A new spectral tau method to study the numerical simulations of the solution, Commun Nonlinear Sci Numer Simul 53 (2017), 44–64.

41.

Sin

, Chen

, Choi

and Ri

, Fractional Jacobi operational matrix for solving fuzzy fractional differential equation, J Intell Fuzzy Syst 33 (2017), 1041–1052.

42.

Salahshour

, Ahmadian

, Ismail

, Baleanu

and Senu

, A New fractional derivative for differential equation of fractional order under interval uncertainty, Adv Mech Eng 7(12) (2015), 1–11.

Application of a spectral method to Fractional Differential Equations under uncertainty 1

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Space of fuzzy numbers and Fuzzy calculus

4.1 Generalized shifted fractional-order Chebyshev functions of first and second kind

4.1.1 Generalized shifted fractional-order Chebyshev functions of first kind T n ( λ ) ( x )

5.1 Approximation of the fuzzy-valued function

References

4.1.1 Generalized shifted fractional-order Chebyshev functions of first kind $T_{n}^{(λ)} (x)$