A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion

: In this ﬁeld of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efﬁciency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the ﬂexibility and the capability of the presented method. considering the problem of fractional order instead of integer order (here, ﬁrst order), we obtain some wider interval than the ﬁrst order case; on the other hand, when α = 1 , the valid interval that the given exact solution verify the assumption FC [( i ) − gH ] -differentiability is ( 0, 0.7101 ) , while for α = 0.9 and α = 0.7 , the valid interval are ( 0, 0.7381 ) and ( 0, 0.7949 , respectively. this is ﬁrst in the is


Introduction
Fuzzy set theory is a powerful tool for modeling uncertain problems. Therefore, large varieties of natural phenomena have been modeled using fuzzy concepts. Particularly, the fuzzy fractional differential equation is a common model in different scientific fields, such as population models, evaluating weapon systems, civil engineering, and modeling electro-hydraulics. Therefore, the concept of the fractional derivative is a very important topic in fuzzy calculus. Therefore, fuzzy fractional differential equations have attracted much attention in mathematics and engineering fields. The first work devoted to the subject of fuzzy fractional differential equations is the paper by Agarwal et al. [1]. They have defined the Riemann-Liouville differentiability concept under the Hukuhara differentiability to solve fuzzy fractional differential equations.
In recent years, fractional calculus has been introduced as an applicable topic to produce the accurate results of mathematical and engineering problems such as aerodynamics and control systems, signal processing, bio-mathematical problems, and others [1][2][3][4][5].
is called the space of fuzzy numbers or the fuzzy numbers set. The r-level set is [u] r = {x ∈ R n | u(x) ≥ r, 0 < r ≤ 1}. Then, from I to IV, it follows that, the r-level sets of u ∈ R F are nonempty, closed, and bounded intervals.

Definition 2.
A triangular fuzzy number is defined as a fuzzy set in R F , that is specified by an ordered triple u = (a, b, c) ∈ R 3 with a ≤ b ≤ c such that u − (r) = a + (b − a)r (or lower bound of u) and u + (r) = c − (c − b)r (or upper bound of u) are the endpoints of r-level sets for all r ∈ [0, 1].
A crisp number k is simply represented by u − (r) = u + (r) = k, 0 ≤ r ≤ 1 and called singleton. For arbitrary u, v ∈ R F and scalar k, we might summarize the addition and the scalar multiplication of two fuzzy numbers by addition : H(u ⊕ w, v ⊕ w) = H(u, v), ∀u, v, w ∈ R F . II.
H(λu, λv) =| λ | H(u, v), ∀λ ∈ R, ∀u, v ∈ R F . III. H(u ⊕ v, w ⊕ z) ≤ H(u, w) + H(v, z), ∀u, v, w, z ∈ R F . Definition 3. Let u, v ∈ R F , if there exists w ∈ R F , such that u = v + w, then w is called the Hukuhara difference (H-difference) of u and v, and it is denoted by u v. Furthermore, the generalized Hukuhara difference (gH-difference) of two fuzzy numbers u, v ∈ R F is defined as follows, it is easy to show that conditions (i) and (ii) are valid if and only if w is a crisp number. The conditions of the existence of u gH v ∈ R F are given in [22]. Through the whole of the paper, we suppose that the gH-difference exists.
In this paper, the meaning of fuzzy-valued function is a function f : A → R F , A ∈ R where R is the set of all real numbers and [ f (t)] r = [ f − (t; r), f + (t; r)] is the so-called r-cut or parametric form of the fuzzy-valued function f .

Definition 4. A fuzzy-valued function
Throughout the rest of this paper, the notation Definition 5. Let f : [a, b] → R F , t 0 ∈ (a, b) with f − (t; r) and f + (t; r) both differentiable at t 0 for all r ∈ [0, 1] and D f gH (gH-derivative) exists:

Definitions and Properties of Fractional gH-Differentiability
In this section, let us focus on some definitions and properties related to the fuzzy fractional generalized Hukuhara derivative which are useful in the sequel of this paper.

Definition 6 ([26]
). Let f (t) be a fuzzy Lebesque integrable function. The fuzzy Riemann-Liouville fractional (for short (F.RL)-fractional) integral of order α > 0 is defined as follows, Definition 7 ([26]). Let f : [a, b] → R F . The fuzzy fractional derivative of f (t) in the Caputo sense is in the following form, where ∀m ∈ N, D m f gH (s) (gH-derivatives of f ) are integrable. In this paper, we consider fuzzy Caputo generalized Hukuhara derivative (for short FC [gH]-derivative) of order 0 < α ≤ 1, for fuzzy-valued function f , so the FC [gH]-derivative will be expressed by (1) respectively. Here, t m−1 , t m−2 , ..., t ≥ a, and they are real numbers.
Proof. This lemma is a fairly straightforward generalization of Lemma 1. The proof will be done by introducing on m ∈ N. Assume that the lemma holds for (m)-times applying operator (F.RL)-fractional integrating for function f , we will prove it will correct for (m + 1)-times applying operator (F.RL)-fractional integrating for function f . By Lemma 1, as  -differentiability of f , if in any neighborhood V of t 0 there exist points t 1 < t 0 < t 2 such that .
. Suppose that f : [a, b] → R F be a fuzzy-valued function and D f gH is exist, then for The principal significance of this lemma is in the following theorem.
is the Riemann-Liouville integrable, and by using Lemma 3 for 0 ≤ r ≤ 1 moreover, by combining Equations (2) with (3) the lemma is proved.
II. If FC D jα * f , j = 1, ..., n are FC [(ii) − gH]-differentiable and the type of their differentiability does not change in the interval [a, b], then .., n we give the proof only for parts II and III. Proving the other parts are similar.

II. Our proof starts with the observation that
Therefore, using the properties of fuzzy Caputo derivative and Theorem 2, we have

III. Under the conditions stated in the part III
In the sense of Section 2 and by Theorem 2, we get which completes the proof.

Fuzzy Generalized Taylor Theorem
provided that type of fuzzy Caputo differentiability has no change. Then, provided that type of fuzzy Caputo differentiability has no change, then .., n, exist and in each order the type of FC [gH]-differentiability changes on T On the other hand, the type of differentiability for FC D jα * f , j ≤ k on [ξ, b] does not change. Therefore, Proof. Here, we prove parts II, III, and IV because the proof of part I is similar to the part II.
Under the assumptions that FC D Under the hypotheses of Theorem, type of differentiability does not change, so by Theorem 3 and by attention to (F.RL)-integrability of FC D α * f ii.gH on T, we obtain Lemma 2 implies that the last double (F.RL)-fractional integral belongs to R f . Therefore, by repeating the above argument, we get Therefore, we find that By Lemma 2, the last triple (F.RL)-fractional integral belongs to R f . Therefore, substituting above equation into Equation (4), we find that the high order of the last formula by Lemma 2 is a continuous function in terms of t so it belongs to R f . With the same manner, we can demonstrate that part II is satisfied.
Lemma 2 now leads to the last double (F.RL)-fractional integral belongs to R f . Therefore, .
with simple and similar method, the proof for this type of differentiability will be completed.
and in the interval According to the hypothesis, we know that ξ is a switching point for differentiability f ; thus, by substituting Equation (5) into Equation (6), we obtain Consider the first (F.RL)-fractional integral on the right side of the Equation (7): By noting the hypothesis of theorem, the fuzzy Caputo derivative of the function f has the switching point ξ 1 of type I. Therefore, , then type of differentiability can be changed. By these conditions, the Theorem 3, admits that On the other hand, we know that FC D α * f ii.gH is FC [(ii) − gH]-differentiable on [ξ 1 , ξ] and the type of differentiability does not change. Thus, for t 1 ∈ [ξ 1 , ξ] from Theorem 3, it follows that substituting Equation (8) into Equation (9) gives that follows from Theorem 3 and also the insertion of Equations (11) and (12) in Equation (10) allows us to obtain Finally, the first (F.RL)-fractional integral on the right side of the Equation (7) obtains as follows the only point remaining concerns the behavior of the second (F.RL)-fractional integral on the right side of the Equation (7). We can now proceed analogously to the first (F.RL)-fractional integral: By noting the hypothesis of theorem, FC D jα * f i.gH , j = 2, 3 are FC [(i) − gH]-differentiable on [ξ, b], and the type of differentiability does not change. By Theorem 3 we deduce that and Thus, the second (F.RL)-fractional integral on the right side of the Equation (7) is as follows, having disposed of this preliminary step, we can now return to the Equation (7). By substituting Equations (13) and (16), in Equation (7), the desired result is achieved.

Fuzzy Generalized Euler's Method
In this section, we will touch on only a few aspects of the fuzzy generalized Taylor theorem and restrict the discussion to the fuzzy generalized Euler's method. This case is important enough to be stated separately. We consider the following fuzzy fractional initial value problem, where f : [0, T] × R F → R F is continuous and y(t) is an unknown fuzzy function of crisp variable t. Furthermore, FC D α * y gH (t) is the fuzzy fractional derivative y(t) in the Caputo sense of order 0 < α ≤ 1, with the finite set of switching points. Now, by dividing the interval [0, T] with the step length of h, we have the partition I N = {0 = t 0 < t 1 < ... < t N = T} where t k = kh for k = 0, 1, 2, ..., N.
Case I. Unless otherwise stated we assume that the unique solution of the fuzzy fractional initial value problem (17) for some points k lie between t k and t k+1 . As h = t k+1 − t k , we have and, y(t) satisfies in problem (5.1), so Thus, for sufficiently small h we find that and finally we get -differentiable such that the type of differentiability does not change on [0, T]. Therefore, the fractional Taylor's series expansion of y(t) about the point t k at t k+1 is according to the process described in Case I, the generalized Euler's method takes the form Case III. Let us suppose that t 0 = 0, t 1 , ..., t j , ζ, t j+1 , ..., t N = T is a partition of interval [0, T] and y(t) has a switching point in ζ ∈ [0, T] of type I. Therefore, according to Equations (18) and (19), we have Case IV. Consider y(t) has a switching point type II in ζ ∈ [0, T] such that t 0 , t 1 , ..., t j , ζ, t j+1 , ..., t N is a partition of interval [0, T]. Therefore, by Equations (18) and (19), we conclude that Our next concern will be the behavior of the fuzzy generalized Euler method.

Analysis of the Fuzzy Generalized Euler's Method
In this section, the local and the global truncation errors of the fuzzy generalized Euler's method are illustrated. Therefore, by applying them the consistency, the convergence, and the stability of the presented method are proved. Furthermore, several definitions and concepts of the fuzzy generalized Euler's method are presented under FC [gH]-differentiability [27].

Local Truncation Error, Consistent
Consider the unique solution of the fuzzy fractional initial value problem (17) : on the other hand, the local truncation error (LTE) (τ k ) is defined as and the fuzzy generalized Euler's method is said to be consistent if therefore, due to the type of differentiability of y(t) for η k ∈ [t k , t k+1 ], the residual (R k ) and the LTE (τ k ) are defined as follows.
Investigating the consistence of the fuzzy generalized Euler's method: For this purpose, assume that H( FC D 2α * y ii.gH (η k ), 0) ≤ M. We have two following steps: note that only one of the steps is proved and the proof of another one is similar. Step .M = 0.
Step II. The same conclusion can be drawn for the FC [(i) − gH]-differentiability of y(t). Thus, note that we have actually proved that the fuzzy generalized Euler's method is consistent as long as the solution belongs to C f ([0, T], R F ).
Definition 11 ([29]). The global truncation error is the agglomeration of the local truncation error over all the iterations, assuming perfect knowledge of the true solution at the initial time step.
In the fuzzy fractional initial value problem (17), assume that y(t) is FC [(i) − gH]-differentiable, then the global truncation error is and for the FC [(ii) − gH]-differentiability of y(t), we have

Definition 12.
If global truncation error leads to zero as the step size goes to zero, the numerical method is convergent, i.e., lim in this case, the numerical solution converges to the exact solution.
Investigating the convergence of the fuzzy generalized Euler's method: To suppose that FC D 2α * y(t) exists and f (t, y) satisfies in Lipschitz condition on the {(t, y) | t ∈ [0, p], y ∈ B(y 0 , q), p, q > 0}, the research on this subject will be divided into two steps: Step I. Suppose that y(t) is FC [(ii) − gH]-differentiable, now by using Equation (18) and assumption , the exact solution of the Equation (17) satisfies .
[H f (t k , y k ) gH f (t k , y(t k )), 0 ] + H(r k , 0), the inequality which is the conclusion of Lipschitz condition, implies that . )H (y(t k ), y k ) + r, as the inequality holds for all k, we get . )H (y(t k−1 ), y k−1 ) + r + r . ) .
Repeated application of the above inequality enables us to write H (y(t k+1 ), y k+1 ) obviously, this sum is a geometric series, so we have , that resulted to with z = − h α Γ(α+1) . in Lemma 5 concludes that where 0 ≤ (k + 1)h α ≤ T for (k + 1) ≤ (N − 1). Thus in Equation (23), we obtain Moreover, H( FC D 2α * y ii.gH (t), 0), and the accuracy of the initial value, concludes that H (y(t 0 ), y 0 ) = 0, so now, letting h → 0 then H (y(t k+1 ), y k+1 ) → 0, which is the desired conclusion, and we can say that the fuzzy generalized Euler's method is convergent in this step.
Step II. To estimate the step II, consider y(t) is FC [(i) − gH]-differentiable, by using Equation (19) and let r k = h 2α Γ(2α+1) FC D 2α * y i.gH (t k ) the proof of this step is similar to step I, so the fuzzy generalized Euler's method is convergent.

Stability
Now, the stability of the presented method is illustrated. For this aim, the following definition is presented.
Definition 13. Assume that y k+1 , k + 1 ≥ 0 is the solution of fuzzy generalized Euler's method where y 0 ∈ R F and also z k+1 is the solution of the same numerical method where z 0 = y 0 ⊕ δ 0 ∈ R F shows its perturbed fuzzy initial condition. The fuzzy generalized Euler's method is stable if there exists positive constant h and K such that Investigating the stability of the fuzzy generalized Euler's method: The proof falls naturally into two steps: Step I. If y(t) is FC [(ii) − gH]-differentiable, by using Equation (18) the perturbed problem is in the following form, According to the Equations (19) and (24), we have which we have been working under the assumption that specifications of the Hausdorff metric are satisfied. Using the Lipschitz condition, it can be concluded that repeating with the inequality and applying Lemma 5 lead us to the following inequality where K = e − T Γ(α + 1) and for k + 1 < N − 1 ⇒ h α (k + 1) ≤ T. In this case, it is obvious the stability of the fuzzy generalized Euler's method.
Step II. For FC [(i) − gH]-differentiability of y(t) the same process can be used. In general, the above-mentioned analysis, points out that the fuzzy generalized Euler method is a stable approach.

Remark 1.
Although we have obtained the solution under FC [(i) − gH]-differentiability, it is easy to check that it is not FC [(i) − gH]-differentiable on (0, 1). Actually, due to obtained results (see Table 5), we can consider the proper interval that the given exact solution and its approximation is FC [(i) − gH]-differentiable. Moreover, note that, we have computed the approximation of the solution of Example 4 at point t = 1, which is clearly this point take place out of proper domain of FC [(i) − gH]-differentiability. In fact, the computed error at point t = 1, just obtained based on the lower-upper approximation of lower-upper of exact solution. For more clarification, we determined switching points regarding each order of differentiability.  Table 5, we in fact deduce that by considering the problem of fractional order instead of integer order (here, first order), we obtain some wider interval than the first order case; on the other hand, when α = 1, the valid interval that the given exact solution verify the assumption FC [(i) − gH]-differentiability is (0, 0.7101), while for α = 0.9 and α = 0.7, the valid interval are (0, 0.7381) and (0, 0.7949), respectively. Actually, this is the first time in the literature that this new result, i.e., extending the length of valid interval that the type of differentiability remains unchanged, is investigated.

Conclusions
Fractional differential equations are one of the important topics of fuzzy arithmetic which have many applications in sciences and engineering. Thus finding the numerical and analytical methods to solve these problems is very important. This paper was presented based on the two main topics. First, proving the generalized Taylor series expansion for fuzzy valued function based on the concept of generalized Hukuhara differentiability. Second, introducing the fuzzy generalized Euler's method as an application of the generalized Taylor expansion and applying it to solve the fuzzy fractional differential equations. The capabilities and abilities of the presented method were shown by presenting several theorems about the consistency, the convergence, and the stability of the generalized Euler's method. Furthermore, the accuracy and efficiency of the method were illustrated by considering the local and global truncation errors. The numerical results especially in the switching point case showed the precision of the generalized Euler's method to solve the fuzzy fractional differential equations.